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Canyons School District Secondary II

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CANYONS SCHOOL DISTRICT

SECONDARY II & II H SCOPE AND SEQUENCE

2014 – 2015


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Secondary II Unit 1: Extending the Number System

Regular: 5 weeks Honors: 6 weeks

Honors Advanced: 3 weeks A.APR.1 Understand that polynomials form a system analogous to the integers—namely, they are closed under the

operations of addition subtraction, and multiplication; add, subtract, and multiply polynomials. Concepts and Skills to Master: • I can add and subtract polynomials. • I can multiply polynomials using the distributive property, and

then simplify. • I can understand closure of polynomials for addition,

subtraction, and multiplication.

Sample Task (DOK 1) Multiply 𝑥! + 3𝑥 − 5 𝑥 + 4 and determine if the result is a polynomial Sample Task (DOK 3) Jane owns three rectangular pieces of land, as shown in the diagram. Find two representations for the area of the land.

Curriculum Supports Walch Unit 1 Lesson 1: Adding and Subtracting Polynomials Walch Unit 1 Lesson 1: Multiplying Polynomials


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N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. (For

example, we define 5!! to be the cube root of 5 because we want 5

!!!= 5

!! ∙! to hold, so 5

!!!must equal 5).

Concepts and Skills to Master: • I can define the meaning of a rational exponent.

Sample Task (DOK 1) Compute 25! ∙ 5!

Curriculum Supports Walch Unit 1 Lesson 2: Defining, Rewriting, and Evaluating Rational Exponents N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Concepts and Skills to Master: • I can convert radical notation to rational exponent notation,

and vice-‐versa. • I can extend the properties of integer exponents to rational

exponents and use them to simplify expressions.

Sample Task (DOK 2) What is 𝑥! ? Sample Task (DOK 2) What is the area of a rectangle with a length 7 and a width 7! ? Sample Task (DOK 3)

What is 𝑥!!!

Curriculum Supports Walch Unit 1 Lesson 2: Defining, Rewriting, and Evaluating Rational Exponents Walch Unit 1 Lesson 2: Rational and Irrational Numbers and Their Properties


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Simplifying Radicals (HONORS)

Write radical expressions in equivalent forms.

Concepts and Skills to Master: • I can write radical expressions in equivalent forms using

radical symbols. • I can perform operations on radical expressions.

Sample Task (DOK 1) Multiply 2 3 5+ 6 Sample Task (DOK 1) Write 32 in an equivalent form Sample Task (DOK 3) Are each of the following always true, sometimes true, or never true? Justify your answers. 𝑥

!= 𝑥!

𝑥 − 𝑦 = 𝑥 − 𝑦 1𝑥=

𝑥𝑥

2 𝑥 = 2𝑥 Curriculum Supports: Walch Unit 1 Lesson 2: HONORS: Writing Radical Expressions in Equivalent Forms


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N.RN.3 Explain why sums and products of rational numbers are rational, why the sum of a rational number and an irrational number is irrational, and why the product of a nonzero rational number and an irrational number is irrational.

Concepts and Skills to Master: • I can simplify radical expressions. • I can add, subtract, and multiply real numbers. • I can explain why adding and multiplying two rational

numbers results in a rational number. • I can explain why adding a rational number to an irrational

number results in an irrational number. • I can explain why multiplying a nonzero number to an

irrational number results in an irrational number.

Sample Task (DOK 1) Simplify 3 2+ 6 Sample Task (DOK 2) What type of number is the product of 3 and 3? Sample Task (DOK 2) Given a right triangle whose hypotenuse is irrational, find measure for legs where:

• Both legs are rational • Both legs are irrational • One leg is irrational and one leg is rational

Curriculum Supports: Walch Unit 1 Lesson 2: Rational and Irrational Numbers and Their Properties


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N.CN.1 Know that there is a complex number i such that 12 −=i , and every complex number has the form a+bi with a and b real.

Concepts and Skills to Master: • I can understand that the set of complex numbers includes the

set of all real numbers and the set of imaginary numbers. • I can express numbers in the form a+bi.

Sample Task (DOK 1) Write −25+ 9 as a complex number in the form of 𝑎 + 𝑏𝑖 Sample Task (DOK 3) Given 𝑝𝑥! + 𝑞 = 0 find values of p and q that result in:

• A real number solution. • An imaginary number solution.

Generalize the relationship between p and q that would result in each type of solution.

Curriculum Supports: Walch Unit 1 Lesson 3: Defining Complex Numbers, i, and i2 N.CN.2 Use the relation 12 −=i and the commutative, associative, and distributive properties to add, subtract, and

multiply complex numbers. Concepts and Skills to Master:

• I can add, subtract, and multiply complex numbers. Sample Task (DOK 1) Perform the following operation and simplify the solutions. 3− 5𝑖 2+ 4𝑖 Sample Task (DOK 3) Under what circumstances does (𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) result in the following?

• A real number • An imaginary number • A non-‐real complex number

Curriculum Supports: Walch Unit 1 Lesson 3: Adding and Subtracting Complex Numbers Walch Unit 1 Lesson 3: Multilplying Complex Numbers


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N.CN.3 (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. Concepts and Skills to Master: • I can determine the conjugate of a complex number. • I can define the modulus of a complex number as the positive

square root of the sum of the squares of the real and imaginary parts of the complex number.

• I can use conjugates to express quotients of complex numbers in standard form.

Sample Task (DOK 3) Determine if the following statement is true or false using complex conjugates: The modulus of z and the modulus of 𝑧 are equal. Justify your answer with both verbal and algebraic arguments.

Curriculum Supports: Walch Unit 1 Lesson 3: HONORS: Finding the Conjugate 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.

Concepts and Skills to Master: • I can convert between the rectangular form, z = x + yi , and

polar form, z = r(cos θ+i sin θ) , of a complex number. • I can graph complex numbers on a complex plane in both

rectangular and polar form. • I can justify rectangular and polar forms of a complex number

as representing the same number.

Sample Task (DOK 3) Given the complex number in polar form 𝑧 = 𝑟(cos𝜃 +𝑖 sin𝜃), what is the polar form of –z? Justify your answer using verbal and algebraic arguments.

Curriculum Supports: Walch Unit 1 Lesson 3: HONORS: Representing Complex Numbers on the Complex Plane


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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 because (-‐1 + √3i) has modulus 2 and argument 120°.)

Concepts and Skills to Master: • I can represent geometrically the sum, difference, product, and

conjugation of complex numbers on the complex plane. • I can show that the conjugate of a complex number in the

complex plane is the reflection across the x-‐axis. • I can evaluate the power of a complex number, in rectangular

form, using the polar form of the complex number.

Sample Task (DOK 3) Find two sets of complex numbers whose differences are equal. Justify graphically.

Curriculum Supports: Walch Unit 1 Lesson 3: HONORS: Representing Complex Numbers on the Complex Plane 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. Concepts and Skills to Master: • I can show that the distance between two complex numbers is

equivalent to the modulus of the difference by applying the distance formula.

• I can find the midpoint of a segment between two complex numbers by taking the average of the numbers at its endpoints using the midpoint formula.

Sample Task (DOK 1) Find the distance and the midpoint between −2+ 3𝑖 and 1− 5𝑖. Sample Task (DOK 2) A treasure is hidden in the complex plane. Follow the sequence of events: From the origin, travel to 1+ 3𝑖, then travel to Point A located at 2+ 5𝑖, noting the distance and direction traveled. Now return to the origin. Traven the same distance and direction to find Point B. The treasure will be halfway between point A and point B. Give the coordinate location of the treasure.

Curriculum Supports: Walch Unit 1 Lesson 3: HONORS: Representing Complex Numbers on the Complex Plane


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Secondary II Unit 2: Quadratic Functions and Modeling

Regular: 5-‐6 weeks Honors: 6 weeks

Honors Advanced: 5 weeks

F.IF.7a, b Graph function expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★

a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-‐defined functions, including step functions and absolute

value functions. Concepts and Skills to Master: • I can graph quadratic functions expressed in various forms

by hand. • I can use technology to model quadratic functions, when

appropriate. • I can graph and find key features of piecewise-‐defined

functions, including step functions and absolute value functions.

Sample Task (DOK 1) Graph the function and identify the key features:

𝑓 𝑥 = 𝑥 + 2, 𝑥 ≤ 1𝑥! − 3, 𝑥 > 1

Sample Task (DOK 3) Write and graph three different functions whose minimum is (-‐1, 5).

Curriculum Supports: Walch Unit 2 Lesson 1: Graphing Quadratic Functions Walch Unit 2 Lesson 1: Interpreting Various Forms of Quadratic Equations Walch Unit 2 Lesson 7: Absolute Value and Step Functions Walch Unit 2 Lesson 7: Piecewise Functions


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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.) ★

Concepts and Skills to Master: • I can distinguish linear, quadratic, and exponential

relationships based on equations, tables, and verbal descriptions.

• Given a function in a table or in algebraic or graphical form, I can identify key features such as x-‐ and y-‐intercepts, intervals where the function is increasing, decreasing, positive, or negative, relative maximums and minimums, symmetries and end behavior.

• I can use key features of an algebraic function to graph the function.

Sample Task (DOK 3) Time f(t) 0 300 5 777.5 10 1010 15 997.5 20 740 25 237.5

Create a situation that could have produced the given data. Use appropriate vocabulary and key features to tell the story.

Curriculum Supports: Walch Unit 2 Lesson 2: Interpreting Key Features of Quadratic Functions


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F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (For example, if the function h(n) gives the number of person-‐hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.) ★

Concepts and Skills to Master: • I can identify domains of functions given a graph. • I can identify a domain in a particular context.

Sample Task (DOK 2) If a function describes the area of an enclosure made with 100 ft. of fence, what would be an appropriate domain for the function? Sample Task (DOK 3) Describe a context where the domain of the function would be:

• All real numbers. • Whole numbers. • Rational numbers. • Integers. • Even numbers from 2 to 10 inclusive.

Curriculum Supports: Walch Unit 2 Lesson 2: Identifying the Domain of a Quadratic Function


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F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. ★

Concepts and Skills to Master: • I can calculate the rate of change in a quadratic function

over a given interval from a table or equation. • I can compare rates of change in quadratic functions with

those in linear or exponential functions.

Sample Task (DOK 1) Given the function 𝑓 𝑥 = 𝑥! − 11𝑥 + 24, find and interpret the average rate of change over each interval:

a. (0, 3) b. (4, 7) c. (6, 8)

Curriculum Supports: Walch Unit 2 Lesson 2: Identifying the Average Rate of Change F.BF.1a, b Write a function that describes a relationship between two quantities. ★

a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. (For example, build a function that

models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these to the model.)

Concepts and Skills to Master: • I can, given a linear, exponential, or quadratic context, find

an explicit algebraic expression or series of steps to model the context with mathematical representations.

• I can combine linear, exponential, or quadratic functions using addition, subtraction, or multiplication.

Sample Task (DOK 2) The total revenue for a company is found by multiplying the price per unit by the number of units sold minus the production cost. The price per unit is modeled by 𝑝(𝑛) = −0.5𝑛! + 6. The number of units sold is n. Production cost is modeled by 𝑐 𝑛 = −3𝑛 + 7. Write the revenue function.

Curriculum Supports: Walch Unit 2 Lesson 3: Building Functions from Context Walch Unit 2 Lesson 3: Operating on Functions


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F.IF.8b Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. (For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.)

Concepts and Skills to Master: I can use the properties of exponents to interpret expressions for exponential functions.

Sample Task (DOK 2) Identify the percent rate of change in y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

Curriculum Supports: Walch Unit 2 Lesson 4: Analyzing Exponential Functions F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically,

numerically in tables, or by verbal descriptions). (For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)

Concepts and Skills to Master: • I can compare intercepts, maxima and minima, rates of

change, and end behavior of two quadratic functions, where one is represented algebraically, graphically, and numerically in tables, or by verbal descriptions, and the other is modeled using a different representation.

Sample Task (DOK 2) Which has the greater average rate of change over the interval [5, 10]?

Time f(t) 0 300 5 777.5 10 1010 15 997.5 20 740 25 237.5

𝑓 𝑥 = 𝑥! + 4 or

Curriculum Supports: Walch Unit 2 Lesson 4: Comparing Properties of Functions Given in Different Forms


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

Concepts and Skills to Master: • I can use a table to observe that exponential functions grow

more quickly than quadratic functions. • I can use a graph to observe that exponential functions grow

more quickly than quadratic functions.

Sample Task (DOK 2) Graph the functions 𝑦 = 𝑥! and 𝑦 = 2! on the same coordinate axes. Compare the values of the functions over various intervals. Sample Task (DOK 3) Find a quadratic and exponential function that:

• Do not intersect. • Intersect once. • Intersect twice. • Intersect more than twice.

Curriculum Supports: Walch Unit 2 Lesson 4: Comparing Properties of Functions Given in Different Forms


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F.BF.3 Identify the effect on the graph of replacing 𝑓(𝑥) by 𝑓 𝑥 + 𝑘, 𝑘𝑓 𝑥 , 𝑓 𝑘𝑥 , and 𝑓(𝑥 + 𝑘) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. (Include recognizing even and odd functions from their graphs and algebraic expressions for them.)

Concepts and Skills to Master: • I can perform transformation on quadratic and absolute

value functions with and without technology. • I can describe the effect of each transformation on functions

(e.g., If (f(x) is replaced with f(x + k)). • I can, given the graph of a function, describe the

transformations using a specific value of k. • I can recognize which transformations take away the even

nature of a quadratic or absolute value of a function.

Sample Task (DOK 3) Sore the functions into the following categories: even, odd, and neither. Justify your work. For any function in the “neither” category, describe how you could transform it into an even or odd function. 𝑓 𝑥 = 𝑥 + 3 𝑗 𝑥 = 5𝑥 ℎ 𝑥 = 𝑥 − 4 ! 𝑔 𝑥 = 2 𝑥 + 1 𝑚 𝑥 = −7𝑥! 𝑝 𝑥 = 2!

Curriculum Supports: Walch Unit 2 Lesson 5: Replacing 𝑓(𝑥) with 𝑓 𝑥 + 𝑘 and 𝑓(𝑥 + 𝑘) Walch Unit 2 Lesson 5: Replacing 𝑓(𝑥) with 𝑘 ∙ 𝑓(𝑥) and 𝑓(𝑘 ∙ 𝑥) F.BF.4a Find inverse functions.

Solve an equation of the form 𝑓 𝑥 = 𝑐 for a simple function f that has an inverse and write an expression for the inverse. For example, 𝑓 𝑥 = 2𝑥! or 𝑓 𝑥 = (!!!)

(!!!), for 𝑥 ≠ 1.

Concepts and Skills to Master: • I can determine whether or not a function has an inverse,

and find the inverse if it exists. • I can understand that creating an inverse of a quadratic

function requires a restricted domain.

Sample Task (DOK 2) Give an example of a function that does not have an inverse function and explain how you know it does not. Sample Task (DOK 4) Prove that the inverse of a non-‐horizontal linear function is also linear and that the slopes are reciprocals.

Curriculum Supports: Walch Unit 2 Lesson 6: Finding Inverse Functions


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Secondary II Unit 3A: Expressions and Equations



A.SSE.1a, b Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. (For

example, interpret P(1 + r)n as the product of P and a factor not depending on P.) Concepts and Skills to Master: • I can identify the parts of an expression, such as terms,

factors, and coefficients, bases, exponents, and constant.

• I can explain the meaning of the part in relationship to the entire expression and to the context of the problem.

• I can understand that the product of two binomials is the sum of monomial terms. For example the product of (3x + 2) and (x – 5) is the sum of 3x2, -‐13x, and -‐10.

Sample Task (DOK 2) Use what you know about square roots to rewrite 𝑥! − 6 as a difference of two squares. Sample Task (DOK 3) A frame of width w surrounds a 4 in. by 6 in. picture. Express the area of the frame and the picture. Identify the constant, coefficient, and terms of the area expression and explain how each relates to the dimensions of the picture.

Curriculum Supports: Walch Unit 3A Lesson 1: Identifying Terms, Factors, and Coefficients Walch Unit 3A Lesson 1: Interpreting Complicated Expressions


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A.CED.1 Create equations and inequalities in one variable and use them to solve problems. (Include equations arising from linear and quadratic function, and simple rational and exponential functions.)

Concepts and Skills to Master: • I can create one-‐variable linear, exponential, quadratic,

and inequalities from contextual situations (stories). • I can solve and interpret the solution to linear,

exponential, quadratic, and inequalities in context. • I can solve compound inequalities. • I can use interval notation to represent inequalities.

Sample Task (DOK 1) Tran is doing a physics experiment with a steel ball. He throws it upwards with a velocity of 11m/s from a height of 1.2m. When is the height of the steel ball greater than 3m? Sample Task (DOK 2) Write an explicit expression to represent the number of dots in step n.

Curriculum Supports: Walch Unit 3A Lesson 2: Taking the Square Root of Both Sides Walch Unit 3A Lesson 2: Solving Quadratic Equations by Factoring Walch Unit 3A Lesson 2: Completing the Square Walch Unit 3A Lesson 2: Applying the Quadratic Formula Walch Unit 3A Lesson 2: Solving Quadratic Inequalities


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A.REI.4a, b Solve quadratic equations in one variable a. Use the method of completing the square to transform any quadratic equation in x into an equation of the

form qpx =− 2)( that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for 492 =x ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as bia ± for real numbers a and b.

Concepts and Skills to Master: • I can complete the square. • I can solve quadratic equations, including complex

solutions, using completing the square, quadratic formula, factoring, and by taking the square root.

• I can derive the quadratic formula from completing the square.

• I can recognize when one method is more efficient than the other.

• I can interpret the discriminant. • I can understand the zero product property and use it

to solve a factorable quadratic equation.

Sample Task (DOK 3) Solve the quadratic equation 49𝑥! − 70𝑥 + 37 = 0 using two methods. Describe the advantages of each method.

Curriculum Supports: Walch Unit 3A Lesson 2: Taking the Square Root of Both Sides Walch Unit 3A Lesson 2: Solving Quadratic Equations by Factoring Walch Unit 3A Lesson 2: Completing the Square Walch Unit 3A Lesson 2: Applying the Quadratic Formula Walch Unit 3A Lesson 2: Solving Quadratic Inequalities


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A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see ,)()( 222244 yxasyx −−thus recognizing it as a difference of squares that can be factored as ))(( 2222 yxyx +− .

Concepts and Skills to Master: • I can understand that an expression has different forms. • I can justify the different forms based on mathematical

properties. • I can interpret different symbolic notation.

Sample Task (DOK 3) Explain how you can use the quadratic formula to solve 𝑥! − 2𝑥! + 35 = 0

Curriculum Supports: Walch Unit 3A Lesson 2: Factoring Expressions by the Greatest Common Factor Walch Unit 3A Lesson 2: Factoring Expressions with A = 1 Walch Unit 3A Lesson 2: Factoring Expressions with A > 1 Walch Unit 3A Lesson 2: Solving Quadratic Equations by Factoring Walch Unit 3A Lesson 2: Completing the Square Walch Unit 3A Lesson 2: Solving Quadratic Inequalities F.IF.8a Write a function defined by an expression in different but equivalent forms to reveal and explain different

properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme

values, and symmetry of the graph, and interpret these in terms of a context. Concepts and Skills to Master: • I can factor quadratics and complete the square to find

intercepts, extreme values, and symmetry of the graph. • I can transition between different forms of quadratic

functions and identify the advantages of each

Sample Task (DOK 2) Transform 𝑓 𝑥 = 𝑥! + 𝑥 − 12 into another form to identify the zeros and vertex.

Curriculum Supports: Walch Unit 3A Lesson 2: Completing the Square Walch Unit 3A Lesson 2: Applying the Quadratic Formula Walch Unit 3A Lesson 2: Solving Quadratic Inequalities ★-‐ modeling standard


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Secondary II

Unit 3B: Creating and Graphing Equations Regular: 4-‐5 weeks Honors: 4 weeks


A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Concepts and Skills to Master: • I can write and graph an equation to represent a quadratic

relationship between two quantities. • I can model a data set using an equation including quadratic

relationships. • I can choose appropriate scale for the variables.

Sample Task (DOK 2) Given a rectangle with a perimeter of 100 feet, determine the units and the scales that would represent the length of the rectangle as the independent variable and the area of the rectangle as the dependent variable. Graph this situation. Sample Task (DOK 3) Create a problem situation where a curved line or graph could misrepresent the given data.

Curriculum Supports: Walch Unit 3B Lesson 1: Creating and Graphing Equations Using Standard Form Walch Unit 3B Lesson 1: Creating and Graphing Equations Using the x-‐intercepts Walch Unit 3B Lesson 1: Creating and Graphing Equations Using Vertex Form


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A.SSE.3a, b, c

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★

a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function

it defines. c. Use the properties of exponents to transform expressions for exponential functions. (For example the

expression 1.15t can be rewritten as (1.151/12)12t – 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.)

Concepts and Skills to Master: • I can rewrite expressions in different forms using mathematical

properties. • I can, given a context, determine the best form of an expression.

Sample Task (DOK 1) Once of the factors of 0.2𝑥! − 1.2𝑥! − 0.6𝑥 is (𝑥 − 2). Find the other factors. Sample Task (DOK 2) Find multiple ways to rewrite 𝑥! − 𝑦!

Curriculum Supports: Walch Unit 3B Lesson 1: Creating and Graphing Equations Using Standard Form Walch Unit 3B Lesson 1: Creating and Graphing Equations Using the x-‐intercepts Walch Unit 3B Lesson 1: Creating and Graphing Equations Using Vertex Form Walch Unit 3B Lesson 3: Writing Exponential Expressions in Equivalent Forms A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (For

example, rearrange Ohm’s law V = IR to highlight resistance R). Concepts and Skills to Master: • I can solve a quadratic formula for a variable of interest.

Sample Task (DOK 2) You are packaging an official game ball for women’s professional basketball that has a volume of 130𝜋 cubic inches. What must be the minimum dimensions for the box?

Curriculum Supports Walch Unit 3B Lesson 1: Rearranging Formulas


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N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.

Concepts and Skills to Master: • I can understand the meaning of a complex number. • I can solve a quadratic equation and understand the nature of

the roots.

Sample Task (DOK 2) Graph and find the solutions to the function 𝑓 𝑥 = 𝑥 − 3 ! + 5. Reflect the parabola across the line y = 5 at the vertex. Compare and contrast the graphs and solutions. Sample Task (DOK 3) Create a quadratic function without x-‐intercepts and verify that its solutions are complex.

Curriculum Supports: Walch Unit 3B Lesson 2: Solving Quadratic Equations with Complex Solutions Walch Unit 3B Lesson 2: HONORS: Applying the Fundamental Theorem of Algebra N.CN.8 (+) Extend polynomial identities to the complex numbers. (For example, rewrite 42 +x as (x+2i)(x-‐2i).)

Concepts and Skills to Master: • I can express a quadratic as a product of two complex

factors.

Sample Task (DOK 2) Expand the expression 𝑥 + 3 𝑥 − 5𝑖 𝑥 + 5𝑖 two ways:

A. 𝑥 + 3 𝑥 − 5𝑖 (𝑥 + 5𝑖) B. (𝑥 + 3) (𝑥 − 5𝑖)(𝑥 + 5𝑖)

Compare and contrast the methods. Curriculum Supports: Walch Unit 3B Lesson 2: HONORS: Extending Polynomial Identities to Include Complex Numbers


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N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Concepts and Skills to Master: • I can know that the Fundamental Theorem of Algebra guarantees

that any quadratic function will have a solution in the complex number system.

Sample Task (DOK 2) In the system of integer numbers, explain why there is no answer to the equation 3x = 5. In the system of rational numbers, explain why there is no answer to the equation 𝑥! + 5 = 0. Sample Task (DOK 3) Why is it better to solve quadratic equations in the complex number system rather than in the real number system?

Curriculum Supports: Walch Unit 3B Lesson 2: HONORS: Applying the Fundamental Theorem of Algebra A.REI.7 Solve a simple system consisting of a linear equations and a quadratic equation in two variables algebraically and

graphically. (For example, find the points of intersection between the line y = -‐3x and the circle x2 + y2 = 3). Concepts and Skills to Master: • I can solve a simple system consisting of a linear equation and a

quadratic equation (i.e., parabolas and circles) in two variables graphically.

• I can solve a simple system consisting of a linear equation and a quadratic equation (i.e., parabolas and circles) in two variables algebraically.

• I can recognize that the solutions of a system that includes a unit circle centered at the origin and a line with a y-‐intercept of 0 are points on a unit circle.

Sample Task (DOK 1) Find the intersection of the circle with a radius of 1 centered at the origin and the line 𝑦 = −3(𝑥 − 2). Show your work both graphically and algebraically. Sample Task (DOK 4) For a system consisting of a linear equation and a quadratic equation, how many possible solutions are there? Give and example for each possibility and include the graph and system.

Curriculum Supports: Walch Unit 3B Lesson 4: Solving Systems Graphically Walch Unit 3B Lesson 4: Solving Systems Algebraically


Scope and Sequence

24

A.REI.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable Concepts and Skills to Master: • I can rewrite a system of linear equations in matrix form as

AX=B, where X is the vector of variables. • I can solve a system of linear equations using matrices.

Sample Task (DOK 3) When asked to represent the following system as a matrix equation, a student produced the following. Will the student’s process result in a correct answer? Justify your answer. 7𝑥 − 𝑦 + 𝑧 = 2 2𝑥 + 2𝑦 − 3𝑥 = −34𝑥 + 𝑦 + 5𝑧 = 6

→ 7 2 4−1 2 11 −3 5

𝑥𝑦𝑧=

2−36

Curriculum Supports: Walch Unit 3B Lesson 5: HONORS: Representing a System of Linear Equations as a Single Matrix A.REI.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (use technology for matrices of

dimension 3x3 or greater). Concepts and Skills to Master: • I can use the determinant to determine whether an inverse

exists. • For 2 x 2 matrices, apply the following to find the inverse: For

A = a bc d

!

"#

$

%&, A−1 =

1det(A)

d −b−c a

!

"#

$

%&=

1ad − bc

d −b−c a

!

"#

$

%&.

Sample Task (DOK 2) As a professional code cracker, you receive an encoded two-‐digit ATM pin 𝐸 = 2 5 that was encoded by multiplying the original pin number by the matrix 𝐾 = 2 3

5 8 . Find the decoding key and use it to find the original pin number P. Teacher Hint: PK=E

Curriculum Supports: Walch Unit 3B Lesson 5: HONORS: Finding the Inverse of a Matrix and Using it to Solve a System of Equations Modeling Standards


Scope and Sequence

25

Secondary II Unit 4: Applications of Probability



S.CP.1 Describe events as the subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or” , “and”, “not”).

Concepts and Skills to Master: • I can use correct set notation, with appropriate symbols and

words, to identify sets and subsets within a sample space. • I can identify an event as a subset of a set of outcomes (a

sample space). • I can draw Venn Diagrams that show relationships (unions,

intersections, or complements) between sets within a sample space.

Sample Task (DOK 3) Describe the event that the summing two rolled dice is larger than 7 and even, and contrast it with the event that the sum is larger than 7 or even.

Curriculum Supports: Walch Unit 4 Lesson 1: Describing Events S.CP.2 Understand that two events A and B are independent if the probability of A and B occurring together is the

product of their probabilities, and use this characterization to determine if they are independent. Concepts and Skills to Master: • I can use appropriate probability notation for individual events as

well as their intersection (joint probability). • I can calculate probabilities for events, including joint

probabilities, using various methods (e.g. Venn diagram, frequency table).

• I can compare the product of probabilities for individual events ( )B(P)A(P ⋅ ) with their joint probability ( )BA(P ∩ ).

• I can understand that independent events satisfy the relationship ( )BA(P)B(P)A(P ∩=⋅ ).

Sample Task (DOK 3) Roll a pair of dice 100 times and keep track of the outcomes. Find pairs of events that are independent and pairs that are not. Justify your conclusions. (For example, the probability of rolling double and the probability of rolling 7 vs. the probability of rolling doubles and the probability of rolling a sum that is less than 4.)

Curriculum Supports: Walch Unit 4 Lesson 1: Understanding Independent Events


Scope and Sequence

26

S.CP.3 Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Concepts and Skills to Master: • I can understand conditional probability and how it applies to real-‐

life events. • I can use

)B(P)BA(P)B|A(P ∩

=to calculate conditional probabilities.

• I can understand that events A and B are independent if and only if they satisfy )A(P)B|A(P = or satisfy )B(P)A|B(P = .

• I can apply the definition of independence to a variety of chance events.

Sample Task (DOK 1) Is participation in sports independent of participation in the arts?

Curriculum Supports: Walch Unit 4 Lesson 2: Introducing Conditional Probability S.CP.4 Construct and interpret two-‐way frequency tables of data when two categories are associated with each object

being classified. Use the two-‐way table as a sample space to decide if events are independent and to approximate conditional probabilities. (For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same of other subjects and compare the results.)

Concepts and Skills to Master: • I can model real-‐life data using two-‐way frequency tables. • I can recognize that the conditional probability, P(A|B), represents

the joint probability for A and B divided by the marginal probability of B.

• I can use )B(P)BA(P)B|A(P ∩

= to calculate conditional probabilities from a

two-‐way frequency table. • I can apply the definition of independence to a variety of chance

events as represented by a two-‐way frequency table.

Sample Task (DOK 4) Select two categorical variables and conduct research to answer various probability questions and determine independence. Write a “newsworthy” article for the school newspaper that interprets the interesting relationships between the events.

Curriculum Supports: Walch Unit 4 Lesson 2: Using Two-‐Way Frequency Tables


Scope and Sequence

27

S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. (For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.)

Concepts and Skills to Master: • I can interpret conditional probabilities and independence in

context.

Sample Task (DOK 1) Is owning a smart phone independent from grade level? Own

smart phone

Do not own smart phone

10th grade 204 170 11th grade 192 160 12th grade 198 165

Curriculum Supports: Walch Unit 4 Lesson 2: Introducing Conditional Probability Walch Unit 4 Lesson 2: Using Two-‐Way Frequency Tables S.CP.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret

the answer in terms of the model. Concepts and Skills to Master: • I can find and interpret conditional probabilities using a two-‐way

table, Venn diagram, or tree diagram. • I can understand the difference between compound and

conditional probabilities.

Sample Task (DOK 2) Life is like a box of chocolates. Suppose your box of 36 chocolates have some dark and some milk chocolate, divided into cream or nutty centers. Out of the dark chocolates, 8 have nutty centers. Out of the milk chocolates, 6 have nutty centers. One-‐third of the chocolates are dark chocolate. What is the probability that you randomly select a chocolate with a nutty center? Given that it has a nutty center, what is the probability you chose a dark chocolate? Show how you determined your answers.

Curriculum Supports: Walch Unit 4 Lesson 2: Introducing Conditional Probability Walch Unit 4 Lesson 2: Using Two-‐Way Frequency Tables


Scope and Sequence

28

S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Concepts and Skills to Master: • I can define the probability of event (A or B) as the probability of

their union. • Understand and use the formula • P(A or B)=P(A)+ P(B) -‐P(A and B)

Sample Task (DOK 2) Given the following table, which includes data regarding boating preferences of boys and girls, use the Addition rule to find 𝑃(𝐿 ∪ 𝐺). Lake (L) River (R) Girls (G) 21 29 Boys (B) 32 18

Curriculum Supports: Walch Unit 4 Lesson 1: The Addition Rule S.CP.8 (+) Apply the general Multiplication Rule in a uniform probability model, x and interpret the answer in terms of the

model. Concepts and Skills to Master: • I can define the probability of event (A and B) as the probability

of the intersection of events A and B. • I can understand P(B|A) to mean the probability of event B

occurring when A has already occurred. • I can use the Multiplication rule, P(A and B) = P(A)P(B|A) =

P(B)P(A|B), to determine P(A and B). • I can determine the probability of dependent and independent

events in real contexts.

Sample Task (DOK 2) The probability that a student passes the written portion of a driving test is 62%. The probability that a student passes the driving part of the test is 86%. Draw a diagram to clearly demonstrate the probability that a student passes both tests.

Curriculum Supports: Walch Unit 4 Lesson 2: HONORS: The Multiplication Rule


Scope and Sequence

29

S.CP.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems. Concepts and Skills to Master: • I can define the probability of event (A and B) as the probability

of the intersection of events A and B. • I can understand P(B|A) to mean the probability of event B

occurring when A has already occurred. • I can use the Multiplication rule, P(A and B) = P(A)P(B|A) =

P(B)P(A|B), to determine P(A and B). • I can determine the probability of dependent and independent

events in real contexts.

Sample Task (DOK 1) Given the set of ice cream flavors {chocolate, strawberry, and vanilla}, list all possible two-‐scoop cones, and find the probability that a randomly selected cone includes chocolate. Sample Task (DOK 2) Referring to the above task, consider all possible sets of two-‐scoop cones. How would you define “two-‐scoop cone” in order to be a permutation? What part of your definition would you change to define the cones as a combination? How do the probabilities of getting chocolate change in each setting?

Curriculum Supports: Walch Unit 4 Lesson 3: HONORS: Combinations and Permutations Walch Unit 4 Lesson 3: HONORS: Probability with Combinations S.MD.6 (+) Use probability to make fair decisions (e.g., drawing by lots, using a random number generator). Concepts and Skills to Master: • I can simulate random outcomes using various tools. • I can analyze the fairness of games by determining the

probabilities of the possible outcomes.

Sample Task (DOK 2) Dice #1 has three 1’s and three 6’s. Dice #2 has two 2’s and four 5’s. When the dice are tossed, the set of dice with the highest number wins. Which set of dice is more likely to win? Sample Task (DOK 3) Vicki and Joyce are playing a dice game with two dice. Vicki gets a point if the sum of the numbers on the dice is even, and Joyce gets a point if the sum is odd. Is this game fair? Explain your reasoning.

Curriculum Supports: Walch Unit 4 Lesson 4: HONORS: Making Decisions


Scope and Sequence

30

S.MD.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).

Concepts and Skills to Master: • I can recognize that data based on random processes are subject

to variability. • I can analyze experimental designs and sampling strategies. • I can use the results of experiments and data samples to evaluate

decisions. • I can recognize the limitations of decisions drawn from sample

data, based on how the data were produced.

Sample Task (DOK 4) You have to get 65% right on a 20-‐question true/false quiz in order to pass a class. Can you pass by guessing alone, or do you need to study like crazy? Design a simulation that would test your answer.

Curriculum Supports: Walch Unit 4 Lesson 4: HONORS: Analyzing Decisions


Scope and Sequence

31

Secondary II Unit 5: Similarity, Right Triangle Trigonometry, and Proof



G.GPE.6 Find the point on a directed line segment between two given points that partitions that segment in a given ratio.

Concepts and Skills to Master: • I can use coordinate geometry to divide a segment into a given

ratio.

Sample Task (DOK 1) A segment with endpoints A(3,2) and B(6,11) is partitioned by a point C such that AC and CB form a 2:1 ratio. Find C.

Curriculum Supports: Walch Unit 5 Lesson 1: Midpoints and Other Points on Line Segments G.SRT.1a, b Verify experimentally the properties of dilations given by a center and a scale factor.

a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Concepts and Skills to Master: • Given a line segment, a point not on the line segment, and a

dilation factor, I can construct a dilation of the original segment. • I can recognize that the length of the resulting image is the length

of the original segment multiplied by the scale factor and that the original and dilated images are parallel to each other.

Sample Task (DOK 1) Create a dilation of segment AB through C with a scale factor of 2 to create segment EF. Find the lengths of EF, AC, BC, CE, and CF.

Curriculum Supports: Walch Unit 5 Lesson 2: Investigating Properties of Parallelism and the Center Walch Unit 5 Lesson 2: Investigating Scale Factors


Scope and Sequence

32

G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Concepts and Skills to Master: • I can decide whether two figures are similar using properties of

transformations. • I can understand that in similar triangles, corresponding sides are

proportional and corresponding angles are congruent.

Sample Task (DOK 3) Under what conditions do two lines intersected by two transversals form similar triangles? Justify your answer.

Curriculum Supports: Walch Unit 5 Lesson 3: Defining Similarity G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Concepts and Skills to Master: • I can prove that if two angles of one triangle are congruent to two

angles of another triangle, the triangles are similar (AA) using the properties of similarity transformations.

Sample Task (DOK 3) Determine whether the two triangles are congruent. Justify your answer. Sample Task (DOK 4) Write an argument to justify that the AA criterion for two triangles guarantees similarity.

Curriculum Supports: Walch Unit 5 Lesson 3: Applying Similarity Using the Angle-‐Angle (AA) Criterion


Scope and Sequence

33

G.SRT.4 Prove theorems about triangles. (Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.)

Concepts and Skills to Master: • I can prove that a line constructed parallel to one side of a triangle

intersecting the other two sides of the triangle divides the intersected sides proportionally.

• I can prove that a line that divides two sides of a triangle proportionally is parallel to the third side.

• I can prove that if three sides of one triangle are proportional to the corresponding sides of another triangle, the triangles are similar.

• I can prove the Pythagorean Theorem using similarity.

Sample Task (DOK 3) Prove the Pythagorean Theorem using similarity.

Curriculum Supports: Walch Unit 5 Lesson 4: Proving Triangle Similarity Using Side-‐Angle-‐Side (SAS) and Side-‐Side-‐Side (SSS) Similarity Walch Unit 5 Lesson 4: Working with Ratio Segments Walch Unit 5 Lesson 4: Proving the Pythagorean Theorem Using Similarity G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric

figures. Concepts and Skills to Master: • I can find lengths of measures of sides and angles of congruent and

similar triangles. • I can solve problems in context involving sides or angles of

congruent of similar triangles. • I can prove conjectures about congruence or similarity in

geometric figures using congruence and similarity criteria.

Sample Task (DOK 3) The length of George Washington’s face at Mt. Rushmore is 60 feet. Describe a method for determining the length of his nose using similar triangles. Justify your reasoning. Sample Task (DOK 3) Prove that the base angles of an isosceles triangles are congruent.

Curriculum Supports: Walch Unit 5 Lesson 4: Solving Problems Using Similarity and Congruence


Scope and Sequence

34

G.CO.9 Prove theorems about lines and angles. (Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.)

Concepts and Skills to Master: • I can prove and use theorems about lines and angles, including but

not limited to: • Vertical angles are congruent. • When parallel lines are cut by a transversal, congruent angle pairs are created. • When parallel lines are cut by a transversal, supplementary angle pairs are created. • Points on the perpendicular bisector of a line segment are equidistant from the segment’s endpoints.

Sample Task (DOK 2) Find as many angle relationships as possible in this pattern.

Curriculum Supports: Walch Unit 5 Lesson 5: Proving the Vertical Angles Theorem Walch Unit 5 Lesson 5: Proving Theorems About Angles in Parallel Lines Cut by a Transversal


Scope and Sequence

35

G.CO.10 Prove theorems about triangles. (Theorems include: measures of interior angles of a triangle sum to 180°, base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.)

Concepts and Skills to Master: • I can prove and use theorems about triangles including, but not

limited to: • Prove that the sum of the interior angles of a triangles = 180° • Prove that the base angles of an isosceles triangle are congruent. Prove that if two angles of a triangle are congruent, the triangle is isosceles. • Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. • Prove the medians of a triangle meet at a point.

Sample Task (DOK 3) Write a paragraph explaining why the segment joining two midpoints of two sides of a triangle is parallel to the third side.

Curriculum Supports: Walch Unit 5 Lesson 6: Proving the Interior Angle Sum Theorem Walch Unit 5 Lesson 6: Proving Theorems About Isosceles Triangles Walch Unit 5 Lesson 6: Proving the Midsegment of a Triangle Walch Unit 5 Lesson 6: Proving Centers of Triangles G.CO.11 Prove theorems about parallelograms. (Theorems include: opposite sides are congruent, opposite angles are

congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.)

Concepts and Skills to Master: • I can prove and use theorems about parallelograms including, but

not limited to: • Opposite sides of a parallelogram are congruent. • Opposite angles of a parallelogram are congruent. • The diagonals of a parallelogram bisect each other. • Rectangles are parallelograms with congruent diagonals.

Sample Task (DOK 3) Write a two-‐column proof showing that opposite sides of a parallelogram are congruent. Sample Task (DOK 3) Write a paragraph proof showing that a rectangle is a parallelogram with congruent diagonals.

Curriculum Supports: Walch Unit 5 Lesson 7: Proving Properties of Parallelograms Walch Unit 5 Lesson 7: Proving Properties of Special Quadrilaterals


Scope and Sequence

36

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Concepts and Skills to Master: • I can understand that the ratio of two sides in one triangle is

equal to the ratio of the corresponding two sides of all other similar triangles.

• I can define sine, cosine, and tangent as the ratio of sides in a right triangle.

Sample Task (DOK 3) Explain why the sine of x is the same regardless of which triangle is used to find it in the figure.

Curriculum Supports: Walch Unit 5 Lesson 8: Defining the Trigonometric Ratios HONORS Define trigonometric ratios and write trigonometric expressions in equivalent forms. Concepts and Skills to Master: • I can show how sine, cosine, and tangent are related using

trigonometric identities. • I can define secant, cosecant, and cotangent in terms of sine,

cosine and tangent. • I can define the six trigonometric functions using the unit circle.

Sample Task (DOK 1) Find the sin, cos, tan, sec, csc, cot of a 45-‐45-‐90 triangle. Sample Task (DOK 3) Prove that sin𝜃 = cos(90°− 𝜃) using congruent triangles.

Curriculum Supports: Walch Unit 5 Lesson 8: HONORS: Writing Trigonometric Expressions in Equivalent Forms G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Concepts and Skills to Master: • I can demonstrate the relationship between sine and cosine in

the acute angles of a right triangle. • I can explain the relationship between the sine and cosine in

complementary angles.

Sample Task (DOK 1) Find the second acute angle of a right triangle given that the first acute angle has measure of 39o. Complete the following statement: If sin 30o = ½, then the cos _____ = ½.

Curriculum Supports: Walch Unit 5 Lesson 8: Exploring Sine and Cosine as Complements


Scope and Sequence

37

G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Concepts and Skills to Master:

• I can use the Pythagorean Theorem and trigonometric ratios to find missing measures in triangles in contextual situations.

Sample Task (DOK 2) A teenager whose eyes are 5’ above ground level is looking into a mirror on the ground and can see the top of a building that is 30’ away from the teenager. The angle of elevation from the center of the mirror to the top of the building is 75°. How tall is the building? How far away from the teenager’s feet is the mirror? Sample Task (DOK 3) While traveling across flat land, you see a mountain directly in front of you. The angle of elevation to the peak is 3.5°. After driving 14 miles closer to the mountain, the angle of elevation is 9°24’36”. Explain how you would set up the problem, and find the approximate height of the mountain.

Curriculum Supports: Walch Unit 5 Lesson 9: Calculating Sine, Cosine, and Tangent Walch Unit 5 Lesson 9: Calculating Cosecant, Secant, and Cotangent Walch Unit 5 Lesson 9: Problem Solving with the Pythagorean Theorem F.TF.8 Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin (θ), cos (θ), or tan (θ), given sin

(θ), cos (θ), or tan (θ), and the quadrant of the angle. Concepts and Skills to Master: • I can prove sin 2 (θ) + cos 2 (θ) = 1 for right triangles in the first

quadrant. • I can, if given sin (θ), cos (θ), or tan (θ) for 0< θ <90, find sin (θ),

cos (θ), or tan (θ).

Sample Task (DOK 1) Given: sin𝜃 = !

!, find cos𝜃 if 𝜃 is in the first quadrant.

Sample Task (DOK 3) Show that the sin of an angle is constant regardless of the size of a triangle.

Curriculum Supports: Walch Unit 5 Lesson 9: Proving the Pythagorean Identity


Scope and Sequence

38

HONORS Prove trigonometric identities using definitions, the Pythagorean Theorem, or other relationships and use the relationships to solve problem is.

Concepts and Skills to Master: • I can prove trigonometric identities based on the Pythagorean

Theorem. • I can simplify trigonometric expressions and solve trigonometric

equations using identities. • I can justify half angle and double angle formulas for

trigonometric values.

Sample Task (DOK 3) Prove: sec! 𝜃 + csc! 𝜃 = !

(!!"! !)(!!"! !)

Sample Task (DOK 3) Develop a formula for sin(𝑥 + 𝑦 + 𝑧)

Curriculum Supports: Walch Unit 5 Lesson 9: HONORS Proving Trigonometric Identities F.TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Concepts and Skills to Master: • I can prove sin 2 (θ) + cos 2 (θ) = 1 for right triangles in the first

quadrant. • I can, if given sin (θ), cos (θ), or tan (θ) for 0< θ <90, find sin (θ),

cos (θ), or tan (θ).

Sample Task (DOK 3) Prove or disprove: tan 𝑥 + !

!= 1+ tan 𝑥. Explain

your answer verbally and algebraically.

Curriculum Supports: Walch Unit 5 Lesson 9: HONORS: Proving the Addition and Subtraction Formulas


Scope and Sequence

39

Secondary Strand II Unit 6: Circles With and Without Coordinates



G.C.1 Prove that all circles are similar. Concepts and Skills to Master: • I can define a circle as the set of points equidistant to a given center

point. • I can prove that all circles are similar.

Sample Task (DOK 2) Given a circle of a radius of 3 and another circle with a radius of 5, compare the ratios of the two radii, the two diameters, and the two circumferences.

Curriculum Supports: Walch Unit 6 Lesson 1: Similar Circles and Central and Inscribed Angles G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. (Include the relationship

between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.)

Concepts and Skills to Master: • I can use circle relationships to find the measures of central, inscribed, and

circumscribed angles of a circle. • I can use circle relationships to show that the measure of the inscribed

angle on a diameter is a right angle. • I can use circle relationships to show that the radius of a circle is

perpendicular to a tangent line where the radius intersects the circle.

Sample Task (DOK 2) Given the measure of a central angle of a circle is 100 degrees, find the measures of an inscribed angle that intersects the circle at the same points as the central angle. Sample Task (DOK 3) Why are all inscribed angles that intersect the same points equal regardless of where the vertex is on the circle?

Curriculum Supports: Walch Unit 6 Lesson 1: Similar Circles and Central and Inscribed Angles Walch Unit 6 Lesson 1: Chord Central Angles Conjecture Walch Unit 6 Lesson 1: Properties of Tangents of a Circle


Scope and Sequence

40

G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

Concepts and Skills to Master: • I can inscribe a circle in a triangle. • I can circumscribe a circle about a triangle. • I can prove that opposite angles in a quadrilateral inscribed in a circle

are supplementary.

Sample Task (DOK 1) Find the other two angles. Sample Task (DOK 3) Find the unique relationships between the angles of a quadrilateral inscribed within a circle if the quadrilateral is: • A square. • A rectangle. • An Isosceles trapezoid.

Curriculum Supports: Walch Unit 6 Lesson 2: Constructing Inscribed Circles Walch Unit 6 Lesson 2: Constructing Circumscribed Circles Walch Unit 6 Lesson 2: Proving Properties of Inscribed Quadrilaterals G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle.

Concepts and Skills to Master: • I can construct a line from a point tangent to a point on the circle.

Sample Task (DOK 2) Construct a line that will be tangent to both circles.

Curriculum Supports: Walch Unit 6 Lesson 3: Constructing Tangent Lines


Scope and Sequence

41

G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Concepts and Skills to Master: • I can use the concept of similarity to understand that arc length

intercepted by a central angle is proportional to the radius. • I can develop the definition of radians as a unit of measure by relating

to arc length. • I can discover that the measure of the central angle in radians is the

constant of proportionality. • I can derive the formula for the area of a sector.

Sample Task (DOK 2) Complete the table and consider the ratio of arc length to radius for different radii. Angle Radius Arc

Length Arc Length/Radius

600 3 inches 600 600 600 Sample Task (DOK 3) Construct an arc on a different circle whose length is five times the length of arc AB with the same central angle.

Curriculum Supports: Walch Unit 6 Lesson 4: Defining Radians Walch Unit 6 Lesson 4: Deriving the Formula for the Area of a Sector


Scope and Sequence

42

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.)

Concepts and Skills to Master: • I can develop the formulas for the circumference of a circle, area of a

circle, and volume of a cylinder, pyramid and cone using a variety of arguments.

Sample Task (DOK 1) Find the volume of the Great Pyramid of Giza. Sample Task (DOK 2) Explain why the volume of a cylinder is 𝑉 = 𝜋𝑟!ℎ.

Curriculum Supports: Walch Unit 6 Lesson 5: Circumference and Area of a Circle Walch Unit 6 Lesson 5: Volumes of Cylinders, Pyramids, Cones, and Spheres G.GMD.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other

solid figures.

Concepts and Skills to Master: • I can show understanding of Cavalieri’s Principle. • I can use Cavalieri’s Principle to find volumes of solid figures.

Sample Task (DOK 3) Use a visual model to represent how to use Cavalieri’s Principle to find the volume of a sphere from the volume of a cone.

Curriculum Supports: Walch Unit 6 Lesson 5: HONORS: Cavalieri’s Principle


Scope and Sequence

43

G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★

Concepts and Skills to Master: • I can find the volume of cylinders, cones, and spheres in

contextual problems.

Sample Task (DOK 2) Given a three-‐dimensional object, compute the effect on volume of doubling or tripling one or more dimension(s). (For example, how is the volume of a cone affected by doubling the height?)

Curriculum Supports: Walch Unit 6 Lesson 5: Volumes of Cylinders, Pyramids, Cones, and Spheres Walch Unit 6 Lesson 5: HONORS: Cavalieri’s Principle G.GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square

to find the center and radius of a circle given by an equation. Concepts and Skills to Master: • I can use the Pythagorean Theorem to find the distance between

two points. • I can find the center of a circle, given its equation.

Sample Task (DOK 2) A circle is tangent to the x-‐axis and y-‐axis in the first quadrant. A point of tangency has coordinates (4,0). Find the equation of the circle. Sample Task (DOK 3) A circle is inscribed in an equilateral triangle. The equilateral triangle lies in the first quadrant with one vertex at the origin and a second vertex at 4 3, 0 . Find the equation of the circle.

Curriculum Supports: Walch Unit 6 Lesson 6: Deriving the Equation of a Circle


Scope and Sequence

44

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

Concepts and Skills to Master: • I can develop the geometric definition of a parabola, including a

focus and directrix. • I can use the distance formula to derive the equation of a

parabola.

Sample Task (DOK 3) A parabola has focus (-‐2,1) and directrix y = -‐3. Determine whether or not the point (2,1) is part of the parabola. Justify your response.

Curriculum Supports: Walch Unit 6 Lesson 6: Deriving the Equation of a Parabola G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. (For example, prove or disprove that a

figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).)

Concepts and Skills to Master: • I can use coordinates to prove simple geometric theorems

algebraically.

Sample Task (DOK 3) Given a circle with center (-‐2,3), determine whether or not the points (-‐4,-‐1) and (3,5) are on the same circle. Justify your response.

Curriculum Supports: Walch Unit 6 Lesson 7: Using Coordinates to Prove Geometric Theorems About Circles and Parabolas Modeling Standards

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