Arizona Mathematic Standards

Mathematics – PreCalculus

Arizona Department of Education Chandler Unified School District #80 State Board Approved December 2016 Page 1 of 22 Revised: August 2017

Arizona Mathematic Standards

Mathematics Curriculum Map

PreCalculus

ARIZONA DEPARTMENT OF EDUCATION HIGH ACADEMIC STANDARDS



Chandler Unified School District Standards

PreCalculus – At a Glance NOTE: Mathematical standards are interwoven and should be addressed throughout the year in as many different units and tasks as possible in

order to stress the natural connections that exist among mathematical topics.

Pre-calculus Map

Semester 1

Semester 2

Optional Flex Units

Unit 1 Functions and

Graphs

Unit 2 Polynomial Functions, Rational

Functions

Unit 3 Exponential

And Logarithmic Functions

Unit 4 Systems of Equations

Unit 5 Trigonometric Functions and

Graphs

Unit 6 Trigonometric

Identities

Unit 7 Polar

Coordinates

and Graphs

Unit 8 Sequences

and Series

Flex Unit 1 Vectors/

Parametric

Functions

Flex Units 2/3

Conics/

Matrices

A2.F-IF.B.4 A2.F-IF.B.6 A2.F-IF.C.7 A2.F-IF.C.9 A2.F-BF.A.1 A2.F-BF.B.3 A2.F-BF.B.4 P.F-BF.A.1 P.F-BF.B.4

A2.A-APR.B.2 A2.A-APR.B.3 A2.A-APR.C.4 A2.A-APR.D.6 A2.F-IF.B.4 A2.F-IF.C.7

A2.F-IF.B.4 A2.F-IF.C.7 A2.F-IF.C.8 A2.F-IF.C.9 A2.F-BF.A.1 A2.F-BF.B.3 A2.F-LE.A.4 A2.F-LE.B.5 A2.A-REI.D.11

A2.A-REI.C.7 A2.A-REI.D.11

A2.F-TF.A.1 A2.F-TF.A.2 P.F-TF.A.3 A2.F-TF.B.5 P.F-TF.A.4 P.F-TF.B.6 P.F-TF.B.7

P.F-TF.C.9 P.G-SRT.D.9 P.G-SRT.D.10 P.G-SRT.D.11

P.N-CN.B.4 P.N-CN.B.5

A2.A-SSE.B.4 A2.F-BF.A.2 P.A-APR.C.5

P.N-VM.A.1 P.N-VM.A.2 P.N.VM.A.3 P.N-VM.B.4 P.N-VM.B.5 P.N-VM.C.11 P.N-VM.C.12

P.N-VM.C.6 P.N-VM.C.7 P.N-VM.C.8 P.N-VM.C.9 P.N-VM.C.10 P.G-GPE.A.2 P.G-GPE.A.3

Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

All units will include the Mathematical Practices



PreCalculus Overview

MCCCD Official Course Competencies

1. Find real and complex zeros of polynomial functions. 2. Calculate and interpret average rate of change. 3. Determine the inverse of a relation when represented numerically, analytically, or graphically. 4. Analyze and interpret the behavior of functions, including domain and range, end behavior, increasing and decreasing intervals, extrema,

asymptotic behavior, and symmetry. 5. Determine whether a function is one-to-one when represented numerically, analytically, or graphically. 6. Determine whether a relation is a function when represented numerically, analytically, or graphically. 7. Graph polynomial, rational, exponential, logarithmic, power, absolute value, piecewise-defined, and trigonometric functions. 8. Perform operations, including compositions, on functions and state the domain of the resulting function. 9. Solve polynomial, rational, exponential, logarithmic, and trigonometric equations analytically and graphically. 10. Use transformations to graph functions. 11. Communicate process and results in written and verbal format. 12. Compare alternative solution strategies. 13. Justify and interpret solutions to application problems. 14. Model and solve real-world problems. 15. Read and interpret quantitative information when presented numerically, analytically, or graphically. 16. Find and evaluate inverse trigonometric functions. 17. Use the definition and properties of trigonometric functions and formulas to solve application problems. 18. Verify trigonometric identities.

Standards for Mathematical Practices (MP)

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.



Semester 1 Unit 1- Functions and Their Graphs Essential Question(s):

How do various functions behave? How do two quantities change together?

Topic Arizona Mathematics Standard Mathematical Practices

Resources

Functions and Their Properties

A2.F-IF.C.7 Graph functions expressed symbolically and show key features

of the graph, by hand in simple cases and using technology for more

complicated cases.

Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions.

A2.F-IF.B.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. Include problem-solving opportunities

utilizing a real-world context. Key features include: intercepts; intervals

where the function is increasing, decreasing, positive, or negative; relative

maximums and minimums; symmetries; end behavior; and periodicity.

MP 5 MP 6

Blitzer 1.3 Demana, Waits, Foley, Kennedy, Bock 1.2 Larson 1.3, 1.4, 1.5, 1.6

Building Functions

from Functions

P.F-BF.A.1 Write a function that describes a relationship between two quantities. 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.

MP 2 MP 7

Blitzer 1.7 Demana et al.1.4 Larson 1.8



Inverse Functions

A2.F-BF.B.4 Find inverse functions.

a. Understand that an inverse function can be obtained by expressing the dependent variable of one function as the independent variable of another, recognizing that functions f and g are inverse functions if and only if f(x) = y and g(y) = x for all values of x in the domain of f and all values of y in the domain of g. b. Understand that if a function contains a point (a,b), then the graph of the inverse relation of the function contains the point (b,a). c. Interpret the meaning of and relationship between a function and its inverse utilizing real-world context. P.F-BF.B.4 Find inverse functions. a. Verify by composition that one function is the inverse of another. b. Read values of an inverse function from a graph or a table, given that the function has an inverse. c. Produce an invertible function from a non-invertible function by restricting the domain.

MP 6 MP 7

Blitzer 1.8

Demana et al.1.5 Larson 1.9 Illustrative Mathematics https://www.illustrativemathematics.org/content-standards/tasks/1279

Graphical Transformations

A2.F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k,

kf(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.

MP 1 MP 5

Blitzer 1.6 Demana et al.1.3, 1.6 Larson 1.7

Piece-Wise

Functions

A2.F-IF.B.6 Calculate and interpret the average rate of change of a continuous function (presented symbolically or as a table) on a closed interval. Estimate the rate of change from a graph. Include problem-solving opportunities utilizing real-world context.

A2.F-IF.C.9: Compare properties of two functions each represented in a

different way (algebraically, graphically, numerically in tables, or by verbal

descriptions.).

MP 4 MP 7

Blitzer 1.3, 1.5

Demana et al.1.3

Larson 1.4

https://www.illustrativemathematics.org/content-standards/tasks/1279




Modeling with Functions

A2.F-BF.A.1 Write a function that describes a relationship between two quantities. Functions include linear, quadratic, exponential, polynomial, logarithmic, rational, sine, cosine, tangent, square root, cube root and piecewise-defined functions. Include problem-solving opportunities utilizing real-world context. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine function types using arithmetic operations and function composition.

A2.F-IF.B.6 Calculate and interpret the average rate of change of a

continuous function (presented symbolically or as a table) on a closed

interval. Estimate the rate of change from a graph. Include problem-solving

opportunities utilizing real-world context.

MP 1 MP 4

Blitzer 1.10

Demana et al.1.1, 1.7

Larson 1.8,1.10



Semester 1 Unit 2- Polynomial and Rational Functions Essential Question(s):

How do you find and interpret zeros of various polynomial and rational functions?

How do you graph and interpret various polynomial and rational functions? Topic Arizona Mathematics Standard Mathematical

Practices Resources

Quadratic Functions and Models

A2.A-APR.B.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.

A2.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

MP 3

MP 4

Blitzer 2.2

Demana et al.2.1

Larson 2.1

Polynomial Functions of Higher Degree

A2.A-APR.B.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. Focus on quadratic, cubic, and quartic polynomials including polynomials for which factors are not provided

A2.A-APR.C.4 Prove polynomial identities and use them to describe numerical relationships.

A2.F-IF.B.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. Include problem-solving opportunities utilizing a real-world context. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

MP 3 MP 4 MP 5 MP 6

Blitzer 2.3

Demana et al.2.3

Larson 2.2



Polynomial Operations and Synthetic Division

A2.A-APR.B.2 Know and apply the Remainder and Factor Theorem: For a polynomial p(x) and a number a, the remainder on division by (x – a) is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

A2.A-APR.D.6 Rewrite rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or for the more complicated examples, a computer algebra system.

MP 1 MP 2 MP 6

Blitzer 2.4

Demana et al.2.4

Larson 2.3

Zeros of Polynomial Functions

A2.A-APR.B.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.

MP 5 MP 6

Blitzer 2.5

Demana et al.2.4,2.5

Larson 2.5

Rational Functions

A2.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MP 3 MP 4

Blitzer 2.6

Demana et al.2.6

Larson 2.6

Nonlinear Inequalities

A2.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MP 5 MP 6

Blitzer 2.7

Demana et al.2.8

Larson 2.7



Semester 1 Unit 3- Exponential and Logarithmic Functions Essential Question(s):

How can you use exponential and logarithmic functions to model real world situations? How are exponential and logarithmic functions the same? How are they different?


Resources

Graphing Exponential

and Logarithmic

Functions

A2.F-IF.C.7 Graph functions expressed symbolically and show key

features of the graph, by hand in simple cases and using technology for

more complicated cases.

A2.F-IF.C.9 Compare properties of two functions each represented in a

different way (algebraically, graphically, numerically in tables, or by

verbal descriptions.).

A2.F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k,

kf(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.

A2.F-IF.B.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. Include problem-solving opportunities

utilizing a real-world context. Key features include: intercepts; intervals

where the function is increasing, decreasing, positive, or negative;

relative maximums and minimums; symmetries; end behavior; and

periodicity.

MP 1 MP 5 MP 8

Blitzer 3.1, 3.2


Larson 3.1, 3.2

Illustrative Math









Evaluating and Solving Logarithmic Equations

A2.F-IF.C.8 Write a function defined by an expression in different but

equivalent forms to reveal and explain different properties of the function.

b. Use the properties of exponents to interpret expressions for exponential functions and classify those functions as exponential growth or decay.

A2.F-BF.A.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 function types using arithmetic operations and function composition.

MP 7 Blitzer 3.1, 3.2, 3.3, 3.4, 3.5


Larson 3.1, 3.2, 3.3, 3.4, 3.5

Evaluate Exponential and Logarithmic Problems

A2.F-LE.A.4 For exponential models, express as a logarithm the solution

to abct = d where a, c, and d are numbers and the base b is 2, 10, or e;

evaluate the logarithms that are not readily found by hand or observation

using technology.

MP 5 Blitzer 3.4

Demana et al.3.3

Larson 3.4

Applications of

exponential and

logarithmic problems

A2.F-LE.B.5 Interpret the parameters in an exponential function with

rational exponents utilizing real-world context.

A2.A-REI.D.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).

MP 8 Blitzer 3.4, 3.5


Larson 3.5 Illustrative Math https://www.illustrativemathematics.org/content-standards/tasks/1551






Semester 1 Unit 4– Systems of Equations Essential Question(s):

Can you solve a system of equation algebraically, graphically, and with technology? Topic Arizona Mathematics Standard Mathematical

Practices Resources

Find the solution to a system of equations algebraically and graphically.

A2.A-REI.C.7: Solve a system consisting of a linear equation and a

quadratic equation in two variables algebraically and graphically. For

example, find the points of intersection between the line y = -3x and the

circle x2 + y2 = 3.

MP 1 MP 4

Blitzer 7.4 Demana et al. 7.1 Larson 7.1, 7.2, 7.3

Use technology to solve systems of equations

A2.A.REI.D.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 problems in real-world context.

MP 4 MP 5

Blitzer 7.4

Demana et al.7.1

Larson 7.1, 7.2, 7.3



Semester 2 Unit 5 – Trigonometric Functions and Their Graphs Essential Question(s):

How do we relate angular measure to periodic behavior? How can we use trigonometric functions to model the world around us?


Resources

Right Triangle Trigonometry

P.F-TF.A.3: Use special triangles to determine geometrically the values

of sine, cosine, tangent for π /3, π/4 and π/6, and use the unit circle to

express the values of sine, cosine, and tangent for π-x, π+x, and 2π-x

in terms of their values for x, where x is any real number.

MP 1 Blitzer 4.2


Larson 4.3

Circular Functions (Radian and Degree Measurement, Arc Length, Area of a Sector)

A2.F-TF.A.1: Understand radian measure of an angle as the length of

the arc on any circle subtended by the angle, measured in units of the

circle's radius.

MP 2 Blitzer 4.1

Demana et al.4.1

Larson 4.1

The Unit Circle A2.P.F-TF.A.4: Use the unit circle to explain symmetry (odd and even)

and periodicity of trigonometric functions.

A2.F-TF.A.2: Explain how the unit circle in the coordinate plane enables

the extension of sine and cosine functions to all real numbers,

interpreted as radian measures of angles traversed counterclockwise

around the unit circle.

MP 2 MP 6 MP 8

Blitzer 4.2

Demana et al.4.3

Larson 4.2

Graphs of Trig Functions

A2.F-TF.B.5: Create and interpret sine, cosine and tangent functions

that model periodic phenomena with specified amplitude, frequency,

and midline.



Larson 4.5, 4.6

Inverse Trig Functions P.F-TF.B.6: Understand that restricting a trigonometric function to a

domain on which it is always increasing or always decreasing allows its

inverse to be constructed.

P.F-TF.B.7: Use inverse functions to solve trigonometric equations

utilizing real world context; evaluate the solution and interpret them in

terms of context.

MP 1 MP 3 MP 6

Blitzer 4.7

Demana et al.4.7

Larson 4.7



Semester 2 Unit 6-Trig Identities Essential Question(s):

How are trigonometric identities used to solve real-world problems? Topic Arizona Mathematics Standard Mathematical

Practices Resources

Fundamental Identities

MCCCD Course Outline V.A. Fundamental identities A2.F-TF.C.8: Use the Pythagorean identity sin2(θ) + cos2(θ) = 1 and the quadrant of the angle θ to find sin(θ), cos(θ), or tan(θ) given sin(θ) or cos(θ).

MP 6 Blitzer 5.1 Demana et al.5.1 Larson 5.1

Verifying Trig Identities

MCCCD Course Competencies #18 Verify trigonometric identities. MP 3 Blitzer 5.1

Demana et al.5.2

Larson 5.2

Sum and Difference Identities

P.F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

MP 6 Blitzer 5.2

Demana et al.5.3

Larson 5.4

Double Angle and Half Angle Identities

MCCCD Course Outline V.D. Double-angle identities MCCCD Course Outline V.E. Half-angle identities

MP 6 Blitzer 5.3

Demana et al.5.4

Larson 5.5

Area of Non-Right Triangles

P.G-SRT.D.9 Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

MP 6 Blitzer 6.1

Demana et al. 5.6

Larson 6.1, 6.2

Law of Sines and Cosines

P.G-SRT.D.10 Prove the Laws of Sines and Cosines and use them to solve problems P.G-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

MP 2 MP 6

Blitzer 6.1, 6.2


Larson 6.1, 6.2



Semester 2 Unit 7- Complex Numbers Essential Question(s):

How do you convert points (and complex numbers) from rectangular to polar form? How do you convert points (and complex numbers) from polar to rectangular form?


Resources

Trigonometric Form of Complex Numbers

P.N-CN.B.4 Represent complex numbers on the complex plane in rectangular and polar form, including real and imaginary numbers, and explain why the rectangular and polar forms of a given complex number represent the same number.

MP 2 Blitzer 6.3 Demana et al.6.6 Larson 10.7, 10.8

Basic Operations with Complex Numbers

P.N-CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + i)3 = 8 because (-1 + i) has modulus 2 and argument 120°.

MP 6 Blitzer 6.5 Demana et al. 6.6 Larson 6.5

De Moivre's Theorem

MCCCD Course Outline VIII.B. De Moivre`s theorem MP 6 Blitzer 6.5 Demana et al.6.6 Larson 6.5

Roots of Complex Numbers

MCCCD Course Competencies #11 Find nth roots of complex numbers MP 6 Blitzer 6.5 Demana et al.6.6 Larson 6.5



Semester 2 Unit 8- Sequences and Series Essential Question(s):

How do you write a formula for the nth term of a sequence? How can you find the sum of a series? How can you determine if/when an infinite series has a sum?


Resources

Write expressions in

equivalent forms to

solve problems.

A2.A-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.


Build a function that

models a relationship

between two

quantities.

A2.F-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

MP 7 Blitzer 10.1, 10.2, 10.3 Demana et al.9.3 Larson 9.1, 9.2

Use polynomial

identities to solve

problems.

P.A-APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.




Flex Unit

Flex Unit 1- Vectors Essential Question(s):

How are vectors used in real world situations? Topic Arizona Mathematics Standard Mathematical

Practices Resources

Definition of Vector

P.N-VM.A.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes.


Component for of a

vector

P.N-VM.A.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.


Velocity in vectors representation

P.N-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.

MP 1 Blitzer 6.6 Demana et al.6.1,6.2 Larson 6.3

Vector properties

P.N-VM.B.4 Add and subtract vectors. a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

MP 6 MP 7




Vector Multiplication by a scalar

P.N-VM.B.5 Multiply a vector by a scalar. a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise e.g., as c(vx, vy) = (cvx, cvy). b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).


Vector Multiplication

using matrices

P.N-VM.C.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

MP 7 Blitzer 6.6, 8.3 Demana et al.6.1 Larson 8.2

Parametric Equations and Graphs

MCCCD Course outline IX.D Parametric Equations and Graphs MP 1 MP 2


Flex Unit Flex Unit 2- Conics Essential Question(s):

How can the understanding of conic sections make more sense of the constructions and designs in our world? What determines the type of conic section you will be using? Why are there key vital coordinates, points and axis and how do they help me use and apply the conic section to solve

problems? Topic Arizona Mathematics Standard Mathematical

Practices Resources

Equations of Parabolas

P.G-GPE.A.2 Derive the equation of a parabola given a focus and directrix.

MP 6 Blitzer 9.3

Demana et al. 8.1

Larson 10.2

Equations of Ellipses and Hyperbolas

P.G-GPE.A.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.


Demana et al. 8.2,8.3

Larson 10.3, 10.4



Flex Unit Flex Unit 3– Matrices Essential Question(s):

How do you use matrices to solve systems of equations? Topic Arizona Mathematics Standard Mathematical

Practices Resources

Matrices and Systems of Equations

P.N-VM.C.6: Use matrices to represent and manipulate data.

MP 1 MP 4 MP 5


Operations with Matrices

P.N-VM.C.7: Multiply matrices by scalars to produce new matrices.

P.N-VM.C.8: Add, subtract, and multiply matrices of appropriate

dimensions.


Square Matrices P.N-VM.C.9: Understand that, unlike multiplication of numbers, matrix

multiplication for square matrices is not a commutative operation, but

still satisfies the associative and distributive properties.

P.N-VM.C.10: Understand that the zero and identity matrices play a

role in matrix addition and multiplication similar to the role of 0 and 1 in

the real numbers. The determinant of a square matrix is nonzero if and

only if the matrix has a multiplicative inverse.

P.N-VM.C.12: Work with 2 x 2 matrices as transformations of the plane,

and interpret the absolute value of the determinant in terms of area.

MP 1 MP 2

Blitzer 8.3 Demana et al.6.1 & 7.2 Larson 8.3, 8.4



The Mathematical Practices: Narratives and Questions

Mathematics Practices Narratives Related Questions

Overa

rch

ing

hab

its o

f m

ind

of

a p

rod

ucti

ve m

ath

th

inker

P.MP.1 Make sense of problems and persevere in solving them

Mathematically proficient students explain to themselves the meaning of a problem, look for entry points to begin work on the problem, and plan and choose a solution pathway. While engaging in productive struggle to solve a problem, they continually ask themselves, “Does this make sense?" to monitor and evaluate their progress and change course if necessary. Once they have a solution, they look back at the problem to determine if the solution is reasonable and accurate. Mathematically proficient students check their solutions to problems using different methods, approaches, or representations. They also compare and understand different representations of problems and different solution pathways, both their own and those of others.

How would you describe the problem in your own words?

How would you describe what you are trying to find?

What do you notice about...?

What information is given in the problem?

Describe the relationship between the quantities.

Describe what you have already tried. What might you change?

Talk me through the steps you’ve used to this point.

What steps in the process are you most confident about?

What are some other strategies you might try?

What are some other problems that are similar to this one?

How might you use one of your previous problems to help you begin?

How else might you organize...represent... show...?

P.MP.6 Attend to precision

Mathematically proficient students clearly communicate to others using appropriate mathematical terminology, and craft explanations that convey their reasoning. When making mathematical arguments about a solution, strategy, or conjecture, they describe mathematical relationships and connect their words clearly to their representations. Mathematically proficient students understand meanings of symbols used in mathematics, calculate accurately and efficiently, label quantities appropriately, and record their work clearly and concisely.

What mathematical terms apply in this situation?

How did you know your solution was reasonable?

Explain how you might show that your solution answers the problem.

What would be a more efficient strategy?

How are you showing the meaning of the quantities?

What symbols or mathematical notations are important in this problem?

What mathematical language...,definitions..., properties can you use to explain...?

How could you test your solution to see if it answers the problem?

Actions and dispositions from NCSM Summer Leadership Academy, Atlanta, GA • Draft, June 22, 2011)

Most questions from all Grades Common Core State Standards Flip Book




Mathematics Practices Narratives Related Questions

Reaso

nin

g a

nd

Exp

lain

ing

P.MP.2 Reason abstractly and quantitatively

Mathematically proficient students make sense of quantities and their relationships in problem situations. Students can contextualize and decontextualize problems involving quantitative relationships. They contextualize quantities, operations, and expressions by describing a corresponding situation. They decontextualize a situation by representing it symbolically. As they manipulate the symbols, they can pause as needed to access the meaning of the numbers, the units, and the operations that the symbols represent. Mathematically proficient students know and flexibly use different properties of operations, numbers, and geometric objects and when appropriate they interpret their solution in terms of the context.

What do the numbers used in the problem represent?

What is the relationship of the quantities?

How is _______ related to ________?

What is the relationship between ______and ______?

What does_______mean to you? (e.g. symbol, quantity, diagram)

What properties might we use to find a solution?

How did you decide in this task that you needed to use...?

Could we have used another operation or property to solve this task? Why or why not?

P.MP.3 Construct viable arguments and critique the reasoning of others

Mathematically proficient students construct mathematical arguments (explain the reasoning underlying a strategy, solution, or conjecture) using concrete, pictorial, or symbolic referents. Arguments may also rely on definitions, assumptions, previously established results, properties, or structures. Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. Mathematically proficient students present their arguments in the form of representations, actions on those representations, and explanations in words (oral or written). Students critique others by affirming or questioning the reasoning of others. They can listen to or read the reasoning of others, decide whether it makes sense, ask questions to clarify or improve the reasoning, and validate or build on it. Mathematically proficient students can communicate their arguments, compare them to others, and reconsider their own arguments in response to the critiques of others.

What mathematical evidence would support your solution?

How can we be sure that...? / How could you prove that...?

Will it still work if...?

What were you considering when...?

How did you decide to try that strategy?

How did you test whether your approach worked?

How did you decide what the problem was asking you to find?

Did you try a method that did not work? Why didn’t it work? Could it work?

What is the same and what is different about...?

How could you demonstrate a counter-example?








Mathematics Practices

Narratives Related Questions

Mo

delin

g a

nd

Usin

g T

oo

ls

P.MP.4 Model with mathematics

Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. When given a problem in a contextual situation, they identify the mathematical elements of a situation and create a mathematical model that represents those mathematical elements and the relationships among them. Mathematically proficient students use their model to analyze the relationships and draw conclusions. They interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

What number model could you construct to represent the problem?

What are some ways to represent the quantities?

What is an equation or expression that matches the diagram, number line, chart, table, and your actions with the manipulatives?

Where did you see one of the quantities in the task in your equation or expression? What does each number in the equation mean?

How would it help to create a diagram, graph, table...?

What are some ways to visually represent...?

What formula might apply in this situation?

P.MP.5 Use appropriate tools strategically

Mathematically proficient students consider available tools when solving a mathematical problem. They choose tools that are relevant and useful to the problem at hand. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful; recognizing both the insight to be gained and their limitations. Students deepen their understanding of mathematical concepts when using tools to visualize, explore, compare, communicate, make and test predictions, and understand the thinking of others.

What mathematical tools can we use to visualize and represent the situation?

Which tool is more efficient? Why do you think so?

What information do you have?

What do you know that is not stated in the problem?

What approach are you considering trying first?

What estimate did you make for the solution?

In this situation would it be helpful to use...a graph..., number line..., ruler..., diagram..., calculator..., manipulative?

Why was it helpful to use...?

What can using a ______ show us that _____may not?

In what situations might it be more informative or helpful to use...?




Mathematics Practices

Narratives Related Questions

Seein

g s

tru

ctu

re a

nd

gen

era

lizin

g

P.MP.7 Look for and make use of structure

Mathematically proficient students use structure and patterns to assist in making connections among mathematical ideas or concepts when making sense of mathematics. Students recognize and apply general mathematical rules to complex situations. They are able to compose and decompose mathematical ideas and notations into familiar relationships. Mathematically proficient students manage their own progress, stepping back for an overview and shifting perspective when needed.

What observations do you make about...?

What do you notice when...?

What parts of the problem might you eliminate..., simplify...?

What patterns do you find in...?

How do you know if something is a pattern?

What ideas that we have learned before were useful in solving this problem?

What are some other problems that are similar to this one?

How does this relate to...?

In what ways does this problem connect to other mathematical concepts?

P.MP.8 Look for and express regularity in repeated reasoning

Mathematically proficient students look for and describe regularities as they solve multiple related problems. They formulate conjectures about what they notice and communicate observations with precision. While solving problems, students maintain oversight of the process and continually evaluate the reasonableness of their results. This informs and strengthens their understanding of the structure of mathematics which leads to fluency.

Explain how this strategy works in other situations?

Is this always true, sometimes true or never true?

How would we prove that...?

What do you notice about...?

What is happening in this situation?

What would happen if...?

Is there a mathematical rule for...?

What predictions or generalizations can this pattern support?

What mathematical consistencies do you notice?



Documents

Arizona Mathematic Standards