SD Common Core State Standards Disaggregated Math Template · SD Common Core State Standards...

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SD Common Core State Standards Disaggregated Math Template Domain: Reasoning with Equations and Inequalities Cluster: Solve systems of equation Grade level: 9-12 Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year 9-12.A.REI.8(+) Represent a system of linear equations as a single matrix equation in a vector variable. Student Friendly Language: I can represent a system of linear equations as a single matrix equation. Know (Factual) Understand (Conceptual) The Students will understand that: Do (Procedural, Application, Extended Thinking) Matrix dimension The variable matrix is called a vector because it has one column. Systems of equations can be represented using matrices. Rewrite a system of linear equations as a single matrix equation. Key Vocabulary: Vector variable Matrix dimensions Rows Columns Coefficient Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”? Introduction to transformations and animation. Introduction to solving systems using inverse operations.

Transcript of SD Common Core State Standards Disaggregated Math Template · SD Common Core State Standards...

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SD Common Core State Standards Disaggregated Math Template

Domain: Reasoning with Equations and Inequalities

Cluster: Solve systems of equation Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.A.REI.8(+) Represent a system of linear equations as a single matrix equation in a vector variable.

Student Friendly Language:

I can represent a system of linear equations as a single matrix equation.

Know (Factual)

Understand (Conceptual)

The Students will understand that:

Do (Procedural, Application, Extended

Thinking)

● Matrix dimension

The variable matrix is called a vector because it has one column. Systems of equations can be represented using matrices.

Rewrite a system of linear equations as a single matrix equation.

Key Vocabulary:

Vector variable Matrix dimensions Rows Columns Coefficient

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Introduction to transformations and animation. Introduction to solving systems using inverse operations.

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SD Common Core State Standards Disaggregated Math Template

Domain: Reasoning with Equations and Inequalities

Cluster: Solve systems of equation Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.A.REI.9(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Student Friendly Language:

I can put a system of linear equations into a matrix equation. I can solve a matrix equation by using the inverse of a matrix (if it exists) and applying it to equality properties of equations. I can use a graphing device to solve a matrix equation using inverse matrices (if they exist).

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● matrix equation ● inverse matrix

Properties of equality can be applied to a matrix equation to solve for the variables. If an inverse of a matrix exists, it can be found using graphing technology and used to solve a system of equations. It is possible that the inverse of a matrix does not exist and that this means that a unique solution does not exist.

Demonstrate that a system of equations can be solved using a matrix equation and inverse matrices. Communicate understanding of solving systems of equations using matrices through the use of graphing technology.

Key Vocabulary:

matrix equation inverse matrix unique solution

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

This can be used any time that a solving a system of equations is necessary: For example: A stadium can fit only 400 seats. Each student seat costs $5 and each adult seat costs $12. How many of each seat should be sold to have an income of $3,666?

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SD Common Core State Standards Disaggregated Math Template

Domain: Building Functions Cluster: Build new functions from existing functions Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

F.BF.9-12.4 Find inverse functions. a. 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)=2x

3 for x>0 or

f(x)=(x+1)/(x-1) for x≠1.

F.BF.9-12.4 Find inverse functions. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

NA

Student Friendly Language:

I can use composition of functions to verify that two functions are inverses of each other.

○ [f∘ g](x)=x and [g∘ f](x)=x

I can read values of an inverse function off of a table by switching the x and f(x) values (or switching the x and y

coordinates).

I can read values of an inverse function off of a graph by reflecting the graph over the function f(x)=x (or over the line

y=x).

I can identify restrictions on the domain when writing the inverse function.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● Both compositions of inverse functions

produce the identity function: [f∘ g](x) and

[g∘ f](x) =x.

● When given a table of values of a function,

the inverse function transposes the given x

and f(x) values (or x and y values).

● When a function and its inverse is graphed,

they are reflections over f(x)=x or y=x.

● One-to-one functions have inverse functions

● Inverse functions must pass the horizontal

line test to be a function.

● The domains of some functions must be

restricted in order to write inverse functions.

● If two functions are composed both ways to produce the identity function, they are inverse functions.

● When given a function in table form, the inverse of that function simply transposes the values.

● When given a function in graphed form, the inverse is the reflection of the graph over the line y=x.

● If a function is not one-to-one, an inverse may still be written if the domain is restricted so that the inverse passes the horizontal line test.

● Use composition of functions to verify that two functions are inverses of each other.

● Transpose the x and f(x) values in a function to read values of the inverse function.

● Reflect a function over the line y=x to produce the graph of its inverse and read values off of the inverse function.

● Decide whether an inverse function exists by determining if the original function is one-to-one.

● Use the horizontal line test to determine whether the inverse of a function is a function.

● If the inverse of a function does not exist, restrict the domain so that it does exist.

Key Vocabulary:

inverse inverse relation inverse function domain range vertical line test one-to-one functions

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Basic idea of inverse function: When someone calls you on the phone, he or she looks up your number in a phone book (a function from names to phone numbers). When Caller ID shows who is calling, it has performed the inverse function, finding the name corresponding to the number.

Inverse functions can be used to convert from one measurement unit to another. ex. If C(x)=5/9(x-32) can be used to convert from Fahrenheit to Celsius. C-1(x) can be used to convert from Celsius to Fahrenheit.

Inverse functions can be used to model and solve real-life problems. ex. If a function gives the factory sales of digital cameras over a period of years an inverse function can be used to determine the year in which a certain dollar amount worth of digital cameras was sold.

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SD Common Core State Standards Disaggregated Math Template

Domain: Building Functions

Cluster: Build new functions from existing functions

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

F.BF.9-12.1, 3, 4a

(+)9-12.F.BF.5 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Student Friendly Language:

I will be to find inverses for logarithmic and exponential forms. I will be able to solve problems involving exponents and logarithms.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended

Thinking)

● Logarithmic and exponential functions can be solved using inverses.

● There are real-world problems related to logarithms, exponents, and their inverses.

How to find inverses of functions. How to determine if two functions are inverses. How to verify that exponential and logarithmic functions are inverses.

Solve logarithmic functions using exponentials as the inverse. Solve exponential functions using logarithms as the inverse. Solve problems using logarithms and exponents.

Key Vocabulary:

function logarithm exponential inverse

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

banking; calculating values for compounding continuous interest

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SD Common Core State Standards Disaggregated Math Template

Domain: Functions Cluster: Analyze functions using different representations Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in

Following Year

8.F.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph

(e.g., where the function is

increasing or decreasing, linear or

nonlinear). Sketch a graph that

exhibits the qualitative features of a

function that has been

described verbally.

9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

● 9-12.F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. (Algebra I)

● 9-12.F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions (Algebra I and Algebra II)

● 9-12.F.IF.7c- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (Algebra II)

● 9-12.F.IF.7d (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (4th course)

● 9-12.F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude (Algebra I and Algebra II)

Algebra I - Linear, exponential, quadratic, absolute value, step, piecewise defined

Algebra II - Focus on using key features to guide selection of appropriate type of model function.

4th course - Logarithmic and trigonometric functions.

Student Friendly Language:

I can graph functions and identify the specific features (including zeros, maxima, minima, intercepts, and end behavior).

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application,

Extended Thinking)

● Key features of functions can be used to graph them. ● Characteristics of functions include maxima, minima,

increasing, decreasing, end behavior, intercepts, zeros,asymptotes,period, midline, and amplitude.

● Key features of graphs can be used to predict the behavior of functions.

● Different types of functions (linear, quadratic, square root, cube root, piecewise, absolute value, polynomial, exponential, logarithmic, trigonometric) can be used to solve problems.

Leading coefficients and degree determine end behavior.

Zeros of the function can be used to help determine factors.

Factors can be used to find zeros.

Maxima and minima values of a function are used for optimization problems.

Graph functions.

Identify and describe key features and characteristics of graphs.

Use features of equations and graphs to predict the behavior of a function.

Use functions to solve problems.

Key Vocabulary:

maxima minima increasing decreasing linear function quadratic function square root function cube root function piecewise function step function absolute value function zeros of functions end behavior rational functions asymptotes exponential function logarithmic functions trigonometric functions period midline amplitude

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Given a rate, such as cost per minute of a phone plan, graph the function.

Given an interest free loan and constant payments, use a graph to find the amount of time needed to pay off the loan.

Model projectile motion using quadratic functions; use key features of graphs to find maximum height and when the projectile reaches certain heights.

Use functions to solve optimization problems (maximum area/volume, etc).

Calculate compound interest. Apply it to loan or investment situations.

Model tides with trigonometric functions.

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SD Common Core State Standards Disaggregated Math Template

Domain: Trigonometric Functions Cluster: Extend the domain of trigonometric functions using the unit circle

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles 9-12.G.SRT.8 Use Trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

9-12.F.TF.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 x, π+x, and 2π–x in terms of their values for x, where x is any real number.

9-12.F.TF.4 (+) Use the

unit circle to explain

symmetry (odd and

even) and periodicity of

trigonometric functions.

Student Friendly Language:

I can calculate the sine, cosine, or tangent of special angles using a unit circle by turning the central angle into a right triangle and using the Pythagorean theorem to solve for either leg knowing that my hypotenuse is always 1. I can calculate the sine, cosine, or tangent of a 30 or 60 degree angle by remembering that the short leg of that special triangle is always half the hypotenuse and the long leg is the short leg times sqrt(3). I can calculate the sine, cosine, or tangent of a 45 degree angle by remembering that both legs of that special triangle are always the same and they are hypotenuse divided by the sqrt(2).

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application,

Extended Thinking)

● special-case triangles (30-60-90 triangle and 45-45-90 triangle)

● sine ● cosine ● tangent ● unit circle ● quadrants of Cartesian

coordinate plane ● Coordinates (cosine, sine) ● reference angles

Relationships in special-case triangles can be used to find the trigonometric values of angles that are multiples of 30° or 45°. Each coordinate pair in Quadrant I can be reflected across the y-axis,

changing the sign of the cosine and tangent (π- . Each coordinate pair in Quadrant I can be reflected across the origin,

changing the sign of the cosine and sine (π . Each coordinate pair in Quadrant I can be reflected across the x-axis,

changing the sign of the sine and tangent (2π- . The meaning of the acronym ASTC is: All trigonometric values positive in the first quadrant, only Sine positive in the second quadrant, only Tangent positive in the third quadrant, and only Cosine positive in the fourth quadrant.

Create special-case right triangles within a unit circle in order to use trigonometric ratios to solve for unknown sides or angles. Calculate values of sine, cosine, and tangent for angles that are multiples of pi/3, pi/4, and pi/6.

Key Vocabulary:

sine cosine tangent unit circle reference angles

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Lay out plans for a stained-glass window, with the requirements of using the special-case triangles as a part of the design.

Special case triangles can be used in designing bridge supports.

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SD Common Core State Standards Disaggregated Math Template

Domain: Trigonometric Functions Cluster: Extend the domain of trigonometric functions using the unit circle

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

9.12.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.

9-12.F.TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Student Friendly Language:

I can solve for any angle in a unit circle by reflecting the point across the x-axis or y-axis and simply changing the signs. I can find the period of trigonometric functions.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● reference angles

● periodic function ● period ● trigonometric

function ● sine function ● cosine function ● tangent function ● unit circle

Reference angles can be used to find trigonometric values of any angle. One revolution of the unit circle is equivalent to one period for sine and

cosine. The period is 2 radians or 360°.

As one continues with multiple revolutions of the unit circle, coterminal angles share the same sine and cosine values.

Explain how to use reference angles to find trigonometric values of any angle. Explain why coterminal angles have the same sine and cosine values. Distinguish between even functions and odd functions.

Key Vocabulary:

periodic function period reference angle trigonometric function sine cosine tangent unit circle

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Determine several times when a point on a rotating bicycle tire is at a given height. Determine the exact height of a ferris wheel seat as it travels the entire way around during a ride.

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SD Common Core State Standards Disaggregated Math Template

Domain: Trigonometric Functions Cluster: Model periodic phenomena with trigonometric functions

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

9.12.F.BF.4 Find inverse functions. a. 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 for x > 0 or f(x) = (x+1)/(x–1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

9-12.F.TF.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.

9-12.F.TF.7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*

Student Friendly Language:

In order to be invertible a function must pass the horizontal line test. I can limit a trigonometric function to a domain which passes the horizontal line test so that its inverse will then pass the vertical line test necessary for functions. A sine function could be limited from -90 to 90 degrees because the values are always increasing on this interval. A cosine graph could be limited from 0 to 180 degrees because the values are always decreasing on this interval. A tangent function could be limited from -90 to 90 degrees because the values are always increasing on this interval.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application,

Extended Thinking)

● inverse cosine function ● arccosine ● inverse sine function ● arcsine ● inverse tangent function ● arctangent

The domain must be restricted in a way such that every value of the range is unique. The domain of the original function will become the range of the inverted function, and the range of the original function will become the domain of the inverted function.

Limit the domain of a given trigonometric function in order to construct its inverse.

Key Vocabulary:

inverse cosine function arccosine inverse sine function arcsine inverse tangent function arctangent

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Trigonometric functions are used in many real-world applications. The use of inverse functions will enable a student to solve for whatever unknown quantity is in a given situation. Solve for an unknown angle measure when any two sides of a right triangle are known. Example: Find the angle of a ramp.

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SD Common Core State Standards Disaggregated Math Template

Domain: Trigonometric Functions Cluster: Model periodic phenomena with trigonometric functions

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.F.TF.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.

9-12.F.TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.*

Student Friendly Language:

I can model situations using trigonometric equations. I can use inverse functions to solve trigonometric equations with and without technology.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● inverse cosine function

● arccosine

● inverse sine function

● arcsine

● inverse tangent

function

● arctangent

Inverse functions can be used to solve for an unknown value in a trigonometric equation. Trigonometric equations can be used to model certain real-world problems. Technology can be used to evaluate inverse functions and relate the results back to a real-world context.

Utilize inverse trigonometric functions to solve for unknowns in real-world problems. Evaluate results using technology, interpret these results, and relate them back to the real-world context.

Key Vocabulary:

inverse cosine function arccosine inverse sine function arcsine

inverse tangent function arctangent inverted function

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Design and build a skateboard ramp regardless of the unknown quantity through the use of inverse trigonometric functions. They could figure out the angle measure necessary, appropriate ramp length, or height. Students can reconstruct an accident scene using arctan to figure out the speed of the vehicle at the time of the crash.

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SD Common Core State Standards Disaggregated Math Template

Domain: Trigonometric Functions Cluster: Prove and apply trigonometric identities

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

9-12.G.SRT.8 Use trigonometric ratios and the

Pythagorean Theorem to solve right triangles in

applied problems.

9-12.G.SRT.8 Use trigonometric ratios and the

Pythagorean Theorem to solve right triangles in

applied problems.

9-12.F.TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Student Friendly Language:

I can prove the addition and subtraction formulas for sine, cosine, and tangent. I can use the addition and subtraction formulas to solve problems.

Know (Factual)

Understand (Conceptual)

The students will understand

that:

Do (Procedural, Application, Extended Thinking)

● Supplements Theorem ● Complements Theorem ● Half-turn Theorem ● Opposites Theorem ● Sine ● Cosine ● Tangent ● symmetry ● unit circle

The addition and subtraction formulas for sine, cosine, and tangent are the Supplements Theorem, Complements Theorem, Half-turn Theorem, and Opposites Theorem.

Prove the following theorems and use them to solve problems: Supplements Theorem, Complement Theorem, Half-Turn Theorem, Opposites Theorem. Prove the addition and subtraction formulas for sine, cosine, and tangent. Use the addition and subtraction formulas to solve problems.

Key Vocabulary:

Supplements Theorem Complements Theorem Half-turn Theorem Opposites Theorem Sine Cosine Tangent symmetry unit circle

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Given one trigonometric value, the students can find many other trigonometric values using the addition and subtraction formulas for sine, cosine and tangent. They can apply these skills to real-world situations to find necessary information (ie. converting between formulas to find heights based on shadows)

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SD Common Core State Standards Disaggregated Math Template

Domain: Geometric Measurement and Dimension

Cluster: Explain volume formulas and use them to solve problems

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

9-12.G.GMD.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. (Algebra I standard)

9-12.GMD.2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

NA

Student Friendly Language:

I can make an informal argument to develop the volume of a sphere and other solid figures using Cavalieri’s Principle.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● Informal arguments can be used to derive the formula of a sphere.

● Cavalieri's Principle ● Cavalieri's Principle can

also be used to determine and compare volumes of solids.

Two-dimensional relationships are

connected to the properties of three-

dimensional figures.

If two solids have the same height and the same cross-sectional area at every level then the two solids have the same height. There is a connection between the volumes of a cylinder, cone, and sphere.

Informally derive the formula for the volume of a sphere.

Demonstrate Cavalieri’s Principle

concretely. (Ex: Using a deck of cards,

stack of pennies, or stack of CDs).

Use Cavalieri’s Principle to compare

volumes of solids and find volumes of

oblique solids.

Key Vocabulary:

Cavalieri’s Principle informal argument sphere solid cross-section volume

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Understanding areas and volume enables design and building. For example, a container that maximizes volume and minimizes surface area will reduce costs and increase efficiency. Mathematicians use geometry to model the physical world. Studying properties and relationships of geometric objects provides insights into the physical world that would otherwise be hidden. Illustration of Cavalieri’s Principle; Volume of Solids http://www.jimloy.com/cindy/cavalier.htm Illustration of Cavalieri’s Principle: Volume of a Sphere http://www.matematicasvisuales.com/english/html/history/cavalieri/cavalierisphere.html

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SD Common Core State Standards Disaggregated Math Template

Domain: Expressing Geometric Properties with Equations

Cluster: Translate between the geometric description and the equation for a conic section

Grade level:

9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures.

9-12.G.GPE.3 (+) Derive the equations of ellipses and hyperbolas given foci and directrices.

NA

Student Friendly Language:

I can write an equation of an ellipse and a hyperbola using the foci and directrices.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended

Thinking)

● Equation of ellipse ● Equation of hyperbola ● Relationship between foci and

directrices and the equation of an ellipse.

● Relationship between foci and directrices and the equation of a hyperbola.

The only difference between the equation of an ellipse and a hyperbola is that an ellipse is a “+” and a hyperbola is a “-” in the equation. The relationship between the foci and the directrices.

Write an equation of an ellipse using the foci and directrices. Write an equation of a hyperbola using the foci and directrices. Find the foci and directrices of an ellipse given the formula. Find the foci and directrices of a hyperbola given the formula.

Key Vocabulary:

ellipse hyperbola focus (plural foci) directrix (plural directrices or directrices)

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Find the elliptical orbit of planets or comets.

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SD Common Core State Standards Disaggregated Math Template

Domain: The Complex number System

Cluster: Perform arithmetic operations with complex numbers.

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

N.CN.1 Know there is a complex number i such that i

2 = –1, and

every complex number has the form a + bi with a and b real. N.CN.2 Use the relation i

2= –1

and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers

N.CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

N.CN.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.

Student Friendly Language:

I can find the conjugate of a complex number. I can use the conjugate to divide complex numbers and find a moduli for them.

Know (Factual)

Understand (Conceptual)

I want students to understand

that:

Do (Procedural, Application, Extended Thinking)

● Definition of complex numbers

● Complex number graph

● Parts of Complex number

● Complex number operations of addition, subtraction, and multiplication

● Complex number conjugate

● Powers of the number i

A complex number consists of a real and imaginary part. That the conjugate of a complex number is used to simplify or divide complex numbers. Complex numbers are written in a + bi form

Graph complex numbers in the complex coordinate plane Use conjugates to divide and simplify complex numbers Find moduli for complex numbers by using conjugates

Key Vocabulary:

Conjugate Moduli Complex Plane Complex Number Argand Diagram

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

electromagnetics, engineering, chaos theory

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SD Common Core State Standards Disaggregated Math Template

Domain: The Complex Number System

Cluster: Represent complex numbers and their operations on the complex plane.

Grade level:

9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.CN.1 Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. N.CN.2 Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N.CN.3(+) Find the conjugate of a complex number;

use conjugates to find moduli and quotients of

complex numbers.

N.CN.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.

N.CN.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 – √3i) 3= 8

For example, (1 – √3i)3 = 8 because (1 –

√3i) has modulus 2 and argument 120°.

Student Friendly Language:

I can plot complex numbers on a complex plane. I can plot complex numbers on a polar graph. I can explain why the rectangular and polar forms of a given complex number represent the same number.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● Properties of quadratic equation/formula.

● Polar form of a Complex Number

● Relationships between polar and cartesian forms

● Parts to a complex plane ● Parts of a polar plane

A complex number can be graphed using a coordinate axis labeled real axis and imaginary axis. Coordinates on a polar coordinate system represent a distance and an angle. Complex numbers can be written in a +bi form or r cis A form.

Plot complex numbers on a complex cartesian coordinate system. Plot coordinates on a polar coordinate system. Explain the difference between a coordinate on a cartesian plane and a polar plane, and why they represent the same number.

Key Vocabulary:

Complex number Imaginary number Complex plane Argand diagram Cartesian graph Complex conjugate Modulus Discriminant Quadratic formula

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Complex numbers are used to analyze the flow of alternating current in electrical circuits.

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SD Common Core State Standards Disaggregated Math Template

Domain: The Complex Number System

Cluster: Represent complex numbers and their operations on the complex plane.

Grade level:

9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.CN.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. N.VM. 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 (e.g., v, |v|,||v||, v). N.VM. 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.

N.CN.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)

3 = 8 because (1 – i√3)

has modulus 2 and argument 120°.

N.CN.6. (+) Calculate the

distance between

numbers in the complex

plane as the modulus of

the difference, and the

midpoint of a segment as

the average of the

numbers at its endpoints.

Student Friendly Language:

I will be able to perform mathematical operations of addition, subtraction, multiplication, as well as conjugation using complex numbers.

I can use what I know about adding, subtracting, and multiplying vectors on a coordinate plane to help me show adding, subtracting, and multiplying complex numbers on a complex plane.

I will be able to produce a visual of the mathematical operations being performed with complex numbers on a complex plane.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● Parts to a complex plane

● There is a complex number i such that i

2=-1 that can be

written in the form a + bi. ● Complex Conjugates

Graphing complex numbers on a complex plane is similar to vector representation.

Addition, subtraction, multiplication, and conjugation of complex numbers using both algebra and graphing on a complex plane. Relate the properties of adding, subtracting, and multiplying complex numbers on a complex plane to finding resultant vectors on a Cartesian coordinate plane.

Key Vocabulary:

Complex conjugates Complex plane Complex number Resultant Vector Imaginary number

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Complex numbers are used with electricity. In a circuit with alternating current, the voltage, current, and impedance can be represented by complex numbers. Electrical Engineering would be a job where complex numbers would be relevant.

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SD Common Core State Standards Disaggregated Math Template

Domain: The Complex Number System

Cluster: Represent complex numbers and their operations on the complex plane.

Grade level:

9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

N.CN.1, N.CN.2,N.CN.3, N.CN.4

N.VM.1-4

N.CN.6 (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Student Friendly Language:

I can apply the distance formula used on a Cartesian coordinate plane to complex numbers on a complex plane. I can apply the midpoint formula used on a Cartesian coordinate plane to complex numbers on a complex plane

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended

Thinking)

● Parts to a complex plane

● Complex number i such that

i2=-1 that can be written in the

form a + bi.

● Euclidean midpoint and

distance formulas

The distance formula can be applied

to complex numbers.

The midpoint formula can be applied

to complex numbers.

Prove that the distance between two points z and w in the complex plane is |z-w|. Find the distance between two points in the complex plane. Find the midpoint of a line segment between two points in the complex plane.

Key Vocabulary:

Distance formula Midpoint formula Imaginary number Complex number Complex plane Argand diagram Modulus of the difference

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

It is relevant in the engineering field, mainly electrical engineering. Complex numbers can also be used to represent vectors and their operations.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Represent and model with vector quantities.

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.CN.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.

N.VM.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 (e.g., v, |v|, ||v||, v).

N.VM.2. (+) Find the

components of a vector by

subtracting the

coordinates of an initial

point from the coordinates

of a terminal point.

Student Friendly Language:

I can represent vectors graphically. I can use appropriate symbols for vectors and their magnitudes.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● vector ● magnitude ● direction ● direction angle ● directed line segment ● initial point ● terminal point ● scalar

Vectors represent quantities that have both magnitude and direction. A vector can be modeled using a directed line segment. Appropriate symbols for vectors and magnitudes are used to represent the actual vector and vector magnitude

Use vectors to model quantities that have both magnitude and direction, such as velocity and force. Represent vector quantities graphically with directed line segments. Use appropriate symbols for vectors and their magnitudes. Calculate the magnitude and direction of a vector.

Key Vocabulary:

vector magnitude direction direction angle directed line segment initial point terminal point scalar compass bearing or heading

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Model velocities of objects using vectors. Graph vector quantities to establish graphical models that depict the size and direction of the quantities. For example, use a vector to model the velocity of a tractor driving 8 mph at a bearing of 170º.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.VM.8 (+) Add, subtract and multiply matrices of appropriate dimensions. N.VM.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.

N.VM.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.

N.VM.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

Student Friendly Language:

I can find the determinant of a square matrix. I can determine if a square matrix has an inverse by identifying its determinant. I can explain how the zero matrix in matrix addition is similar to 0 when adding real numbers. I can explain how the identity matrix in matrix multiplication is similar to 1 when multiplying real numbers.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● square matrix ● determinant ● zero matrix ● identity matrix ● multiplicative inverse

The adding of any square matrix, A, with the zero matrix will result in matrix A. Multiplying any square matrix, A, by the identity matrix will result in matrix A. If the determinant of a square matrix is non-zero, then the matrix will have a multiplicative inverse.

Find the determinant of a square matrix and determine if the matrix has a multiplicative inverse. Explain the similarity between the zero matrix and the real number zero. Explain the similarity between the identity matrix and the real number 1.

Key Vocabulary:

determinant matrix identity matrix square matrix zero matrix multiplicative inverse

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Students may be given a set of real world data that could be represented by a system of equations. They would be asked to identify the coefficient matrix and its determinant so they can explain if there is a solution to the system of equations.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

N.VM.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.

N.VM.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.

N.VM.12 (+) Work with 2

× 2 matrices as a

transformations of the

plane, and interpret the

absolute value of the

determinant in terms of

area.

Student Friendly Language:

I can multiply a vector by a matrix. I can determine if I can multiply a vector by a matrix based on the dimensions of the matrix. I can use a matrix to transform a vector.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● matrix ● dimensions ● transformations

A vector can only be multiplied by a matrix if the inner dimensions match. A vector can be transformed by multiplying it by a matrix.

Multiply a vector by a matrix to produce a new vector. Describe the transformation of a vector after it has been multiplied by a matrix.

Key Vocabulary:

vector matrix dimensions transformations

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.VM.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.

N.VM.12 (+) Work with 2 X 2 matrices as a transformation of the plane, and interpret the absolute value of the determinant in terms of area.

NA

Student Friendly Language:

I can use a 2 X 2 matrix to transform a polygon in the coordinate plane. I can find the absolute value of the determinant of a 2 X 2 matrix and interpret in terms of area in the coordinate plane

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● Coordinates of figures in the coordinate plane can be put into a 2 x 2 matrix.

● The absolute value of the determinant can be used to determine the area of a figure.

A 2 x 2 matrix can be used to transform figures in the coordinate plane. The absolute value of the determinant of a 2 x 2 matrix used to transform a figure in the plane can be used as a scalar multiple of the area of the image of the figure

Transform figures in the coordinate plane using 2 x 2 matrices. Use the absolute value of the determinant of a 2 x 2 matrix to determine the area of a figure in the plane transformed by the 2 x 2 matrix.

Key Vocabulary:

matrix dimensions absolute value transformations determinant area coordinate plane

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

A company is designing a very large rectangular logo for another company. They then learn later that the logo must also be made into a very large banner. The company producing the logo and banner must use a matrix to transform the logo into the size of the banner and determine the area of the banner so they can minimize the amount of material needed to produce the banner.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Represent and model with vector quantities.

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.VM.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

(e.g., v, |v|, ||v||, v)

N.VM.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N.VM.3. (+) Solve

problems involving

velocity and other

quantities that can be

represented by vectors.

Student Friendly Language:

I can find the components of a vector.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended

Thinking)

● components of a vector ● initial point ● terminal point

The components of a vector are found by subtracting the coordinates of the initial point from the coordinates of the terminal point.

Find the components of a vector when given the coordinates of the initial and terminal points. Use components of vectors to solve problems. For example, find the vertical and horizontal components of a vector describing the initial velocity of a projectile.

Key Vocabulary:

vector components of a vector horizontal component vertical component initial point component form of a vector terminal point unit vector standard unit vector

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

A projectile is launched having an initial velocity of 100 meters per second at an angle of elevation of 40º. Find the vertical and horizontal components of the initial velocity. These will be used to find the maximum height of the projectile, the amount of time it spends in the air, and how far the projectile travels. A 80 pound box is resting on a ramp having a 20º incline. Find the component forces applied perpendicular to the ramp and parallel to the ramp.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Represent and model with vector quantities.

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

N.VM.2. (+) Find the components

of a vector by subtracting the

coordinates of an initial point from

the coordinates of a terminal

point.

N.VM.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

N.VM.4 (+) Add and subtract vectors.

Student Friendly Language:

I can solve problems involving quantities that can be represented by vectors.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● vector ● magnitude ● direction ● direction angle ● bearing ● displacement vector

Vectors are used to solve problems involving quantities, such as velocity, that can be represented by vectors.

Interpret information to describe vectors used to model problems. Use vectors to solve problems involving vector quantities.

Key Vocabulary:

vector magnitude direction direction angle bearing angle of elevation angle of depression resultant horizontal component vertical component displacement vector

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

A baseball is driven off of a bat 3 feet above the ground at an initial velocity of 90 mph at an angle of elevation of 30º. Find the vertical and horizontal components of the initial velocity. How far from home plate will the ball hit the ground? A 10,000 pound truck is parked on a 15º incline. What amount of force is required to keep the truck from rolling or sliding down the incline?

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities Cluster: Perform operations on vectors Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

G.SRT.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).

N.VM.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.

N.VM.5(+) Multiply a scalar by a vector.

Student Friendly Language:

I can add and subtract vectors. I can determine the magnitude and direction of the sum or difference of two vectors.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● vector ● vector addition ● vector subtraction ● additive inverse ● parallelogram rule ● magnitude ● direction ● components ● resultant ● displacement vector

Vectors can be added end-to-end, component-wise, and by the parallelogram rule.

Vector addition is used to find the magnitude and direction of the resultant; they should understand that the magnitude of the resultant is typically not the same as the sum of the magnitudes.

The additive inverse can be used for vector subtraction.

Add vectors end-to-end, component-wise, and by the parallelogram rule.

Calculate the magnitude and direction of the sum of two vectors.

Subtract vectors using an additive inverse.

Key Vocabulary:

vector vector addition vector subtraction additive inverse parallelogram rule magnitude direction components resultant displacement vector bearing or heading

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

A hiker first walks 2 miles east, then 3 miles north, and finally 1 mile NW. How far is the hiker from the starting point? If the hiker wants to walk in a straight line from the current location to the starting point, what direction must the hiker travel? Given sufficient information about the forces acting on an object, use vector addition to find the sum of the vectors. Describe the magnitude and direction of the resultant derived from combining the forces.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities Cluster: Perform operations on vectors Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

N.VM.4. (+) Add and subtract vectors.

N.VM.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(v x , v y ) = (cv x , cv y ). 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).

Student Friendly Language:

I can multiply a vector by a scalar. I can represent scalar multiplication graphically. I can compute the magnitude and direction of scalar multiples.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● vector ● scalar ● scalar multiplication ● magnitude ● direction ● components of a

vector

When a vector, v, is multiplied by a scalar, c, the magnitude of cv is |c| times as large as the magnitude of v. If a scalar c is positive cv has the same direction as v, but if c is negative cv has the opposite direction of v.

Multiply a vector by a scalar. Represent scalar multiplication graphically. Solve problems involving scalar multiplication.

Key Vocabulary:

vector scalar scalar multiplication magnitude direction components of a vector

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Each dog in a team of six sled dogs exerts a 80N force on the sled. If two dogs are unharnessed and the other four dogs continue exerting the same force, what is the total magnitude of the force exerted by the team on the sled? A large 50-pound container is placed on a 30° incline and is secured by a rope parallel to the plane. If the rope has a 400-pound breaking strength and the coefficient of static friction between the container and the ramp is 0.5, calculate the amount of weight that can be placed in the container.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

8.F.4. Construct a function to model a linear relationship between

two quantities. Determine the rate of change and initial value of the

function from a description of a relationship or from two (x, y) values,

including reading these from a table or from a graph. Interpret the

rate of change and initial value of a linear function in terms of the

situation it models, and in terms of its graph or a table of values (x,

y) values, including reading these from a table or from a graph.

Interpret the rate of change and initial value of a linear function in

terms of the situationit models, and in terms of its graph or a table of

values

N.VM.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

N.VM.7 (+) Multiply

matrices by scalars to

produce new matrices,

e.g., as when all of the

payoffs in a game are

doubled.

Student Friendly Language:

I can use matrices to organize data and use addition, subtraction, and multiplication to answer questions about the data.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended Thinking)

● Matrix dimensions ● Matrix addition and

subtraction ● Scalar Multiplication of a

matrix ● Matrix multiplication ● Determinant of a matrix ● Inverse of a matrix

Data can be organized in a matrix. Matrix dimensions are defined as rows by columns. Addition, subtraction and multiplication can be performed on matrices. Apply matrices to problem situations to determine answers to questions.

Use matrices to organize data. Use matrices to manipulate data. Apply matrices to real world problems to interpret the data.

Key Vocabulary:

Matrix Dimension Scalar element member inverse determinant matrix multiplication matrix addition matrix subtraction array

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

A school needs to purchase bats, balls, and helmets for the baseball and softball team. Use matrices to organize two matrices and use matrix multiplication to determine the total cost of the equipment

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.VM.6 (+) Use matrices to represent

and manipulate data, e.g., to

represent payoffs or incidence

relationships in a network

9-12.N.VM.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

N.VM.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

Student Friendly Language:

I can multiply a matrix by a real number, called a scalar, to produce a new matrix.

Know (Factual)

Understand (Conceptual)

I want students to understand that:

Do (Procedural, Application, Extended

Thinking)

● Matrix dimensions ● Multiply real numbers ● Scalar ● Matrix addition ● Matrix subtraction

A scalar is different than a matrix. A matrix can be multiplied by a scalar. The new matrix produced by scalar multiplication represents the result of the scalar multiplication.

Multiply matrices by scalars. Interpret the meaning of the new matrix obtained by scalar multiplication.

Key Vocabulary:

scalar matrix multiplication real numbers matrix dimensions

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Use scalar multiplication of a matrix to determine the amount of money in various accounts when interest is compounded into an account. Use scalar multiplication of a matrix to show the change in cost of items based on inflation.

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

N.VM.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

N.VM.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

N.VM.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

N.VM.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.

Student Friendly Language:

I can represent and manipulate data using matrices.

I can add matrices and understand that the associative, commutative, and distributive properties apply.

I can subtract matrices and understand that the distributive property applies.

I can multiply matrices and understand that the associative and distributive properties apply.

I can analyze two matrices to know whether their dimensions allow them to be added, subtracted, or multiplied.

I can use matrices to solve problems and interpret the results.

Know (Factual)

Understand (Conceptual)

I want students to understand:

Do (Procedural, Application, Extended

Thinking)

A matrix is an ordered array of d or subtract the corresponding elements of each matrix to get the new matrix sum or difference. Matrix multiplication requires that the number of columns in the first matrix match the number of rows in the second matrix. The product of two matrices with dimensions m x n and n x k is a matrix with dimensions m x k.numbers in rows and columns

● The dimensions of a matrix are determined by its number of rows and columns

● A matrix is named by a capital letter

● associative property

● commutative property

● distributive property

● scalar

● spreadsheet

The associative, commutative, and distributive properties apply to matrix addition.

The associative and distributive properties apply to matrix subtraction, but the commutative property does NOT apply.

Matrices can only be added or subtracted if the dimensions of each matrix are exactly the same.

Add matrices.

Subtract matrices.

Multiply matrices.

Use technology (graphing calculators, spreadsheets, etc.) to create and perform operations on matrices.

Use matrices to solve application problems and interpret the results in the context of the problem.

Key Vocabulary:

matrix scalar commutative property associative property dimensions row column distributive property

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Matrices can help manipulate growing information. Computational technology and computer programs can store and organize a large amount of information, often using matrix notation. Spreadsheets (matrices) can be used by businesses in areas such as budgeting, sales projections and cost estimation. Scientists can use spreadsheets to analyze the results of an experiment. Teachers can use spreadsheets to record and average grades.

Matrices are connected to linear transformations and to the process of solving matrix equations that provides an alternative method of solving linear systems with the same number of variables as equations.

Matrices can be used to solve systems of equations.

Matrix operations may be used in cryptography (example: data encryption).

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SD Common Core State Standards Disaggregated Math Template

Domain: Vector and Matrix Quantities

Cluster: Perform operations on matrices and use matrices in applications

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

N.VM.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

N.VM.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.

N.VM.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.

Student Friendly Language:

I can accurately compute matrix multiplication problems that require the use of the associative and distributive property. I understand that I can not apply the commutative property to matrix multiplication problems. I can explain why matrix multiplication is not commutative, but is associative and distributive.

Know (Factual)

Understand (Conceptual)

I want students to understand

that:

Do (Procedural, Application, Extended

Thinking)

● Matrices may be multiplied using the associative and distributive property.

● Matrices can not be multiplied using the commutative property.

The associative and distributive

properties apply to matrix

multiplication, but the

commutative property does NOT

apply.

Multiply matrices using the associative and distributive properties. Prove that given matrices A and B, AB is not equal to BA.

Key Vocabulary:

matrix, associative property commutative property distributive property

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Use cryptography Given the cost of items for a softball team, use matrices to find the total cost of equipment for both men’s and women’s teams.

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SD Common Core State Standards Disaggregated Math Template

Domain: Using Probability to Make Decisions

Cluster: Calculate expected values and use them to solve problems

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

9-12.S.MD.1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

NA

Student Friendly Language:

I can look at the events in a sample space and identify the random variable I can assign numerical values to each event in the sample space further defining the random variable I can determine the probability distribution of the sample space I can graphically display probability distributions using methods such as frequency distributions, grouped frequency distributions, histograms, frequency polygons, stem-and-leaf plots, bar charts, and normal probability distributions.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● Random variable

● Probability distributions

● Graphical displays of probability distributions

The random outcomes of an experiment of chance are called random variables.

Each value of the random variable can be associated with a probability to indicate the chance of that potential outcome

A probability distribution shows the probabilities associated with each event in the sample space.

Some possible ways to graphically display these distributions are frequency distributions, grouped frequency distributions, histograms, frequency polygons, stem-and-leaf plots, bar charts, and normal probability distributions.

Identify the numerical values of events in a sample space that define the random variable.

Determine a the probabilities associated with the events in the sample space (probability distribution).

Graphically display the probability distributions.

Key Vocabulary:

event sample space probability frequency distribution grouped frequency distribution histogram frequency polygon stem-and-leaf plot bar chart normal probability distribution

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Some ideas of random variables: ● the number of heads when tossing two coins (zero, one or two) is a random variable. ● the grade level (9, 10, 11, 12) of students on the football team. ● the number of Monday absences (0, 1, 2,...) of a group of employees, ● the number of defective lightbulbs produced per hour at a manufacturing company.

Use of graphical probability distributions can be helpful to visualize the probabilities of the random variables.

Ot her Information and Resources: http://www.khanacademy.org/math/statistics/v/introduction-to-random-variables Watch this video to better understand this standard.

I have also found that introductory college statistics books do a good job explaining the basics of these ideas. Of course, these books go much further than is necessary in a high school math course.

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SD Common Core State Standards Disaggregated Math Template

Domain: Using Probability to Make Decisions

Cluster: Calculate expected values and use them to solve problems

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.S.MD(+). Analyze decisions and strategies using probability concepts.

9-12.S.MD.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

NA

Student Friendly Language:

I can calculate the expected value of a random variable by finding the mean of the probability distribution. I can understand that the expected value is the same as the mean of the probability distribution.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application,

Extended Thinking)

● Expected value ● Mean of the

probability distribution

The expected value of a random variable is the mean of the probability distribution. The mean of the probability distribution is a weighted average. The mean represents the central location of the probability distribution. The mean is the also the long-run average of the random variable.

Calculate the expected value of the random variable by finding the mean of the probability distribution. Find the mean of the probability distribution by using weighted averages.

Key Vocabulary:

mean probability distribution random variable weighted average

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Using the expected value can help any business determine their long-run average which can help in planning expenses, projecting sales, etc. Specifically, if a car salesman determines an expected value of 2.1 for Saturday sales using data from past Saturday sales, he can determine how many cars he will sell over the 50 Saturdays he may work in the course of a year. Other Information and Resources: Can watch this video to better understand the standard. http://www.khanacademy.org/math/probability/v/expected-value--e-x I have also found that introductory college statistics books do a good job explaining the basic concept of an expected value. Of course, these books go into much further detail than is probably necessary in a high school math course.

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SD Common Core State Standards Disaggregated Math Template

Domain: Using Probability to Make Decisions

Cluster: Calculate expected values and use them to solve problems

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in Following Year

9-12.S.MD.1(+)Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions 9-12.S.MD.2(+)Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

9-12.S.MD.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

NA

Student Friendly Language:

I can create a probability distribution of theoretical probabilities of an experiment. I can use a probability distribution model to find the probability and expected values of a specific outcome.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● probability distribution ● probability histogram and

other models ● expected value

A probability distribution can be created in various forms using theoretical probabilities. Expected values of an event can be found using multiplication and distribution probabilities.

Make a table of values to show how the probability of an experiment are distributed. Make a histogram to describe the theoretical probabilities an experiment. Find the expected value of an occurrence using the distribution probabilities.

Key Vocabulary:

random variable sample space theoretical probability probability distribution probability histogram expected value

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Expected values of survival of patients infected with a certain virus using the theoretical probabilities of success and failure. Expected values of success on a test of M/C or T/F questions when a student merely guesses. Expected values can be used for quality control for assembly lines and manufacturing.

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SD Common Core State Standards Disaggregated Math Template

Domain: Using Probability to Make Decisions

Cluster: Calculate expected values and use them to solve problems

Grade level: 9-12

Correlating Standard in Previous Year Number Sequence & Standard Correlating Standard in

Following Year

9-12.S.MD.1(+)Define a random variable for a quantity of interest by

assigning a numerical value to each event in a sample space; graph the

corresponding probability distribution using the same graphical displays as

for data distributions

9-12S.MD.2(+)Calculate the expected value of a random variable; interpret

it as the mean of the probability distribution.

9-12.S.MD.3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.

9-12.S.SMD.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

N/A

Student Friendly Language:

I can create a probability distribution of empirically assigned probabilities of an experiment.

I can use a probability distribution model to find the probability and expected values of a specific outcome.

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended Thinking)

● Probability distribution for empirically assigned probabilities of an experiment can be created.

● Theoretical probabilities can be described with a histogram.

● A probability distribution model can be used to find the probability and expected values of a specific outcome.

A probability distribution can be created in various forms using empirically assigned probabilities.

Expected values of an event can be found using multiplication and distribution probabilities.

Make a table of values to show how the

probabilities of an experiment are distributed.

Make a histogram to describe the theoretical

probabilities an experiment.

Find the expected value of an occurrence using

the distribution probabilities

Key Vocabulary:

probability distribution random variable sample space expected value empirically assigned probabilities theoretical probability

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context?

Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Expected values of survival of patients infected with a certain virus using the empirically assigned probabilities Expected values of success on a test of M/C or T/F questions when a student merely guesses. Expected values can be used for quality control for assembly lines and manufacturing. Possible solution to example given in standard: a random sample of 150 households needs to be taken and the number of children per household determined. Suppose that the values are C=(0,1,2,3,4,5,6,7,8} with frequencies f={50,35,70,27,7,2,4,0,5}. The empirical assignment is the relative frequencies given by f/150 for each value. The expected value is the weighted average of the values with their probabilities. Expected value = (0*50+1*35+....+7*0+8*5)/150

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SD Common Core State Standards Disaggregated Math Template

Domain: Using Probability to Make Decisions

Cluster: Use Probability to evaluate outcomes of decisions

Grade level: 9-12

Correlating Standard in Previous Year

Number Sequence & Standard Correlating Standard in Following Year

9-12.SMD.4(+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?

9-12.SMD.5(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fastfood restaurant. b. Evaluate and compare strategies on the basis of expected values.For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

NA

Student Friendly Language:

I can determine the chances of winning a game or lottery by examining the probabilities of success .

Know (Factual)

Understand (Conceptual)

The students will understand that:

Do (Procedural, Application, Extended

Thinking)

● Strategies to compare chances of success.

● Expected payoffs.

Odds can be used to determine success in gaming. A favorable outcome probability can be used to rate chance of winning.

Assign probabilities to outcomes of a game or lottery and compare the chance of success to loss. Determine payoff of an event.

Key Vocabulary:

probability odds chance favorable outcomes theoretical probability actual (observed or experimental) probability

Relevance and Applications: How might the grade level expectation be applied at home, on the job or in a real-world, relevant context? Include at least one example stem for the conversation with students to answer the question “why do I have to learn this”?

Use cost of lottery tickets and chance of winning to determine if playing the lottery is sensible.