High School Math Element Cards

September 2014Revised December 2016

Elements CardFLS: MAFS.912.N-RN.1.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5√3 to be the cube root of 5 because we want (5√3)3 = 5(√3)3 to hold, so (5√3)3 must equal 5.

Access Point NarrativeMAFS.912.N-RN.1.AP.1a

Extend the properties of exponents to justify that (51/2)2.=.5.

MAFS.912.N-RN.1.AP.1b

Understand that the denominator of the rational exponent is the root index and the numerator is the exponent of the radicand (e.g., 51/2 = √5).

Essential Understandings:Access Point Concrete

UnderstandingsRepresentation

MAFS.912.N-RN.1.AP.1a

Identify a fractional exponent.

Use manipulatives to demonstrate laws of exponents.

Understand the following concepts, symbols and vocabulary: radicand, root index, root, numerator, denominator, radical, exponent.Click here

Understand how to multiply fractions and whole numbers.

Understand that a fractional exponent is a root. Understand that “radicals” (roots) are the

“opposite” operation of applying exponents. Understand that you can “undo” a power with a

radical, and a radical can “undo” a power.MAFS.912.N-RN.1.AP.1b

Identify a fractional exponent.

Use manipulatives to demonstrate laws of exponents.

Understand the following concepts, symbols and vocabulary: radicand, root index, root, numerator, denominator, radical, exponent.

Understand the parts of a fraction: Click here. Understand how to multiply fractions and whole

numbers. Understand that a fractional exponent is a root. Understand that a root is the inverse of a

fractional exponent. Understand that “radicals” (roots) are the

“opposite” operation of applying exponents. Understand that you can “undo” a power with a

radical, and a radical can “undo” a power.

Suggested Instructional Strategies: Demonstrate that a root is the inverse of a fractional exponent (opposite) Introduce 51/2= , and 81/3= Introduce (51/2)2=5, and (81/3)3 = 8

Supports and Scaffolds: Math is Fun−Exponent Laws: Click here

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FLS: MAFS.912.N-RN.1.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Access Point NarrativeMAFS.912.N-RN.1.AP.2a

Convert from radical representation to using rational exponents and vice versa.

Essential Understandings:Concrete Understandings Representation

Identify expressions with exponents. Create a model with objects to show

that the exponent of a number says how many times to use the number in a multiplication (substitute a chip for each “a” – a7 = a × a × a × a × a × a × a = aaaaaaa).

Use a calculator to compute the expressions.

Calculate using online calculator. Click here

Understand the concepts, symbols, and vocabulary for: expression, exponent, raising to a power, radicand, root index, root, numerator, denominator, radical and exponent.

Simplify expression into expanded form: (x⁴)(x³) =(xxxx)(xxx).

Simplify expression into the simplest form: (x⁴)(x³) = (xxxx)(xxx)= (xxxxxxx)= x7.

Rewrite radicals as fractional exponents (E.g., ∛x = x 1/3).

Rewrite fractional exponents as radicals (E.g., x 1/3 = ∛x).

Suggested Instructional Strategies: Task analysis

o Identify 10 as the place valueo Identify the exponento Multiply by the coefficient

*Model/Lead/Test through the steps of the task analysis Video resource: Click here

Supports and Scaffolds: Internet converters: Click here Graphic organizer Calculator Website support: Click here

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FLS: MAFS.912.N-RN.2.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Access Point NarrativeMAFS.912.N-RN.2.AP.3a

Know and justify that when adding or multiplying two rational numbers the result is a rational number.

MAFS.912.N-RN.2.AP.3b

Know and justify that when adding a rational number and an irrational number the result is irrational.

MAFS.912.N-RN.2.AP.3c

Know and justify that when multiplying of a non-zero rational number and an irrational number the result is irrational.

Essential Understandings:Access Point Concrete

UnderstandingsRepresentation

MAFS.912.N-RN.2.AP.3a

Recognize rational numbers (numbers you can write as a fraction).

Recognize irrational numbers (approximations like square root of 2, or pi).

Identify the patterns in multiplying rational numbers by rational numbers.

Identify the patterns in adding rational numbers by rational numbers

Understand rational number – any number you can write as a fraction.

Understand irrational number – non-repeating, non-terminating decimal number (various square roots, pi).

Understand that you can represent irrational numbers as rational numbers by rounding (e.g., pi).

The sum of two rational numbers is a rational number (e.g., 1/2 c sugar plus 1/4 c sugar = 3/4 c sugar).

The product of two rational numbers is a rational number.

MAFS.912.N-RN.2.AP.3b

Recognize rational numbers (numbers you can write as a fraction).

Recognize irrational numbers (approximations like square root of 2, or pi).

Identify the patterns in adding irrational numbers by rational numbers.

Understand rational number – any number you can write as a fraction.

Understand irrational number – non-repeating, non-terminating decimal number (various square roots, pi).

The sum of a rational number and an irrational number is an irrational number (e.g., 2 + √3 = 2 + √3; 8 + 2π = 8 + 2π )

MAFS.912.N-RN.2.AP.3c

Recognize rational numbers (numbers you can write as a fraction).

Recognize irrational numbers (approximations like square root of 2, or pi).

Identify the patterns in multiplying rational numbers by irrational numbers.

Understand rational number – any number you can write as a fraction.

Understand irrational number – non-repeating, non-terminating decimal number (various square roots, pi).

The product of a rational number and an irrational. number is an irrational number. E.g., 8(2π ¿ = 16πE.g., finding the circumference of a pizza multiplies a rational and irrational number (pi) – when you use the calculator/extended version of pi.

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Suggested Instructional Strategies: Demonstrate that a rational number is any number that you can write as a

whole number or a fraction. E.g., 2 = 21 Understand that when adding two rational numbers, the sum is a rational

number. 2+3=5. 2 +

31 = 5,

12 +

14 =

34

Demonstrate that multiplying a rational number by a rational number will result in a rational number. E.g., 2 x 3= 6, 2 x

31 = 6,

12x 14=18 .

Demonstrate that adding a rational number with an irrational number will result in an irrational number. E.g., 2 + √3 = 2 + √3; 8 + 2π = 8 + 2π Demonstrate that multiplying a rational number with an irrational number will result in an irrational number. E.g., 23 ∙ π =

2π3 , 8(2π ¿ = 16π , -5(2π ¿= -10π

Supports and Scaffolds:

Legos, Base Ten Block, and Pattern Blocks

Scholastic: using Legos for Math Concepts: Click here Math worksheets land lesson: Click here LearnZillion: distinguish between rational and irrational numbers: Click here LearnZillion: predict the result of adding and subtracting rational and

irrational numbers: Click here LearnZillion: multiply rational and irrational numbers: Click here

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FLS: MAFS.912.N-Q.1.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Access Point NarrativeMAFS.912.N-Q.1.AP.1a Interpret units in the context of the problem.MAFS.912.N-Q.1.AP.1c Choose the appropriate units for a specific formula and

interpret the meaning of the unit in that context.Essential Understandings:

Access Point Concrete Understandings Representation MAFS.912.N-Q.1.AP.1a

Identify the unit used in the given problem.

Match the given unit with the appropriate type of measurement.

Identify the type of unit (e.g., milliliters and gallons for liquids. Miles and meters for distance, etc.).

MAFS.912.N-Q.1.AP.1c

Determine what units are used in a problem (e.g., money, time, units of measurement).

Match the unit to the issue within a real-world problem (e.g., How long did the trip take? Solve in hours/ minutes.)

Match the unit to the symbol in a formula.

Determine the appropriate unit for the answer, within the context of the problem.

Suggested Instructional Strategies: Matching assignment: list the different types of units and match to the

correct unit of measure. E.g., Grams, pounds, ounces can be used to measure weight. Centimeters, inches. Feet can be used to measure length. Ounces, Gallons, Liters can be used to measure volume, Seconds, minutes, hours to measure time, Cents, dollars to measure money.

Learn the abbreviations for the units of measurement. Centimeter = cm, Liter = L, Pound = lb., ounces = oz., dollars = $, cents = ȼ, etc.

Supports and Scaffolds: Measurements worksheet: Click here Best measuring units worksheet: Click here Choosing units of measure: Click here Math is Fun−Measure Units: Click here

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FLS: MAFS.912.N-Q.1.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Access Point NarrativeMAFS.912.N-Q.1.AP.1b

Choose and interpret both the scale and the origin in graphs and data displays.

Essential Understandings:Concrete Understandings Representation

Identify the x-axis and y-axis. Identify the origin, starting point

or zero point in a data display

Determine the scale by using addition or multiplication.

Explain the scale in the context of the given problem. (E.g., in the graph below, each block represents 2 students).

Suggested Instructional Strategies: Determine the x-axis and y-axis Demonstrate that the origin of the coordinate plane will always be (0,0). Determine the scale, by examining the coordinate points. E.g., (10,100),

(50,300), (40,600). The scale for the x-axis would increase by 10’s (10, 20, 30, 40, etc.) and the y-axis scale would increase by 100’s (100, 200, 300, 400, etc.)

Match appropriate scale to coordinate points Tape floor to make life size coordinate plane, or spaghetti on sand paper

Supports and Scaffolds: Graph paper, calculator, base ten blocks, ruler, tape, spaghetti, and sandpaper Math is Fun−Cartesian Coordinates: Click here Math Planet−Visualizing Linear Functions: Click here

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FLS: MAFS.912.N-Q.1.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Access Point NarrativeMAFS.912.N-.1.AP.1d When solving a multi-step problem, use units to evaluate

the appropriateness of the solution.

Essential Understandings:Concrete Understandings Representation

Determine what units are used in problem (e.g., money, time, units of measurement).

Match the action of combining with vocabulary (i.e., in all; altogether) or the action of decomposing with vocabulary (i.e., have left; take away, difference) in a word problem.

Apply conversions of units while solving problems (e.g., recognize that monetary units can be combined to equal other monetary units).

Translate wording into numeric equation.

Translate wording into an algebraic equation.

Suggested Instructional Strategies: Task analysis *Model/Lead/Test *Least-to-Most Prompts Create relevant, story-based problems. E.g., the story may be used to solve a

problem about money and shopping at the grocery store. Use graphic organizers to provide students a means for organizing their work. Break down and isolate each step in solving the math task.

Suggested Supports and Scaffolds: $1.00, $5.00, and $10.00 bills Number Line labeled with $1.00/unit, $5.00/unit, and $10.00/unit Calculator, software that counts, or other means of hand tallying Graph Paper where each square equals a unit

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FLS: MAFS.912.N-Q.1.2 Define appropriate quantities for the purpose of descriptive modeling.

Access Point NarrativeMAFS.912.N-Q.1.AP.2a

Determine and interpret appropriate quantities when using descriptive modeling.

Essential Understandings:Concrete Understandings Representation

Match the unit to the issue within a real-world problem (e.g., How long did the trip take? Solve in hours/ minutes.)

Match the equation to the problem.

Understand written representation of time, money, temperature, weight, speed, mass, volume, distance.

Enter data from a problem into the provided equation.

Translate wording into a numeric equation.

Translate wording into an algebraic equation.

Suggested Instructional Strategies: Read word problem. Break down word problem to determine what needs to be solved.

(E.g., Is it asking how long, how far, how many, etc.) Cross off information not needed Highlight important information Select/determine equation

Supports and Scaffolds: Key words for mathematical operations: Click here Equation bank Highlighters

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FLS: MAFS.912.N-Q.1.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Access Point NarrativeMAFS.912.N-Q.1.AP.3a

Describe the accuracy of measurement when reporting quantities (you can lessen your limitations by measuring precisely).

Essential Understandings:Concrete Understandings Representation

Match units of measurement with appropriate quantities (i.e., distance would be measured with inches, feet, miles; volume would be measured with cubic liters).

Order units of measure from smallest to largest and vice versa (i.e., inches, feet, yards, meters).

Round and estimate numbers to a specific digit.

Measure an object using different units (i.e., millimeters and centimeters).

Understand the following concepts and vocabulary: accuracy, place value, precision, significant digits, and significant figures.

Choose a unit of measure from a list of units to determine the most appropriate for the task.

Select a larger or smaller unit of measure given a situation (i.e., the distance from the student to the classroom door – measured in feet; selecting inches or yards).

Describe the difference between an exact measurement and a rounded measurement.

Suggested Instructional Strategies: Determine/match which unit of measurement is appropriate Select/match the appropriate tool to measure distance/weight, etc. Determine/match when it is appropriate to use an exact measure or a

rounded measure.

Supports and Scaffolds: Rulers, measuring cups, scale, tape measure, protractor, and objects to

measure. Measurement tools: Click here Choosing appropriate units of measure: Click here SlideShare tool: Click here

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FLS: MAFS.912.N-CN.1.1 Know there is a complex number i such that i² = –1, and every complex number has the form a + bi with a and b real.

Access Point NarrativeMAFS.912.N-CN.1.AP.1a

Identify i as the square root of -1

MAFS.912.N-CN.1.AP.1b

Identify a number in the form a + bi as a complex number.

Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.N-CN.1.AP.1a

Match i with the √−1

For teacher information: Click here

Understand the following vocabulary and concepts: real number, imaginary number, complex number, squares, square roots.

Understand the concept of a variable.

Understand that i always represents √−1.

MAFS.912.N-CN.1.AP.1b

Match i with the √−1

Match the parts of a complex number with a label. I.e.,

A + bi (a is the real part and bi is the imaginary part) i ꞊꞊ √−1

For teacher information: Click here

Understand the following vocabulary and concepts: like terms, positive integer, negative integer, real number, imaginary number, complex number, squares, square roots.

Understand that a complex number is a combination of a real number and an imaginary number. I.e.,

A + bi (a is the real part and bi is the imaginary part) i ꞊꞊ √−1

Identify the parts of the complex number. For example:

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Suggested Instructional Strategies: Understand that i=√−1 or i2=−1 Understand that the simplified imaginary numbers follow a pattern.

Example activity: Have students make flash cards using the imaginary numbers chart shown above. For instance, on one card students will writei2. On another card students will write−1, and so on. Have student then match the appropriate imaginary number to the simplified version.

Example activity: Using the flash cards made above, have students play a memory game with a partner. Note: It is possible to match an imaginary number to numerous simplified numbers. For example, −1 can be matched to both i2 and i6 as shown in the chart above.

Understand that complex numbers are a combination of a real number and an imaginary number.

Example activity for complex numbers: have students create flash cards with a “real” number or an “imaginary” number on each card. Have each student pick a flash card. Designate one side of the room as “imaginary” numbers and one side as “real” numbers. Students should take their card to the corresponding side of the room. Students in each group will check each other’s cards to insure appropriate placement.

Example activity for complex numbers: teacher will give each student a flash card that has either a real number or an imaginary number on it. Students will then find a partner that is holding a flash card that will create a complex number when combined.

Example activity for complex numbers: teacher will provide students with complex numbers and students will complete the real number and imaginary number chart. For example:

Supports and Scaffolds: Index cards, markers, complex numbers, real numbers, imaginary numbers,

imaginary number chart, or complex number chartResources:

Definition: Click here Introduction to imaginary numbers video: Click here Definitions for complex numbers: Click here Introduction to complex numbers video: Click here

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FLS: MAFS.912.N-CN.1.2 Use the relation i² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Access Point NarrativeMAFS.912.N-CN.1.AP.2a

Use properties to combine like terms of complex numbers in the form of a + bi with a and b being real numbers.

Essential Understandings:Concrete Understandings Representation

Use manipulatives to represent terms for the purpose of simplifying and combining like terms.

Use manipulatives to combine like terms (i.e., demonstrate 5i + 3i by combining like manipulatives).

Create an array of objects for the mathematical equation and match the answer symbol (+ or -) following addition rules for an equation.

Create an array of objects for the mathematical equation and match the answer symbol (+ or -) following subtraction rules for an equation.

Understand the following concepts, symbols, and vocabulary for: like terms, combine, variables, positive integer, negative integer, real number, imaginary number, complex number, squares, square roots.

Use tools, as needed, to add and subtract like terms. For example: (a+bi )+ (c+di )=(a+c )+ (b+d ) i

(2 + 5i) + (5 + 3i) =(7 + 8i)

Suggested Instructional Strategies: Understand that in the formula below (also shown in representation above), a

and c represent real numbers and bi and di represent imaginary numbers.

(a+bi )+ (c+di )=(a+c )+ (b+d ) i

Understand that when you combine like terms the real numbers are combined and the imaginary numbers are combined separately. For example:

4+2 i−3+6 i

(4−3 )+ (2i+6 i )

1+8 i

Example activity: use visuals to represent real numbers and imaginary numbers in complex numbers. E.g., real numbers are designated with a white index card and imaginary numbers are designated with a yellow index card. If the card represents a positive number a happy face is on the card. If the card represents a negative number, a sad face is on the card:

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Given the following problem, have students use the cards to complete the expression:

o Students will need 4 white index cards with happy faces, 3 white index cards with sad faces, 2 yellow index cards with happy faces, and 6 yellow index cards with sad faces. Place all the white index cards in one group and all the yellow index cards in another group. Pair up a white happy face index card with a white sad face index card to cancel each other out. Remove those cards from the group. Continue with the remaining white index cards until no more pairs can be made. If the remaining cards have a happy face it is a positive number. If the remaining cards have a sad face it is a negative number. Repeat the process with the yellow index cards. Given the above expression, the students should have a result of 1 white happy face index card and 4 yellow sad face index cards. This represents the answer.

Supports and Scaffolds: White index cards, yellow index cards, happy faces, sad faces, calculator, and

computer software program

Resources: Complex number calculator: Click here Worksheets: Click here

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FLS: MAFS.912.A-SSE.1.1 Interpret expressions that represent a quantity in terms of its context.

a. Interpret parts of an expression, such as terms, factors, and coefficients.b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

For example, interpret as the product of P and a factor not depending on P.Access Point Narrative

MAFS.912.A-SSE.1.AP.1a

Identify the different parts of the expression and explain their meaning within the context of a problem.

MAFS.912.A-SSE.1.AP.1b

Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts.

Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.A-SSE.1.AP.1a

Match items from a problem with variables (e.g., In the expression 6x + 7y, students explain that Bill had 6 times as many apples and 7 times as many oranges as Sam, with x representing the number of apples and y representing the number of oranges).

Understand the following related vocabulary and symbols: add (+), subtract (-), multiply (x), divide (÷), equal (=), variables, unknown.

MAFS.912.A-SSE.1.AP.1b

Use manipulatives (pattern blocks, two-way counters) to represent portions of the problem.

Use a tool (such as a mat, table or graphic organizer) to separate the expression into parts.

Use algebra tiles to represent a part of the expression.

Use virtual manipulatives to represent the problem. teacher support: Click herestudent manipulative: Click here)

Suggested Instructional Strategies: Identify the variables in a given problem:

o It is $15.00/hour to rent a canoe plus a $4.00 flat fee for life vests. Write/match an expression 15x + 4.

o If Bill wants to rent the canoe himself for 5 hours, the expression would now be 15(5) + 4.

o The policy at the canoe rental has changed and now it cost $4.00/life vest. Write/match expression 15x + 4y.

o If Bill’s brother Jim decides to come and they canoed for 3 hours, the expression would now be 15(3) + 4(2).

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Supports and Scaffolds: Paper money, preferred edibles, blocks, calculators, marbles, pattern blocks,

graphic organizers, etc. Purple Math−translating word Problems: Click here

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FLS: MAFS.A.SSE.2.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or

minimum value of the function it defines.c. Use the properties of exponents to transform expressions for exponential

functions. For example the expression can be rewritten as ≈ to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Access Point NarrativeMAFS.912.A-SSE.2.AP.3a

Write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the meaning of the zeros.

Essential Understandings:Concrete Understandings Representation

Identify expressions with exponents Create a model with objects to show that

the exponent of a number says how many times to use the number in a multiplication (e.g., substitute a chip for each “a” –

a7 = a × a × a × a × a × a × a = aaaaaaa).

Simplify an expression into expanded form (x⁴)(x³) =

(xxxx)(xxx). Simplify expression into the simplest form:

(x⁴)(x³) = (xxxx)(xxx)= (xxxxxxx)=x7.

Understand the following concepts, symbols, and vocabulary for: expression, exponent, raising to a power.

Suggested Instructional Strategies: Explicitly teach rules for simplification. *Multiple-Exemplars (*Example/Non-Example) expression with exponents.

Supports and Scaffolds: Templates Calculator

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FLS: MAFS.912.A-SSE.2.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.b. Complete the square in a quadratic expression to reveal the maximum or

minimum value of the function it defines.c. Use the properties of exponents to transform expressions for exponential

functions. For example the expression can be rewritten as ≈ to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

Access Point NarrativeMAFS.912.A-SSE.2.AP.3b

Given a quadratic function, explain the meaning of the zeros of the function (e.g., if f(x) = (x – c) (x – a) then f(a) = 0 and f(c) = 0).

MAFS.912.A-SSE.2.AP.3c

Given a quadratic expression, explain the meaning of the zeros graphically (e.g., for an expression (x – a) (x – c), a and c correspond to the x-intercepts (if a and c are real).

Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.A-SSE.2.AP.3b

Use a tool to determine whether the quadratic function crosses the x-axis: Click here

Use a graphing tool or graphing software to find the roots (where the function intersects the x-axis) of a function.

Understand the concepts and vocabulary: root, factor, quadratic, integer, real number, quadratic equation, quadratic formula, square root, solution, terms, coefficient, intercept, intersect, zero.

Understand that a root (zero) is the point where a function intersects the x-axis.

MAFS.912.A-SSE.2.AP.3c

Use a tool to determine whether the quadratic function crosses the x-axis: Click here

Use a graphing tool or graphing software to find the roots (where the function intersects the x-axis) of a function.

Match the quadratic equation with its corresponding graph.

Match the graph of a quadratic equation with its roots (zeros).

Understand the concepts and vocabulary: root, factor, quadratic, integer, real number, quadratic equation, quadratic formula, square root, solution, terms, coefficient, intercept, intersect, zero.

Understand that a quadratic function that intersects the x-axis has real roots (zeros)

Teacher tool:Click here

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Suggested Instructional Strategies: Introduce Factored Form: F(x) = (x-a)(x-c) Example:

o (x-2) (x-3) o Set each factor equal to zero: x-2 = 0, x-3=0o Solve for X: x – 2 = 0, x – 3 = 0o Zeros of the quadratic: x=2, x=3 o When graphing this quadratic, it crosses the x-axis at 2 and 3.

Another example:o f(x)= 2x+1o Understand that a root is when y = 0o F(x) = 2x + 1, is the same as y = 2x +1o Replace y with 0 and solve for xo 0 = 2x +1 o 2x = -1o X = -1

2.

o When graphing this quadratic, it crosses the x-axis at -12

. Example:

o Use a number line and a happy face (), two sad faces (), and an equal sign (=) placed above the zero on the number line.

o Using the equation x + 2 = 0, place a happy face over the 2 on the number line.

o To solve the equation, -2 needs to be added to each side of x.o Place a sad face under the 2 on the number line. o In order to demonstrate the factored equation x = -2, have the student

indicate where the next sad face should be placed on the negative side of the number line to indicate the –2.

o Indicate that the happy face and sad face on the positive side cancel each other out, so they can be removed.

Supports and Scaffolds: Algebra tiles, computer software program, Hands-On Equations

Manipulatives, graph paper, and number line. Math Playground–Algebra Equations: Click here Video: solving one-step equations equal to zero with manipulatives:

Click here Video: finding zeros of quadratic functions on graphs Click here

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FLS: MAFS.912.A-APR.1.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Access Point NarrativeMAFS.912.A-APR.1.AP.1a

Understand the definition of a polynomial.

MAFS.912.A-APR.1.AP.1b

Understand the concepts of combining like terms and closure.

MAFS.912.A-APR.1.AP.1c

Add, subtract, and multiply polynomials and understand how closure applies under these operations.

Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.A-APR.1.AP.1a

Identify examples of polynomials. Identify non-examples of

polynomials. Given a field of two, polynomial

and distractor, the student will match to the polynomial.

Understand the following vocabulary and symbols: variable, exponent, constant, and coefficient.

Understand that a polynomial is an expression consisting of variables and coefficients that involves only the operations addition, subtraction, multiplication and non-negative integer exponents.

MAFS.912.A-APR.1.AP.1b

Identify examples of polynomials. Identify non-examples of

polynomials. Sort variables into like terms (e.g.,

sort all the x’s and y’s).

Understand that variable terms can be added, subtracted, and multiplied.

Understand when you multiply, add, or subtract a polynomial you will get a polynomial (which is closure).

MAFS.912.A-APR.1.AP.1c

● Align variables of like terms (e.g., 4x - 0y + z3x + 2y + 2z

● Add, subtract, and multiply coefficients.

● Use manipulatives to combine like terms.

Understand the following vocabulary and symbols: +, -, ×, ÷, =, variable, equation.

Understand that variable terms can be added, subtracted, and multiplied.

Understand that when you add, subtract, or multiply a polynomial as your answer this makes it closed.

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Suggested Instructional Strategies: Explain the rules of a polynomial. Give examples of polynomials:

A polynomial has variables, exponents, and constants. Exponents must be positive and cannot involve division.

Combine Like Terms:

Like Terms are when variables and exponents are the same. The coefficients can be different.

E.g., (5x) x (-3x) are like terms because the variable x is the same. When given the polynomials, (x + 3) (x - 2) you must perform the

multiplication before combining like terms: x2 + 3x – 2x -6 = x2 + x – 6. Understand that closure means that if you add, subtract, and multiply

polynomials the result is a polynomial. Example Activity: give students some of each of the different manipulatives,

e.g., apples, oranges, and carrots. Have students combine like manipulatives so that all the oranges, apples, and carrots are combined into like groups.

Supports and Scaffolds: Manipulatives, computer software program, calculator, and algebra tiles. Multiplying Polynomials: Click here Explains closure: Click here Polynomial definition: Click here Adding and subtracting Polynomials: Click here Multiplying Polynomials: Click here

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FLS: MAFS.912.A-APR.2.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Access Point NarrativeMAFS.912.A-APR.2.AP.3a

Find the zeros of a polynomial when the polynomial is factored (e.g., If given the polynomial equation y = x2 + 5x + 6, factor the polynomial as y = (x + 3)(x + 2). Then find the zeros of y by setting each factor equal to zero and solving. x = –2 and x = –3 are the two zeroes of y.).

MAFS.912.A-APR.2.AP.3b

Use the zeros of a function to sketch a graph of the function.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.A-APR.2.AP.3a

Use a number line to find the opposite number for a given constant (e.g., in(x + 3), the “zero” for the factor is -3.

Identify x-intercept(s) on a graph

Understand the coordinate plane. Understand coordinate pairs. Understand related vocabulary:

factor, x-intercept.

MAFS.912.A-APR.2.AP.3b

Use a graphing tool or graphing software to sketch the graph given the roots (where the function intersects the x-axis) of a function. Click here

Trace the graph of a function given a template.

Identify the zeros of a function on a coordinate plane

Match the polynomial function with its corresponding graph.

Match the graph of the polynomial function with its roots (zeros).

Understand the following concepts and vocabulary: root, factor, quadratic, integer, real number, quadratic equation, quadratic formula, polynomial, degree, exponent, end behavior, square root, solution, terms, coefficient, intercept, intersect, zero.

Understand that a root is where a function crosses the x-axis.

Understand that the degree (largest exponent) of a polynomial determines the type and shape of the graph.

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Concrete Understandings Representation

Determine the direction (end behavior) of the sketch of the graph.

For example:

Teacher tools: Click here and Click here

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Suggested Instructional Strategies: Review rules for polynomials Review factored form: y = (x-a)(x-c) Example:

o (x-2) (x-3) o Set each factor equal to zero: x-2 = 0, x-3=0o Solve for X: x – 2 = 0, x – 3 = 0o Zeros of the polynomial: x=2, x=3 o When graphing this polynomial, it crosses the x-axis at 2 and 3.

Another example:o y = 2x+1o Understand that a root is when y = 0o F(x) = 2x + 1 is the same as y = 2x +1o Replace y with 0 and solve for xo 0 = 2x +1 o 2x = -1o X = -1

2.

o When graphing this polynomial, it crosses the x-axis at -12

. Example:

Example Activity:o Using a number line and a happy face (), two sad faces (), and an

equal sign (=) placed above the zero on the number line. o Using the equation x + 2 = 0, place a happy face over the 2 on the

number line.o To solve the equation, -2 needs to be added to each side of x.o Place a sad face under the 2 on the number line. o In order to demonstrate the factored equation x = -2, have the student

indicate where the next sad face should be placed on the negative side of the number line to indicate the – 2.

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each other out, so they can be removed.

Supports and Scaffolds: Algebra tiles, computer software program, and Hands-On Equations

Manipulatives, graph paper, and number line Math Playground–Algebra Equations: Click here Solving one-step equations equal to zero with manipulatives: Click here Video: Click here Math is Fun–Solving Polynomials: Click here

When x is 0, y = -12Click here

Tutorial: Click here

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FLS: MAFS.912.A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions.Access Point Narrative

MAFS.912.A-CED.1.AP.1a Create linear, quadratic, rational, and exponential equations and inequalities in one variable and use them in a contextual situation to solve problems.

Essential Understandings:Concrete Understandings Representation

Match an equation with one variable to the real-world context.

Use tools, (i.e. manipulatives, algebra tiles, software, equation calculators, etc.) to solve equations and inequalities in one variable.

Create a pictorial array of a simple equation to translate wording.

Understand the following vocabulary and symbols: +, -, ×, ÷, =, linear, variable, inequality, equation, exponent, rational, quadratic.

Use tools, (i.e. manipulatives, algebra tiles, software, equation calculators, etc.) to solve equations and inequalities in one variable.

Create linear equations and inequalities in one variable.

Create quadratic equations and inequalities in one variable.

Create rational equations and inequalities in one variable.

Create exponential equations and inequalities in one variable.

Suggested Instructional Strategies: Task analysis

o Present the story problem based on a real world, relevant context and provide a template for recording facts/operation to solve the real world problem.

o Highlight key information in the problem; strike through irrelevant information.o Identify what question is being asked (define x).o Identify the facts.o Fill-in the facts in the order presented in the story problem on the template.o Determine the operation(s) (+, -, x, ÷).o Identify what operation should be completed first.o Fill in the operation.o State the equation.o Solve for x.o Answer the problem statement.

Suggested Supports and Scaffolds: Counters Multiplication chart Calculator

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FLS: MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Access Point Narrative

MAFS.912.A-CED.1.AP.2a Graph equations in two or more variables on coordinate axes with labels and scales.

Essential Understandings:Concrete Understandings Representation

Match the equation to its graph. Identify point of intersection between two

graphs (of a two-variable equation). Use tools to graph equations in two

variables (i.e., manipulatives, calculators, equation calculators, software, etc.)

Understand the following related vocabulary: axis, labels, scales.

Graph linear equations and inequalities in two variables.

Graph quadratic equations and inequalities in two variables.

Graph rational equations and inequalities in two variables.

Graph exponential equations and inequalities in two variables

Suggested Instructional Strategies: Understand the difference between an equation (linear, quadratic, rational,

and exponential) and inequalities in two variables. If it is an inequality, the equation will contain (<, >, ≤, ≥) and shading is used

within the appropriate area of the graph. Determine whether the equation is linear, quadratic, rational, or exponential. Linear equation: y=x+4, Linear inequality : y<x+4 Quadratic equation: y=x2– 4x-2 Quadratic inequality: y≥ x2– 4x-2 Rational equation: y= Rational inequality: y≤ Exponential equation: y= 2x

Exponential inequality: y>2x

Complete a T-Chart using the appropriate equation. Choose at least three points to create the T-Chart

(E.g., x=-1,0,1). So our three points are: (-1,3), (0,4), (1,5) Create a coordinate plane and plot the coordinate points (x,y). Connect the points. How to plot a line using a T-Chart: Click here

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With inequalities, if the inequality is <, shade below the dashed line. If the inequality is >, shade above the dashed line. If the inequality is ≤, shade below the solid line. If the inequality is ≥, shade above the solid line.

Examples of Graphs:

Linear equation: y=x+4

Linear inequality: y<x+4

Quadratic equation: y=x2– 4x-2

Quadratic inequality: y≥ x2– 4x-2

Rational Equation: y=

Rational Inequality: y≤

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Exponential equation: y= 2x Exponential inequality: y>2x

Supports and Scaffolds: Graph paper, ruler, calculator, and computer software program T-Charts: Click here Math is Fun−Graphing Linear Inequalities: Click here How to plot a line using a T-Chart: Click here How to plot a quadratic equation using a T-Chart: Click here How to plot a rational equation using a T-Chart: Click here How to plot a exponential equation using a T-Chart: Click here Graphing Calculator: Click here

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FLS: MAFS.912.A-CED.1.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Access Point NarrativeMAFS.912.A-CED.1.AP.3a Identify and interpret the solution of a system of linear equations

from a real-world context that has been graphed.

Essential Understandings:Concrete Understandings Representation

Identify the point where the two lines cross.

Identify the two lines on the graph.

Match the solution with its meaning.

Understand the following related vocabulary: more than, less than, equal, equation, inequality.

Given a graph:o Identify the solution within context o Interpret what the solution means within contexto Understand that if the two lines don’t cross there is

no solution

Suggested Instructional Strategies: Bill wants to rent a canoe and has found two companies that will rent the

canoe with life vest. One company rents their canoe for $15.00/hour plus a $4.00 flat fee for life

vests. y=15x + 4 Another company rents the canoe for $19.00/hour and life vest rental is free.

y=19x How many hours in a canoe would result in the same price for both

companies? In this example, both companies would charge the same price for a one-hour rental.

In this graph, when renting the canoes for one hour, the total amount of money spent would match.

Supports and Scaffolds: Computer software programs, and calculator Examples of graphs with one solution, no solutions, and infinite solutions:

Click here

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FLS: MAFS.912.A-CED.1.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Access Point NarrativeMAFS.912.A-CED.1.AP.4a Solve multi-variable formulas or literal equations for a specific

variable.Essential Understandings:

Concrete Understandings Representation Match literal equation with its

function (e.g., area = base × height).

Identify the variables in a literal equation.

Substitute numbers for variables. Isolate one variable in a

multivariate equation.

Understand algebraic rules (e.g., what you do to one side of the equation you must do to the other).

Understand vocabulary related to literal equations (e.g., perimeter, triangle).

Understand related symbols (e.g., A = area,B = Base, etc.)o perimetero p=2l×2w literal equation o solve for lo area of a triangleo A=1/2 ×bho solve for bo 2A= 1/2 × 2bho 2A=1/2 ×2o 2A=bho 2A over h =bo interest = principle x rate x timeo I=prto pi=3.14o rounding skillso calculator skillso approximation

Suggested Instructional Strategies: Given a formula, for example Area = Length x Width, the student will solve

for width (w) by isolating the indicated variable.A¿ lwAl=lwl - The student will need to isolate the w by dividing each side by l.

Al=w - The formula is now solved for w.

Students take cards with sequenced steps that are out of order and put them in order.First card: A¿ lw

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Second card: Al =lwl

Third card: Al =w

Supports and Scaffolds: Dry erase boards, markers, magnets, and cards Algebra Class–Literal Equations: Click here Purple Math–Solve Literal Equations: Click here

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FLS: MAFS.912.A-REI.1.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Access Point NarrativeMAFS.912.A-REI.1.AP.1a Solve equations with one or two variables and explain the

process.Essential Understandings:Concrete Understandings Representation

Match the equation to its solution. Match the step of an equation

problem to its operation or property.

Identify the point of intersection between two graphs (of a two variable equation).

Understand related vocabulary: axis, labels, scales, equation, multi-step equation.

Solve one- and two step equations. Solve multi-step equations. Explain the operation or step used to solve an

equation.

Suggested Instructional Strategies: When given an equation, solve for the missing variable (x). For example:

o Multiply both sides by 4 (Multiplication Property of Equality)5x + 3 = 28

o Subtract 3 from each side of the equation (Subtraction Property of Equality)5x + 3 - 3= 28 – 35x = 25

o Divide both sides of the equation by 5 (Division Property of Equality)x = 5

Create a puzzle piece for each step of the equation, then have students put puzzle pieces together to create the correct sequence of steps to solve for the missing variable.

Supports and Scaffolds: Calculator, dry erase boards, algebra tiles, Hands-On Equations, puzzle

pieces, and markers. Video on sequence steps: Click here Written directions: Click here Equation solver: Click here Game: Click here

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FLS: MAFS.912.A-REI.1.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Access Point NarrativeMAFS.912.A-REI.1.AP.2a Solve simple rational and radical equations in one variable.

Essential Understandings:Concrete Understandings Representation

Identify the variables in an equation.

Substitute numbers for variables.

Understand the following related vocabulary: variable, rational numbers, radical.

Solve radical equations. For example, y=3 √100+√100y=3 (10 )+10

y=40 Solve simple rational equations in one variable.

Suggested Instructional Strategies: Tiling/fill-in space and count. Sequence: 1; Area 2: Volume; 3: Missing attribute. “If the area of a rectangle is 24cm² and it has a base of 6cm, what would the

height be?” Task analysis with *System of Least to Most Prompts Replace a letter (variable representing an unknown quantity) with a number

or representation of a number (symbols, manipulatives). Provide a labeled prism and the equation V = L x W x H. Ask the student to

draw/indicate the label on the prism to the letter in the equation. Break down and isolate each step in solving the math task.

Provide nets to be taken apart (unfolding) to illustrate three-dimensional objects. This process can also be used for the study of the surface area of prisms.

Suggested Supports and Scaffolds: Pre-made formula Use of calculator Manipulatives (2-D shapes, prism, cube (e.g., box)) Counters (e.g., tally counter) and counting mechanism (e.g., number line)

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FLS: MAFS.912.A-REI.2.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Access Point NarrativeMAFS.912.A-REI.2.AP.3a Solve linear equations in one variable, including coefficients

represented by letters.MAFS.912.A-REI.2.AP.3b Solve linear inequalities in one variable, including coefficients

represented by letters.Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.A-REI.2.AP.3a Using manipulatives represent variables.

Identify the unknown quantity when given an equation and labeled figure.

Use manipulatives to solve linear equations.

Understand formula representation (e.g., “h” in the equation means height).

Use letters to represent numbers.

Recognize symbols for equations and operations.

Solve equations and one variable. For example:

2x+4=10 x=3

MAFS.912.A-REI.2.AP.3b Using manipulatives to represent variables.

Identify the unknown quantity when given an inequality and labeled figure.

Use manipulatives to solve linear inequalities

Understand formula representation (e.g., “h” in the equation means height).

Use letters to represent numbers.

Solve inequalities and one variable. For example:2x+4<10 x<3

Recognize symbols for equals (=), addition (+), and multiplication (×), less than (<), greater than (>), less than or equal to (≤), greater than or equal to (≥).

Suggested Instructional Strategies: When given an equation, solve for the missing variable (x). For example:

o Multiply both sides by 4 (Multiplication Property of Equality): bx + 3 = 28

o Subtract 3 from each side of the equation (Subtraction Property of Equality):bx + 3 - 3= 28 – 3bx = 25

o Divide both sides of the equation by b (Division Property of Equality): x =

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o Multiply both sides by 4: bx + 3 = 28

o Subtract 3 from each side of the inequality: bx + 3 - 3 28 – 3bx 25

o Divide both sides of the inequality by b: x Create a puzzle piece for each step of the equation, then have students put

puzzle pieces together to create the correct sequence of steps to solve for the missing variable.

Supports and Scaffolds: Calculator, dry erase boards, algebra tiles, Hands On Equations, puzzle

pieces, and markers Algebra Class–Literal Equations: Click here Purple Math–Solve Literal Equations: Click here Video: Click here Literal equation: Click here

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Elements CardFLS: MAFS.912.A-REI.2.4 Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)² = q that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Access Point NarrativeMAFS.912.A-REI.2.AP.4a Solve quadratic equations by completing the square.MAFS.912.A-REI.2.AP.4b Solve quadratic equations by using the quadratic formula.MAFS.912.A-REI.2.AP.4c Solve quadratic equations by factoring.

Essential Understandings:Access Point Concrete Understandings Representation MAFS.912.A-REI.2.AP.4a

Count the number of terms in a quadratic equation.

Identify the numbers in the equation that would be used to complete the square.

Multiply and divide numbers using appropriate tools.

Use appropriate tools to calculate squares and square roots.

Order the steps for solving an equation using completing the square.

Perform operations on fractions. Simplify fractions.

Simplify radicals. Calculate using multiple

operations. Substitute numbers from the

equation into a template for completing the square.

Calculate squares and square roots using appropriate tools.

Determine the root from adding the terms.

Determine the root from subtracting the terms.

Understand the following vocabulary and concepts: + symbol, factor, coefficient, terms, exponent, base, constant, variable, binomial, monomial, polynomial, square, square root.

Move the constants to the same side of the equation.

Take half the middle term, square it, and add to both sides of equal sign.

Factor the left side into a perfect square.

Take the square root of both sides

Solve for x For example:

x2+6 x−16=0

x2+6 x=16

x2+6 x+9=16+9

x2+6 x+9=25

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(x+3)2= 25

√(x+3)2=√25x + 3 = ± 5x = -5 – 3, x = 5 – 3x = -8, 2

MAFS.912.A-REI.2.AP.4b

Identify the quadratic formula. Identify the numbers in the

equation that would be used in the quadratic equation.

Use appropriate tools to calculate squares and square roots.

Order the steps for solving an equation using the quadratic formula.

Perform operations under a radical.

Calculate using multiple operations.

Substitute numbers from the equation into the quadratic formula.

Calculate using the quadratic formula using appropriate tools.

Determine the root from adding the terms.

Determine the root from subtracting the terms.

Understand the following vocabulary and concepts: + symbol, factor, coefficient, terms, exponent, base, constant, variable, binomial, monomial, polynomial, square, square root.

MAFS.912.A-REI.2.AP.4c

Use the distributive property to simplify expressions with manipulatives.

Use a template to expand and simplify the product of binomials.

Use algebra tiles to multiply binomials.

Use algebra tiles to factor quadratic equations.

Order the steps for multiplying binomials.

Order the steps for factoring quadratic equations.

Calculate the squares and square roots of rational numbers using appropriate tools.

Identify the factors of rational numbers using appropriate tools.

Use the distributive property to simplify equations.

Use models to determine the factors of quadratics.

Use tools appropriately to determine factors for constants and coefficients (i.e., multiplication tables, calculators, multiplication matrixes).

Understand the following concepts and vocabulary: factor, coefficient, terms, exponent, base, constant, variable, binomial, monomial, polynomial.

Find the zeros of the equation.

Suggested Instructional Strategies: Completing the Square:

o Middle term is 6x and the constant is -16.x2+6 x−16=0

o Move the constants to the same side of the equation.2

Elements Cardx2+6 x−16+16=0+16x2+6 x=16

o Take half the middle term, square it, and add to both sides of the equal sign (half of 6 equals 3 and 32 = 9, therefore 9 is added to both sides of the equation.)

x2+6 x+9=16+9

x2+6 x+9=25o Factor the left side.

x2+6 x+9 factors as (x+3) (x+3) = (x + 3)2(x+3)2= 25

o Take the square root of both sides√(x+3)2=±√25x + 3 = ± 5

o Solve for x x = -5 – 3, x = 5 – 3x = -8, 2

Quadratic Formula

Example: x2+6 x−16=0ax2 + bx + ca = 1, b = 6, c = -16

o Solve for addition, solve for subtraction

x = 2, -8 Factoring

o Factor out a common term if needed. E.g., 4x2 – 20 = 4(x2 – 5) Rules for factoring:

o Factoring Trinomials1. Factor out any common terms first

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2. Look at the third term (constant). If it is positive, then you are adding to find the middle term. The middle term then determines if the signs in the parentheses are subtraction signs or both addition signs.

x2+5x+6 (x + 2) (x + 3)x2−5 x+6 (x - 2) (x - 3)

3. Look at the third term (constant). If it is negative, then you will subtract to find the middle term. One sign is positive and the other is negative in the parentheses. Think about what two numbers multiple together to give the last term but add (subtract) to give the last term.

x2−5 x−6 -6 * 1 = -6 -6 + 1 = -5 (x-6) (x+1)x2+5 x−6 6 * -1 = -6 -1 + 6 = 5 (x+6) (x-1)x2−x−6 -3 * 2 = -6 -3 + 2 = 1 (x-3) (x+1)

Example:x2+6 x−16=0(x + 8) (x – 2) = 0x + 8 = 0, x – 2 = 0x = -8, x = 2

Example:o Use manipulatives to represent the factors of 16 in the above

example and place into groups. (1 Skittle, 2 Skittles, 4 Skittles, 4 Skittles, 8 Skittles, 16 Skittles.)

o For instance, use straws to represent x, use edibles or manipulatives (Skittles) to represent the amounts in the formula (1 straw+ 8 Skittles) (1 straw– 2 Skittles) = 0.

o Students will determine if by adding or subtracting will result in 6 Skittles.

Supports and Scaffolds: Calculator, scientific calculator, software program with scientific calculator,

algebra tiles, index cards for matching activity, multiplication table, and edibles or manipulatives.

Video of quadratic song: Click here Explanation of quadratic formula

o Purple Math: Click hereo Regents Prep: Click here

Explanation of completing the square o Math is Fun: Click hereo Regents Prep: Click here

Completing the square calculator: Click here Factoring calculator: Click here Quadratic calculator: Click here

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FLS: MAFS.912.A-REI.3.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Access Point NarrativeMAFS.912.A-REI.3.AP.5a Create a multiple of a linear equation showing that they are

equivalent (e.g., x + y = 6 is equivalent to 2x + 2y = 12).MAFS.912.A-REI.3.AP.5b Find the sum of two equations

Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.A-REI.3.AP.5a Duplicate original equation and add like terms.

Multiply each term by the same number.

Identify coefficients and variables in a system of two equations.

Click here

Understand the following related vocabulary: variable, coefficient, copy, duplicate, terms.

Understand the distributive property.

MAFS.912.A-REI.3.AP.5b Use a tool to represent a system of two equations (e.g., transfer the coefficients from the equations to a matrix template, equation calculator, algebra tiles, etc.).

Identify coefficients and variables in a system of two equations.

Combine like terms.

Suggested Instructional Strategies: Create a multiple of a linear equation showing that they are equivalent Duplicate original equation and add like terms: 2x +3y = 10 Multiply each term by the same number: 4(2x +3y = 10); 8x + 12y = 40 Find the sum of two equations.

o Example 1: adding by elimination Solving systems of equations with addition

o x + y = 6o 3x - y = -2

Add the two equations togethero x + y = 6o +3x - y = -2o 4x = 4

Solve for x1

Elements Cardo 4x = 4o x = 1

Plug 1 into x in either equation and solve for yo 1 + y = 6o y = 5

The solution to the system of equations is (1,5) Solving systems of equations by multiplying

o 2x + 4y = 3o x – 3y = 1

You can add these two system of equations together because nothing will cancel. Multiple the second equation by -2 so that the x’s will cancel. You will need to multiple every term in the second equation by -2.

o 2x + 3y = 3o -2(x – y = 9)o 2x + 4y = 3o -2x + 2y = -18

Now add the two equationso 2x + 3y = 3o -2x + 2y = -18o 5y = -15

Solve for xo 5y = -15o y = -3

Plug -3 into y in either equation and solve for xo 2x + 3(-3) = 3o 2x – 9 = 3o 2x = 12 o x = 6

The solution to the system of equations is (6,-3) Example Activity:

o Supply students with a predetermined amount of nickels and quarters. o Sally has three times as many quarters as nickels. She has $5.60. How

many quarters and nickels does she have?o Student will slide down one nickel and three quarters. Student adds

the money to find sum. o Student continues to slide down one more nickel and three more

quarters until the sum equals $5.60. o Student will then determine how many nickels and how many quarters

are required to equal $5.60.

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Supports and Scaffolds: Calculator, money calculator, nickels, quarters, and algebra tiles, Online calculator: Click here Purple Math: Click here Elimination Method: Click here Video–solving systems of equations with manipulatives: Click here Solving systems with manipulatives: Click here Candy activity sheet: Click here

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FLS: MAFS.912.A-REI.3.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Access Point Narrative

MAFS.912.A-REI.3.AP.6a Given a graph, describe or select the solution to a system of linear equations.

Essential Understandings:Concrete Understandings Representation

Manipulate lines on a graph to show no solution (parallel).

Manipulate lines on a graph to show one solution (point of intersection).

Manipulate lines on a graph to show infinite solutions (the same line).

Locate coordinate pairs.

Understand ordered pairs. Understand the following vocabulary:

intersection, parallel, and infinite. Describe the solutions to the linear

equations.

Suggested Instructional Strategies:

One Solution

No Solution

Infinite Solutions

Give students a graph with two lines intersecting at one point. Provide students with a field of three cards, one with the solution

(point of intersection), and two distractors. Students must match the correct solution (point of intersection) to the graph. Have students glue a colored string across a line on a graph. Glue different colored string across the other line.

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have students make parallel (no solution), intersecting (solution), and infinite solutions (lines on top of each other).

Supports and Scaffolds: Pre-made graphs, solution cards, spaghetti, paper strips, straws, colored

string, glue, and graph paper Math is Fun−Systems of Linear Equations: Click here Examples of graphs with one solution, no solutions, and infinite solutions:

Click here Example and video (right side of screen): Click here

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FLS: MAFS.912.A-REI.4.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Access Point NarrativeMAFS.912.A-REI.4.AP.10a Identify and graph the solutions (ordered pairs) on a graph of an

equation in two variables.

Essential Understandings:Concrete Understandings Representation

Create a table of values from an equation. Graph an equation using a table of values. Locate coordinate pairs on a graph.

Understand ordered pairs. Understand the following vocabulary:

solution, variable, graph, and coordinate plane.

Understand that all solutions to an equation in two variables are contained on the graph of that equation.

Suggested Instructional Strategies: Understand the difference between an equation (linear, quadratic, rational,

and exponential) in two variables. Determine whether the equation is linear, quadratic, rational, or exponential.

o Linear equation: y=x+4o Quadratic equation: y=x2– 4x-2o Rational Equation: y= x+1x+2o Exponential equation: y= 2x

Complete a T-Chart using the appropriate equation, Choose at least three points to create the T-Chart. (E.g., x=-1,0,1)

So our three points are: (-1,3), (0,4), (1,5)

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Elements Card Create a coordinate plane and plot the coordinate points (x, y). Connect the points. How to plot a line using a T-Chart: Click here Examples of Graphs:

Linear equation: y=x+4

Quadratic equation: y=x2– 4x-2

Rational Equation: y= x+1x+2

Exponential equation: y= 2x Have students glue a colored string across a line or curve on a graph to

indicate a linear, quadratic, rational and/or exponential equation. Using spaghetti, laminated thin strips of paper, or straws to represent lines or

curves. Have students create linear, quadratic, rational, exponential graphs. Give students an ordered pair and three different graphs (linear, quadratic,

rational, or exponential). Have students determine which graph represents the ordered pair.

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Supports and Scaffolds: Graph paper, ruler, calculator, and computer software program Example:

Lesson and video: Click here T-Charts: Click here How to plot a line using a T-Chart: Click here How to plot a quadratic equation using a T-Chart: Click here How to plot a rational equation using a T-Chart: Click here How to plot a exponential equation using a T-Chart: Click here Graphing Calculator

Click here - Desmos Calculator

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FLS: MAFS.912.A-REI.4.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Access Point NarrativeMAFS.912.A-REI.4.AP.11a Understand the solution to a system of two linear equations in

two variables corresponds to a point(s) of an intersection of their graphs, because the point(s) of intersection satisfies both equations simultaneously.

Essential Understandings:Concrete Understandings Representation

Manipulate lines on a graph to show one solution (point of intersection).

Manipulate lines on a graph to show infinite solutions (the same line).

Locate coordinate pairs.

Understand ordered pairs. Understand the transitive property.

For example: If y = f(x) and y = g(x), then f(x) = g(x)

Understand the following vocabulary: substitution, intersection, solution, variables.

Suggested Instructional Strategies: Understand the transitive property, e.g., If y = f(x) and y = g(x), then f(x) = g(x)

One Solution Infinite Solutions Give students a graph with two lines intersecting at one point. Provide students with a field of three cards, one with the solution (point of

intersection), and two distractors. Students must match the correct solution (point of intersection) to the graph. Have students glue a colored string across a line on a graph. Glue different colored string across the other line. Have students indicate the point of intersection.

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Elements Card Using spaghetti, laminated thin strips of paper, or straws to represent lines,

have students make an intersecting (solution), and infinite solutions (lines on top of each other).

Create a coordinate plane and points of intersection using sidewalk chalk.

Supports and Scaffolds: Pre-made graphs, solution cards, spaghetti, paper strips, straws, colored

string, glue, graph paper, and sidewalk chalk Math is Fun–Linear Equations: Click here Examples of graphs with one solution, and infinite solutions: Click here Example and video (right side of screen): Click here

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FLS: MAFS.912.A-REI.4.12 Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Access Point NarrativeMAFS.912.A-REI.4.AP.12a Graph a linear inequality in two variables using at least two

coordinate pairs that are solutions.MAFS.912.A-REI.4.AP.12b Graph a system of linear inequalities in two variables

using at least two coordinate pairs for each inequality.Essential Understandings:

Access Point Concrete Understandings Representation MAFS.912.A-REI.4.AP.12a Identify the y-intercept (where

the line crosses the y-axis, or x = 0).

Graph the linear inequality. Understand that a linear

inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality.

Identify above and below the boundary line.

Locate coordinate pairs. Identify or shade the half-plane

that is a solution to the inequality.

Understand ordered pairs.

Understand coordinate planes.

Understand <, >, =. Understand the following

related vocabulary: boundary line and linear inequality).

MAFS.912.A-REI.4.AP.12b Identify the y-intercept (where the line crosses the y-axis, or x = 0).

Graph the linear inequality. Understand that a linear

inequality divides the coordinate plane into two halves by a boundary line where one half represents the solutions of the inequality.

Understand ordered pairs.

Understand coordinate planes.

Understand <, >, =. Understand the following

related vocabulary: boundary line and linear inequality.

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Elements CardAccess Point Concrete Understandings Representation

Identify above and below the boundary line.

Locate coordinate pairs. Identify or shade the half-plane

that is a solution to the inequality.

Graph the second inequality on the same coordinate grid. The overlapping shaded area is the solution to the system.

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Suggested Instructional Strategies: Complete a T-Chart using the appropriate equation

o Choose at least three points to create the T-Chart (e.g., x = -1,0,1)

o So our three points are: (-1,3), (0,4), (1,5)o Create a coordinate plane and plot the coordinate points (x,y)o Connect the points

How to plot a line using a T-Chart: Click here Identify the y-intercept (where the line crosses the y-axis or x = 0) Linear Inequality

o With Inequalities, if the inequality is <, shade below the dashed line. If the inequality is >, shade above the dashed line. If the inequality is ≤, shade below the solid line. If the inequality is ≥, shade above the solid line.

o The line is dashed because the equation is < instead of ≤− Example: on graph paper, have students glue red yarn on one

linear inequality, shade the solution of the linear inequality with a red crayon.

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Elements Card System of Linear Inequalities

o The solution will be the part of the graph that is shaded from both lines, darkest shading. (E.g., similar to a Venn diagram where the solution would be the section where the circles overlap.)

o The solution to the system of linear inequalities is where the two shaded regions overlap.

− Example: using two transparent sheet protectors, have students create the coordinate plane, dashed or solid lines, and shading to represent the appropriate inequality. Have students graph the second inequality on the other sheet protector. Have students overlap the two graphs (two sheet protectors) and shade in where the two graphs overlap to indicate the solution.

− Example: on graph paper, have students glue red yarn on one linear inequality, shade the solution with a red crayon. Then glue yellow yarn on the second inequality and shade the solution with a yellow crayon. The overlapping area (solution of the system) will be the orange area.

Supports and Scaffolds: Graph paper, ruler, calculator, computer software program, sheet protectors,

markers, yarn, crayons, and glue T-Charts: Click here Math is Fun–Graphing Linear Inequalities: Click here How to plot a line using a T-Chart: Click here Purple Math–Systems of Linear Inequalities: Click here Systems of Inequalities: Click here Graphing Linear Inequalities: Click here Videos: Click here Graphing calculator: Desmos Calculator

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Calculator to find the solution to a linear inequality

Calculator to find the solution to the system of inequalities

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FLS: MAFS.912.F-IF.1.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).FLS: MAFS.912.F-IF.1.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Access Point Narrative

MAFS.912.F-IF.1.AP.1a Demonstrate that to be a function, from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range.

MAFS.912.F-IF.1.AP.1b Map elements of the domain sets to the corresponding range sets of functions and determine the rules in the relationship.

MAFS.912.F-IF.1.AP.2a Match the correct function notation to a function or a model of a function (e.g., x f(x) y).

Essential Understandings:Access Point Concrete

UnderstandingsRepresentation

MAFS.912.F-IF.1.AP.1a Evaluate an expression using substitution (e.g., find the value of 4x = y, when x = 2, y = 8 using manipulatives).

Input - domain (x), Output (y)

Recognize in an ordered pair that the first number represents the domain (x-value) and the second number represents the range (y-value).

Understand function notation (e.g., in y = f(x) is another way to write y is f(x), also stated f of x).

Understand x as the input and y as the output (cause and effect).

Understand that the graph of f is the graph of the equation y = f(x).

MAFS.912.F-IF.1.AP.1b Evaluate an expression using substitution (ex: find the value of x + 1 = y, for example when x = -1, y = 0 using manipulatives).

Recognize in an ordered pair that the first number represents the domain (x-value) and the second number represents the range (y-value).

Understand function notation (e.g., in y = f(x) is another way to write y is f(x), also stated f of x).

Understand x as the input and y as the output (cause and effect).

Understand that the graph of f is the graph of the equation y = f(x).

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Elements CardAccess Point Concrete

UnderstandingsRepresentation

MAFS.912.F-IF.1.AP.2a Use concrete cause and effect examples (e.g., add blue (x) to yellow [f(x)] to get green (y).

Use distance (y)/time (x) scenarios where movement f(x) is the function of time (e.g., how long does it take to cross the room).

Use a function box, e.g.,

Recognize in an ordered pair that the first number represents the domain (x-value) and the second number represents the range (y-value).

Understand function notation (e.g., in y = f(x) is another way to write y is f(x) – read f of x).

Understand x as the input and y as the output (cause and effect).

Understand that the graph of f is the graph of the equation y = f(x).

Suggested Instructional Strategies: Introduce that in an ordered pair, the first number (domain) represents the x-

value. The second number (range) represents the y-value. Explain that the x-value is the input (cause) and the y-value is the output (effect). Understand that the function notation is y=f(x) and another way to write y is

f(x), which is stated f of x.

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Elements Card In order for the equation to be a function, the x-value can never repeat. The

graph below represents an equation that is a function.For example: y=2x, Function notation: f(x)=2x.

The graph below represents an equation that is not a function because thex-value 2, is repeated.

When mapping an equation, the T-chart is replaced with

Hands-On activity:o Supply students with a predetermined amount of nickels and quarters

and a blank T-Chart. o Sally has two times as many quarters as nickels. y=2x. y = quarters,

x = nickels

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Elements Cardo Student will slide down one nickel and two quarters. Student will fill in

the appropriate line on T-chart.o Student continues to slide down one more nickel and two more quarters

until the T-Chart is complete.

o Given a real life scenario, the student will match the function notation to the situation.− For Example: a pack of gum is on sale for “buy one get one free”. If

Sally buys one pack of gum she gets a total of two packs. If she buys 2 packs of gum, she gets 4 packs, etc.

− How many packs of gum would she get if she buys “x” number of packs of gum? f(x)=2x.

Supports and Scaffolds: Graph paper, calculator, ruler, vocabulary index cards, nickels, quarters, and

T-Chart. Graphing domain and range functions: Click here Math is Fun–Domain, Range, Codomains: Click here Video: Click here Domain and range from context: Click here

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FLS: MAFS.912.F-IF.1.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.Access Point Narrative

MAFS.912.F-IF.1.AP.3a Recognize that the domain of a sequence is a subset of the integers.

Essential Understandings:Concrete Understandings Representation

Use a T-chart and manipulatives to pair the values in the domain to the values in the range.

Identify what is and is not a function when given a set of values in a relation.

Use a T-chart to predict the output (y-value).

Understand the following related vocabulary: domain, range, and sequence.

Suggested Instructional Strategies: Understand that an integer is any positive or negative whole number. Understand that a sequence is a list of numbers in order.

E.g., -2, 0, 2, 4, 6, 8 etc. Activities:

o Students will order numbers in a sequence.o Human number line. Give students each a number from the sequence

and have them line up in the correct sequence. o Give each student a set of magnetic numbers and have them sequence

them in order.o Use a T-Chart to make a sequence. (0, 2, 4, 6, 8)

o The subset of the domain (x-value) is 0, 1, 2, 3, 4

Supports and Scaffolds: Number Line, numbers, magnetic numbers, T-Chart, and calculator Math is Fun–Functions: Click here

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FLS: MAFS.912.F-IF.2.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Access Point NarrativeMAFS.912.F-IF.2.AP.4a Recognize and interpret the key features of a function.MAFS.912.F-IF.2.AP.4b Select the graph that matches the description of the

relationship between two quantities in the function.Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.F-IF.2.AP.4a Use objects to demonstrate individual key features on a number line or graph.

Identify the x- and y-axis, data points.

Understand the following related vocabulary: increasing, decreasing, positive, negative; maximum, minimums, and symmetry.

MAFS.912.F-IF.2.AP.4b Match individual key features with the relationship between x and y values in a graph.

Understand the following related vocabulary: increasing, decreasing, positive, negative; maximum, minimums, and symmetry.

Suggested Instructional Strategies:

Example graph above: Has a positive slope, Increasing from negative infinity to infinity and a line symmetric with respect to the origin (0,0).

Example graph above: Minimum at (0,0), decreases from negative infinity to zero and increases from zero to infinity. Symmetric with respect to the y-axis.

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Elements Card Example activity: give students three different graphs, provide students with

key features (increasing, decreasing, positive, negative, maximum, minimum, and symmetry) then have students choose the graph that satisfies the key features. For example: Have students select the graph that shows a positive slope and increases from negative infinity to infinity.

Example activity: provide students with a cue card of a graph and a field of three additional. (One cue card, two distractors.) Have students match their cue card to the identical cue card.

Supports and Scaffolds: Graphs, cue cards with graphs, and distractors Math is Fun–Functions: Click here Symmetry: Click here Increasing/decreasing: Click here Maximum and minimum: Click here Key features of a graph video: Click here Example graphs and activities: Click here Graph matching activity: Click here

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FLS: MAFS.912.F-IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

Access Point NarrativeMAFS.912.F-IF.2.AP.5a Given the graph of a function, determine the domain.

Essential Understandings:Concrete Understandings Representation

Pair domain numbers to positions on the x-axis of a coordinate plane.

Label the domain as positive or negative.

Understand related vocabulary: positive, negative.

Understand coordinate planes. Understand the subsets of numbers (i.e., integers,

whole numbers, natural numbers) within the real number system.

Suggested Instructional Strategies: The Domain is the set of all x-values.

For Example, in the graph above, the domain is {1,2,3,5,6}.

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Elements Card For Example, in the graph above, the domain is the set of all real numbers

(negative infinity to infinity).

For Example, in the graph above, the domain is the set of all real numbers (negative infinity to infinity).

Example activity: provide the graph and have students provide the domain, all the x-values.

Example activity: provide the graphs and have students match to the matching graph.

Example activity: provide the domain (x-values) and have the students match to the correct graph.

Supports and Scaffolds: Graphs, and x-values Graphing the domain and range: Click here Graph matching activity: Click here Graph matching activity: Click here

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Elements CardFLS: MAFS.912.F-IF.2.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Access Point Narrative

MAFS.912.F-IF.2.AP.6a Describe the rate of change of a function using words.MAFS.912.F-IF.2.AP.6b Describe the rate of change of a function using numbers.MAFS.912.F-IF.2.AP.6c Pair the rate of change with its graph.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.F-IF.2.AP.6a

Manipulate lines on a graph to show steepness.

Manipulate lines on a graph to show rise or fall.

Manipulate lines on a graph to show positive or negative.

Identify the concepts of steepness, rise, and fall in real-life contexts (e.g., ramps, roofline, stairs, escalators).

Define rate of change (describes the average rate at which one quantity is changing with respect to something else changing).

Identify common rate of change o Miles per gallon – calculated by

dividing the number of miles by the number of gallons used.

o Cost per kilowatt – calculated by dividing the cost of the electricity by the number of kilowatts used.

o Miles per hour.

Understand related vocabulary (domain, range, rise, fall, steepness, increase, decrease, positive, negative).

Identify the concepts of steepness, rise, and fall in visual images (e.g., pictures of ramps, roofline, stairs, escalators).

MAFS.912.F-IF.2.AP.6b

Locate coordinate pairs. Identify the concepts of steepness,

rise and fall in real-life contexts (e.g., ramps, roofline, stairs, escalators).

Pair domain with “run” and range with “rise.”

Understand the following related vocabulary: domain, range, rise, rise over run, fall, steepness, ratio, increase, decrease, positive, negative, y-intercept, and x-intercept).

Identify coordinate pairs on a coordinate plane.

Identify the concepts of steepness, rise, and fall in visual images (e.g., pictures of ramps, roofline, stairs, escalators).

Understand that “rise over run” means vertical change over horizontal change (∆y / ∆x).

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Elements CardAccess Point

Concrete Understandings Representation

m=y2− y1x2−x1

MAFS.912.F-IF.2.AP.6c

Match rate of change with the graph.

Understand the following related vocabulary: domain, range, rise, rise over run, fall, steepness, ratio, increase, decrease, positive, and negative.

Identify coordinate pairs on a coordinate plane.

Identify the concepts of steepness, rise, and fall in visual images (e.g., pictures of ramps, roofline, stairs, escalators).

Understand that “rise over run” means vertical change over horizontal change (∆y / ∆x).

Suggested Instructional Strategies: Understand that rate of change is slope. Given a real life scenario, the student will match the function notation to the

situation.o For Example: a pack of gum is on sale for “buy one get one free”. If

Sally buys one pack of gum she gets a total of two packs. If she buys 2 packs of gum, she gets 4 packs, etc.

o How many packs of gum would she get if she buys “x” number of packs of gum? f(x)=2x

The rate of change in words: 2 packs of gum for the price of 1 pack of gum. The rate of change in numbers: 21 or 2:1, or 2

The rate of change in graph form: 21 or 2:1, or 2

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o Example activity: using the graph above, teach the students to count from 1 point on the graph to the next by counting up and over. (Up 2 lines, over 1 line.)

Slope = riserun

In the graph above the rate of change (slope) is 21

In the graph above, there is a negative slope: −31 . Understand that in rise over run formula, if you move to the left, the result is a negative slope.

o Example activity: tape a coordinate plane on the floor. Have one student stand at (1,1). Have another student stand at (3,6). Have a third student stand by first student and count their steps up and over to get to the second student. The steps up indicate the rise and the steps over indicate the run.

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Elements Cardo Example activity: give students a field of three graphs and have them

select the graph that has a slope of 21Supports and Scaffolds:

Graph paper, tape, number line, geoboards and geobands Slope and y-intercept: Click here Slope and rate of change: Click here Multi-sensory activities: Click here Graph activity: Click here Slope Rummy Game: Click here Slope manipulative: Click here Definitions: Click here

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Elements CardFLS: MAFS.912.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

FLS: MAFS.912.F-IF.3.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such asy = , y = , y = , y = , and classify them as representing exponential growth or decay.

Access Point NarrativeMAFS.912.F-IF.3.AP.7a Select a graph of a function that displays its symbolic

representation (e.g., f(x) = 3x + 5).MAFS.912.F-IF.3.AP.7b Locate the key features of linear and quadratic equations.MAFS.912.F-IF.3.AP.8b Describe the properties of a function (e.g., rate of change,

maximum, minimum).Essential Understandings:

Access Point Concrete Understandings Representation MAFS.912.F-IF.3.AP.7a

Indicate the point on the line that crosses the y-axis(y-intercept).

Indicate “rise over run” of a line on the coordinate plane from the point of origin (0,0).

Substitute “rise over run” for m and y-intercept for b in the formula y = mx + b.

Convert function notation (e.g., f(x) = 3x + 5) to slope intercept form(y = mx + b) and vice versa.

Identify the y-intercept and slope on a graph.

MAFS.912.F-IF.3.AP.7b

Use objects to demonstrate individual key features on a graph.

Understand the following related vocabulary: increasing, decreasing, positive, negative, maximum,

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Elements CardAccess Point Concrete Understandings Representation

Identify x- and y-axis, data points.

minimums, symmetry, roots and zeros.

MAFS.912.F-IF.3.AP.8b

Point to the highest point on the graph as the maximum.

Point to the lowest point on a graph as the minimum.

Identify rise/run. (rate of change)

Understand the following related vocabulary: slope, rate of change, y-intercept, rise, run, high, low, maximum and minimum.

Understand when y = mx + b thatm = rate of change and b =y-intercept.

Understand the coordinate plane.Suggested Instructional Strategies:

Example of linear graph above: has a positive slope, increasing from negative infinity to infinity. It has a root (zero) when x = 3. (Where it crosses the x-axis) E.g., plug 3 into f(x)=x - 3, then y = 0.

Example of quadratic graph above: minimum at (-0.5,6.25), decreases from negative infinity to -0.5 and increases from -0.5 to infinity. Symmetric with respect to the line x = -0.5. It has two roots, so the roots of this function are x = -3, and x = 2.

o E.g., plug -3 or 2 into f(x) = x2 +x – 6, then y = 0. o Example activity: give students three different graphs, provide students with

key features (increasing, decreasing, positive, negative, maximum, minimum, symmetry, and roots (zero).) Then have students choose the graph that satisfies the key features. E.g., have students select the graph that shows a positive slope and increases from negative infinity to infinity.

o Example activity: provide students with a cue card of a graph and a field of three additional. (One cue card, two distractors.) Have students match their cue card to the identical cue card.

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Elements CardSupports and Scaffolds:

Graphs, cue cards with graphs, and distractors Function: Click here Symmetry: Click here Increasing/decreasing: Click here Maximum and minimum: Click here Key features of a graph video: Click here Example graphs and activities: Click here Graph matching activity: Click here Roots (zeros): Click here

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FLS: MAFS.912.F-IF.3.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = , y = , y = , y = , and classify them as representing exponential growth or decay.

Access Point NarrativeMAFS.912.F-IF.3.AP.8a Write or select an equivalent form of a function (e.g., y = mx + b,

f(x) = y, y – y1 = m(x – x1), Ax + By = C).

Essential Understandings:Concrete Understandings Representation

Match equivalent forms of a function.

Understand similarities in functions. Understand point slope form (y – y1 = m(x – x1). Understand slope intercept form (y=mx + b). Understand function form (f(x) = 3x + 5). Understand standard form (Ax + By = C).

Suggested Instructional Strategies: Start with a point and a slope, e.g., (1,2) m = 3 Point slope y-intercept form: y – y1 = m(x – x1). E.g., y – 2 = 3(x – 1) Slope intercept form: (y=mx + b). E.g., y = 3x – 1 Function form: (f(x) = 3x + 5). E.g., F(x) = 3x - 1 Standard form: (Ax + By = C). E.g., 3x – y = 1 Activity: have students select or match equations. Activity: have students plug numbers into the equation when given the point and

slope.

Supports and Scaffolds: Equations, pencils, and paper Math is Fun–Linear Equations: Click here Slope manipulative: Click here Definitions: Click here Game: Click here

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FLS: MAFS.912.F-IF.3.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Access Point NarrativeMAFS.912.F-IF.3.AP.9a Compare the properties of two functions.

Essential Understandings:Concrete Understandings Representation

Identify properties of a function on a graph (for example, slope, increasing or decreasing, where does it cross the x- and y-axis).

Identify if a function exists given a table. Match the symbolic representation of the same function. Identify the properties of a function using a table. Compare the properties of one function to the properties

of another function.

Understand the following concept, vocabulary, and symbols of: function.

Identify properties of a function given a graph, table, or equation.

Suggested Instructional Strategies: Supply the students with two different graphs.

o Example: two linear graphs.

Example of linear graph above: has a positive slope, increasing from negative infinity to infinity. It has a root (zero) when x = 3 (where it crosses the x-axis.) E.g., plug 3 into f(x) = x - 3, then y = 0.

Example of linear graph above: has a positive slope, increasing from negative infinity to infinity. It has a root (zero) when x = 0 (where it crosses the x-axis.) E.g., plug 0 into f(x) = 2x, then y = 0.

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Elements Card Example Activity: using a Venn diagram, demonstrate the similarities and

difference between the two graphs.

Example Activity: create a Venn diagram on the floor using hula hoops, tape or something similar. Give each student a card with a key feature (attribute) on it. Have students stand in the appropriate area of the Venn diagram with their key feature card.

Example Activity: supply students with one graph and three different tables/charts. Students then will match the appropriate table to the corresponding graph.

Supports and Scaffolds: Venn diagram, functions, tables or graphs, hula hoops, tape, and key features

on cards Compare and contrast linear functions: Click here Linear equations: Click here Slope manipulator: Click here Video−Features of a line: Click here

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FLS: MAFS.912.F-BF.1.1 Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Access Point NarrativeMAFS.912.F-BF.1.AP.1a Select a function that describes a relationship between two

quantities (e.g., relationship between inches and centimeters, Celsius Fahrenheit, distance = rate × time, recipe for peanut butter and jelly – relationship of peanut butter to jelly f(x)=2x, where x is the quantity of jelly, and f(x) is peanut butter.

Essential Understandings:Concrete Understandings Representation

Describe representations of model proportional relationships.

Understand proportional relationships. Understand the following related vocabulary:

quantity, function and relationship.

Suggested Instructional Strategies: Example Activities: Give students a real life scenario:

o Carpet needs to be purchased for a 12 foot by 11 foot bedroom. Students need to find the area (A = l x w). When given a field of three different formulas, one being the correct formula for area, students will then choose/match the correct formula for the given problem. Students may use a geoboard to show the area.

o When purchasing a new shirt at Walmart, the price tag reads $7.99. How much will be the total cost including sales tax? Total Cost = Price of Shirt + (sales tax rate x Price of shirt). When given a field of three different formulas, one being the correct formula, students will then choose/match the correct formula for the given problem. Therefore:

− $7.99 + (0.065 x $7.99) = Total Cost− $7.99 + (0.52) = Total Cost− $7.99 + 0.52 = $8.51− $8.51 is the total cost of shirt including sales tax of 6.5%.

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Elements Card Example Activity: use construction paper cut into 1 inch squares to represent

square inches. Have students find area of a rectangle. If rectangle is 3 inches by 4 inches they can place three rows of 4 across one inch squares and then count the squares to check their multiplication of 3 x 4.

Example Activity: have students create a “Formula Book” that they may use to reference or find the correct formulas for the scenario given.E.g., A = l x w, Slope = riserun etc.

Supports and Scaffolds: Calculator, real life scenario’s, geoboards, geobands (rubber bands),

construction paper cut in 1 inch squares, construction paper, and completed formula book.

Area: Click here Perimeter and circumference: Click here Volume: Click here Slope: Click here Basic Algebra formulas: Click here Cue words that determine operations: Click here Function: Click here Example formula books: Click here

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FLS: MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.Access Point Narrative

MAFS.912.F-BF.2.AP.3a Write or select the graph that represents a defined change in the function (e.g., recognize the effect of changing k on the corresponding graph).

Essential Understandings:Concrete Understandings Representation

Identify the y-intercept on a graph.

Given the graph of the new and parent functions, describe the change when k is introduced.

Use graphing software to discriminate between skinny and wide graphs (k f(x)).

Use graphing software to discriminate directionality (right/left) (f(x + k) ) (up/down) f(x) + k.

Identify the y-intercept in a function. Identify k in a function (f(x) = x + 5), k = 5. Given a graphic template, predict the changes k

will have on the function. Describe the effects k has on a given equation. Understand the coordinate plane. Understand related vocabulary (y-intercept, y-axis, x-

axis, function, positive, negative, k, translation, function, stretch, shrink, quadratic function, exponential function).

Suggested Instructional Strategies: Provide students with a bank of graphs with explanations. Explain that

positive and negative k-values determine the direction the graphs is shifted (moved) up, down, right or left when using addition or subtraction. When using multiplication with a parabola, the parabola is skinnier when multiplied by a whole number and wider when multiplied by a fraction. When using multiplication with a line, the line is steeper when multiplied by a whole number and less steep when multiplied by a fraction. When the slope is positive, the line increases from left to right. When multiplying by a negative the parabola opens downwards and when the slope is negative, the line is decreasing from left to right.

f(x) = x + 0, k = 0, since k = 0, the line does not move. Since the slope is positive, the line is increasing from left to right.

f(x) = x +3, Since k = 3, the line is moved over 3 units to the left. Since the slope is positive, the line is increasing from left to right.

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f(x) = x -3, Since k = -3, the line is moved right 3 units. Since the slope is positive, the line is increasing from left to right.

f(x) = 2 x, Since k = 2, the line has a steeper slope. Since the slope is positive, the line is increasing from left to right.

f(x) = 12 x, since k = 12 , the line has a less steep slope. Since the slope is positive, the line is increasing from left to right.

f(x) = -x +0, k = 0, since k = 0, the line does not move. Since the slope is negative, the line decreases from left to right.

f(x) = x2 + 0, Since k = 0, the parabola does not move up or down.

f(x) = x2 + 3, Since k = 3, the parabola is moved up 3 units.

f(x) = x2 -3, since k = -3, the parabola is moved down 3 units.

f(x) = (x + 2)2, k = 2, since we are only adding the 2 to the x and not the whole function (x2), it shifts the parabola to the left.

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f(x) = (x - 2)2, k = -2, since we are only subtracting the 2 from the x and not the whole function (x2), it shifts the parabola to the right.

f(x) = 4x2, k = 4, since the 4 is multiplied by the whole function, it makes the parabola skinnier.

f(x) = 14 x2, k = 14 , since the14 is multiplied by the whole function, it makes the parabola wider.

f(x) = x2 + 0, Since k = 0, the parabola does not move up or down. The parabola will open downwards because of the negative in front of the function.

Example Activity: teacher gives students a transformation of a function (f(x) = x + 5, k = 5), student will match/select the correct graph from a field of three graphs.

Example Activity: memory game–place index cards with pictures of graphs face down in an orderly fashion. Place index cards with the function transformation rules (equations) face down in another orderly fashion. At this point you will have two groups of turned over index cards. Students will then take turns turning over one card from one group and one card from the other group to see if the equation matches the graph. Continue playing, (like the memory game) until all the cards are matched correctly.

Supports and Scaffolds: Index cards with graphs, and index cards with function transformation rules

(equations) Graph dance video: Click here Function transformations: Click here Notes on function transformations: Click here Graphing Tool: Click here Transformation rules for functions: Click here

(In this example, they use a “c” in place of the “k.”)

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FLS: MAFS.912.F-LE.1.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Access Point NarrativeMAFS.912.F-LE.1.AP.1a Select the appropriate graphical representation of a linear model

based on real-world events.

Essential Understandings:Concrete Understandings Representation

Match a point on a line as being part of a data set for a given line.

Determine if a point is or is not on a line.

Identify coordinates (points) on a graph and in a data table.

Select a graph that represents a simple linear equation.

Understand the following concepts and vocabulary:x axis, y axis, x intercept, y intercept, line, slope.

Suggested Instructional Strategies: Model lines, graphs, and coordinates of varying slopes; match coordinates to

graphs. Explicitly teach the relationship between positive slope and a line that slopes

up left to right and negative slope and a line that goes down left to right. Task analysis:

o Present a story problem and a simple equation (e.g., y = 5x.)o Create a formula template and substitute x for at least three values to

determine y.o Create a table (T-Chart) listing coordinates (x,y).o Plot points on a coordinate grid; connect the points.o Identify the coordinates on the line graph.o Reverse the steps and begin with a line graph; identify the coordinates

of at least three points, create a table listing the x and y coordinates; write a simple linear equation to represent the line graph.

Suggested Supports and Scaffolds: Grid Paper with raised perpendicular lines (horizontal and vertical lines) and points Models T-Chart, or graphic organizer Rulers, or straight edge Graphing calculator Interactive Whiteboard

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FLS: MAFS.912.G-CO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Access Point Narrative

MAFS.912.G-CO.1.AP.1a Identify precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

Essential Understandings:Concrete Understandings Representation

Match the definitions to manipulatives of the terms (circle, line, etc.).

Match the definitions to visual representations of the terms.

Match the definitions to the terms.

Suggested Instructional Strategies: Have students match definitions to correct vocabulary or pictures of

geometric terms. Example Activity: memory game−place cards with pictures of geometric

terms or vocabulary upside down. Have a student select two cards. If both cards go together, i.e., definition matches picture or term, student keeps the pair. If the cards don’t go together student returns both cards to original position.

Example Activity: provide students with magazines. Have students create a geometric terms scrap book with pictures of parallel lines, perpendicular lines, angles, etc.

Example Activity: play the Jenga game−tell students to place three blocks parallel next to each other. Tell students to take three more blocks and place perpendicular on top of the parallel blocks. Continue to the build the tower, alternating parallel and perpendicular rows.

Example Activity: scavenger hunt−have students go through school looking for parallel lines, perpendicular lines, angles, circles, etc. E.g., if available, students can take pictures on iPads.

Supports and Scaffolds: Magazines or catalogs, index cards, graphs, vocabulary words, glue sticks,

pictures of terms, construction paper, scissors, Jenga blocks, or iPads Video on geometric terms: Click here Definitions: Click here Matching game: Click here Definitions of angles: Click here Circles and arcs: Click here Definitions: Click here Video definition of circles: Click here

FLS: MAFS.912.G-CO.1.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as

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Elements Cardfunctions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).FLS: MAFS.912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.FLS: MAFS.912.F-IF.3.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = , y = , y = , y = , and classify them as representing exponential growth or decay.

Access Point NarrativeMAFS.912.G-CO.1.AP.2a Represent transformations in the plane using, e.g., transparencies

and geometry software.MAFS.912.G-CO.1.AP.2b Compare transformations that preserve distance and angle to

those that do not (e.g., translation versus horizontal stretch).MAFS.912.G-CO.1.AP.4a Using previous comparisons and descriptions of

transformations, develop and understand the meaning of rotations, reflections, and translations based on angles, circles, perpendicular lines, parallel lines, and line segments.

Essential Understandings:Access Point Concrete Understandings Representation

MAFS.912.G-CO.1.AP.2a

Use manipulatives to demonstrate a transformation on a graph.

Use graph paper or geometry software to demonstrate a transformation.

Identify a rotation, reflection, or translation when it occurs on the coordinate plane.

MAFS.912.G-CO.1.AP.2b

Use manipulatives or geometry software to demonstrate a transformation on a graph.

Use manipulatives or geometry software to compare and contrast transformations and horizontal or vertical stretches.

Use graph paper or geometry software to demonstrate a transformation.

Compare and contrast transformations and horizontal or vertical stretches.

Identify pairs of points that are collinear and those that are not.

MAFS.912.G-CO.1.AP.4a

Match a model to the term rotations, reflections and translations.

Use manipulatives or geometry software to demonstrate rotation, reflections and translations.

Make a list of the characteristics

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observed in the demonstration of each term.

Use the list of characteristics to develop a definition for rotations, reflections and translations.

Suggested Instructional Strategies: Show a visual of what a reflection, rotation, translation, vertical stretch and a

horizontal stretch looks like.

Reflection (Flip)

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Elements Card Translation (slide)

Rotation (Turn)

Vertical Stretch – The thin lined triangle is the vertical stretch.

Horizontal Stretch–the thin lined triangle is the horizontal stretch.

Example Activity: use transparencies with coordinate grids. Draw on the first transparency a 90 degree right triangle. Place a second transparency on top of the first one. Trace the 90 degree right triangle. Then turn the transparency to correctly demonstrate a reflection, rotation or translation.

Example Activity: using geometric shapes of different sizes, show reflection, rotation, and translation, horizontal and vertical stretch on coordinate plane.

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Supports and Scaffolds: Transparencies with coordinate grids on them, dry erase markers, erasers, or

geometric shapes of different sizes Video of transformation dance: Click here Transformation: Click here Transformation games: Click here Definition: Click here Vertical stretch of a polygon: Click here Horizontal stretch of a polygon: Click here Symmetry: Click here Rotation activity: Click here Reflection activity: Click here Translation activity: Click here

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FLS: MAFS.912.G-CO.1.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.Access Point Narrative

MAFS.912.G-CO.1.AP.3a Describe the rotations and reflections of a rectangle, parallelogram, trapezoid, or regular polygon that maps each figure onto itself.

Essential Understandings:Concrete Understandings Representation

Use coordinates to draw plane figures in a coordinate plane.

Distinguish between orientations of plane figures.

Distinguish between translations, rotations, and reflections.

Suggested Instructional Strategies:

Polygon that is a reflection across the y-axis.

Polygon reflected over a line.

Trapezoid that is rotated 90 degrees.

Trapezoid that is rotated.

Example Activity: give student a picture of a geometric figure that shows either a reflection, rotation, or two geometric figures randomly placed (neither). Student must indicate whether the figure is a reflection, rotation, or neither.

Example Activity: use transparencies with coordinate grids. Draw on the first transparency a trapezoid. Place a second transparency on top of the first one.

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Elements CardTrace the trapezoid. Then turn the transparency to correctly demonstrate a reflection or a rotation. Repeat process using parallelogram, rectangle, and polygons.

Example Activity: use geometric shapes of the same size to show reflection or rotation.

Supports and Scaffolds: Transparencies, dry erase markers, erasers, geometric shapes, pictures of

geometric shapes either reflected, rotated, or neither. Transformations: Click here Video of transformation dance: Click here Transformation games: Click here Definition: Click here Symmetry: Click here Rotation activity: Click here Reflection activity: Click here

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FLS: MAFS.912.G-CO.1.5 Given a geometric figure and a rotation, reflection or translation, draw the transformed figure using, e.g., graph paper, tracing paper or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Access Point Narrative

MAFS.912.G-CO.1.AP.5a Transform a geometric figure given a rotation, reflection, or translation using graph paper, tracing paper, or geometric software.

Essential Understandings:Concrete Understandings Representation

Use coordinates to draw plane figures in a coordinate plane.

Match or identify transformations of plane figures.

Distinguish between orientations of plane figures. Distinguish between translations, rotations, and

reflections.

Suggested Instructional Strategies: *Model/Lead/Test: use math tools (e.g., tangrams, Legos, or stickers) to

demonstrate the transformation of the shape. Demonstrate one transformation at a time.

Use *Most-to-Least Prompting to teach students to demonstrate transformations. Given a picture or drawing of a shape, students use whatever tool is appropriate

to transform the shape. Label the sides of a cube (dice) with letters or stickers (whichever is more

recognizable to the student), rotate the cube and note the change.

Suggested Supports and Scaffolds: Manipulatives such as geoboards, tangram shapes, pattern blocks, or

magnetic pattern blocks Legos to construct then manipulate the object Graphic organizer Provide an arrow to show the direction of the movement of the object to

create a flip, a turn, or a slide (transformation). Assistive technology Virtual manipulatives

Additional Resources: Math is Fun−point & click to transform the shape (Click here ) Transform shapes to create a robot: Click here

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FLS: MAFS.912.G-CO.1.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Access Point Narrative

MAFS.912.G-CO.1.AP.5b Create sequences of transformations that map a geometric figure on to itself and another geometric figure.

Essential Understandings:Concrete Understandings Representation

Translation:

Reflection:

Rotation:

Click here

A transformation is a copy of a geometric figure, where the copy holds certain properties.  Think of when you copy/paste a picture on your computer. The original figure is called the pre-image; the new (copied) picture is called the image of the transformation. 

A rigid transformation is one in which the pre-image and the image both have the exact same size and shape.

Translation: The formal definition of a translation is "every point of the pre-image is moved the same distance in the same direction to form the image." Take a look at the picture at the left labeled translation for some clarification. Each translation follows a rule.  In this case, the rule is "5 to the right and 3 up."  You can also translate a pre-image to the left, down, or any combination of two of the four directions. The transformation for this example would be T(x, y) = (x+5, y+3).Translation: T(x,y) = (x + a, y + b)

Reflection: is a "flip" of an object over a line.  Let's look at two very common

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reflections:  a horizontal reflection and a vertical reflection. Refer to image labeled Reflection on the left. Notice the colored vertices for each of the triangles.  The line of reflection is equidistant from both red points, blue points, and green points.  In other words, the line of reflection is directly in the middle of both points.  Examples of transformation geometry in the coordinate plane...Reflection over x-axis: T(x, y) = (x, -y) Reflection over y-axis: T(x, y) = (-x, y)

Rotation: is a transformation that is performed by "spinning" the object around a fixed point known as the center of rotation.  You can rotate your object at any degree measure, but 90° and 180° are two of the most common.  Also, rotations are done counterclockwise! The figure shown at the left labeled Center of Rotation is a rotation of 90°. Notice that all of the colored lines are the same distance from the center or rotation than are from the point.  Also all the colored lines form 90° angles.  That is what makes the rotation a rotation of 90°. Examples of transformation geometry in the coordinate plane... Rotation 90° T(x,y) = (-y, x)

Suggested Instructional Strategies: Example activities for Transformations: supply students with geoboards and

rubber bands. Have students create a geometric figure on the geoboard. Next, have students create a translation (slide), a reflection (flip), and/or a rotation (turn) of the geometric figure.

Example activity for Reflection: have students place a geometric figure on top of a piece of paper. Student will trace the sides of the figure. Place a reflective device near the figure and trace a line along the bottom. Note: The reflection should show through the reflective device. Students will then trace the reflective figure located on the other side of the reflective device. For example: click here .

Example Foldable for Transformations: students will need construction paper, scissors, and shapes (student cut or templates). Have students fold

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Elements Cardconstruction paper to create folder (fold in half hamburger style), on the left hand side have students write definitions for transformations, translations, reflections, and rotations. Next students will cut out various shapes from another piece of construction paper.

o To show translation, have students attach the shape to the right hand side of the paper using a paperclip. This shows that the shape is able to be moved around the paper.

o To show reflection, have students attach a shape to the right hand side of the paper using a piece of tape alongside one edge of the shape. This shows that the shape can be moved at the crease of the tape to indicate a reflection.

o To show rotation, have students attach a shape to the right hand side of the paper using a brad fastener. This shows that the shape can be rotated on the paper.

o Geometric shapes to print: Click here

Supports and Scaffolds: Geoboards, geoboard software, interactive geoboard app, rubber bands,

reflective device, computer software programs, construction paper, scissors, tape, brad fasteners, shape templates, pencils, markers, graph paper, or paper clips.

Resources: Examples of transformations: Click here Steps and videos for transformations: Click here Videos: Click here Transformation song: Click here Interactive game: Click here Transformation interactive game: Click here Reflection activity: Click here Transformation foldable activity: Click here Interactive geoboard app: Click here

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FLS: MAFS.912.G-CO.2.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Access Point Narrative

MAFS.912.G-CO.2.AP.6a Use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on figures in the coordinate plane.

MAFS.912.G-CO.2.AP.6b Knowing that rigid transformations preserve size and shape or distance and angle, use this fact to connect the idea of congruency and develop the definition of congruent.

Essential Understandings:Access Point Concrete Understandings Representation MAFS.912.G-CO.2.AP.6a

Use manipulatives to model congruent.

Identify congruent figures when presented with a model.

A rigid transformation is one in which the pre-image and the image both have the exact same size and shape.

If one shape can become another using Turns, Flips and/or Slides, then the shapes are congruent. Congruent is two objects same shape and same size.

MAFS.912.G-CO.2.AP.6b

Using manipulatives, demonstrate the following transformations:

Teacher resource: Click here

A rigid transformation is one in which the pre-image and the image both have the exact same size and shape.

If one shape can become another using Turns, Flips and/or Slides, then the shapes are congruent. Congruent is two objects same shape and same size.

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Suggested Instructional Strategies:

Reflection (flip) is a rigid transformation because it does not change its shape or size.

Translation (slide) is a rigid transformation because it does not change its shape or size.

Rotation (turn) is a rigid transformation because it does not change its shape or size.

Introduce to the student that rigid motion pertains to reflections, rotations, and transformations. Rigid translations preserves the size and the shape regardless of the transformation on the coordinate grid.

Example Activity: give the students a field of three different transformations. One transformation is rigid and the other two contain different sizes or shapes (distractors). Student will choose the transformation that is rigid.

Example Activity: give the student pattern blocks and have them rotate, reflect and translate them. Students can then see that the size and shape have been preserved even though the pattern blocks have moved to a different location on the coordinate plane.

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Supports and Scaffolds: Pattern blocks, coordinate grids with and without shapes on them, geometric

manipulatives of the same size and shape to demonstrate a rigid transformation. Visuals of rigid reflections, rotations and translations.

Definitions and visuals: Click here Video: Click here Pattern block video: Click here Word Wall: Click here Rotation activity: Click here Reflection activity: Click here Translation activity: Click here

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FLS: MAFS.912.G-CO.2.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Access Point Narrative

MAFS.912.G-CO.2.AP.7a Use definitions to demonstrate congruency and similarity in figures.

Essential Understandings:Concrete Understandings Representation

Match sides on two congruent triangles using manipulatives.

Match angles on two congruent triangles using manipulatives.

Match sides on two congruent triangles using pictures or geometry software.

Match angles on two congruent triangles using pictures or geometry software.

Suggested Instructional Strategies: Introduce that the term “congruent” means the same as rigid, whereas the

figure remains the same size and shape regardless of placement on the coordinate plane.

Congruent Triangles Similar Triangles–Not Congruent (same shape, different sizes)

Example Activity: give students a coordinate plane with three triangles of different sizes. Give student a triangle that matches one of the three on the coordinate plane. Have student match their triangle to the one on the coordinate plane by laying it on top of the correct triangle to ensure congruency.

Example Activity: use geoboards and geobands to create congruent triangles.

Supports and Scaffolds: Coordinate plane, triangles, geoboards, or geobands How to find if triangles are congruent: Click here

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FLS: MAFS.912.G-CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions.Access Point Narrative

MAFS.912.G-CO.2.AP.8a Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria; ASA, SSS, and SAS.

Essential Understandings:Concrete Understandings Representation

Given two triangles, match the corresponding sides on each triangle.

Given two triangles, match the corresponding angles on each triangle.

Mark the pieces of the triangle that are congruent. Identify which triangles are and are not congruent.

Using graphs or geometry software, mark the pieces of the triangle that are congruent.

Suggested Instructional Strategies: Introduce that the term “congruent” means the same as rigid, whereas the

figure remains the same size and shape regardless of placement on the coordinate plane.

Understand Theorem abbreviations: ASA–(angle-side-angle), SSS–(side-side-side), SAS–(side-angle-side), and HL–(hypotenuse-leg).

Example Activity: have students match congruent triangles based on sides and angles. Refer to visuals above.

Example Activity: give students two congruent triangles with appropriate side markings (like above) and/or side lengths and angles. Then have students determine which theorem is used to prove congruency.

Example Activity: have students measure the sides and angles to determine congruency.

Example Activity: use geoboards and geobands to create congruent triangles.

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Supports and Scaffolds: Coordinate planes, triangles, manipulatives, protractor, ruler, geoboards, or

geobands Theorems for congruent triangles: Click here How to find if triangles are congruent: Click here Congruent song: Click here Game: Click here Memory game: Click here

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FLS: MAFS.912.G-CO.3.9 Prove theorems about lines and angles; use theorems about lines and angles to solve problems. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.Access Point Narrative

MAFS.912.G-CO.3.AP.9a Measure lengths of line segments and angles to establish the facts about the angles created when parallel lines are cut by a transversal and the points on a perpendicular bisector.

Essential Understandings:Concrete Understandings Representation

Given an angle measure, draw an angle. Use a protractor to show that angles are

supplementary (equals 180 degrees). Use a protractor to show that angles are

equal. Recognize that the angle measure of a

straight line is 180 degrees. Identify vertical angles, adjacent angles,

and supplementary angles, given a set of parallel lines cut by a transversal.

Identify congruent angles, given a set of parallel lines cut by a transversal.

Identify alternate interior angles, given a set of parallel lines cut by a transversal.

Construct parallel lines crossed by a transversal.

Measure line segments on a line. Given a line segment, use paper folding to

determine the perpendicular bisector.

Match or identify angle measurements. Understand the following concepts and

vocabulary: acute, obtuse, right, straight line, vertical angles, parallel lines, alternate interior angles, perpendicular bisector, equal distance, transversal.

Describe parallel lines by telling about or showing lines and angles.

Label adjacent angle measurements on a diagram of parallel lines cut by a transversal given one of the angles, using subtraction to find the missing angle.

Identify points on a perpendicular bisector of a line segment are equidistant from the segment’s endpoints.

Identify angle relationships based on position given a diagram of parallel lines cut by a transversal.

Use appropriate tools as needed.Suggested Instructional Strategies:

Introduce vocabulary: acute, obtuse, right, straight line, vertical angles, parallel lines, alternate interior angles, perpendicular bisector, equidistant, transversal, supplementary angles, complimentary angles, adjacent angles, and corresponding angles.

Acute angle: angles that measure less than 90 degrees. Obtuse angle: angles that measure more than 90 degrees. Right angle: angles that measure 90 degrees. Straight line: angles that measure 180 degrees. Parallel lines: two lines that are on the same plane that never cross. Supplemental angles: two angles that added together equal 180 degrees. Complimentary angles: two angles that added together equal 90 degrees. Equidistant: measures the same distance. Transversal: line that cuts through two other lines.

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Corresponding angles: two angles that are on the same side of the transversal, one being an interior angle and one being an exterior angle. The angle measures of these two angles are equal to each other. There are four pairs of corresponding angles in the picture above. They are angles 1 and 5, angles 3 and 7, angles 2 and 6, and angles 4 and 8.

Alternate interior angles (angles in between the parallel lines and diagonal to each other): there are two pairs of alternate interior angles. The angle measures of these two angles are equal to each other. Angles 3 and 6, and angles 4 and 5.

Vertical angles: angles that are diagonals to each other and on opposite sides of the transversal. The angle measures of these pairs of angles are equal to each other. There are 4 pairs of verticals angles; angles 1 and 4, angles 2 and 3, angles 5 and 8, and angles 6 and 7.

Adjacent angles: angles that share a common side. There are 8 pairs of adjacent angles; angles 1 and 2, angles 2 and 4, angles 1 and 3, angles 3 and 4, angles 5 and 6, angles 6 and 8, angles 5 and 7, and angles 7 and 8.

Perpendicular bisector: a line segment that crosses another line segment, cutting the line segment in half, forming four 90 degree angles.

Example Activity: have students create a reference book with above definitions and examples.

Example Activity: have students match the definitions to the appropriate visual.

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Elements Card Example Activity: have students measure line segments and angles using a

ruler and protractor.

Supports and Scaffolds: Construction paper, protractors, rulers, visuals, definitions, pencils, or markers Adjacent angles: Click here Definitions of angles: Click here Transversal song: Click here Parallel lines and angles: Click here Angles with parallel lines: Click here Equidistant lines: Click here

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FLS: MAFS.912.G-CO.3.10 Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Access Point NarrativeMAFS.912.G-CO.3.AP.10a Measure the angles and sides of equilateral, isosceles, and

scalene triangles to establish facts about triangles.

Essential Understandings:Concrete Understandings Representation

Given an angle measure, draw an angle. Recognize that a triangle consists of three

angles that total 180 degrees. Recognize the measures of base angles of an

isosceles triangle are equal (same). Use paper folding to demonstrate angle

relationships. Use a protractor to show that angles are

supplementary (equals 180 degrees). Use a protractor to measure interior and

exterior angles of a triangle. Recognize that the angle measure of a

straight line is 180 degrees. Construct a triangle. Construct a triangle given angle

measurements. Given a paper triangle, tear the angles off

and place them together to create a straight angle.

Given a triangle, draw the exterior angles. Measure the length of a line segment.

Construct the median of the sides of a triangle.

Match or identify angle measurements. Match or label angle measurements. Given an angle measurement,

determine the interior or exterior angle supplementary to it.

Find the sum and difference of pairs of interior angles in a triangle.

Use appropriate tools as needed. Understand the following concepts and

vocabulary: acute, obtuse, right, straight line, interior angles, exterior angles, perpendicular bisector, equal distance, angle sum theorem, supplementary angles, base angles, isosceles triangles, midpoints, median, congruent, centroid.

Given a triangle, determine the median of each side.

Identify the centroid of the triangle, given the median of each side.

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Suggested Instructional Strategies:

Equilateral Triangle: all sides of equal length and all angles of equal measurement.

Isosceles Triangle: two sides of equal length and two angles of equal measurement.

Scalene Triangle: each angle and side length is a different measurement.

Median of a Triangle: goes from the midpoint of one side of a triangle to the opposite angle.

Understand Vocabulary: equilateral, isosceles, scalene triangles, and median Example Activity: have students match a given triangle to the same triangle

in a field of two or three. Example Activity: have students label whether a given triangle is an

equilateral, isosceles, or scalene triangle. Example Activity: have students use a ruler to measure the side lengths and

a protractor to measure the angles of triangles. Students will determine if the triangle is equilateral, isosceles, or scalene based on the measurements taken.

Example Activity: have a student describe a triangle to a partner using the established facts of different triangles. The partner must determine if the described triangle is equilateral, isosceles, or scalene.

Supports and Scaffolds: Triangles, protractors, rulers, cards with the triangle names, or visuals of

triangles Triangles: Click here Medians of a triangle: Click here Medians of a triangle: Click here Game: Click here Virtual geoboard: Click here Virtual protractor: Click here

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FLS: MAFS.912.G-CO.3.11 Prove theorems about parallelograms; use theorems about parallelograms to solve problems. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Access Point NarrativeMAFS.912.G-CO.3.AP.11a Measure the angles and sides of parallelograms to establish facts

about parallelograms.

Essential Understandings:Concrete Understandings Representation

Draw or construct a parallelogram. Measure the angles and sides of a

parallelogram to show congruence. Draw diagonals on a parallelogram. Use paper folding to determine

congruent angles and sides, diagonals, medians, and midpoints of a parallelogram.

Identify characteristics of parallelograms. Understand the following concepts and

vocabulary: acute, obtuse, right, parallelogram, rectangle, rhombus, square, base angles, midpoints, median, congruent, bisect, diagonals.

Label and describe congruent parts of parallelograms.

Construct diagonals and medians of parallelograms.

Use descriptions of characteristics of figures to identify and classify figures (i.e., parallelograms with congruent diagonals are rectangles).

Suggested Instructional Strategies: Acute angle: angles that measure less than 90 degrees. Obtuse angle: angles that measure more than 90 degrees. Right angle: angles that measure 90 degrees.

Parallelogram: opposite sides are parallel and the same length. Opposite angles are equal in measurement.

Rectangle: opposite sides are parallel and are equal in length to each other. All four angles measure 90 degrees.

Rhombus: all four side are the same length and opposite angles are equal in measurement.

Square: all four sides are equal in length and all four angles measure 90 degrees. All squares are rectangles.

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Elements Card Understand that squares, rectangles, and rhombuses are parallelograms

because they meet all the requirements for a parallelogram (opposite sides are equal and opposite angles are equal.)

Base Angles: the angles at the base of the shape. Midpoints: point at the middle of the line segment of the shape. Congruent: regardless of where the shape is placed on a coordinate plane it

retains its size and shape. Bisect: divide into two equal parts. Diagonals: a line segment that connects opposite corners of a parallelogram. Understand Vocabulary: acute, obtuse, right, parallelogram, rectangle,

rhombus, square, base angles, midpoints, congruent, bisect, and diagonals. Example Activity: measure the sides and angles (right, obtuse, acute) of a

parallelogram with a ruler and protractor. Based on the measurements, determine what kind of parallelogram is shown (rhombus, rectangle, square.)

Example Activity: have a student describe a parallelogram to a partner using the established facts of different parallelograms. The partner must determine if the described parallelogram is a square, rhombus, or rectangle.

Example Activity: have students cut parallelograms out of foam sheets. Next have them measure the sides and angles and write the measurements on the foam. Students will then label the type of parallelogram that they have created.

Example activity: have students use playdoh or geo boards to create parallelograms. Example activity: have students play a memory game using a rhombus, square,

rectangle, equilateral triangle, isosceles triangle and a scalene triangle.

Supports and Scaffolds: Play-Doh, geoboards, foam sheets, ruler, protractor, markers, scissors, visuals

of parallelograms, or shapes Parallelogram: Click here Virtual geoboard: Click here Game: Click here Virtual protractor: Click here

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FLS: MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Access Point NarrativeMAFS.912.G-O.4.AP.12a Copy a segment.MAFS.912.G-CO.4.AP.12b Copy an angle

Essential Understandings:Access Point Concrete Understandings Representation MAFS.912.G-O.4.AP.12a

When given a segment, use manipulatives (e.g., compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, tracing paper, patty paper, etc.) to create another segment of equal length.

Draw a line with a straightedge. Place a starting point on the line. Place the point of the compass on point A. Stretch the compass so that the pencil is

exactly on B. Without changing the span of the compass,

place the compass point on the starting point on the reference line and swing the pencil so that it crosses the reference line.

Label the new line segment.Click here

Understand the following concepts and vocabulary: segment, reference line and endpoint.

A segment is what you would ordinarily think of when you draw a line on a piece of paper. E.g., a segment is a piece of a line. It begins at one point on the line, and ends at another. These points are known as the endpoints of the segment. Click here

MAFS.912.G-CO.4.AP.12b

When given an angle, use manipulatives (e.g., compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, tracing paper, patty paper, etc.) to create another angle of equal degrees or measure.

Draw a reference line with your straightedge. Place a starting point on the

Understand the following concepts and vocabulary: angle, reference line, straightedge, arc, intersection point, rays, compass and vertex.

A shape formed by two lines or rays diverging from a common point.

o Formally : An angle is formed by the intersection of two rays with a common endpoint.

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Elements CardAccess Point Concrete Understandings Representation

reference line. Place the point of the compass on the

vertex of ∢BAC (point A). Stretch the compass to any length so long

as it stays ON the angle. Swing an arc with the pencil that crosses

both sides of ∢BAC. Without changing the span of the compass,

place the compass point on the starting point of the reference line and swing an arc that will intersect the reference line and go above the reference line. (Refer to Click here)

Go back to ∢BAC and measure the width (span) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle.

With this width, place the compass point on the reference line where the new arc crosses the reference line and mark off this width on the new arc.

Connect this new intersection point to the starting point on the reference line.

Click here

o Informally : When two segments (or lines or rays) intersect, they form an angle.

o Click here

Suggested Instructional Strategies: Example Activity to copy a segment: have students draw a segment with a

straight edge (ruler). Lay a piece of patty paper (tracing paper) on top of the segment and trace it.

Example Activity to copy a segment: have students draw a segment with a straight edge (ruler). Using a dry spaghetti noodle, have students place the noodle above or below the line segment. Students will then place their finger at the end point of the line segment on the noodle and break the noodle to create the same length as the line segment. A pipe cleaner may also be used in place of the dry spaghetti noodle by bending the pipe cleaner at the length of the line segment.

Example activity for copying an angle: have students draw an angle with a straight edge (ruler). Lay a piece of patty paper (tracing paper) on top of the angle and trace it.

Example activity for copying an angle: have students draw an angle using a straight edge (ruler). Using a pipe cleaner, have students copy the angle by bending the pipe cleaner to match the angle.

Supports and Scaffolds: Ruler, dry spaghetti noodles, pipe cleaners, patty paper, pencil, and paper

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Resources: Copy a line segment example video: Click here Directions for copying a line segment: Click here Copy an angle example video: Click here Steps for copying an angle: Click here

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Elements CardFLS: MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Access Point NarrativeMAFS.912.G-CO.4.AP.12c Bisect a segment.MAFS.912.G-CO.4.AP.12d Bisect an angle.MAFS.912.G-CO.4.AP.12e Construct perpendicular lines, including the perpendicular

bisector of a line segment.Essential Understandings:

Access Point Concrete Understandings Representation MAFS.912.G-CO.4.AP.12c

When given a line segment, use manipulatives (ruler, compass, string, reflective devices, paper folding, dynamic geometric software, etc.) to bisect the given segment.

Place compass point on A and stretch the compass MORE THAN half way to point B, but not beyond B.

With this length, swing a large arc that will go BOTH above and below AB.

Without changing the span on the compass, place the compass point on B and swing the arc again.  The two arcs that have been created should intersect.

With a straightedge, connect the two points of intersection.

This new straight line bisects AB.  Label the point where the new line and

AB cross as C. AB has now been bisected and AC = CB.

(It could also be said that the segments are congruent, AC ≅CB.)

Click here Click here

Understand the following concepts and vocabulary: line segment, compass point, arc, intersect, bisect and congruent.

Identify the mid-point of a line segment.

Bisect means to divide into two equal parts.

Click here and Click here

MAFS.912.G-CO.4.AP.12d

When given an angle, use manipulatives (e.g., compass, protractor, reflective devices, paper folding, dynamic

Bisect means to divide into two equal parts.

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Elements CardAccess Point Concrete Understandings Representation

geometric software, etc.) to bisect the given angle.

Place point of compass on the vertex of ∢BAC (point A). Stretch the compass to any length so

long as it stays ON the angle. Swing an arc so the pencil crosses both

sides of ∢BAC. This will create two intersection points with the sides (rays) of the angle.

Place the compass point on one of these new intersection points on the sides of ∢BAC.

Without changing the width of the compass, place the point of the compass on the other intersection point on the side of the angle and make the same arc.  The two small arcs in the interior of the angle should be crossing.

Connect the point where the two small arcs cross to the vertex A of the angle.

The two new angles created are of equal measure (and are each ½ the measure of ∢BAC).

Click here

Click here

MAFS.912.G-CO.4.AP.12e

When given a line segment, use manipulatives (e.g., ruler, compass, string, reflective devices, paper folding, dynamic geometric software, etc.) to bisect the given segment forming perpendicular lines.

Place compass point on A and stretch the compass MORE THAN half way to point B, but not beyond B.

With this length, swing a large arc that will go BOTH above and below AB.

Understand the following concepts and vocabulary: perpendicular, bisector, line segment and midpoint.

Perpendicular lines are two lines that cross forming 90° angles.

A perpendicular bisector is a perpendicular line or a segment that passes through the midpoint of a line.

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Elements CardAccess Point Concrete Understandings Representation

Without changing the span on the compass, place the compass point on B and swing the arc again. The two arcs that have been created should intersect.

With a straightedge, connect the two points of intersection.

This new straight line bisects AB. Label the point where the new line and

ABcross as C. ABhas now been bisected and AC = CB.

(It could also be said that the segments are congruent, AC ≅CB.)

Click here Click here Click here

Click here Click here

Suggested Instructional Strategies: Example activity to bisect a segment: have students draw a line segment

with end points on patty paper (tracing paper). Have students fold the patty paper so that the end points overlap. Next have students use their thumb to create a crease on the fold. Students then open the patty paper and trace the crease using a pencil, pen, or marker and a straight edge if desired. This creates a bisected segment.

Example activity to bisect a segment: have students bend a pipe cleaner in half, end to end. Next have students use their thumb to create a crease in the pipe cleaner. Then have students lay another pipe cleaner on the crease creating a perpendicular bisector.

Example activity to bisect an angle: have students draw an angle on patty paper (tracing paper). Have students fold the patty paper so that the lines overlap. Next have students use their thumb to create a crease on the fold. Students then open the patty paper and trace the crease using a pencil, pen, or marker and a straight edge if desired. This creates a bisected angle. This activity can also be done by cutting an angle out of construction paper and folding the paper in half to create the bisected angle.

Supports and Scaffolds: Pipe cleaners, patty paper, pencil, pen, marker, ruler, construction paper, and

compassResources:

Directions for bisecting angles and line segments: Click here Videos on bisecting angles and line segments: Click here Video for constructing line segment bisectors: Click here Directions for bisecting angles and line segments: Click here Directions for bisecting angles and line segments: Click here Video for bisecting an angle: Click here Directions for bisecting angles and line segments using patty paper: Click here

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FLS: MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Access Point NarrativeMAFS.912.G-CO.4.AP.12f Construct a line parallel to a given line through a point not on the

line.

Essential Understandings:Concrete Understandings Representation

Using manipulatives (e.g., ruler, compass, protractor, straight edge, dynamic geometric software, etc.) to construct a line through a point that is parallel to the given line.

Understand the following concepts and vocabulary: parallel and plane.

Parallel lines are two lines on the same plane that never meet. They are always the same distance apart.Click here

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Suggested Instructional Strategies: Example activity: have students draw a line segment onto a piece of patty

paper. Next have students draw a dot above or below the line. Then students fold the paper so that the fold goes through the dot and the line overlaps itself. Have students make a crease with their thumb. Then students unfold the paper and refold the paper perpendicular so that the creased line overlaps itself as well going through the dot. Have students create another crease using their thumb. Have students trace the new crease with a straight edge to create parallel lines. (Refer to video in resources.)

Supports and Scaffolds: Patty paper, ruler, and marker

Resources: Video of constructing parallel lines using patty paper: Click here Video for constructing a parallel through a point: Click here Video for parallel line through a point: Click here

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FLS: MAFS.912.G-CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Access Point NarrativeMAFS.912.G-O.4.AP.13a Construct an equilateral triangle, a square, and a regular

hexagon inscribed in a circle.Essential Understandings:Concrete Understandings Representation

Identify a triangle, a square, and a hexagon.

Construct a triangle, a square, and a hexagon inside a circle using manipulatives.

Construct a circle using appropriate tools (e.g., protractor, compass, geometry software, patty paper).

Construct a triangle, a square, and a hexagon using appropriate tools (e.g., ruler, protractor, compass, geometry software, patty paper).

Suggested Instructional Strategies: Example activity: provide students with a circle drawn on a sheet of paper

placed inside a sheet protector. Students may then use manipulatives (dry erase markers, Play-Doh, molding clay, etc.) to construct equilateral triangles, squares, and/or hexagons within the circle.

Example activity: provide students with a hula hoop and painters tape. Place the hula hoop on the floor and have students construct equilateral triangles, squares, and/or hexagons within the hula hoop using painters tape.

Example activity: using a computer software app for a geoboard (see resources below) have students choose the circular geoboard. Have students use the rubber bands to construct equilateral triangles, squares, and/or hexagons.

Example activity: have students construct a circle with a triangle, square, and/or hexagon within the circle using various sensory items (shaving cream, sand, rice, etc.)

Supports and Scaffolds: Sheet protector, paper with a circle drawn on it, dry erase markers, Play-Doh,

molding clay, hula hoop, painters tape, computer software program, shaving cream, sand, or rice

Resources Geoboard computer software app: Click here Construction directions for square, equilateral triangle and hexagon video:

Click here Equilateral triangle in circle directions: Click here Step by step directions with visual for constructing a square inside a circle:

Click here Step by step directions with visual for constructing a hexagon inside a circle:

Click here

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Elements CardFLS: MAFS.912.G-SRT.1.1 Verify experimentally the properties of dilations given by a center and a scale factor:

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Access Point NarrativeMAFS.912.G-SRT.1.AP.1a

Given a center and a scale factor, verify experimentally that when dilating a figure in a coordinate plane, a segment of the pre-image that does not pass through the center of the dilation, is parallel to its image when the dilation is performed. However, a segment that passes through the center remains unchanged.

MAFS.912.G-SRT.1.AP.1b

Given a center and a scale factor, verify experimentally that when performing dilations of a line segment, the preimage, the segment which becomes the image, is longer or shorter based on the ratio given by the scale factor.

Essential Understandings:Access Point Concrete Understandings Representation MAFS.912.G-SRT.1.AP.1a

Use manipulatives or geometry software to model dilation.

Identify if the dilation makes the figure bigger or smaller.

Determine if a figure has been dilated.

Online activity manipulating figures with a scale factor. Click here

Use appropriate tools to multiply or divide something by the scale factor.

Use a computer program to scale an object to a larger or smaller size.

Use appropriate tools to find the dimensions of a figure.

MAFS.912.G-SRT.1.AP.1b

Using manipulatives (i.e., patty paper, snap cubes) demonstrate how the line segment pre-image could increase or decrease by the scale factor, for example, a line segment that is 2 snap cubes long could increase by a scale factor of two by putting it next to a line segment that has four snap cubes.

Online activity manipulating line segments with a scale factor. Click here

A dilation is a transformation (notation DK) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure.  

The description of a dilation includes the scale factor (or ratio) and the center of the dilation. The center of dilation is a fixed point in the plane about which all points are expanded or contracted.  It is the only invariant point under a dilation.

The pre-image PQ is three times bigger than the image P ' Q' .Click here

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Elements CardSuggested Instructional Strategies:

Understand that a dilation is a change from the pre-image to the image where the shape is the same but the size is different by a scale factor, expanded (larger) or contracted (smaller). The dilation is around an invariant (fixed) point called the center of dilation. (Refer to the dilation definition and video links in resources below.)

Understand that a scale factor is the ratio of corresponding sides of two similar figures. The sides of the smaller figure is multiplied by the same number to create the larger figure.

Understand that collinear means that the points lie on the same line. Understand that parallel lines are lines that have the same slope. Understand that a dilation of a line segment the line segment will be longer or

shorter based on the scale factor. Example activities: using pattern blocks of the same shape but different sizes, have

students place the smaller shape on top of the larger one, lining up two sides and an angle. This will create a center of dilation for the two shapes. Note: line segment AB is parallel to line segment A ' B' .

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Elements Card Example activity: using computer software app for a geoboard (see resources

below), have students construct the same shape two different sizes using the center point (peg) as the center of dilation. In this example, the segments in the image are twice the size as the segments in the pre-image. Therefore the scale factor is 2. Note: line segment CB is parallel to line segment C ' B '.

Supports and Scaffolds: Pattern blocks, computer software program, and shapes

Resources: Video: Click here Dilation of a polygon: Click here Dilation definition: Click here Geoboard computer software app: Click here Notes and worksheets: Click here Dilations and line segments: Click here PowerPoint on dilations: Click here Video: Click here

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FLS: MAFS.912.G-SRT.1.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.FLS: MAFS.912.G-SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Access Point NarrativeMAFS.912.G-SRT.1.AP.2a Determine if two figures are similar.MAFS.912.G-SRT.1.AP.2b Given two figures, determine whether they are similar and

explain their similarity based on the equality of corresponding angles and the proportionality of corresponding sides.

MAFS.912.G-SRT.1.AP.3a Apply the angle-angle (AA) criteria for triangle similarity on two triangles.

Essential Understandings:Access Point Concrete Understandings Representation MAFS.912.G-SRT.1.AP.2a

Select two objects that are the same shape.

Use appropriate tools as needed to duplicate a shape (e.g., wiki sticks, computers, interactive white boards, markers).

Use geometry software to create dilations.

Identify congruent angles of similar figures.

Describe the characteristics of the two figures that are similar.

Use geometry software to construct and compare figures based on angle measurements and side lengths.

Use proportions to compare figures based on side lengths to determine similarity.

Understand the following vocabulary: similar, congruent, angles, corresponding, transformation.

MAFS.912.G-SRT.1.AP.2b

Select two objects that are the same shape.

Select two objects that have different shapes.

Use appropriate tools as needed to duplicate a shape (e.g., wiki sticks, computers, interactive white boards, markers).

Given two shapes, label (identify, point to, mark,) the parts of the figures that are congruent.

Describe the characteristics of two figures that are the same.

Describe the characteristics of two figures that are different.

Understand the following vocabulary: similar, congruent, angles, corresponding, transformation.

MAFS.912.G-SRT.1.AP.3a

Given two triangles, match the corresponding angles on each triangle.

Mark the pieces of the triangle that are similar.

Using graphs or geometry software, mark the pieces of the triangle that are similar.

Given two triangles, determine if the triangles are similar using AA.

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Suggested Instructional Strategies:

Similar: not congruent (same shape, same angle measurements, different side lengths, different sizes.) If comparing two triangles, the criteria of AA (angle-angle) must be met in order for them to be similar.

AA (angle-angle): two angles of one triangle have to be the same as two angles in the second triangle.

Understand Vocabulary: similar, congruent, angles, corresponding, and transformation

Congruent: regardless of where the shape is placed on a coordinate plane it retains its size and shape.

Angles: two rays that meet at a common point called the vertex.

Corresponding sides: xcb = dffex10 = 45

Cross multiply: 5x = 40, x = 8 Corresponding angles: measure of angle a = measure of angle d measure of angle b = measure of angle e measure of angle c = measure of angle f Transformation: includes rotations, reflections, translations, horizontal and

vertical stretches Example Activity: use geoboards and geobands to create similar shapes.

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Elements Card Example Activity: give students a field of 4 shapes, where two shapes are the

same shape, different size and two are completely different shapes. Student then must indicate which two shapes are similar.

Example activity: teacher provides a visual of a shape, student replicates similar shapes using Play-Doh, clay, geoboards, markers and paper, tape, etc.

Supports and Scaffolds: Play-Doh, clay, geoboards, geobands, markers, paper, tape, and geometric shapes Game: Click here Definitions: Click here Definitions: Click here Activity: see pages 37 and 38 Click here Game: Click here

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FLS: MAFS.912.G-SRT.2.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

Access Point NarrativeMAFS.912.G-SRT.2.AP.4a Establish facts about the lengths of segments of sides of a

triangle when a line parallel to one side of the triangles divides the other two sides proportionally.

Essential Understandings:Concrete Understandings Representation

Label the parts of a triangle. Identify what each variable in the

Pythagorean theorem represents (two sides and the hypotenuse).

Use squares to explain the Pythagorean theorem. Click here

Measure the angles and segments of sides of a triangle sliced by a line parallel to one side of the triangle.

Understand the following concepts and vocabulary: Pythagorean theorem, length, right triangle.

Use a graphic organizer to calculate a missing side using the Pythagorean theorem, using appropriate tools as needed.

Enter information into the formula for the Pythagorean theorem.

Use the Pythagorean theorem to calculate missing side lengths.

Explain the steps used to solve using the Pythagorean theorem.

Use the angle and segment measurements to determine proportionality.

Suggested Instructional Strategies: Understand vocabulary: Pythagorean theorem, right triangle, hypotenuse

Parallel = proportional Non parallel = not proportional Understand that when a line is drawn parallel to one of the sides, the smaller

triangle created is proportional to the larger triangle. If the line is not parallel, then the two triangles are not proportional.

Understand that the Pythagorean Theorem is only used when the triangle is a right triangle.

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Elements Card Pythagorean Theorem: a2 + b2 = c2

Understand the mid-segment theorem: segment that is parallel to another side and cuts through a mid-point of another side.

Understand that a parallel line to one of the sides of the triangle divides the other two sides proportionally.

o 1510

= 12x

o Cross multiply: 15x = 120x = 8

Example Activity: given a triangle, have students label sides and angles. Have students draw a line parallel to the base of the triangle. Students then measure each section of the triangle. Students will determine proportionality using the formula aeed = abbc

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Elements Card Example Activity: have students create a triangle on their desk using shaving

cream. Students then measure with a ruler two sides of the triangle. Students will indicate the mid-point of those sides and connect the two mid-points across the triangle using shaving cream. Students will then fill in the smaller triangle formed with shaving cream. This will show proportionality between the smaller triangle within the large triangle.

Example Activity for Pythagorean Theorem: give students a right triangle with the measurements of the sides that create the right angle. Have students draw a line parallel to the base of the triangle. Students need to understand that if the line is parallel to the base, it is proportional. Provide students with the Pythagorean Theorem formula a2 + b2 = c2.Hypotenuse2 = (side 1)2 + (side 2)2 or Hypotenuse = . Provide students with the rules of proportionality: aeed = abbc

Supports and Scaffolds: Rulers, shaving cream, markers, dry erase boards and dry erase markers,

clay, Play-Doh, formulas, triangles, geoboards, geobands, geometric shapes, calculator, paper, pencil, or sand

The Triangle Midsegment Theorem: Click here Parallel lines and triangles: Click here PowerPoint parallel lines and triangles: Click here Midsegment of a triangle: Click here Midsegment Theorem: Click here Pythagorean Theorem: Click here Pythagorean Theorem calculator: Click here Triangle proportionality: Click here Proportional calculator: Click here

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FLS: MAFS.912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Access Point NarrativeMAFS.912.G-SRT.2.AP.5a Apply the criteria for triangle congruence and/or similarity (angle-

side-angle [ASA], side-angle-side [SAS], side-side-side [SSS], angle-angle [AA]) to determine if geometric shapes that divide into triangles are or are not congruent and/or can be similar.

Essential Understandings:Concrete Understandings Representation

Given two triangles, match the corresponding sides on each triangle.

Given two triangles, match the corresponding angles on each triangle.

Mark the pieces of the triangle that are congruent and/or similar.

Using graphs or geometry software, mark the pieces of the triangle that are congruent and/ or similar.

Given two triangles, determine if the triangles are congruent and /or similar using AA, ASA, SAS, and SSS.

Use appropriate tools (e.g., calculator, paper pencil) to solve one-step equations.

Use appropriate tools to find the dimensions of a figure.

Suggested Instructional Strategies: Understand that the term “congruent” means the two figures would be the

same size and shape. Understand that similar is not congruent. Similar has the same shape, same

angle measurements, different side lengths (the sides are proportional), and different sizes. If comparing two triangles, the criteria of AA (angle-angle) must be met in order for them to be similar.

Understand Theorem abbreviations: ASA–(angle-side-angle), SSS–(side-side-side), SAS–(side-angle-side), AA–(angle-angle).

Example Activity: have students match similar triangles based on sides and angles. Refer to visuals above.

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Elements Card Example Activity: give students two congruent triangles with appropriate side

markings (like above) and/or side lengths and angles. Then have students determine which theorem is used to prove congruency.

Example Activity: have students measure the sides and angles to determine congruency.

Example Activity: use geoboards and geobands to create congruent triangles.

Supports and Scaffolds: Coordinate planes, triangles, manipulatives, protractor, ruler, geoboards, and

geobands Theorems for Congruent Triangles: Click here How to find if triangles are congruent: Click here Congruent song: Click here Game: Click here Memory game: Click here

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FLS: MAFS.912.G-SRT.3.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.FLS: MAFS.912.G-SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles.Access Point Narrative

MAFS.912.G-SRT.3.AP.6a Using a corresponding angle of similar right triangles, show that the relationships of the side ratios are the same, which leads to the definition of trigonometric ratios for acute angles.

MAFS.912.G-SRT.3.AP.7a Explore the sine of an acute angle and the cosine of its complement and determine their relationship.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.G-SRT.3.AP.6a

Given an angle, identify the adjacent sides, opposite side, and hypotenuse.

Given trigonometric ratio, identify the parts of the triangle that relate.

Identify the appropriate ratio, given a formula sheet.

Understand the following concepts and vocabulary: adjacent sides, opposite side, hypotenuse, sine, cosine, tangent, secant, cosecant, cotangent and trigonometric ratio.

Set up the fraction for the trigonometric ratio.

MAFS.912.G-SRT.3.AP.7a

Given an angle, identify the adjacent sides, opposite side, and hypotenuse.

Given the sine, identify the cosine of the other angle.

Given an angle, identify the angle complementary to it.

Given trigonometric ratio, identify the parts of the triangle that relate.

Understand the following concepts and vocabulary: adjacent sides, opposite side, hypotenuse, sine, cosine and tangent.

Set up the fraction for the trigonometric ratio.

Suggested Instructional Strategies: Understand that the sum of the degrees of all three angles in a triangle

equals 180 degrees. In a right triangle, the right angle is equal to 90 degrees therefore the sum of the other two angles must equal 90 degrees and therefore complement each other. In other words, two angle measurements that when added together equal 90 degrees are called complementary angles.

Understand that θ (theta) is an angle measurement. Understand that sin θ = cos (90o -θ) and cosθ = sin (90o -θ).

E.g., sin 30o = cos 60o. Sin 30o = 0.5 and cos 60o = 0.5.

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Elements Card Understand that opposite sides are sides that are across from the angle that is

being measured. E.g., if you are measuring angle A the opposite side is side A. If you are measuring angle B the opposite side is side B. The adjacent side is the side that is connected to the angle that is being measured. E.g., the adjacent side to angle A is side B and the adjacent side to angle B is side A. (See figure below.)

Understand that the hypotenuse is the side opposite the right angle of the triangle. It is the longest side of the triangle.

Understand that sine (sin) is equal to the ratio of the length of the opposite side to the length of the hypotenuse (sin θ= opposite

hypotenuse ) Understand that cosine (cos) is equal to the ratio of the length of the adjacent

side to the length of the hypotenuse (cosθ= adjacenthypotenuse )

Understand that tangent (tan) is equal to the ratio of the opposite side to the length of the adjacent side (tanθ=oppositeadjacent )

Understand that cosecant (csc) is equal to the ratio of the length of the hypotenuse to the length of the opposite side (csc θ=hypotenuseopposite )

Understand that secant (sec) is equal to the ratio of the length of the hypotenuse to the length of the adjacent side (secθ=hypotenuseadjacent )

Understand that cotangent (cot) is equal to the ratio of the adjacent side to the length of the opposite side (cotθ=adjacentopposite )

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Elements Card Example Activity: provide students with index cards labeled right angle,

hypotenuse, adjacent, θand opposite. Provide students with a paper that has a premade right triangle already on it. Have students match and identify the areas of the right triangle using the given index cards (understand that the right angle and the hypotenuse never change, but θ can be either of the other angles and then the adjacent and opposite would correspond toθ).

Example activity: provide students with a right triangle that has two of the angle measurements given. Have students determine the measurement of the missing angle. For instance if the given measurements are 90 degrees and 35 degrees, the missing measurement is 55 degrees because the three angle measurements must equal 180 degrees.

Example activity: memory game−provide students with a stack of red index cards that include an angle measurement. Provide students with a stack of blue index cards that include an angle measurement. Assign the red cards to represent cosine and the blue cards to represent sine. Have students lay out the red and blue index cards face down. Students are to choose one red index card and one blue index card and add the measurements together. If the measurements equal 90 degrees the students are able to keep that pair of cards. Students take turns until all pairs are made. Note: if the cards do not equal exactly 90 degrees, they must be replaced face down in original position.

Example activity: provide students with a scientific calculator or computer software program for a scientific calculator: Click here. We are going to show that the ratio of the sides is equal to the angle measure.E.g., if angle A = 53.1o, angle B = 36.9o, side a = 4, side b = 3, and side c = 5.

o Use the scientific calculator to prove this relationship: sin 36.9o = 35

o Use the scientific calculator to find sin 36.9o

(the calculator will give you the approximate value of 0.6.)o Use the scientific calculator to find the decimal value for 3

5

(the calculator will give you the value of 0.6.)o Therefore 0.6 = 0.6

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o Use the scientific calculator to prove this relationship: cos 36.9o = 45

o Use the scientific calculator to find cos 36.9o

(the calculator will give you the approximate value of 0.8.)o Use the scientific calculator to find the decimal value for 4

5

(the calculator will give you the value of 0.8.)o Therefore 0.8 = 0.8

o Use the scientific calculator to prove this relationship: tan 36.9o = 34

o Use the scientific calculator to find tan 36.9o

(the calculator will give you the approximate value of 0.75.)o Use the scientific calculator to find the decimal value for 3

4

(the calculator will give you the value of 0.75.)o Therefore 0.75 = 0.75

o Use the scientific calculator to prove this relationship: csc 36.9o = 53

o Use the scientific calculator to find csc 36.9o

(the calculator will give you the approximate value of 1.67.)o Use the scientific calculator to find the decimal value for 5

3

(the calculator will give you the value of 1.67.)o Therefore 1.67 = 1.67

o Use the scientific calculator to prove this relationship: sec 36.9o = 54

o Use the scientific calculator to find sec 36.9o

(the calculator will give you the approximate value of 1.25.)o Use the scientific calculator to find the decimal value for 5

4(the calculator will give you the value of 1.25.)

o Therefore 1.25 = 1.25

o Use the scientific calculator to prove this relationship: cot 36.9o = 43

o Use the scientific calculator to find cot 36.9o

(the calculator will give you the approximate value of 1.33.)o Use the scientific calculator to find the decimal value for 4

3

(the calculator will give you the value of 1.33.)o Therefore 1.33 = 1.33

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Supports and Scaffolds: Trigonometry calculator, formulas, definitions, index cards, markers, right

triangle, right triangles with two angle measurements given, calculator, red and blue index cards

Resources: Trigonometry calculator: Click here Formulas and notes: Click here Videos: Click here Scientific calculator: Click here Video: Click here Interactive practice activity: Click here Formulas and notes: Click here

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FLS: MAFS.912.G-C.1.1 Prove that all circles are similar.

Access Point NarrativeMAFS.912.G-C.1.AP.1a Compare the ratio of diameter to circumference for several

circles to establish all circles are similar.

Essential Understandings:Concrete Understandings Representation

Given two circles and a non-circle (oval, egg shape, etc.), identify the circles as similar.

Using two circles of different sizes, place one on top of the other (translations) to prove the circles are similar by stretching or shrinking (dilations).

Using a compass, protractor, or geometry software, create and compare similar circles of different sizes.

Suggested Instructional Strategies: Understand that π equals approximately 3.14. Understand that the diameter is the line segment that divides the circle into

two equal parts. Understand that the circumference is the distance around the outside of the

circle. Understand that the circumference equals π times the diameter (C = π x d). Example activity: provide students with lids of various diameters, string, and

rulers. Have students wrap the string around the various lids. Then students will measure the string and the length of the string is the circumference. Then have students measure across the middle of the circle to determine the diameter. Have students multiply the diameter by π (3.14) to determine if their measurement of the circumference (string) is accurate. The measurement of the string should approximately equal the solution from the formula (π x d).

o To compare the ratio of the lids, divide the circumference by the diameter of the circle and that should approximately equal 3.14 (π). If the ratio of the measurements of each lid are approximately 3.14, then the circles are similar.

Supports and Scaffolds: Various round lids, string, rulers, calculators, scissors, and formulas

Resources: Definitions video: Click here Recording chart: Click here

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FLS: MAFS.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Access Point NarrativeMAFS.912.G-MG.1.AP.1a:

Describe the relationship between the attributes of a figure and the changes in the area or volume when one attribute is changed.

Essential Understandings:Concrete Understandings Representation

Identify a figure that represents a change in the original figure.

Use descriptive words about two figures (e.g., bigger, smaller, longer, shorter).

Find the area of a figure. Find area and volume of a figure. Identify which attribute has been changed

when shown the original figure.

Suggested Instructional Strategies:Create same shapes of two different sizes. Three shapes the same size, one smaller or larger. Show students one of the shapes and present them with the other shapes, one being smaller or larger. Then ask students to identify which shape is smaller than this shape? Example:

Repeat this with various pictures identifying concepts such as bigger, smaller, longer, shorterModel with students determining volume of a figure. Provide the equation and work through examples. After determining the volume of a provided figure, make a change to the figure. Have students identify what attribute changed with the figure and determine the volume. For example:Identify the length (3cm), height (2cm) and width(2cm) of this rectangular prism. Work with students to determine

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volume(V=L X W X H) V= 3x2x2 = 12 V= 12

cm❑3

Provide students an example of the same rectangular prism with one attribute changed. Have students identify the attribute that has been changed. Have students identify the length, width and height and redetermine volume as a group. For example: Using the figure below determine the volume and answer the questions.

1) What attribute changed? Length2) What is the length, width and height now? l=4cm, w=2cm, h=2cm3) What is the volume? V=L x W x H = 4 x 2 x 2= 16cm❑

3 4) Did the change in the length cause a change in the volume? Yes

How did this change the volume? The volume increased

Repeat questions with surface area. For example, have students count the cubes or use the formula for area to determine the area of the following

Provide students an example of a figure with one attribute changed. Have students identify the attribute that has been changed. Have students identify the length, width and height and re-determine area as a group.

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For example: Using the figure above to determine the area and answer the questions.

1) What attribute changed? Length2) What is the length and width now? l=4cm, w=2cm3) What is the area? A=L x W = 4 x 2 = 8cm❑

2 ❑ 4) Did the change in the length cause a change in the area? Yes5) How did this change the area? The area increased

Supports and Scaffolds: Small group instruction Interactive whiteboard Repeated exposure to content Student collaboration Cut outs of various shapes Manipulatives Calculators

Additional ResourcesKhan Academy:

Volume of a rectangular prism - Click HereSurface area of a box using nets - Click Here

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FLS: MAFS.912.G-GMD.2.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Access Point NarrativeMAFS.912.G-MD.2.AP.4a Identify shapes created by cross sections of two-dimensional and

three-dimensional figures.

Essential Understandings:Concrete Understandings Representation

Identify the shape of a side(s) of a three-dimensional object.

Match a picture of the side with a picture of the shape.

Dissect a two- or three-dimensional shape using a model or geometry software, and identify the new shape that is created.

Suggested Instructional Strategies: Understand that two-dimensional shapes are called figures and three-dimensional

shapes are called solids. Understand that a plane can cut through a solid in any direction. It does not

have to be horizontal or vertical. Understand that the number of intersected faces equals the number of

edges. E.g., if you are slicing a solid with 8 faces (sides) then you will never have more than 8 edges. If you are slicing a solid with 6 faces (sides) then you will never have more than 6 edges.

Example Activity: have students match a card that contains the visual of a two-dimensional figure that has been sliced from a three-dimensional solid, to the corresponding card that contains the visual of the original three-dimensional solid.

Example Activity: have students create various three-dimensional solids (sphere, cylinder, cone, rectangular solid, etc.) Using Play-Doh or molding clay. Students will then slice a cross-section from the three-dimensional solid using a slicing tool (fishing line, plastic knife, string, etc.) Students will then identify the cross-section figure formed by slicing the solid.

Example Activity: students will create various three-dimensional solids using manipulatives (Legos, magna-tiles, cubes, etc.) Students will then separate a cross-section from the three-dimensional solid. Students will then identify the cross-section figure formed.

Supports and Scaffolds: Computer software programs, Play-Doh, string, plastic knife, fishing line,

Legos, magna-tiles, blocks, cards with shapes and cross-sections, andthree-dimensional solids

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Resources: Video for visualizing cross-sections of prisms: Click here Interactive cross-sections: Click here Interactive matching activity: Click here Interactive cross-sections: Click here Cross-sections: Click here Cross-section video and activities: Click here Videos of cross-sections: Click here

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FLS: MAFS.912.S-ID.1.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

Access Point NarrativeMAFS.912.S-ID.1.AP.1a Complete a graph given the data, using dot plots, histograms, or

box plots.

Essential Understandings:Concrete Understandings Representation

Match the source of the values at the bottom of the x-axis with the appropriate category of the related data table.

Describe the elements within a graph (e.g., in a box plot the line is the median, the line extending from each box is the lower and upper extreme, and the box shows the lower quartile and the upper quartile).

Complete the steps to create a box plot, dot plot or histogram.

Understand the following concepts and vocabulary: quartile, median, intervals, upper and lower extremes, box plot, histograms, and dot plots.

Suggested Instructional Strategies: Follow steps of task analysis to complete box plot, dot plots, or histograms

(these can be found on internet or many calculators). *Model/Lead/Test

Supports and Scaffolds: Technology (e.g., computers) Graphing calculators Self-monitoring task analysis for student independence

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FLS: MAFS.912.S-ID.1.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.MAFS.912.S-ID.1.3Access Point Narrative

MAFS.912.S-ID.1.AP.2a Describe a distribution using center and spread.MAFS.912.S-ID.1.AP.2b Use the correct measure of center and spread to describe a

distribution that is symmetric or skewed.MAFS.912.S-ID.1.AP.2c Identify outliers (extreme data points) and their effects on data

sets.MAFS.912.S-ID.1.AP.2d Compare two or more different data sets using the center and

spread of each. MAFS.912.S-ID.1.AP.3a Use statistical vocabulary to describe the difference in shape,

spread, outliers, and the center (mean).

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-ID.1.AP.2a

Given a data display, identify outliers in the data set.

Identify the highest and lowest value in a data set given a number line and matching symbols (concept of range).

Identify the concept of median using concrete representations of data (create a bar graph with an odd number of bars using snap cubes; arrange from shortest to tallest; students place fingers on two outside towers, knock towers over and move inward until they reach the one middle tower left standing).

Find the mean using concrete materials.

Identify the mean and/or median and the spread of the data.

Calculate the mean using a pre-slugged template of data points.

Order the data set. Understand the following

concepts and vocabulary: median, mode, mean, outliers, standard deviation, interquartile range, center, spread, and range.

Read and describe a display of given data using information about the center and spread.

MAFS.912.S-ID.1.AP.2b

Given a data display, identify outliers in the data set.

Identify the highest and lowest value in a data set given a number line and matching symbols (concept of range).

Identify the concept of median using concrete representations of data (create a bar graph with an odd number of bars using snap cubes; arrange from shortest to tallest; students place fingers on two outside towers, knock towers over and move

Identify the median and the mean of the data set.

Calculate the mean using a pre-slugged template of data points.

Order data set. Understand the following

concepts and vocabulary: median, mean and outliers.

When comparing the mean and the median, if the mean is larger than the median, the

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Elements CardAccess Point

Concrete Understandings Representation

inward until they reach the one middle tower left standing).

Find the mean using concrete materials.

data is skewed to the right. If the mean is smaller than the median the data is skewed to the left. If the mean equals the median, the data is symmetric.

MAFS.912.S-ID.1.AP.2c

Arrange all data points from lowest to highest.

Calculate the median (middle number) of the data set. (The median can also be called Q2).

Calculate the lower quartile (Q1).  This is the halfway point of the points in the data set below the median. If there are an even number of values below the median, average the two middle values to find Q1.

Calculate the upper quartile (Q3).  This is the halfway point of the points in the data set above the median. If there are an even number of values below the median, average the two middle values to find Q3.

Find the interquartile range (IQR). IQR = Q3 – Q1

Calculate: 1.5 X IQR Calculate the lower bound for the

outliers Q1- 1.5 X IQR (Any data value lower than the lower bound is considered an outlier.)

Calculate the upper bound for the outliersQ3+ 1.5 X IQR (Any data value higher than the upper bound is considered an outlier.)

Click here Click here

Understand the vocabulary: data points, median, mode, mean, outliers, quartile, box plot, 5 number summary, maximum, minimum, lower bound, upper bound.

Understand that an outlier affects the mean but not the mode or median. For example: When zero is added to the given data set (see figure below) the mean changes but the median does not.

Click here Click here

MAFS.912.S-ID.1.AP.2d

Click here Back to Back Stem Plots: Amount of

money carried by teenage boys and girls

Understand the following concepts and vocabulary: center, spread, data points, median, mean, quartile, 5 number summary (minimum, Q1, median, Q3, maximum), maximum, minimum, lower

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Concrete Understandings Representation

Find the mean.Mean: boys $44.43, girls $34.93

Find the median.Median: boys $42, girls $36

Find the interquartile range (IQR). IQR = Q3 – Q1IQR: boys $59 - $34 = $25, girls$44 - $28 = $16

Find the standard deviation. Standard Deviation: boys $18.43, girls

$10.47. Center: On average, the boys carry

more money than the girls. Spread: The amount of money carried

by boys is more dispersed than the amount of money carried by girls.

Double Bar Charts:

Find the mean.Mean: Pretest 67.5, Post-test 77.5

Find the median.Median: Pretest 67.5, Post-test 80

Find the interquartile range (IQR). IQR = Q3 – Q1IQR: Pretest 77.5 – 57.5 = 20, Post-test 92.5 – 62.5 = 30

bound, upper bound, standard deviation and interquartile range. Click here

Use graphs or graphic organizers to compare the measures of central tendency of two different data sets.

Identify the same measure of central tendency in two different data sets (e.g., the mean in one data set and the mean in another data set).

Read and interpret each display of given data (e.g., bell curve, scatter plot, box plot, stem plot) to draw inferences about the data.

When comparing two standard deviations, understand that the larger standard deviation indicates more variability (spread). For example, in the stem plot, the boys’ standard deviation of $18.43 versus the girls’ standard deviation of $10.47 means that the amount of money carried by teenage boys is dispersed further from the mean than the amount of money carried by girls.

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Concrete Understandings Representation

Find the standard deviation. Standard Deviation: Pretest 11.9,

Post-test 18.48. Center: On average, students scored

higher on the post-test than the pretest.

Spread: The post-test scores were more dispersed than the pretest scores.

MAFS.912.S-ID.1.AP.3a

Given a data set, identify outliers. Identify the highest and lowest value in

a data set given a number line and matching symbols (concept of range).

Find the mean (average) using concrete materials.

Using concrete materials to create the shape that the data set represents.

Match up pictures of data distribution (e.g., normal curve, skewed left or right) to its statistical vocabulary.

Order numbers in a data set from least to greatest.

Understand the following concepts and vocabulary: median, mode, mean, outliers, normal, skewed, symmetric shaped curve and range.

Use pictures of data distributions (e.g., normal curve, skewed left or right) to describe the difference in shape, spread, outliers, and center using statistical vocabulary.

Calculate the mean using a template of data points.

State/show the highest and lowest value in a data set (concept of range – e.g., template of data points).

Suggested Instructional Strategies: Example Activity for measures of Central Tendency (center- mean, median,

mode): have students get into two groups (boys vs girls), and count out the number of letters in everyone’s name and place into a table (graphic organizer). To find the mode, students must list the numbers and determine the number that occurs the most. E.g., if we have the following numbers for girls' names: 5,

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Elements Card7, 5, 6, 9, 5, 11, 2, the mode would be 5. To find the median, have the students list the numbers in numerical order and determine the middle number. In this example the median is 5.5. To find the mean, have the students add up the numbers and divide by the total number of items. In this example the mean is 6.25. Have students repeat these processes for the boys' names in order to compare the results of the two data sets. Using the data collected, have students generalize the results to larger populations. If the data shows that girls’ names tend to be longer than boys, the students can generalize that based on this sample, girls names are usually longer than boys’ names.

Example Activity: Students can compare the height of girls vs. the height of boys using measures of central tendency and generalizing the results of the smaller sample size to larger populations.

To help students understand the definition of mean, have students stack unifix cubes to the number of letters in their name. Using the sample data like the example above, if there is 5 letters in their name, the student will stack 5 unifix cubes, if there are 7 letters, have them stack 7 unifix cubes. After the students have stacked their unifix cubes to the number of letters in their names, then students will level all the stacks so that the stacks are the same height (which gives us the mean of the data.) E.g., unifix cube stacks representing girls' names with the height numbers of 2, 5, 8, 3, 2 will level off to 4 cubes high (mean). The unifix cube stacks for boys 3, 6, 7, 8, 4, 2 will level off to 5 cubes high (mean).

To find the median of the data with unifix cubes: line cubes up numerically (before they are leveled) to find the middle number. E.g., 2, 2, 3, 5, 8 with the median being 3 for girls.

To find the mode with the unifix cubes: line cubes up by height (before they are leveled) to determine which height has the most stacks. E.g., 2, 2, 3, 5, 8 with the mode being 2 for girls.

To help students understand the definition of the range (spread), have students take the highest number from the data set for girls (11) and subtract the lowest number (2) therefore the range (spread) is 11 - 2 =9.

To help students understand the definition of standard deviation (spread).

The larger the spread is around the mean, the larger the standard deviation.

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The smaller the spread is around the mean, the smaller the standard deviation. Given measures of central tendency and measures of variability (range and

standard deviation), have students compare results of two different data sets. Example activity: have students compare given data results from

experiments. E.g., in an experiment that determines the amount of water held by various paper towel brands the following data was collected:

o Paper towel A – Mean = 29.5, Mode = 0, Median = 29.5, range = 2, standard deviation = 1.29.

o Paper towel B – Mean = 11.5, Mode = 0, Median = 11.5, range = 3, standard deviation = 1.29.

o Have students compare the measures of central tendency and measures of variability of the two data sets. In the above example, because the median and mean are the same in both sets, this indicates that the trials are consistent.

o The spread around the mean is the same in both sets of data because the standard deviations are the same.

o The range is small in both data sets, indicating that the data was collected consistently.

o Based on the data given, the experiment indicates that towel A is more absorbent than towel B.

Graphic representation Graphic representation with normal curve (standard deviation)

o Based on the graphic representation above, students can determine that the data given is consistent within the trials conducted however the two different paper towels absorb different amounts of water.

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Elements Cardo Understand that flaws and bias can affect the results of the data sets.

E.g., each brand of paper towels had only four trials, a flaw in this collection could be that the results may differ if more trials were conducted. Another flaw could be human error when squeezing the water out of the paper towel.

o Understand that lurking variables may be present but not accounted for in the data results. E.g., the strength of the student may influence the data results.

o Understand that if the data is skewed to the left then the mean is smaller than the median and if the data is skewed to the right then the mean is larger than the median. The mean is always pulled towards the outlier.

Skewed Left (Negative)

Skewed Right (Positive)

o Understand that when using skewed data, the median should be used to describe the center because the mean is pulled towards the outlier.

o Understand that if the data is symmetric (normal), the mean equals the median.

Supports and Scaffolds: Calculator, graphic organizers, ruler, tape measure, unifix cubes, data sets,

formulas for central tendency, formulas for measures of variability, paper towels, water, beaker, and computer program software

Resources: Mean, median, and mode song: Click here Mean and median tool: Click here Bears in a boat aActivity (hands on activity to help students figure the mean,

median, mode, and range): Click here Examples of data collection: Click here Example of comparing data: Click here Measure of central tendencies: Click here Mean, mode, and median with manipulatives: Click here

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Elements Card Mean, mode, median worksheet: Click here Mean, mode, median learning strategy: Click here Examples of data sets: Click here Example activity: Click here Standard deviation calculator: Click here Comparing data results: Click here Definitions: Click here Questions about measures of central tendency: Click here Measures of central tendency with visuals: Click here When and how to use measures of central tendency: Click here Real world examples of measures of central tendency: Click here Measures of variability: Click here Measures of spread: Click here Examples of skewed data: Click here Symmetrical and skewed data: Click here Types of skewed data with real world examples: Click here

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FLS: MAFS.912.S-ID.1.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Access Point NarrativeMAFS.912.S-ID.1.AP.4a Use descriptive stats like range, median, mode, mean, and

outliers/gaps to describe the data set.

Essential Understandings:Concrete Understandings Representation

Given a scatter plot, identify outliers in the data set.

Identify the highest and lowest value in a data set given a number line and matching symbols (concept of range).

Identify the representation (use plastic snap cubes to represent the tally showing the number of occurrences) of the concept of mode.

Identify the concept of median using concrete representations of data (create a bar graph with an odd number of bars using snap cubes; arrange from shortest to tallest; student place fingers on two outside towers, knock towers over and move inward until they reach the one middle tower left standing).

Find the mean using concrete materials.

Identify the mode and the spread of the data using a line drawing of the distribution.

Calculate the mean using pre-slugged template of data points.

Order data set using numeric symbols. Understand the following concepts and

vocabulary: median, mode, mean, and outliers.

Suggested Instructional Strategies: Task analysis for finding range, median, mode, and mean Explicit vocabulary instruction for outliers *Multiple Exemplars for outliers Model data descriptions Use concrete materials to find the mean (leveled plastic snap cubes: using

the same bar graph with snap cubes, rearrange cubes into equal stacks.)

Supports and Scaffolds: Template for finding mean Assistive technology/voice output devices Interactive Whiteboard Provide a graph of the data set Templates with sentence starters Manipulatives

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FLS: MAFS.912.S-ID.2.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Access Point NarrativeMAFS.912.S-ID.2.AP.5a Recognize associations and trends in data from a two-way table

Essential Understandings:Concrete Understandings Representation

Identify data as categorical or continuous.

Identify a question that uses categorical or continuous data.

Identify a data collection method that gathers categorical or continuous data.

Understand the concepts and vocabulary related to survey: data, categorical data, continuous data, sample, population and discrete data.

Select a sample and data collection plan that matches a provided question.

Suggested Instructional Strategies: Understand Vocabulary: trends and associations are patterns and

relationships between the variables. Example activity: divide students by gender, provide them with three types of

pets (dog, cat, fish). Have students choose which animal they prefer and tally up the votes on a chart. Create a two-way table that represents the completed tally chart.

o Using the results of the two way table above, students will determine if there are any associations or trends. E.g., based on data above, more females tend to pick cats than males.

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by going to the designated corner of the room indicated by visuals of the specific animal. Have class create the two way table from the data determined by the votes of the preferred animals.

Supports and Scaffolds: Understand vocabulary: trend, association, tally table, two-way table, and variable Visuals of animals Two-way table Tally table

Resources: Two-way tables:

o Math Bits Notebook: Click hereo Stat Trek: Click here

Video: Click here LearnZillion video: Click here Lab activity: Click here Example lesson: Click here

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FLS: MAFS.912.S-ID.2.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

b. Informally assess the fit of a function by plotting and analyzing residuals.c. Fit a linear function for a scatter plot that suggests a linear association.

Access Point NarrativeMAFS.912.S-ID.2.AP.6a Create a scatter plot from two quantitative variables.MAFS.912.S-ID.2.AP.6b Describe the form, strength, and direction of the relationship.MAFS.912.S-ID.2.AP.6c Categorize data as linear or not.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-ID.2.AP.6a

Given a scatter plot, identify the two variables.

Predict the variables using a template of data points.

Order the numbers in a data set from least to greatest.

Understand the following concepts and vocabulary: linear, function, data set, variables and predict.

Create a scatter plot given a data set.

MAFS.912.S-ID.2.AP.6b

Given a scatter plot, identify the two variables.

Plot points on a scatter plot.

Draw a line of best fit (i.e., use a transparency to place a line over the data in a scatter plot or a strand of spaghetti or a pipe cleaner to represent the line of best fit).

Match the line with the statement describing the relationship between the variables (are the data points moving up to the right or left, positive or negative correlation).

Identify the relationship (fit) between the data and the function (e.g., linear, non-linear).

Understand the following concepts and vocabulary: scatter plot, variables, positive/negative correlation, linear, non-linear and line of best fit.o e.g., As the age increases (gets larger),

the weight also increases (gets larger).

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MAFS.912.S-ID.2.AP.6c

Draw or point to a line that goes through as many points on a scatter plot or divides the data in half (i.e., use a transparency to place a line over the data in a scatter plot or a strand of spaghetti or a pipe cleaner to represent the line of best fit – a picture of a simple scatterplot with variables and a line of linear fit [line of best fit]).

Determine if the line goes through many points. If not the graph is not linear.

Identify the relationship (fit) between the data and the function (e.g., linear, non-linear).

Non-linear:

Linear:

Click here

Suggested Instructional Strategies: Understand vocabulary: scatter plot, data set, linear, non linear, form, strength,

direction, and variable Example activity: provide students with data sets from two quantitative variables

(arm span and height) and have students create scatter plot from the given data. Extension activity: have students measure their own arm span and height and

graph the data set.

Example Activity for line of best fit: using the data above, draw a line of best fit (i.e., use a transparency to place a line over the data in a scatter plot or a strand of spaghetti or a pipe cleaner to represent the line of best fit.)

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the right or left, positive or negative correlation):

Identify the relationship (fit) between the data and the function (e.g., linear, non-linear):

Non-Linear Linear

Supports and Scaffolds: Graph paper, dry erase boards, ruler, measuring tape, spaghetti, pipe cleaner,

transparency, dry erase markers, wet/dry markers, and graphing resources.

Resources: Tool to create scatter plots: Click here Scatter plot practice worksheets: Click here Scatter plot activity: Click here Video: Click here Notes to construct scatter plots: Click here Interactive notebook: Click here Example activity: Click here Line of best fit: Click here

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FLS: MAFS.912.S-ID.2.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

b. Informally assess the fit of a function by plotting and analyzing residuals.c. Fit a linear function for a scatter plot that suggests a linear association.

Access Point NarrativeMAFS.912.S-ID.2.AP.6d Use algebraic methods and technology to fit a linear function to

the data.MAFS.912.S-ID.2.AP.6e Use the function to predict values. MAFS.912.S-ID.2.AP.6f Explain the meaning of the constant and coefficients in context.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-ID.2.AP.6d

Draw a line between two data points. Find the slope between the two points on the graph by counting the units vertically and horizontally between the two points.

m=y2− y1x2−x1

Click here Click here

Choose two points from the data set, and calculate the slope between the two points.

m=y2− y1x2−x1

Use the calculated slope and one of the points to find the equation of the line.

y – y1 = m(x−x1) Use the line of best fit capability on

the graphing calculator.MAFS.912.S-ID.2.AP.6e

Use manipulatives to model the function with input values. For example, y = x + 3 could be modeled with snap cubes. Given three snap cubes, adding 5 more snap cubes would give you 8.

Generate input values. For example,x = -1, 0, 1, 2, 3, 4, 5, etc.

Use the function to generate output values based on the input values. For example: y = x + 3, when x = 5, y = 8.

MAFS.912.S-ID.2.AP.6f

Use manipulatives to model the function with input values. For example, y = x + 3 could be modeled with snap cubes. Given three snap cubes, adding 5 more snap cubes would give you 8.

Use a template to identify the constant and the coefficient. For example: y = Ax + C where A is the coefficient and C is the constant. In the following example, Jane pays $3 per video

Generate input values. For example,

x = -1,0,1,2,3,4,5 etc. Use the function to generate output

values based on the input values. For example: y = x + 3, when x = 5, y = 8

Given a word problem, identify the parts of the problem that relate to the parts of the function. For example, Jane pays $3 per video

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Concrete Understandings Representation

every time she goes to Movies Plus. Her membership fee is $2 a visit. Given the equation y = 3x + 2, x equals the number of movies, 3 represents the coefficient and 2 represents the constant.

every time she goes to Movies Plus. Her membership fee is $2 a visit. Given the equation y = 3x + 2, x equals the number of movies, 3 represents the cost per movie, 2 represents the fee per visit and y represents the total cost per visit.

Suggested Instructional Strategies: Understand vocabulary and concepts: rate of change (slope), constant,

coefficient, positive, negative, linear, and function Given a real life scenario, the student will match the function notation to the

situation.o E.g., a pack of gum is on sale for “buy one get one free.” If Sally buys

one pack of gum she gets a total of two packs. If she buys 2 packs of gum, she gets 4 packs, etc.

How many packs of gum would she get if she buys “x” number of packs of gum? f(x)=2x

The rate of change in words: 2 packs of gum for the price of 1 pack of gum. The rate of change in numbers: 21 or 2:1, or 2

The rate of change in graph form: 21 or 2:1, or 2

o aa is amount of packs purchasedo bb is amount of packs received (2 for 1)

Example activity: using the graph above, teach the students to count from 1 point on the graph to the next by counting up and over (Up 2 lines over 1 line.)

Using Algebraic method:

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Example activity: tape a coordinate plane on the floor. Have one student stand at (1,1). Have another student stand at (3,6). Have a third student stand by first student and count their steps up and over to get to the second student. The steps up indicate the rise and the steps over indicate the run.

Example activity: give students a field of three graphs and have them select the graph that has a slope of 21

Example activity for constant and coefficients:o Bill wants to rent a canoe and has found two companies that will rent

the canoe with life vest. o One company rents their canoe for $15.00/hour plus a $4.00 flat fee

for life vests y=15x + 4. The constant is the $4.00 flat fee, the coefficient (slope) is $15.00/hour.

o Another company rents the canoe for $19.00/hour and life vest rental is free y=19x. The constant is 0 and the coefficient (slope) is $19.00/hour.

Supports and Scaffolds: Graph paper, tape, number line, geoboards and bands, computer software

programs, calculator, gum, counters, and ruler Resources:

Slope and y-intercept: Click here Slope and rate of change: Click here Graph activity: Click here Slope manipulative: Click here Definitions: Click here Activity: Click here Slope memory game: Click here Foldables: Click here Slope activities: Click here

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FLS: MAFS.912.S-ID.3.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Access Point NarrativeMAFS.912.S-ID.3.AP.7a Interpret the meaning of the slope and y-intercept in context.

Essential Understandings:Concrete Understandings Representation

Match the slope to the appropriate visual (for example, given a positive slope, match the appropriate graph from a field of three: positive, negative, or no slope.)

Interpret the slope as the rate of change in the data. A positive slope is data that is increasing and a negative slope is data that is decreasing.

Interpret the y-intercept in terms of the data.

Suggested Instructional Strategies: Example activity for negative slope: “How Fast Does Ice Cream Melt?” If a

student has an ice cream cone in Florida that is 6 inches tall and is melting at a constant rate of an inch a minute, if none of the ice cream is eaten, determine the following: how tall is the ice cream after 3 minutes? How long it will take for the ice cream to melt completely?

Example activity for positive slope: “canoe rental cost.” Bill wants to rent a canoe and has found two companies that will rent the canoe with life vest.

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$15.00/hour plus a $4.00 flat fee for life vests y=15x + 4. They-intercept is the $4.00 flat fee, the slope is $15.00/hour.

Another company rents the canoe for $19.00/hour and life vest rental is free y=19x. The y-intercept is 0 and the slope is $19.00/hour.

Supports and Scaffolds: Graph paper, computer software, ruler, ice cream, dry spaghetti, geoboard

and geobands, calculator, and painters tape for floor graph

Resources: Examples of graphs with one solution, no solutions, and infinite solutions:

Click here Slope and y-intercept in context: Click here Domino’s pizza activity: Click here Slope and y-intersect activity: Click here Slope-intercept rummy: Click here Linear sorting and matching: Click here Templates: Click here Who Wants To Be A Millionaire slope-intercept: Click here Matching graphs, points and slopes activity: Click here

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FLS: MAFS.912.S-ID.3.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.Access Point Narrative

MAFS.912.S-ID.3.AP.8a Identify the correlation coefficient (r) of a linear fit.MAFS.912. S-ID.3.AP.8b Describe the correlation coefficient (r) of a linear fit (e.g., a strong

or weak positive, negative, perfect correlation).Essential Understandings:

Access Point

Concrete Understandings Representation

MAFS.912.S-ID.3.AP.8a

Identify which representation more likely represents a coefficient of 1, -1, and 0.

Match the words strong with (correlation coefficient “r”) between 1 and -1, and weak with the number 0 – e.g., sorting cards on a number line template (with the words strong written under 1 and -1 and weak under 0) if the correlation coefficient is closer to 1 (-1) the data has a strong correlation to the graph and x and y and if closer to 0 it is weak.

Understand that “r” represents the correlation coefficient.

Understand that the closer “r” is to 1 (-1) the stronger the data fits the relationship of x and y.

Understand that the closer “r” is to 0 the weaker the data fits the relationship of x and y.

Understand the following concepts and vocabulary: linear model, correlation coefficient, linear relationship and linear fit.

MAFS.912. S-ID.3.AP.8b

When given a number (correlation coefficient “r”) between 1 and -1, show by sorting cards on a number line template (with the words strong written under 1 and -1) if the correlation coefficient is closer to 1 (-1) the data has a strong correlation to the graph and x and y.

Match descriptors of a correlation coefficient with its numeric “r” value (e.g., weak = 0.1, strong = 0.9).

Understand that “r” represents the correlation coefficient.

Understand that the closer “r” is to 1 (-1) the stronger the data fits the relationship of x and y.

Understand that the closer “r” is to 0 the weaker the data fits the relationship of x and y.

Describe a correlation coefficient using appropriate vocabulary (e.g., positive correlation, negative correlation, no correlation or perfect [exactly 1 or -1] correlation).

Understand the following concepts and vocabulary: correlation coefficient, linear relationship, positive correlation, negative correlation, no correlation and

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Concrete Understandings Representation

perfect correlation.

Suggested Instructional Strategies: Example Activity: give students visual representations of correlation graphs.

Have students match correlation coefficients (r) to the correct given graph and identify the strength of the correlation.

Sample correlation ranges Example Activity: have students play memory game with correlation graphs

and coefficients (r).

Supports and Scaffolds: Computer software program, correlation graph visuals, index cards, ruler, dry

spaghetti, pipe cleaner, and visual of correlation scale

Resources: Guessing correlations: Click here Correlations guessing game: Click here Correlation and coefficient worksheets (beginning on pg.17): Click here

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FLS: MAFS.912.S-ID.3.9 Distinguish between correlation and causation.Access Point Narrative

MAFS.912.S-ID.3.AP.9a Given a correlation in a real-world scenario, determine if there is causation.

Essential Understandings:Concrete Understandings Representation

State the cause and effect relationship between two variables.

State the cause and effect relationship between two variables in reverse.

Recognize examples of correlation with causation (e.g., If you push an object, force is correlated with the distance it moves and the distance the object moved is caused by the force.).

Recognize examples of correlation without causation (e.g., The distance a rolled ball travels is correlated with how much time passes, but the distance it travels is not caused by time.).

Understand that the cause and effect relationship should be true for the situation and its reverse to have causation.

Understand that a correlation is a relationship between two or more variables.

Understand that a high correlation does not imply causation (i.e., We observe a very strong correlation when comparing US highway fatality rates and lemons imported from Mexico (R2 = 0.97). However, importing lemons does not cause traffic fatalities.).

Suggested Instructional Strategies: Understand Vocabulary: correlation, causation, variables, positive strength,

and negative strength Understand that correlation does not mean there is causation. E.g., just

because there may be a strong correlation between shoe size and test scores does not mean that the test scores caused the shoe size or vice versa.

Example Activity: provide students with real world examples to determine causation. E.g., is there a correlation with possible causation between the amount of money earned compared to the amount of hours worked in a week? If the amount of hours increased and the amount of money earned increased there is a positive correlation with probable causation.

Supports and Scaffolds: Real world scenarios

Resources: Example real world scenario’s (refer to attachment section): Click here Correlation and causation: Click here Differentiation between correlation and causation: Click here Funny examples of real data with no causation: Click here and here

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FLS: MAFS.912.S-IC.1.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Access Point NarrativeMAFS.912.S-IC.1.AP.1a Determine what inferences can be made from statistics.

Essential Understandings:Concrete Understandings Representation

Match a sample to a population – e.g., Count out the total number of red candies in a small sample size to estimate, identify the probability of and/or hypothesize the total number of red candies in a larger sample.

Understand the following concepts and vocabulary: statistics, inference, conclusion, estimation, probability (likelihood), prediction and hypothesis testing (cause/effect).

Identify potential inferences when given data from a sample.

Suggested Instructional Strategies: Example Activity: using the number of brown-haired students in a classroom

of 25 students, estimate how many students in a school with 1000 students has brown hair: Mrs. Lee’s classroom of 25 students has 11 students with brown hair. Based on these results, the students can infer that the school has approximately 440 students that have brown hair.

Supports and Scaffolds: Calculator, assorted manipulatives, and tally table

Resources: Example probability activity: Click here Making inferences about wetlands video: Click here

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FLS: MAFS.912.S-IC.1.2 Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?Access Point Narrative

MAFS.912.S-IC.1.AP.2a Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

MAFS.912.S-IC.1.AP.2b Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

MAFS.912.S-IC.1.AP.2c Determine what inferences can be made from the model. Essential Understandings:

Access Point

Concrete Understandings

Representation

MAFS.912.S-IC.1.AP.2a

Given a chance event, find the probability using a manipulative. For example, the probability of landing on yellow = 1/5 or 0.2.

Understand the following concepts, symbols, and vocabulary for: probability and likelihood.

Given a chance event, find the probability. For example the probability of rolling a 2 with a die is 1/6 or 0.166...

Given a chance event, find the probability. For example the probability of rolling a B on a number die is 0/6 or 0.

Given a chance event, find the probability. For example the probability of pulling a marble out of a bag of 5 marbles is 5/5 or 1.

Click here MAFS.912.S-IC.1.AP.2b

Use items like coins to create events.

Using manipulatives and a chart to capture the outcomes of coin flips or dice rolls.

Use items like coins to determine the probability of an outcome (1/2 heads).

Understand the following concepts, symbols, and vocabulary for: probability and likelihood.

Identify the formula for finding probability of an event (probability of an event happening = number of ways it can happen/total number of outcomes).

MAFS.912.S-IC.1.AP.2c

Use items like coins to determine the

Understand the following concepts, symbols, and vocabulary for: probability, likelihood and

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Concrete Understandings

Representation

probability of an outcome (1/2 heads).

Using manipulatives and a chart to capture the outcomes of coin flips or dice rolls.

Use a picture or manipulative to choose the next possible outcome.

inference. Given a chart of outcomes predict the next

possible outcome. Given an experimental probability, determine

the sub group of the population. E.g., if the experimental probability of a student selecting ice cream is 2/5 and the population is 200. The sub group of students choosing ice cream = 80. Click here

Suggested Instructional Strategies: Understand that a probability of “1” indicates that the chance event is likely

going to happen and a probability of “0” indicates that the chance event is not likely to happen.

Example activity: given a container filled with counters, (5 red, 5 green, 5 blue, 5 yellow, and 5 white). Have students predict the likelihood of pulling out 1 yellow counter from the container (0.2 or 1/5).

Example Activity: given a container filled with counters, (9 red, 1 green). Have students predict the likelihood of pulling out 1 red counter from the container (0.9 or 9/10). To show a probability of “0”, have students predict the likelihood of pulling a blue counter from the container.

Example Spinner Activity: given the following data, have students make inferences about larger sample sizes: a spinner was spun 5 times, which resulted in a probability of 2/5 (or 0.40) that the spinner landed on blue. Have students make inferences on how many times the spinner will land on blue if it is spun 100 times (40 times).

Supports and Scaffolds: Spinner, dice, counters, container, tally chart, calculator, candy,

manipulatives, and computer software programResource:

Guess vs. statistics activities (see attachments section): Click here Probability activities: Click here adjustable spinner: Click here probability notes: Click here Candy applet activity: Click here Experimental probability activity: Click here

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FLS: MAFS.912.S-IC.2.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.Access Point Narrative

MAFS.912.S-IC.2.AP.3a Understand that statistics can be used to gain information about a population by examining a random sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population.

MAFS.912.S-IC.2.AP.3c Use measures of center tendency (mean, median, mode) and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-IC.2.AP.3a

Given a classroom, put all of the student names in a hat. The hat with all of the names in it represents the population. Randomly choosing names out of the hat represents a random sample.

Given a classroom in the school, determine the range of ages in the classroom. Use the data to approximate the range of ages in a school or grade level.

Understand that statistics is collecting, organizing, analyzing, and interpreting data in order to make decisions.

Understand that mean, median, mode, range, and standard deviation are common statistics.

Understand that each item/subject in a random sample has the same chance of being selected.

Understand that generalizations are only valid if they are based on similar characteristics in both the sample and the population.

Understand that decisions about the population can be made based on the information gathered from the random sample.

MAFS.912.S-IC.2.AP.3c

Use manipulatives to add the numbers in a given data set.

Use manipulatives to divide the sum of a data set.

Identify the mean of a data set from manipulatives or pictorial representations.

Identify the lowest to highest value in a data set given a number line and matching

Understand that the measures of center include the mean (average), median (middle value), and mode (most common value).

Understand that the measures of variability include the range (difference between the largest and smallest value) and standard deviation. (The standard deviation is a statistic that tells you how tightly

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symbols. Arrange data from lowest to

highest. Identify the representation

(plastic snap cubes, wiki sticks) of the mode.

Use concrete materials to produce the mean (leveled plastic snap cubes).

Find the object or manipulative in a sequence that represents the middle.

all the various examples are clustered around the mean in a set of data.)

Given a measure of center or variability from two populations, make a generalization for the population based on the data.

Suggested Instructional Strategies: Example activity for measures of central tendency: have students get into

two groups (boys vs. girls), and count out the number of letters in everyone’s name and place into a table (graphic organizer). To find the mode, students must list the numbers and determine the number that occurs the most. E.g., if we have the following numbers for girls' names: 5, 7, 5, 6, 9, 5, 11, 2, the mode would be 5. To find the median, have the students list the numbers in numerical order and determine the middle number. In this example the median is 5.5. To find the mean, have the students add up the numbers and divide by the total number of items. In this example the mean is 6.25. Have students repeat these processes for the boys’ names in order to compare the results of the two data sets. Using the data collected, have students generalize the results to larger populations. If the data shows that girls’ names tend to be longer than boys, the students can generalize that based on this sample, girls names are usually longer than boys’ names.

Example Activity: students can compare the height of girls vs. the height of boys using measures of central tendency and generalizing the results of the smaller sample size to larger populations.

To help students understand the definition of mean, have students stack unifix cubes to the number of letters in their name. Using the sample data like the example above, if there is 5 letters in their name, the student will stack 5 unifix cubes, if there are 7 letters, have them stack 7 unifix cubes. After the students have stacked their unifix cubes to the number of letters in their names, then students will level all the stacks so that the stacks are the same height. (Which gives us the mean of the data.) E.g., unifix cube stacks representing girls' names with the height numbers of 2, 5, 8, 3, 2 will level off to 4 cubes high (mean). The unifix cube stacks for boys 3, 6, 7, 8, 4, 2 will level off to 5 cubes high (mean).

To find the median of the data with unifix cubes: line cubes up numerically (before they are leveled) to find the middle number. E.g., 2, 2, 3, 5, 8 with the median being 3 for girls.

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Elements Card To find the mode with the unifix cubes: line cubes up by height (before they

are leveled) to determine which height has the most stacks. E.g., 2, 2, 3, 5, 8 with the mode being 2 for girls.

Supports and Scaffolds: Calculator, graphic organizers, ruler, tape measure, and unifix cubes

Resources: Advantages and disadvantages of studies: Click here Mean, median, and mode song: Click here Mean and median tool: Click here Bears in a boat activity: Click here Examples of data collection: Click here Example of comparing data: Click here Measure of central tendencies: Click here Mean, mode, and median with manipulatives: Click here Mean, mode, median worksheet: Click here Mean, mode, median learning strategy: Click here Questions about measures of Central Tendency: Click here Measures of central tendency with visuals: Click here When and how to use measures of central tendency: Click here Real world examples of measures of central tendency: Click here Measures of variability: Click here Measures of spread: Click here

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FLS: MAFS.912.S-IC.2.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.Access Point Narrative

MAFS.912.S-IC.2.AP.3b Identify the purpose of sample surveys, experiments and observational studies.

MAFS.912.S-IC.2.AP.3d Identify the differences between sample surveys, experiments and observational studies.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-IC.2.AP.3b

(Survey) Given a statistical question, record student answers using pictures or manipulatives in a table. For example: What is your favorite lunch menu choice?

(Observational Study) Have students observe other students in the hall during the change of classes. Have students draw a picture of their observations.

(Experiment) Have students drop food coloring into water and observe for 5 minutes. Compare the change in the water with food coloring to water without food coloring.

Understand that an observational study is where the researcher observes and measures characteristics of interest. They do not interact with the subjects. For example: A teacher observes a student and makes antidotal notes.

Understand that an experiment is where a researcher applies a treatment/intervention to the group observed. The researcher may use a control group. They may also use a placebo (fake treatment). For example: A doctor prescribes medicine to one group and a placebo to another group to compare the effectiveness of a medication.

Understand that a survey is an investigation of one or more of the characteristics of a population. The researcher interacts with the subjects. For example: The researcher asks 20 participants for their favorite subject in school.

Given a sample survey, experiment, and observational studies, list the purpose of each.

MAFS.912.S-IC.2.AP.3d

(Survey) Given a statistical question, record student answers using pictures or manipulatives in a table. For example: What is your favorite lunch menu choice?

(Observational Study) Have students observe other students in the hall

Understand that an observational study is where the researcher observes and measures characteristics of interest. They do not interact with the subjects.

Understand that an experiment is where a researcher applies a treatment/intervention to the group observed. The researcher may use a control group. They may also use a placebo (fake treatment).

Understand that a survey is an

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during the change of classes. Have students draw a picture of their observations.

(Experiment) Have students drop food coloring into water and observe for 5 minutes. Compare the change in the water with food coloring to water without food coloring.

investigation of one or more of the characteristics of a population. The researcher interacts with the subjects. For example: The researcher asks 20 participants for their favorite subject in school.

Given a sample survey, experiment, and observational studies, list the characteristics of each.

Compare the differences between the characteristics of a sample survey, experiment and observational study.

Suggested Instructional Strategies: Understand that an observational study is where the researcher observes and

measures characteristics of interest. Understand that an experiment is where a researcher applies a

treatment/intervention to the group observed. Understand that a survey is an investigation of one or more of the

characteristics of a population. Example Activity: provide students with a statistical scenario and have them

decide if the data was collected through an observation, experiment or survey. E.g., if the given scenario is asking for food preference sold in the cafeteria, a survey would be used. If the given scenario is asking if eating chocolate helps them do better on a test, an experiment would be used. If the given scenario is asking who uses umbrellas more when it is raining, girls or boys, an observation would be used.

Supports and Scaffolds: Scenarios, definitions of observational studies, experiments, and surveys

Resources: Distinguish between observational studies, surveys, and experiments:

Click here Sample surveys, experiments and observations: Click here Advantages and disadvantages of studies: Click here

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FLS: MAFS.912.S-IC.2.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Access Point NarrativeMAFS.912.S-IC.2.AP.4a Understand that the margin of error produces a range of values.

Essential Understandings:Concrete Understandings Representation

Use manipulatives and a ruler to measure the lengths of an object. Record the measurement from several different students. (The measurements will be slightly different.) Determine an acceptable range of measurement that would still be considered correct.

Understand that the margin of error describes the level of accuracy of an experiment. The margin of error describes a range that helps determine the trustworthiness of results. For example: An election results poll has a +/– 3% margin of error.

Understand that a smaller margin of error indicates trustworthy results and a larger margin of error means the results are not considered as accurate.

Suggested Instructional Strategies: Understand that when using the results from a sample, to generalize to a

larger population, there will always be some room for error. This is called the margin of error. E.g., if we are using the results from a voting poll, the data will not match exactly to a larger population because we are using a “sample.” Therefore, there will always be a margin of error present.

Understand that the larger the sample size the smaller the margin of error which means that the data is more accurate.

Understand that if the pollsters are predicting a candidate to currently have 55% of the votes with a margin of error of ±3% that the candidate’s votes could range from 52% to 58%.

Resources: Margin of error: Click here Margin of error explanation: Click here

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FLS: MAFS.912.S-IC.2.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.Access Point Narrative

MAFS.912.S-IC.2.AP.4b Use the sample data to create a proportional relationship to find the population data. For example, if there are 10 squirrels living in a 200 square foot area, how many squirrels are in a 2000 square foot area?

Essential Understandings:Concrete Understandings Representation

Identify the proportional relationship using visualsWhat is the proportional relationship below?

8:3 | 1:2 | 25:3 Use a table with visuals to represent proportions

to solve ratio problems.

Understand that proportions can be used to find characteristics of the population based on sample data. For example: 10200 = x

2000 (equation for the example above)

Suggested Instructional Strategies: Introduce Proportion Formula: Number of SuccessesTotal Sample¿¿¿ = Unknown(x )

Total Population ¿¿¿ Example Activity: to determine the total amount of female students in a

school, have students tally up the number of female students in a classroom sample. Then place this number as the numerator and the total number students in the classroom as the denominator. The total population of the school is the Total Population Size. Using the proportion formula, have students calculate the number of female students in the entire school. E.g., if there are 10 female students in a classroom of 25 students, how many female students are there in a school with a population of 2000 students?

o 1025

= x2000

o 10 x 2000 = 25xo 20000 = 25xo 20000 ÷ 25 = 800 females in the total school population

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from a large bag by the results of the number of colored candy in a smaller bag. E.g., give each student an individual sized bag of colored candy. Have the students count the number of total pieces of candy and the number of “green” candies in the bag. Using the proportion formula, have the students determine how many candies are green in a bag of 300 candies.

Supports and Scaffolds: Calculator, manipulatives, graphic organizers, proportion formula, and

colored candy

Resources: Capture recapture activity: Click here Skittles activity: Click here Video on proportions: Click here

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FLS: MAFS.912.S-IC.2.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.Access Point Narrative

MAFS.912.S-IC.2.AP.4c Use the sample data to estimate the population mean. Essential Understandings:

Concrete Understandings Representation Use manipulatives to add the numbers in a

given data set. Use manipulatives to divide the sum of a data

set. Identify the mean of a data set from

manipulatives or pictorial representations.

Given the mean from a sample, estimate the mean of the population based on the data. For example: If the mean of the sample is 12, you could estimate the population’s mean is around 12.

Suggested Instructional Strategies: Understand that the larger the sample size the more representational the

sample mean is to the population mean. Example Activity to determine the mean of a larger population: if we want to

find out the average number of letters in names of students in an entire school, first have students from one randomly selected classroom count the number of letters in their names and place into a table (graphic organizer). E.g., if we have the following numbers for names: 5, 7, 5, 6, 9, 5, 11, 2 to find the mean, have the students add up the numbers and divide by the total number of items. In this example the mean is 6.25. Using the data collected, have students generalize the results to larger populations. If the numbers are representational to the larger population (entire school) the mean will be the same as the sample (classroom).

Supports and Scaffolds: Graphic organizers, calculator

Resources: Video sample mean vs. population mean: Click here Sample mean calculator: Click here Mean definition: Click here Mean, mode, median learning strategy: Click here

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FLS: MAFS.912.S-IC.2.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. FLS: MAFS.912.S-IC.2.6 Evaluate reports based on data.

Access Point NarrativeMAFS.912.S-IC.2.AP.5a Use measures of center tendency (mean, median, mode) and

measures of variability (range and standard deviation) for numerical data from random experiment to compare two treatments.

MAFS.912.S-IC.2.AP.6a Make or select an appropriate statement(s) about findings.MAFS.912.S-IC.2.AP.6b Describe the data collection process. Describe potential biases or

flaws in simple scenarios.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-IC.2.AP.5a

Use manipulatives to add the numbers in a given data set.

Use manipulatives to divide the sum of a data set.

Identify the mean of a data set from manipulatives or pictorial representations.

Identify the lowest to highest value in a data set given a number line and matching symbols.

Arrange data from lowest to highest. Identify the representation (plastic

snap cubes, wiki sticks) of the mode. Use concrete materials to produce the

mean (leveled plastic snap cubes). Find the object or manipulative in a

sequence that represents the middle.

Understand that the measures of center include the mean (average), median (middle value), and mode (most common value).

Understand that the measures of variability include the range (difference between the largest and smallest value) and standard deviation. (The standard deviation is a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data.)

Given a measure of center or variability from two populations, make a generalization for the population based on the data.

MAFS.912.S-IC.2.AP.6a

Match findings (e.g., graphic representation, ratio, percentage) with appropriate statement – e.g., Identify a pictograph that shows 4 out of the 10 candies are red.

Read a graph to identify correct statement of findings.

Understand the following concepts and vocabulary: data, summary and statement.

Understand the concept of ratio or percentage (e.g., 4 red out of 10 candies = 40 out of 100).

MAFS.912.S-IC.2.AP.6b

Identify the data collected. Identify report participants. Identify whether the data would be

Given a list of potential biases and flaws, choose flaws that could apply to the given

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quantitative or qualitative. Identify the sample size.

situation Describe data points that might

have been left out of the survey group (i.e., a survey of students who made an A or B – students who made a D would be left out)

Understand the following concepts and vocabulary: prediction, estimation, hypothesis, bias, flaw, sample size, qualitative data, quantitative data, survey, scale and line of best fit.

Suggested Instructional Strategies: To help students understand the definition of mean, have students stack

unifix cubes to the number of letters in their name. E.g., if we have the following numbers for girls names: 5, 7, 5, 6, 9, 5, 11,2, if there is 5 letters in their name, the student will stack 5 unifix cubes, if there are 7 letters, have them stack 7 unifix cubes. After the students have stacked their unifix cubes to the number of letters in their names, then students will level all the stacks so that the stacks are the same height (which gives us the mean of the data.) E.g., unifix cube stacks representing girls' names with the height numbers of 2, 5, 8, 3, 2 will level off to 4 cubes high (mean). The unifix cube stacks for boys 3, 6, 7, 8, 4, 2 will level off to 5 cubes high (mean).

To help students understand the definition of the median of the data using unifix cubes: line cubes up numerically (before they are leveled) to find the middle number. E.g., 2, 2, 3, 5, 8 with the median being 3 for girls.

To help students understand the definition of the mode using unifix cubes: line cubes up by height (before they are leveled) to determine which height has the most stacks. E.g., 2, 2, 3, 5, 8 with the median being 3 for girls.

To help students understand the definition of the range, have students take the highest number from the data set for girls (11) and subtract the lowest number (2) therefore the range is 11 - 2 = 9.

To help students understand the definition of standard deviation.

The larger the spread is around the mean, the larger the standard deviation.

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The smaller the spread is around the mean, the smaller the standard deviation. Given measures of central tendency and measures of variability, have

students compare results of two different data sets. Example activity: have students compare given data results from experiments.

E.g., in an experiment that determines the amount of water held by various paper towel brands the following data was collected:

o Paper towel A – Mean = 29.5, Mode = 0, Median = 29.5, range = 2, standard deviation = 1.29.

o Paper towel B – Mean = 11.5, Mode = 0, Median = 11.5, range = 3, standard deviation = 1.29.

o Have students compare the measures of central tendency and measures of variability of the two data sets. In the above example, because the median and mean are the same in both sets, this indicates that the trials are consistent.

o The spread around the mean is the same in both sets of data because the standard deviations are the same.

o The range is small in both data sets, indicating that the data was collected consistently.

o Based on the data given, the experiment indicates that towel A is more absorbent than towel B.

Graphic representation Graphic representation with normal curve (standard deviation)

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that the data given is consistent within the trials conducted however the two different paper towels absorb different amounts of water.

Understand that flaws and bias can affect the results of the data sets. E.g., each brand of paper towels had only four trials, a flaw in this collection could be that the results may differ if more trials were conducted. Another flaw could be human error when squeezing the water out of the paper towel.

Understand that lurking variables may be present but not accounted for in the data results. E.g., the strength of the student may influence the data results.

Supports and Scaffolds: Data sets, calculator, formulas for central tendency, formulas for measures of

variability, graphic organizers, paper towels, water, beaker, computer program software, and unifix cubes

Resources: Examples of data sets: Click here Example activity: Click here Standard deviation calculator: Click here Comparing data results: Click here Definitions: Click here Types of bias: Click here Bias in data: Click here Measures of variability: Click here Measures of spread: Click here

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FLS: MAFS.912.S-CP.1.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).Access Point Narrative

MAFS.912.S-CP.1.AP.1a Describe events as subsets of a sample space using characteristics or categories. For example: When rolling a die the sample space is 1, 2, 3, 4, 5, 6. The even numbers would be a subset of the sample space.

MAFS.912.S-CP.1.AP.1b Describe the union of events in a sample space. For example: Event A contains soccer players, event B contains football players. The union of the sets is football players and soccer players all together.

MAFS.912.S-CP.1.AP.1c Describe the intersection of events in a sample space. For example: Event A contains soccer players, event B contains football players. Intersection of the sets is players that participate in both soccer and football.

MAFS.912.S-CP.1.AP.1d Describe the complement of events in a sample space. For example: Event A contains soccer players, event B contains football players. The complement of Event B is all players that are not football players.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-CP.1.AP.1a

Create a subset from a sample space. For example: blue marbles from set of marbles.

Create a subset from a sample space. E.g., even numbers from a list of numbers.

MAFS.912.S-CP.1.AP.1b

Given two sets of objects, combine them to make one set. For example:Let A = {1 orange, 1 pineapple, 1 banana, 1 apple} and B = { 1 spoon, 1 knife, 1 fork}

A∪B = {1 orange, 1 pineapple, 1 banana, 1 apple, 1 spoon, 1 knife,1 fork}

Understand that A∪B represents the union of sets A and B. This is all the items which appear in set A or in set B or in both sets

Given two sets of objects, combine them to make one set. For example:

MAFS.912.S-CP.1.AP.1c

Given two sets of objects, determine what they have in common. For example:(To make it easy, notice that what they have in common is in bold)

Let A = {1 orange, 1 pineapple, 1 banana, 1 apple} and B = { 1

Understand that A∩B represents the

intersection of sets A and B.This is all the items which appear in set A and in set B.

Given two sets of objects, determine what they have in

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spoon, 1 orange, 1 knife, 1 fork,1 apple}A∩B = {1 orange, 1 apple}

common. For example:

MAFS.912.S-CP.1.AP.1d

Given two sets of objects, determine what is not in the given set from the sample space. For example:Let B = {1 orange, 1 pineapple, 1 banana, 1 apple} Let U = {1 orange, 1 apricot, 1 pineapple, 1 banana, 1 mango, 1 apple, 1 kiwifruit }

Again, we show in bold all elements in U, but not in BBc , ~B or B’ = {1 apricot, 1 mango, 1 kiwifruit}

Understand that the compliment is denoted as Bc , ~B or B’

Given two sets of objects, the compliment is everything in the sample space outside of the selected set.

Suggested Instructional Strategies: Understand the definition of a sample space: the sample space is all the

possible outcomes. Understand the definition of a subset: a part of the sample space. E.g., if the

sample space was all the girls in a classroom, the subset could be all the girls who wear glasses within that sample space.

Understand the definition of a union: every outcome in both sets. Understand the definition of an intersection: every outcome that is shared in

both sets. Understand the definition of complement: every outcome that is not in the

event. E.g., in the activity below, a complement could include every student who is not wearing a red shirt. Another example from activity below could be every student not wearing glasses.

Example Activity: create two sub sets, sub set A: students wearing red shirts. Sub set B: students wearing glasses. Create a Venn diagram on the floor using tape. Have the students wearing red shirts and glasses stand in the overlap part of the Venn diagram (intersection). Have the students wearing red shirts without glasses stand in the left side of the Venn diagram. Have the students wearing glasses and any color shirt other than red stand in the right side of the Venn diagram. All of the students standing in the entire Venn diagram makeup the “union.”

Another option: place two hula hoops on floor or desk top to create the Venn diagram. Using the two sets from above, have students write their name on index cards and place their name in the appropriate part of the Venn diagram.

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Supports and Scaffolds: Graphic organizers, manipulatives, tape, hula hoop, index cards, interactive

computer programs, dice, coins, and cards

Resources: Examples of union, intersection, and complement: Click here Interactive Venn diagram: Click here Examples of complement: Click here Examples of sample space: Click here Sets and Venn diagrams: Click here

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FLS: MAFS.912.S-CP.1.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

FLS: MAFS.912.S-CP.1.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Access Point NarrativeMAFS.912.S-CP.1.AP.2a Describe the characteristics that make events independent. MAFS.912.S-CP.1.AP.2b Calculate the probability of events A and B occurring together.

P(A and B) = P(A) × P(B)MAFS.912.S-CP.1.AP.3b Identify when two events are independent. P(A and B) ÷ P(A) =

P(B)

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-CP.1.AP.2a

Use two different manipulatives (e.g., spinner, dice and coins) to demonstrate how events are generated (flipping a coin and rolling a die).

Use pictures (e.g., picture of the coin, picture of die) to record two different events (results).

List the possible events (results) from two different manipulatives (Example die: 1, 2, 3, 4, 5, 6 coin: head, tail)

Use the list of possible events to show how one event does not affect the other.

Understand the concept of independent as two events that generate results without affecting each other.

MAFS.912.S-CP.1.AP.2b

Use two different manipulatives (e.g., spinner, dice and coins) to demonstrate how events are generated (flipping a coin and rolling a die).

Use pictures (e.g., picture of the coin, picture of die) to record two different events (results).

Create a chart of all possible event combinations to determine the probability of a specific event.

List the possible events (results) from two different manipulatives (e.g., die: 1, 2, 3, 4, 5, 6 coin: head, tail)

Use the list of possible events to show how one event does not affect the other.

Understand the concept of independent as two events that generate results without affecting each other.

Represent the probability of each event using numbers. E.g., P(heads) = ½ is the probability of flipping a coin and getting heads; P(2) = 1/6 is the probability of rolling a die and getting a 2

Multiply the probability of two events

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together. E.g., P(heads) = 12, P(2) = 16 ;

P(heads and 2) = 12 × 16 = 112 .MAFS.912.S-CP.1.AP.3b

Use two different manipulatives (e.g., spinner, dice and coins) to demonstrate how events are generated (flipping a coin and rolling a die).

Use pictures (e.g., picture of the coin, picture of die) to record two different events (results).

Create a chart of all possible event combinations to determine the probability of a specific event.

List the possible events (results) from two different manipulatives (e.g., die: 1, 2, 3, 4, 5, 6 coin: head, tail)

Use the list of possible events to show how one event does not affect the other.

Understand the concept of independent as two events that generate results without affecting each other.

Represent the probability of each event using numbers. E.g., P(heads) = ½ is the probability of flipping a coin and getting heads; P(2) = 1/6 is the probability of rolling a die and getting a 2

Multiply the probability of two events together. E.g., P(heads) = 12, P(2) = 16 ;

P(heads and 2) = 12 × 16 = 112 Divide the probability of two events

together by the probability that one of those events will happen.

E.g., P(heads) = 12, P(2) = 16 ; P(heads

and 2) = 12 × 16 = 112

P(heads and 2) ÷ P(2) = 112 ÷ 16 = 12

Suggested Instructional Strategies: Understand that in order for two events to be considered independent from each

other means that the results from one event are not affected by the other event. E.g., flipping a coin and rolling a die are independent events. One event does not affect the other event.

Understand that if one event affects the other event then they are dependent. E.g., the probability of pulling two kings from a deck of cards, one after the other, without replacement, is an example of dependent events. The probability of the first king being removed from the deck affects the probability of another king being

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Example Activity: using a coin and a 6 sided die, find the probability of tossing “heads” on a coin and rolling a 5 on a die. Because the events are independent, use the probability formula, P(A and B) = P(A) × P(B).

o P(heads and 5) = P(heads) X P(5) o P(heads and 5) = 1

2 X 16

o P(heads and 5) = 112

o In this example, the probability of having both “heads” and a “5” is 112

If given the results of an independence rule and one probability, the formula to find

the unknown value is P(A and B) ÷ P(A) = P(B). E.g., if the probability of having rain and lightning today is 50% and the probability of having rain is 70%, then the unknown probability of having lightning is 71%.

o P(A and B) ÷ P(A) = P(B). o P(rain and lightning) ÷ P(rain) = P(lightning)o 0.50 ÷ 0.70 = P(lightning)o 0.71 = P(lightning)

Supports and Scaffolds: Dice, coins, cards, calculator, probability formula

Resource: Definition independent, dependent: Click here Examples of independent events: Click here independent and dependent events: Click here

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FLS: MAFS.912.S-CP.1.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.FLS: MAFS.912.S-CP.1.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.FLS: MAFS.912.S-CP.1.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.FLS: MAFS.912.S-CP.2.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

Access Point NarrativeMAFS.912.S-CP.1.AP.3a Using a two-way table, find the conditional probability of A given B. MAFS.912.S-CP.1.AP.4a Select or make an appropriate statement based on a two-way

frequency table.MAFS.912.S-CP.1.AP.5a Select or make an appropriate statement based on real-world

examples of conditional probabilityMAFS.912.S-CP.2.AP.6a Using a two-way table, find the conditional probability of A given

within the context of the model.Essential Understandings:

Access Point

Concrete Understandings Representation

MAFS.912.S-CP.1.AP.3a

Create a two-way table using objects/manipulatives

List the possible combinations to create a two-way table (e.g., using one male figurine, twelve matchbox cars, and eight toy SUV’s to model the probability of the type of car given the customer is a male).

Understand that a two-way table looks at sub groups within the whole group. For example, looking at how boys scored on a test verses how girls scored on the same test.

Use this two-way table to determine specific probabilities. E.g., the probability of the owner being female given a sports car is 45 out of 84 or 0.54.

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Concrete Understandings Representation

Use this two-way table to determine specific probabilities. E.g., the probability of choosing a sports car given they are a male in the table below is 39 out of 60 or 0.65.

MAFS.912.S-CP.1.AP.4a

Match appropriate statements to the data displayed on a two-way table.

Identify information using a two-way frequency table.

Explain the data within a two-way frequency table.

Understand the following concepts and vocabulary: two-way frequency table, category, classify, rows, columns and relationship.

Understand the relationship between two categories in a two-way frequency table (rows and columns).

MAFS.912.S-CP.1.AP.5a

Distinguish independent events (e.g., a coin toss) from dependent events (e.g., three red chips and two blue – if I remove a red chip, the probability or chance I pull a blue chip next changes).

Demonstrate concepts of more (likely), less (likely), good (probability), high (probability), low (probability), and same (probability).

Understand the following concepts and vocabulary: probability, category, classify, dependent, independent and likelihood.

Understand the difference between dependent and independent events (e.g., flipping the two coins are two independent events).

MAFS.912.S-CP.2.AP.6a

Create a two-way table using objects/manipulatives

Understand that a two-way table looks at sub groups within the

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Elements CardAccess Point

Concrete Understandings Representation

List the possible combinations to create a two-way table (e.g., using one male figurine, twelve matchbox cars, and eight toy SUV’s to model the probability of the type of car given the customer is a male.)

whole group. For example, looking at how boys scored on a test verses how girls scored on the same test.

Use this two-way table to determine specific probabilities. E.g., the probability of the owner being female given a sports car is 45 out of 84 or 0.54.

Use this two-way table to determine specific probabilities. E.g., the probability of choosing a sports car given they are a male in the table below is 39 out of 60 or 0.65.

Suggested Instructional Strategies: Understand that the terminology “given” indicates that you are isolating that event.

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Elements Card Example activity: given the two-way table above, students will determine specific

probabilities. E.g., based on the data above, students will determine the conditional probability of a student liking a cat given they are a female is 7 out of 10. (In the two-way table below the unneeded data is eliminated in order to stress that the only row needed is females.) Therefore 7 out of 10 females, in our survey, prefer cats.

Example activity: given the original two-way table above, students will determine specific probabilities. E.g., based on the data above, students will determine the conditional probability that the student is a male given that the student likes dogs is 6 out of 8. (In the two-way table below the unneeded data is eliminated in order to stress that the only column needed is dogs.) Therefore, dogs are preferred by males in a sample of 6 out of 8 students who prefer dogs.

Conditional Probability Formula (if needed):

Supports and Scaffolds: Two-way table with data provided, strips of paper to eliminate unneeded data, and

interactive software.

Resources: Interactive conditional probabilities: Click here

(Note: it is important that students include the totals for each column and row which is not given in this site.)

Example of conditional probability (begin on page 4): Click here Calculate Conditional Probability: Click here

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FLS: MAFS.912.S-CP.2.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.Access Point Narrative

MAFS.912.S-CP.2.AP.7a Use the addition rule, P(A or B) = P(A) + P(B) – P(A and B),MAFS.912.S-CP.2.AP.7b Interpret the answer to the Addition Rule within the context of the

model.

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-CP.2.AP.7a

Create a table that records student responses to questions like: Do you like to color with crayons? Do you like to color with markers? Do you like to color with both crayons and markers?

Use the data from the table to create a Venn diagram to visually represent the addition rule.

Define the addition rule; When two events occur the probability that you will get events A or B is the probability of A plus the probability of B minus their overlap.

Use the addition rule to compute the probability. For example:

o In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the0 probability of choosing a girl or an A student?

MAFS.912.S-CP.2.AP.7b

Create a table that records student responses to questions like: Do you like to color with crayons? Do you like to color with markers? Do you like to color with both crayons and markers?

Use the data from the table to create a Venn diagram.

The P(A) only or the P(B) only are the responses outside the overlapping area. P(A or B)

Define the addition rule; When two events occur the probability that you will get events A or B is the probability of A plus the probability of B minus their overlap.

Use the addition rule to compute the probability. For example:

o In a math class of 30 students, 17 are boys and 13 are girls. On a unit test, 4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the probability of choosing a girl or an A student?

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Concrete Understandings Representation

The probability of selecting a girl or a student who made an A on the test is 17 out of 30.

Suggested Instructional Strategies: Understand addition rule for probability formula: P(A or B) = P(A) + P(B) – P(A

and B) Example Activity: to determine how many students went to the beach or the

park over the weekend, have students create a tally table with the results. To find the answer, have students use the addition rule for probability formula. E.g., the tally table indicated that 80% went to the beach, 40% went to the park, and 30% went to both the beach and park over the weekend:

o P(beach) = 80%, P(park) = 40%, P(beach and park) = 30%o P(beach or park) = P(beach) + P(park) – P(beach and park)o P(beach or park) = 0.8 + 0.4 – 0.3o P(beach or park) = 0.9o Therefore 90% of the students went to the beach or the park over the

weekend.

Supports and Scaffolds: Graphic organizers, calculators, manipulatives, and software programs

Resources: Examples of addition rule of probability (begin at experiment 4): Click here Addition rule of probability formula explanation: Click here Addition rule of probability (practice worksheet problem #3 has a derogatory

statement): Click here

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FLS: MAFS.912.S-MD.1.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

Access Point NarrativeMAFS.912.S-MD.1.AP.3a Determine the theoretical probability of multistage probability

experiments (e.g., draw or select a representation of the theoretical probability for a sample space).

MAFS.912.S-MD.1.AP.3b Collect data from multistage probability experimentsMAFS.912.S-MD.1.AP.3c Compare actual results of multi-stage experiment with theoretical

probabilities (e.g., make a statement that describes the relationship between the actual results of a multistage experiment with its theoretical probabilities [ex., more, less, same, different, equal]).

Essential Understandings:Access Point

Concrete Understandings Representation

MAFS.912.S-MD.1.AP.3a

Create/recognize combinations of products/pairings. o E.g., Use at least three different

colored counting cubes and determine the number of different pairs that can be created.

o E.g., Use coins to represent the theoretical probability for a sample space – show rows of two coins each, one with heads face up and one with tails up; each row represents another stage of the experiment.

o E.g., Use the "Rock, Paper, Scissors" game to demonstrate a multi-stage probability experiment (i.e., each round of the game represents a stage [trial] in the experiment).

Understand the following concepts and vocabulary: probability, sample space, combine, combination, outcome, trial, stage.

Understand that sample space is the set of all possible outcomes (combinations) of an experiment.

MAFS.912.S-MD.1.AP.3b

Create/recognize combinations of products/pairings. E.g., Use counting cubes to represent data from a multi-stage probability experiment (e.g., use red for rock, blue for paper, and yellow for scissors to represent the winner for

Organize data from a multistage probability experiment into a table.

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Concrete Understandings Representation

each stage [trial]).MAFS.912.S-MD.1.AP.3c

Identify the correct match for probabilities and results. e.g., Select a picture card that describes the relationship between the actual results of a multi-stage experiment with its theoretical probabilities (e.g., more, less, same, different, equal).

Understand related vocabulary (e.g., more, less, same, different, equal, compare).

Compare the actual results from an experiment with its theoretical probability.

Suggested Instructional Strategies: Understand that in a probability distribution the probabilities must add up to

1. E.g., these are the probabilities if a coin was flipped three times: 18 + 38 + 38 + 18 = 1

Example activity: to find the probability distribution of flipping a coin two times, have the students write down all of the possible combinations. In this example, the possibilities are:

o HH, HT, TH, TT. o Have students fill in a chart to determine the theoretical probability:

X = heads 0 1 2P(X) 1

4 = 0.25 24 = 0.50 1

4 = 0.25

o Have students add up the probabilities to insure that the sum is 1. 14 + 24 + 14 = 1

o Next, have students conduct the experiment to find the experimental results. In this experiment have each student flip the coin two times and place tally marks in their individual tally table. Record all the student’s data in the class tally table drawn on the board.

X = heads 0 1 2tally

o Example classroom tally table results:X = heads 0 1 2

tally 3 9 4o Once all data has been entered into the tally table, the students will

calculate the probabilities for each outcome. E.g., if there are 16 students and 3 tally marks for 0 heads the probability would be: 316

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Elements CardStudents will complete the probability table with the data. E.g., there are 3 tally marks for “0”, 9 tally marks for “1”, and 4 tally marks for “2”. Fill in the probability row with the correct fraction that represents the data.

X = heads 0 1 2P(X)

o Example classroom probability distribution table:X = heads 0 1 2

P(X) 316 = 0.1875 9

16 = 0.5625 416 = 0.25

o Have students add the fractions or decimals to insure that the sum of the probabilities equals “1.”

316 + 916 + 416 = 1616 = 1

0.1875 + 0.5625 + 0.25 = 1o Next, have students compare the theoretical probabilities to the

experimental probabilities. E.g., the probability of getting one head, when we flipped the coin twice, the experimental result is 0.5625 which is close to the theoretic probability of 0.5.

o The experimental probability of getting two heads, when we flipped the coin twice, matches our theoretic probability of 0.25.

o The experimental probability of getting zero heads, when we flipped the coin twice, does not match our theoretic probability of 0.25.Note: the more experimental data collected, the closer the experimental probabilities will be to the theoretical probabilities.

Supports and Scaffolds: Graphic organizers, calculators, coins, manipulatives, computer software

programs, dice, and cards

Resources: Probability distribution: Click here Probability distribution video: Click here Probability distribution chart: Click here Probability distribution idea starter: Click here Estimate probability with experiments: Click here Interactive probability experiments: Click here

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FLS: MAFS.912.S-MD.2.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Access Point Narrative

MAFS.912.S-MD.2.AP.7a Identify and describe the degree to which something is rated “good” or “bad”/desirable or undesirable based on numerical information (Identify and describe the degree to which a decision/strategy is rated “good” or “bad"/desirable or undesirable based on numerical Information.).

Essential Understandings:Concrete Understandings Representation

Determine why something was labeled each of the ways given data (e.g., why working more hours a week for a doctor or pilot = more money but is a bad idea; why riding a bicycle to work takes more time but is a good idea).

Describe a good and desirable decision/strategy based on the outcome from a concrete example of a probability concept that illustrates an expected value.

Suggested Instructional Strategies: Understand that when analyzing data results of desirable/undesirable

outcomes, it is situationally based. E.g., if the data from the brakes on a school bus are showing that the brakes have too much wear on them, then the desirable result would be to fix the brakes so that an accident will not occur from the worn brakes. If the data from a potato chip company shows that the salt level is higher than desirable, the potato chip will most likely not adversely affect individuals, however, it may not taste good.

Example activity: to determine desirable/undesirable outcomes, when analyzing the weather, have students refer to the weather forecast to decide what to wear. E.g., if the weather report indicates an 80% chance of rain, the students will need appropriate rain gear (raincoat, rain boots, and umbrella). However, if the forecast also includes a high chance of lightning, using an umbrella would be undesirable.

Example activity: to determine desirable/undesirable outcomes: when analyzing what size bowling ball to use when bowling, the data shows that the heavier the ball, the more power hitting the pins. Knocking down more pins is desirable, however, the weight of the ball could cause damage to the bowler, which is undesirable.

Supports and Scaffolds: Situational events, graphic organizers, and computer software programs

Resources: Videos analyzing data from different scenarios: Click here Videos analyzing data from different scenarios: Click here Interactive probability experiments: Click here Understanding Uncertainty in data: Click here

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