SD Common Core State Standards Disaggregated Math Template

Transcription

1 main: Reasoning with Equations and Inequalities Solve systems of equation Correlating Standard in Previous Year 9-12.A.REI.8(+) Represent a system of linear equations as a single matrix equation in a vector variable. Correlating Standard in I can represent a system of linear equations as a single matrix equation. The Students will understand that: (Procedural, Application, Extended Thinking) Matrix dimension The variable matrix is called a vector because it has one column. Rewrite a system of linear equations as a single matrix equation. Systems of equations can be represented using matrices. Vector variable Matrix dimensions Rows Columns Coefficient Introduction to transformations and animation. Introduction to solving systems using inverse operations.

2 main: Reasoning with Equations and Inequalities Solve systems of equation Correlating Standard in Previous 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). Correlating Standard in 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). matrix equation inverse matrix The students will understand that: 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. (Procedural, Application, Extended Thinking) 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. matrix equation inverse matrix unique solution 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?

3 main: Building Functions Build new functions from existing functions Correlating Standard in Previous Year F.BF 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 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. Correlating Standard in NA 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. The students will understand that: (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. inverse inverse relation inverse function domain range vertical line test one-to-one functions relevant context? Include at least one example stem for the conversation with students to answer the question why do 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.

4 main: Building Functions Build new functions from existing functions Correlating Standard in Previous Year F.BF , 3, 4a (+)9-12.F.BF.5 (+) the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Correlating Standard in I will be to find inverses for logarithmic and exponential forms. I will be able to solve problems involving exponents and logarithms. Logarithmic and exponential functions can be solved using inverses. There are real-world problems related to logarithms, exponents, and their inverses. The students will understand that: How to find inverses of functions. How to determine if two functions are inverses. How to verify that exponential and logarithmic functions are inverses. (Procedural, Application, Extended Thinking) Solve logarithmic functions using exponentials as the inverse. Solve exponential functions using logarithms as the inverse. Solve problems using logarithms and exponents. function logarithm exponential inverse banking; calculating values for compounding continuous interest

5 main: Functions Analyze functions using different representations Correlating Standard in Previous 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 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 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. Correlating Standard in I can graph functions and identify the specific features (including zeros, maxima, minima, intercepts, and end behavior). 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. The students will understand that: 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. (Procedural, Application, Extended Thinking) 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. 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 relevant context? Include at least one example stem for the conversation with students to answer the question why do 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.

6 main: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle Correlating Standard in Previous 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 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. Correlating Standard in 9-12.F.TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 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). special-case triangles ( triangle and triangle) sine cosine tangent unit circle quadrants of Cartesian coordinate plane Coordinates (cosine, sine) reference angles The students will understand that: 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. (Procedural, Application, Extended Thinking) 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. sine cosine tangent unit circle reference angles relevant context? Include at least one example stem for the conversation with students to answer the question why do 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.

7 main: Trigonometric Functions Extend the domain of trigonometric functions using the unit circle Correlating Standard in Previous Year Correlating Standard in 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 F.TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 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. reference angles periodic function period trigonometric function sine function cosine function tangent function unit circle The students will understand that: 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. (Procedural, Application, Extended Thinking) 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. periodic function period reference angle trigonometric function sine cosine tangent unit circle 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.

8 main: Trigonometric Functions Model periodic phenomena with trigonometric functions Correlating Standard in Previous Year Correlating Standard in 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 noninvertible function by restricting the domain F.TF.6 (+) that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed 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.* 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. inverse cosine function arccosine inverse sine function arcsine inverse tangent function arctangent The students will understand that: 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. (Procedural, Application, Extended Thinking) Limit the domain of a given trigonometric function in order to construct its inverse. inverse cosine function arccosine inverse sine function arcsine inverse tangent function arctangent 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.

9 main: Trigonometric Functions Model periodic phenomena with trigonometric functions Correlating Standard in Previous Year 9-12.F.TF.6.(+) that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed 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.* Correlating Standard in I can model situations using trigonometric equations. I can use inverse functions to solve trigonometric equations with and without technology. inverse cosine function arccosine inverse sine function arcsine inverse tangent function arctangent I want students to understand that: 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. (Procedural, Application, Extended Thinking) 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 realworld context. inverse cosine function arccosine inverse sine function arcsine inverse tangent function arctangent inverted function 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.

10 main: Trigonometric Functions Prove and apply trigonometric identities Correlating Standard in Previous Year Correlating Standard in 9-12.G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems F.TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. I can prove the addition and subtraction formulas for sine, cosine, and tangent. I can use the addition and subtraction formulas to solve problems. The students will understand that: (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. Supplements Theorem Complements Theorem Half-turn Theorem Opposites Theorem Sine Cosine Tangent symmetry unit circle 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)

11 main: Geometric Measurement and Dimension Explain volume formulas and use them to solve problems Correlating Standard in Previous Year Correlating Standard in 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 I can make an informal argument to develop the volume of a sphere and other solid figures using Cavalieri s Principle. 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. The students will understand that: Two-dimensional relationships are connected to the properties of threedimensional 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. (Procedural, Application, Extended Thinking) 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. Cavalieri s Principle informal argument sphere solid cross-section volume relevant context? Include at least one example stem for the conversation with students to answer the question why do ing 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 Illustration of Cavalieri s Principle: Volume of a Sphere

12 main: Expressing Geometric Properties with Equations Translate between the geometric description and the equation for a conic section Grade level: 9-12 Correlating Standard in Previous Year Correlating Standard in 7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures G.GPE.3 (+) Derive the equations of ellipses and hyperbolas given foci and directrices. NA I can write an equation of an ellipse and a hyperbola using the foci and directrices. 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 students will understand that: 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. (Procedural, Application, Extended Thinking) 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. ellipse hyperbola focus (plural foci) directrix (plural directrices or directrices) Find the elliptical orbit of planets or comets.

13 main: The Complex number System Perform arithmetic operations with complex numbers. Correlating Standard in Previous Year N.CN.1 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. Correlating Standard in 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. I can find the conjugate of a complex number. I can use the conjugate to divide complex numbers and find a moduli for them. I want students to understand that: (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 Conjugate Moduli Complex Plane Complex Number Argand Diagram electromagnetics, engineering, chaos theory

14 main: The Complex Number System Represent complex numbers and their operations on the complex plane. Grade level: 9-12 Correlating Standard in Previous Year Correlating Standard in N.CN.1 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. 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. 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 I want students to understand that: 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. (Procedural, Application, Extended Thinking) 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. Complex number Imaginary number Complex plane Argand diagram Cartesian graph Complex conjugate Modulus Discriminant Quadratic formula Complex numbers are used to analyze the flow of alternating current in electrical circuits.

15 main: The Complex Number System Represent complex numbers and their operations on the complex plane. Grade level: 9-12 Correlating Standard in Previous Year Correlating Standard in 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. 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. 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. 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. 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 I want students to understand that: Graphing complex numbers on a complex plane is similar to vector representation. (Procedural, Application, Extended Thinking) 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. Complex conjugates Complex plane Complex number Resultant Vector Imaginary number relevant context? Include at least one example stem for the conversation with students to answer the question why do 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.

16 main: The Complex Number System Represent complex numbers and their operations on the complex plane. Grade level: 9-12 Correlating Standard in Previous 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. Correlating Standard in 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 Parts to a complex plane Complex number i such that i 2 =-1 that can be written in the form a + bi. Euclidean midpoint and distance formulas I want students to understand that: The distance formula can be applied to complex numbers. The midpoint formula can be applied to complex numbers. (Procedural, Application, Extended Thinking) 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. Distance formula Midpoint formula Imaginary number Complex number Complex plane Argand diagram Modulus of the difference It is relevant in the engineering field, mainly electrical engineering. Complex numbers can also be used to represent vectors and their operations.

17 main: Vector and Matrix Quantities Represent and model with vector quantities. Correlating Standard in Previous Year Correlating Standard in 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. I can represent vectors graphically. I can use appropriate symbols for vectors and their magnitudes. vector magnitude direction direction angle directed line segment initial point terminal point scalar I want students to understand that: 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 (Procedural, Application, Extended Thinking) 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. vector magnitude direction direction angle directed line segment initial point terminal point scalar compass bearing or heading 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.

18 main: Vector and Matrix Quantities Perform operations on matrices and use matrices in applications Correlating Standard in Previous Year Correlating Standard in N.VM.8 (+) Add, subtract and multiply matrices of appropriate dimensions. N.VM.9 (+) 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 (+) 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 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. square matrix determinant zero matrix identity matrix multiplicative inverse I want students to understand that: 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 nonzero, then the matrix will have a multiplicative inverse. (Procedural, Application, Extended Thinking) 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. determinant matrix identity matrix square matrix zero matrix multiplicative inverse 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.

19 main: Vector and Matrix Quantities Perform operations on matrices and use matrices in applications Correlating Standard in Previous Year N.VM.10 (+) 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. Correlating Standard in 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. 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. matrix dimensions transformations I want students to understand that: 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. (Procedural, Application, Extended Thinking) 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. vector matrix dimensions transformations

20 main: Vector and Matrix Quantities Perform operations on matrices and use matrices in applications Correlating Standard in Previous Year Correlating Standard in N.VM.10 (+) 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 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 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. I want students to understand that: 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 (Procedural, Application, Extended Thinking) 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. matrix dimensions absolute value transformations determinant area coordinate plane 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.

21 main: Vector and Matrix Quantities Represent and model with vector quantities. Correlating Standard in Previous Year Correlating Standard in 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. I can find the components of a vector. I want students to understand that: (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. vector components of a vector horizontal component vertical component initial point component form of a vector terminal point unit vector standard unit vector 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.

22 main: Vector and Matrix Quantities Represent and model with vector quantities. Correlating Standard in Previous 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. Correlating Standard in N.VM.4 (+) Add and subtract vectors. I can solve problems involving quantities that can be represented by vectors. I want students to understand that: (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. vector magnitude direction direction angle bearing angle of elevation angle of depression resultant horizontal component vertical component displacement vector 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?

23 main: Vector and Matrix Quantities Perform operations on vectors Correlating Standard in Previous Year G.SRT.11 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. 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. 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. Correlating Standard in N.VM.5(+) Multiply a scalar by a vector. I can add and subtract vectors. I can determine the magnitude and direction of the sum or difference of two vectors. vector vector addition vector subtraction additive inverse parallelogram rule magnitude direction components resultant displacement vector I want students to understand that: Vectors can be added end-to-end, componentwise, 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. (Procedural, Application, Extended Thinking) 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. vector vector addition vector subtraction additive inverse parallelogram rule magnitude direction components resultant displacement vector bearing or heading 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.

24 main: Vector and Matrix Quantities Perform operations on vectors Correlating Standard in Previous 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). Correlating Standard in I can multiply a vector by a scalar. I can represent scalar multiplication graphically. I can compute the magnitude and direction of scalar multiples. I want students to understand that: (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. vector scalar scalar multiplication magnitude direction components of a vector 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.

25 main: Vector and Matrix Quantities Perform operations on matrices and use matrices in applications Correlating Standard in Previous Year Correlating Standard in 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. I can use matrices to organize data and use addition, subtraction, and multiplication to answer questions about the data. Matrix dimensions Matrix addition and subtraction Scalar Multiplication of a matrix Matrix multiplication Determinant of a matrix Inverse of a matrix I want students to understand that: 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. (Procedural, Application, Extended Thinking) Use matrices to organize data. Use matrices to manipulate data. Apply matrices to real world problems to interpret the data. Matrix Dimension Scalar element member inverse determinant matrix multiplication matrix addition matrix subtraction array 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