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Roswell Independent School District Math Curriculum Map 2013-Algebra II

26

Domain: Algebra Creating Equations* A-CED

Pacing Guide Q2:Week 7

Cluster: Create equations that describe numbers or relationships.

Essential Question: How do I create an equation that will represent the relationship between Numbers or relationships?

CCSS Criteria For Success (Target)

Vocabulary Activity/Assessment/Resources Math Practice Focus

H.S. A-CED.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.

• Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. MP 2, 4, 5, 7

Literal equation Students are to rearrange formulas to solve for a

given variable, such as Ohm’s law, distance/speed,

etc.

Assessment-

Levona and Kathleen are hiking in a state park. They

start at opposite ends of a trail that is 5 miles long

and hike towards each other. Levona is moving at a

speed of 3 miles per hour, whereas Kathleen is

moving at a speed of 6 miles per hour. How much

time will go by before they meet up?

Resources: Ohm’s law PowerPoint presentation- http://wsautter.com/wp-content/uploads/2010/02/current-electricity-ohms-law1.ppt

MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP5: Use appropriate tools strategically MP7: Look for and make use of structure

http://wsautter.com/wp-content/uploads/2010/02/current-electricity-ohms-law1.ppt




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Domain: Algebra Reasoning with Equations & Inequalities A-REI


Cluster: Understand solving equations as a process of reasoning and explain the reasoning.

Essential Question: How does an algebraic expression relate to an algebraic equation?



H.S. A-REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

• Solve single variable rational and radical equations using inverse operations. • Determine whether the solution to a rational or radical equation is extraneous. • Create a rational or radical equation that contains extraneous solutions. MP 2, 3, 7

Inverse Operation Radical Extraneous Solution

Students are to solve rational and radical equations involving one variable. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). Assessment- Students are to complete the examples below-

Resources: What Are the Ways You Can Solve a System of Linear Equations? http://www.prometheanplanet.com/en-us/Resources/Item/139354/what-are-the-ways-you-can-solve-a-system-of-linear-equations

MP2: Reason abstractly and quantitatively MP3: Construct viable arguments and critique the reasoning of others MP7: Look for and make use of structure

http://www.prometheanplanet.com/en-us/Resources/Item/139354/what-are-the-ways-you-can-solve-a-system-of-linear-equations




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Domain: Algebra Reasoning with Equations & Inequalities A-REI


Cluster: Represent and solve equations and inequalities graphically.

Essential Question: How do I graph inequalities on a number line?



H.S .A.REI.4 Solve quadratic equations in one variable H.S. A.REI.4b 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

• Solve a system of equations which may include linear, polynomial, rational, absolute value, exponential or logarithmic functions. • Utilize technology to find the solution to a system of equations by graphing, tables and successive approximations. • 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). MP 2, 4, 5, 6

Absolute Value Function System of Equations Logarithm Logarithmic Function

Students should solve by factoring, completing the square, and using the quadratic formula. The zero product property is used to explain why the factors are set equal to zero. Students should relate the value of the discriminant to the type of root to expect. A natural extension would be to relate the type of solutions to ax2 + bx + c = 0 to the behavior of the graph of y = ax2 + bx + c .

Assessment-

Use the table below to answer the questions-

Value of Discriminant

Nature of Roots

Nature of Graph

b2 – 4ac = 0 1 real roots intersects x-axis once

b2 – 4ac > 0 2 real roots intersects x-axis twice

b2 – 4ac < 0 2 complex roots does not intersect x-axis

● Are the roots of 2x2 + 5 = 2x real or complex? How many roots does it have? Find all solutions of the equation.

● What is the nature of the roots of x2 + 6x + 10 = 0? Solve the equation using the quadratic formula and completing the square. How are the two methods

MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP5: Use appropriate tools strategically MP6: Attend to precision


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related? Resources: How Do You Solve a System of Equations by Graphing? http://www.prometheanplanet.com/en-us/Resources/Item/138999/how-do-you-solve-a-system-of-equations-by-graphing

http://www.prometheanplanet.com/en-us/Resources/Item/138999/how-do-you-solve-a-system-of-equations-by-graphing




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Domain: Interpreting Functions F-IF

Pacing Guide Q1:Week 1-2

Cluster: Interpret functions that arise in applications in terms of the context.

Essential Question: What are the characteristics of a function and how can you use those characteristics to represent the function in multiple ways?



H.S. F-IF.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.*

• Describe the features of the graph of a function including intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. • Sketch graphs showing key features given a verbal description of the relationship. MP 2, 4, 5, 6

Symmetry Intercepts Increasing Decreasing Intervals Interval Notation Relative Maximum and Minimum Introductory Periodicity

Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. Assessment- ● A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. o What is a reasonable domain restriction for t in this context? o Determine the height of the rocket two seconds after it was launched. o Determine the maximum height obtained by the rocket. o Determine the time when the rocket is 100 feet above the ground. o Determine the time at which the rocket hits the ground. o How would you refine your answer to the first question based on your response to the second and fifth questions?

MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP5: Use appropriate tools strategically MP6: Attend to precision


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Resources: Boomerangs-The Flow of a Formative Assessment http://www.gatesfoundation.org/learning/Documents/math-big-card-

flow-of-fal-and-curriculum-pathways.pdf F.1F.4/5/6

http://www.gatesfoundation.org/learning/Documents/math-big-card-flow-of-fal-and-curriculum-pathways.pdf

http://www.gatesfoundation.org/learning/Documents/math-big-card-flow-of-fal-and-curriculum-pathways.pdf


32




Essential Question: How do you analyze and graph linear functions?



H.S. F-IF.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

• Explain any restrictions

on the domain of a graph

of a function both

algebraically and

contextually.

MP 2, 4, 6

Domain Range Intercepts relative maximum relative minimum end behavior periodicity symmetry

Students may explain orally, or in written format, the existing relationships Assessment- Solve the following equation and check your solution.

3x + 7 = 5x + 1 If you were to graph

f (x) =3x + 7 and g(x) = 5x +1 where would f intersect

g?

Explain how you know.

Resources: Exponential growth- http://sst. u.edu/mathns153/notes/exponential_functions.ppt

MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP6: Attend to precision


33




Essential Question: What are the characteristics of a function and how can you use those characteristics to represent the function in multiple ways?



H.S. F-IF.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

• Describe rate of change algebraically and contextually. • 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. MP 2, 4, 5

Rate of Change Interval

Students will find the average rate of change over a

specific interval or time.

Assessment-

Students will find the slope of the line that passes

through (8, 10) and (5, 11).

Simplify your answer and write it as a proper fraction, improper fraction, or integer.

Resources: Interpret Rate of Change of a Quadratic- http://www.cowetaschools.org/nghs/skelton/algebra_II/M2ch3_3.ppt

MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP5: Use appropriate tools strategically

http://www.cowetaschools.org/nghs/skelton/algebra_II/M2ch3_3.ppt

http://www.cowetaschools.org/nghs/skelton/algebra_II/M2ch3_3.ppt


34



Cluster: Analyze functions using different representations.

Essential Question: How do you analyze and interpret the characteristics of root, and rational functions using graphs and tables.



H.S. F-IF.7b. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Graph square root, cube root, and piecewise-defined functions, including step functions.

• Describe characteristics of square root, cube root, piece-wise, step, and absolute value functions. • Graph functions expressed symbolically by hand in simple cases and using technology for more complicated cases. MP 5, 6

Piece-Wise Function Step Function Cube Root Function Square Root Function

Students will graph functions and show key features

of the graph.

Assessment-

Students will graph this function: y = |x| using proper

scales, labeling, and titles. They then need to justify

their answer.

Resources: F.1F.7,8.9 Introduce students to graphing functions and to reading simple functions from graphs- http://www.shodor.org/interactivate/lessons/ReadingGraphs/ Exploring Quadratic Functions- http://www.prometheanplanet.com/en-us/Resources/Item/28826/exploring-quadratic-functions

MP5: Use appropriate tools strategically MP6: Attend to precision

http://www.shodor.org/interactivate/lessons/ReadingGraphs/

http://www.shodor.org/interactivate/lessons/ReadingGraphs/

http://www.prometheanplanet.com/en-us/Resources/Item/28826/exploring-quadratic-functions




35




Essential Question: How do you analyze and interpret the characteristics of root, and rational functions using graphs and tables?



H.S. F-IF.7c. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior

• Factor the polynomial. • Identify the zeros of the polynomial factors. • Determine the behavior of the graph given the multiplicity of the zeros. • Describe the end behavior of the polynomial using degree and leading coefficient. • Construct a graph of the polynomial using the zeros and end behavior. MP 5,6

Intercepts Maximum Minimum Square root Function cube Root function Piecewise-Defined function Step function Absolute value function

Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions.

Assessment- Use the graph above to answer the associated questions.

MP5: Use appropriate tools Strategically MP6: Attend to precision


36

Resources- http://www.shmoop.com/common-core-standards/ccss-hs-f-if-7c.html

http://www.shmoop.com/common-core-standards/ccss-hs-f-if-7c.html

http://www.shmoop.com/common-core-standards/ccss-hs-f-if-7c.html


37







H.S. F-IF.7e. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

• Describe characteristics

of exponential, logarithmic

and trigonometric

functions. • Graph trigonometric

functions including the

midline, period and

amplitude as well as end

behavior when applicable. • Graph logarithmic and

exponential functions

including the intercepts

and end behavior.

MP 5, 6

Period Midline Amplitude

Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Assessment- Use the graph below to answer the questions associated with it-

MP5: Use appropriate tools strategically MP6: Attend to precision


38

Resources: Exponential and logarithmic functions- http://mcs-cmarinas.barry.edu/net/ppt/MAT%20108/explog.ppt

http://mcs-cmarinas.barry.edu/net/ppt/MAT%20108/explog.ppt

http://mcs-cmarinas.barry.edu/net/ppt/MAT%20108/explog.ppt


39







H.S. F-IF.8a. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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

• Factor a quadratic function. • Complete the square on a quadratic function. • Find the zeros of a quadratic function. • Find the extreme values and symmetry of a quadratic function. • Interpret the zeros, extreme values and symmetry of a quadratic function in terms of context MP 2,7

Extreme Value Equivalent form Completing the square Zeros Extreme values Symmetry of the graph

Students write functions in different but equivalent

forms and explain the different properties of a

function. They need to show zeros, extreme values,

and symmetry.

Assessment-

u2 + 14u = 0

Write your answers as whole numbers or as proper

or improper fractions in simplest form.

u = or u =

Solve by completing the square.

j2 – 26j + 45 = 0

Write your answers as integers, proper or improper

fractions in simplest form, or decimals rounded to

the nearest hundredth.

j = or j =

MP2: Reason abstractly and quantitatively MP7: Look for and make use of structure


40

Resources: Factoring and completing the square in a quadratic function PowerPoint- http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/6-3/2006_6_3.ppt

http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/6-3/2006_6_3.ppt

http://teachers.henrico.k12.va.us/math/hcpsalgebra2/Documents/6-3/2006_6_3.ppt


41







H.S. F-IF.8b. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the properties of exponents to interpret expressions for exponential functions.

• Compare and contrast exponential growth and decay. • Classify exponential functions both algebraically and graphically as exponential growth or decay. MP 2, 7

Exponential Growth and Decay :

Students will compare and contrast exponential growth and decay, and classify that growth or decay algebraically and by graphing.

Example:

Assessment-

Would the graph of 0.5x show

exponential growth or exponential

decay?

Would the graph of 1.5x show

exponential growth or exponential

decay? Student graph with

labeling and explain how they got

their answer.

Resources: Exponential Growth and Decay PPT- http://www.klvx.org/DocumentCenter/Home/View/174

MP2: Reason abstractly and quantitatively MP7: Look for and make use of structure

http://www.klvx.org/DocumentCenter/Home/View/174

http://www.klvx.org/DocumentCenter/Home/View/174


42







H.S. F-IF.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.

• Identify the key characteristics of a function algebraically, graphically, numerically in tables, or by verbal descriptions. • Compare properties of two functions each represented in a different way. MP 6, 7

Symmetry of the graph

Students will compare two functions and determine how they can be represented, and which has the larger maximum. Assessment-

● Examine the functions below. Which function has the larger maximum? How do you know?

MP6: Attend to precision MP7: Look for and make use of structure


43

Resources:

Talk or Text-

http://illuminations.nctm.org/LessonDetail.aspx?id=L780

Walk the Plank-





44

Domain: Building Functions F-BF


Cluster: Build a function that models a relationship between two quantities.

Essential Question: How does the domain and range of a function relate to its graphical representation?



H.S. F-BF.1b. Write a function that describes a relationship between two quantities.* 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

• Perform arithmetic operations on functions. • Combine functions in context using arithmetic operations. MP 1, 2, 4, 5, 6, 7, 8

Explicit Recursive Linear Exponential Composition of functions

Students will analyze a given problem to determine the function expressed by identifying patterns in the function’s rate of change. They will specify intervals of increase, decrease, constancy, and, if possible, relate them to the function’s description in words or graphically. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model functions. Assessment- You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Resources:

Modeling Conditional Probabilities: 2- http://map.mathshell.org/materials/download.php?fileid=1200

What's an Exponential Function? http://www.prometheanplanet.com/en-us/Resources/Item/138668/what-s-an-exponential-function

MP1: Make sense of problems and persevere in solving them MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP5: Use appropriate tools strategically MP6: Attend to precision MP7: Look for and make use of structure MP8: Look for and express regularity in repeated reasoning

http://map.mathshell.org/materials/download.php?fileid=1200

http://www.prometheanplanet.com/en-us/Resources/Item/138668/what-s-an-exponential-function

http://www.prometheanplanet.com/en-us/Resources/Item/138668/what-s-an-exponential-function


45



Cluster: Build new functions from existing functions.

Essential Question: How does the vertical translation of a linear function model translations for other functions?



H.S. F-BF.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. Expressions for them.

• 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) with and without using technology. • Find the value of k given the graphs. • Prove that a function is even or odd using their graphs and algebraic expressions. MP 4, 5, 7

Transformations Even Function Odd Function

Students will apply transformations to functions and recognize functions as even and odd. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Assessment- • Is f(x) = x3 - 3x2 + 2x + 1 even, odd, or neither? Explain your answer orally or in written format. • Compare the shape and position of the graphs of f(x) = x2 and g(x) = 2x2, and explain the differences in terms of the algebraic expressions for the functions

MP4: Model with mathematics MP5: Use appropriate tools strategically MP7: Look for and make use of structure


46

Resources: Identifying Even and Odd Functions- http://faculty.uaeu.ac.ae/m.zalzali/ADM/35-mzalzali-adm.pps

http://faculty.uaeu.ac.ae/m.zalzali/ADM/35-mzalzali-adm.pps

http://faculty.uaeu.ac.ae/m.zalzali/ADM/35-mzalzali-adm.pps


47



Cluster: Build new functions from existing functions.

Essential Question: How can you build a completely new function from a existing function?



H.S. F-BF.4b. Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.

• Explain the steps to finding the inverse of a function. • Find the inverse of a function. MP 2, 4, 5, 7

Inverse of a Function Independent variable Dependent variable

Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model functions. Assessment-

● For the function h(x) = (x – 2)3, defined on the domain of all real numbers, find the inverse function if it exists or explain why it doesn’t exist.

● Graph h(x) and h-1(x) and explain how they relate to each other graphically.

Find a domain for f(x) = 3x2 + 12x - 8 on which it has an inverse. Explain why it is necessary to restrict the domain of the function. Resources: Linear Functions- http://www.prometheanplanet.com/en-us/Resources/Item/28828/graphing-and-interpreting-linear-functions Quadratic functions- http://www.prometheanplanet.com/en-us/Resources/Item/28826/exploring-quadratic-functions

MP2: Reason abstractly and quantitatively MP4: Model with mathematics MP5: Use appropriate tools strategically MP7: Look for and make use of structure

http://www.prometheanplanet.com/en-us/Resources/Item/28828/graphing-and-interpreting-linear-functions






48

Domain: Linear, Quadratic, & Exponential Models* F-LE


Cluster: Construct and compare linear, quadratic, and exponential models and solve problems.

Essential Question: How does exponential growth differ from linear growth?



H.S.F-LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Describe how logarithms relate to exponents. • Evaluate a logarithm where base b is 2, 10 or e algebraically or with technology. • Simplify expressions using the properties of logarithms where base b is 2, 10 or e. • Solve a logarithmic equation where base b is 2, 10 or e. express the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e MP 7

Exponential

Logarithm

Base of a Logarithm

e

Students determine if a quantity is increasing

exponentially will eventually exceed a quantity in a

linear, quadratically, or as a polynomial function.

Assessment-

Students contrast the growth of the f(x)=x3 and

f(x)=3x.They need to show their work and explain how

they got their answer.

Resources: Exponential Models- http://www.damien-hs.edu/moodle/file.php/137/Chapter_8_Exponential_and_Logarithms_Power_Point.ppt

MP7: Look for and make use of structure

http://www.damien-hs.edu/moodle/file.php/137/Chapter_8_Exponential_and_Logarithms_Power_Point.ppt




49

Domain: Trigonometric Functions F-TF


Cluster: Extend the domain of trigonometric functions using the unit circle.

Essential Question: How do you use/read a unit circle (using radians)?



H.S. F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

• Explain that the radian measure of an angle is the length of the arc on the unit circle subtended by the angle. MP 6

Arc Radian Measure Unit Circle

Students are to understand the radian measure of an angle is length of the arc on the unit circle subtended by the angle. Assessment- Students answer the following questions- The minute hand on the clock at city hall in Roswell measures 2.2 meters from the tip to the axle.

a. Through what angle does the minute hand pass through between 7:07 and 7:43 am?

b. What distance does the tip of the minute hand travel during this period?

Resources: Trigonometric Functions- F.TF.1,2 http://www.prometheanplanet.com/en-us/Resources/Item/28633/trigonometric-functions

MP6: Attend to precision

http://www.prometheanplanet.com/en-us/Resources/Item/28633/trigonometric-functions

http://www.prometheanplanet.com/en-us/Resources/Item/28633/trigonometric-functions


50

Domain: Trigonometric Functions F-TF


Cluster: Extend the domain of trigonometric functions using the unit circle.

Essential Question: What are the different ways you can represent a trigonometric function?



H.S. F-TF.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

• Convert radians to degrees and degrees to radians. • Label the basic radians on the unit circle. • Approximate all positive real number radian and degree measures on the unit circle. MP 2

Trigonometric functions

Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. Assessments- ● A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h = – 16t2 + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. o What is a reasonable domain restriction for t in this context? o Determine the height of the rocket two seconds after it was launched. o Determine the maximum height obtained by the rocket. o Determine the time when the rocket is 100 feet above the ground. o Determine the time at which the rocket hits the ground. o How would you refine your answer to the first question based on your response to the second and fifth questions?.

MP2: Reason abstractly and quantitatively