content deepening 8 th grade math

Content Deepening8th Grade Math

February 7, 2014

Jeanne Simpson

AMSTI Math Specialist

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Welcome

NameSchoolClasses you teachWhat are you hoping to learn today?

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He who dares to

teach must never

cease to learn.

John Cotton Dana

Goals for Today

Implementation of the Standards of Mathematical Practices in daily lessons

Understanding of what the CCRS expect students to learn blended with how they expect students to learn.

Student-engaged learning around high-cognitive-demand tasks used in every classroom.

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Agenda

Square RootsPythagorean TheoremStatisticsExponents and Scientific Notation

acos2010.wikispaces.com Electronic version of handouts

Links to web resources

http://acos2010.wikispaces.com/Home

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Five Fundamental Areas Required for Successful Implementation of CCSS

Instruction Content

Intervention

Assessment

Collaboration

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How do we teach?

Instruction• Deep conceptual understanding• Collaborative lesson design• Standards for Mathematical

Practice

Content• Fewer standards with greater

depth• Understanding, focus, and

coherence• Common and high-demand tasks

Intervention• Common required response to

intervention framework response• Differentiated, targeted, and

intensive response to student needs

• Student equity, access, and support

Assessment• PLC teaching-assessing-learning

cycle• In-class formative assessment

processes• Common assessment instruments

as formative learning opportunities

Collaboration

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Standards for Mathematical Practice

Mathematically proficient students will:

SMP1 - Make sense of problems and persevere in solving them

SMP2 - Reason abstractly and quantitatively

SMP3 - Construct viable arguments and critique the reasoning of others

SMP4 - Model with mathematics

SMP5 - Use appropriate tools strategically

SMP6 - Attend to precision

SMP7 - Look for and make use of structure

SMP8 - Look for and express regularity in repeated reasoning

Students: (I) Initial (IN) Intermediate (A) Advanced1a Make sense of

problems

Explain their thought processes in solving a problem one way.

Explain their thought processes in solving a problem and representing it in several ways.

Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways.

1b Persevere in solving them

Stay with a challenging problem for more than one attempt.

Try several approaches in finding a solution, and only seek hints if stuck.

Struggle with various attempts over time, and learn from previous solution attempts.

2 Reason abstractly andquantitatively

Reason with models or pictorial representations to solve problems.

Are able to translate situations into symbols for solving problems.

Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.

3a Construct viablearguments

Explain their thinking for the solution they found.

Explain their own thinking and thinking of others with accurate vocabulary.

Justify and explain, with accurate language and vocabulary, why their solution is correct.

3b Critique the reasoning of others.

Understand and discuss other ideas and approaches.

Explain other students’ solutions and identify strengths and weaknesses of the solution.

Compare and contrast various solution strategies and explain the reasoning of others.

4 Model withMathematics

Use models to represent and solve a problem, and translate the solution to mathematical symbols.

Use models and symbols to represent and solve a problem, and accurately explain the solution representation.

Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.

5 Use appropriate toolsstrategically

Use the appropriate tool to find a solution.

Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection.

Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution.

6 Attend to precision

Communicate their reasoning and solution to others.

Incorporate appropriate vocabulary and symbols when communicating with others.

Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas.

7 Look for and make useof structure

Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7.

Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.

See complex and complicated mathematical expressions as component parts.

8 Look for and expressregularity in repeatedreasoning

Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns.

Find and explain subtle patterns.

Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such asdiscovering an underlying function.

SMP Proficiency Matrix

SMP Instructional Implementation Sequence

1. Think-Pair-Share (1, 3)

2. Showing thinking in classrooms (3, 6)

3. Questioning and wait time (1, 3)

4. Grouping and engaging problems (1, 2, 3, 4, 5, 8)

5. Using questions and prompts with groups (4, 7)

6. Allowing students to struggle (1, 4, 5, 6, 7, 8)

7. Encouraging reasoning (2, 6, 7, 8)

Students: (I) Initial (IN) Intermediate (A) Advanced

1a Make sense ofproblems

Explain their thought processes in solving a problem one way.

Explain their thought processes in solving a problem and representing it in several ways.

Discuss, explain, and demonstrate solving a problem with multiple representations and in multiple ways.

1b Persevere in solving them

Stay with a challenging problem for more than one attempt.

Try several approaches in finding a solution, and only seek hints if stuck.

Struggle with various attempts over time, and learn from previous solution attempts.

2 Reason abstractly andquantitatively

Reason with models or pictorial representations to solve problems.

Are able to translate situations into symbols for solving problems.

Convert situations into symbols to appropriately solve problems as well as convert symbols into meaningful situations.

3a Construct viablearguments

Explain their thinking for the solution they found.

Explain their own thinking and thinking of others with accurate vocabulary.

Justify and explain, with accurate language and vocabulary, why their solution is correct.

3b Critique the reasoning of others.

Understand and discuss other ideas and approaches.

Explain other students’ solutions and identify strengths and weaknesses of the solution.

Compare and contrast various solution strategies and explain the reasoning of others.

4 Model withMathematics

Use models to represent and solve a problem, and translate the solution to mathematical symbols.

Use models and symbols to represent and solve a problem, and accurately explain the solution representation.

Use a variety of models, symbolic representations, and technology tools to demonstrate a solution to a problem.

5 Use appropriate toolsstrategically

Use the appropriate tool to find a solution.

Select from a variety of tools the ones that can be used to solve a problem, and explain their reasoning for the selection.

Combine various tools, including technology, explore and solve a problem as well as justify their tool selection and problem solution.

6 Attend to precision

Communicate their reasoning and solution to others.

Incorporate appropriate vocabulary and symbols when communicating with others.

Use appropriate symbols, vocabulary, and labeling to effectively communicate and exchange ideas.

7 Look for and make useof structure

Look for structure within mathematics to help them solve problems efficiently (such as 2 x 7 x 5 has the same value as 2 x 5 x 7, so instead of multiplying 14 x 5, which is (2 x 7) x 5, the student can mentally calculate 10 x 7.

Compose and decompose number situations and relationships through observed patterns in order to simplify solutions.

See complex and complicated mathematical expressions as component parts.

8 Look for and expressregularity in repeatedreasoning

Look for obvious patterns, and use if/ then reasoning strategies for obvious patterns.

Find and explain subtle patterns.

Discover deep, underlying relationships, i.e. uncover a model or equation that unifies the various aspects of a problem such asdiscovering an underlying function.

SMP Proficiency Matrix

Grouping/Engaging Problems



Pair-Share

Showing Thinking

Showing Thinking

Questioning/Wait Time



Questions/Prompts for Groups

Questions/Prompts for Groups

Pair-Share






Allowing Struggle

Allowing Struggle

Allowing Struggle


Showing Thinking

Encourage Reasoning



Showing Thinking

Showing Thinking

Encourage Reasoning

Encourage Reasoning

Encourage Reasoning

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What are we teaching?

Instruction• Deep conceptual understanding• Collaborative lesson design• Standards for Mathematical

Practice

Content• Fewer standards with greater

depth• Understanding, focus, and

coherence• Common and high-demand tasks

Intervention• Common required response to

intervention framework response• Differentiated, targeted, and

intensive response to student needs

• Student equity, access, and support

Assessment• PLC teaching-assessing-learning

cycle• In-class formative assessment

processes• Common assessment instruments

as formative learning opportunities

Collaboration

Critical Focus Areas

Functions

Define, evaluate, compareUse to model relationships

Standards 11-15

Geometry

Transformations, similar triangles, angles formed by parallel lines, Pythagorean

theorem, volume

Standards 16-24

Other

Irrational numbers, radical, integer exponents

Standards 1-6

Expressions and Equations

Represent, analyze, and solve a variety of problems

Linear equations, systems of equations, linear functions,

slope, bivariate data

Standards 7-10, 25-28

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Square Roots and Pythagorean

TheoremChapter 6

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Work with radicals and integer exponents.

8.EE.2 – Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

http://ccssmath.org/?page_id=689

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Connected Mathematics

1. Coordinate Grids

2. Squaring Off

3. The Pythagorean Theorem

4. Using the Pythagorean Theorem

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Looking for Pythagoras

Relate the area of a square to the side length

Estimate the values of square roots of whole numbers

Locate irrational numbers on a number line Develop strategies for finding the distance

between two points on a coordinate grid Understand and apply the Pythagorean

Theorem Use the Pythagorean Theorem to solve

everyday problems

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2.1 Looking for Squares

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HomeworkACE 1-3, 42, 47, 48

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2.2 Square Roots

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Understand and Apply the Pythagorean Theorem

8.G.6 – Explain a proof of the Pythagorean Theorem and its converse.

8.G.7 – Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

8.G.8 – Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.




• Use two different colored squares of paper.

• Student 1 completes Steps 1&2 and Student 2 completes Steps 3&4.

• Students label the

area of rectangles, squares and triangles in terms of a and b.

• Students then cut out shapes. The triangles should fit perfectly on the rectangles leaving the squares a2 and b2 (of one color) = to c2 (of other color).

Pythagorean Tile Proof

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Proofs of the Pythagorean Theorem

1

3

2

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Comparing TVs

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Mathematics Assessment Project

Tools for formative and summative assessment that make knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. These tools comprise:

Classroom Challenges: lessons for formative assessment, some focused on developing math concepts, others on non-routine problem solving.

Professional Development Modules: to help teachers with the new pedagogical challenges that formative assessment presents.

Summative Assessment Task Collection: to illustrate the range of performance goals required by CCSSM.

Prototype Summative Tests: designed to help teachers and students monitor their progress, these tests provide a model for examinations that may replace or complement current US tests.

http://map.mathshell.org/



MARS Tasks

Lines and Linear Equations (8.EE.6)

Identifying Similar Triangles (8.EE.6)

Systems of Equations (8.EE.8)

Interpreting Time-Distance Graphs (8.F.4)

Modeling Situations with Linear Equations (8.F.4)

The Pythagorean Theorem: Square Areas (8.G.6)

P-38

P-39The Pythagorean Theorem: Square AreasProjector Resources

The Pythagorean Theorem: Square Areas

The Pythagorean Theorem: Square AreasProjector Resources

Jason’s Method

P-40

“I drew a square all round the tilted square. I then took away the area of the four right triangles.”


Kate’s Method

P-41

“I divided the tilted squares into four right triangles and little squares inside.”


Simon’s Method

P-42

“I found the area inside the bold line is the same area as the tilted square and used that.”

P-43

xy

0 1 2 3 4 Comments

0

1

2

3

4



What is the gray area in each case?

P-44

x y

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Data Analysis and

Displays

Chapter 7

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Investigate patterns of variability in bivariate

data 8.SP.1 – Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.


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48




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c. Removing the outliers from the data set, make a new scatterplot of the remaining animal body and brain weights.

d. Does there appear to be a relationship between body weight and brain weight? If yes, write a brief description of the relationship.

e. Take a piece of uncooked spaghetti and use that spaghetti to informally fit a line to the data. Attempt to place your line so that the vertical distances from the points to the line are as small as possible.

f. How well does the spaghetti line fit the data?

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Illustrative Mathematics

Illustrative Mathematics provides guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students experience in a faithful implementation of the Common Core State Standards, and by publishing other tools that support implementation of the standards.

http://www.illustrativemathematics.org/

http://www.illustrativemathematics.org/

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data 8.SP.2 – Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.


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data 8.SP.3 – Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

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data 8.SP.3 – Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

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data 8.SP.4 – Understand that patterns of association

can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

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Exponents and Scientific Notation

Chapter 9

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Work with radicals and integer exponents

8.EE.1 – Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

“Ponzi” Pyramid Schemes


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8.EE.1 – Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.

Alien Attack!


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8.EE.3 - Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

Exploring Powers of Ten

8.EE.4


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8.EE.3 - Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

Orders of Magnitude

a. Your age in hours.

b. The number of breaths you take in a year.

c. The number of heart beats in a lifetime.

d. The number of basketballs that would fit in your classroom.


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8.EE.4 – Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Giantburgers


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8.EE.4 – Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Estimating Length Using Scientific Notation


Estimating Length Using Scientific NotationProjector Resources

What’s the Number?

P-62

30,000 × 10−1

3 × 103

0.3 × 104


Scientific Notation or Not?

P-63

30,000 × 10−1

3 × 103

0.3 × 104


Which is Greater?

P-64

8 × 10−3

4 × 10−1


Matching Card Set A

1. Take turns to match a card in scientific notation with a card in decimal notation.

2. Each time you match a pair of cards explain your thinking clearly and carefully. Place your cards side by side on your desk, not on top of

one another, so that everyone can see them.

3. Partners should either agree with the explanation, or challenge it if it is not clear or not complete.

4. It is important that everyone in the group understands the matching of each card.

5. You should find that two cards do not have a match. Write the alternative notation for these measurements on the blank cards to produce a pair.

P-65


Matching Card Set B

1. Match the objects in Card Set B with the corresponding paired measurements from Card Set A. You may want to put the objects in size

order first to help you.

2. It is important that everyone in the group agrees on each match.

3. When you have finished matching the cards, check your work against that of a neighboring group.

4. If there are differences between your matching, ask for an explanation. If you still don’t agree, explain your own thinking.

5. You must not move the cards for another group. Instead, explain why you disagree with a particular match and explain which cards you think need to be changed.

P-66

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Transformations

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8G - Understanding

8.G.2 – Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations…

8.G.4 - Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations…

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High School Geometry

Represent transformations in the place using e.g., transparencies and geometry software; describe transformations as functions 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). [G-CO2]

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO3]

Develop definitions of rotations , reflections, and translations in terms of angels, circles, perpendicular line, parallel lines, and line segments. [G-CO4]

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. [G-CO5]

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. [G-CO6]

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High School Geometry

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. [G-CO7]

Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8]

Verify experientially the properties of dilations given by a center and a scale factor. [G-SRT1]

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. [G-SRT2]

Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

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Fishing for Points

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Stretching and Shrinking

Representing and Combining TransformationsProjector Resources

Translation

P-73

Where will theL-shape be if it is translated by −2 horizontally and +1 vertically?


Reflection

P-74

Where will theL-shape be if it is reflected over the line x = 2?


Rotation

P-75

Where will theL-shape be if it is rotated through 180°around the origin?


Matching Cards

• Take turns to match two shape cards with a word card. Each time you do this, explain your thinking clearly and carefully.

• Your partner should then either explain that reasoning again in his or her own words, or challenge the reasons you gave.

• It is important that everyone in the group understands the placing of a word card between two shape cards .

• Ultimately, you want to make as many links as possible.

• Use all the shape card, and all the word the cards if possible.

P-76


Starting point (1, 4)

P-77

Show me the new coordinates of the point (1, 4) after it is:

• Reflected over the x-axis

• Reflected over the y-axis

• Rotated through 180°about the origin.

• Reflected over the line y = x.

• Reflected over the line y = −x.

• Rotated through 90°clockwise about the origin.

• Rotated through 90°counterclockwise about the origin.


General starting point (x, y)

P-78

Show me the new coordinates of the point (x, y) after it is:

• Reflected over the x-axis

• Reflected over the y-axis

• Rotated through 180°about the origin.

• Reflected over the line y = x.

• Reflected over the line y = −x.• Rotated through 90°clockwise about the origin.

• Rotated through 90°counterclockwise about the origin.

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The Number System

Apply and extend previous understandings of operations with

fractions to add, subtract, multiply, and divide rational numbers.

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Mathematics consists of pieces that make sense; they are not just independent manipulation/skills to be practiced and memorized – as perceived by many students.

These individual pieces progress through different grades (in organized structures we called “flows”) and can/should be unified together into a coherent whole.

Jason Zimba, Bill McCallum

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Fractions

Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens.

National Mathematics Panel Report, 2008

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Fractions

“Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.”

Imm, Fosnot, Uittenbogaard (2012)

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Unit Fractions

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Fraction Multiplication in Grade 5

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5th Grade Division

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5th Grade Division

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5th Grade Division Problems

How much chocolate will each person get if 3 people share ½ pound equally?

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5th Grade Division Problems

How many 1/3 cup servings are in 2/3 cups of raisins?

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Fraction Division in Grade 6

6.NS.1 – Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem.

Examples:

Create a story context…

Use a visual fraction model to show the quotient…

Explain division using its relationship with multiplication

Sample problems

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6th Grade Division

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6th Grade Division

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Contact Information

Jeanne Simpson

UAHuntsville AMSTI

[email protected]

[email protected]

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