CoreConnects Mathematics Sample Pages - High School

Teacher Resource Manual

Professional Development for College and Career Readiness Teaching a deep understanding of math content and how to use math in the real world

HS Algebra & Geometry

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CoreConnects: Mathematics

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Teaching Math Skills

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to Achieve Common Core Outcomes

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Algebra and Geometry

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Table of Contents

Overview Common Core General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Building Performance Character Traits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

How to Use the Teaching Math Skills Sheets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Recommended Manipulatives and Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Skills Sheets at a Glance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

The Standards for Mathematical Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Directed Math Activity/DMA Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Math-At-A-Glance Teacher Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Sequence of Skills Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Skills Sheets

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A.SSE.A Interpret the structure of expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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A.SSE.B Write expressions in equivalent forms to solve problems. . . . . . . . . . . . . . . . . . . . . . 43 A.APR.A Perform arithmetic operations on polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A.APR.B Understand the relationship between zeros and factors of polynomials. . . . . . . . . . 55 A.APR.C Use polynomial identities to solve problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.APR.D Rewrite rational expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 A.CED.A Create equations that describe numbers or relationships. . . . . . . . . . . . . . . . . . . . . . 67 A.REI.A Understand solving equations as a process of reasoning and

explain the reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A.REI.B Solve equations and inequalities in one variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.REI.C Solve systems of equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

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CoreConnects: Mathematics – Teaching Math Skills Algebra and Geometry – Table of Contents A.REI.D Represent and solve equations and inequalities graphically. . . . . . . . . . . . . . . . . . . . 87 G.CO.A Experiment with transformations in the plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 G.CO.B Understand congruence in terms of rigid motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 G.CO.C Prove geometric theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 G.CO.D Make geometric constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 G.SRT.A Understand similarity in terms of similarity transformations. . . . . . . . . . . . . . . . . . . 121 G.SRT.B Prove theorems involving similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 G.SRT.C Define trigonometric ratios and solve problems involving right triangles. . . . . . . . . 131

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G.SRT.D Apply trigonometry to general triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

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G.C.A Understand and apply theorems about circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 G.C.B Find arc lengths and areas of sectors of circles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 G.GPE.A Translate between the geometric description and the equation

for a conic section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

G.GPE.B Use coordinates to prove simple geometric theorems algebraically. . . . . . . . . . . . . 161

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G.GMD.A Explain volume formulas and use them to solve problems. . . . . . . . . . . . . . . . . . . 167 G.GMD.B Visualize relationships between two-dimensional and three-dimensional objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 G.MG.A Apply geometric concepts in modeling situations . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Performance Tasks

Algebra and Geometry Performance Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Test Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Performance Task Rubric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Performance Task Tracking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Algebra: Dream Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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CoreConnects: Mathematics – Teaching Math Skills Algebra and Geometry – Table of Contents Algebra: The Perfect Plan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Algebra: Ancient Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Geometry: Investment Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Geometry: Project Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Geometry: Fancy Fish. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

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How to Use the Teaching Math Skills Sheets The Teaching Math Skills sheets are designed to assist teachers in approaching Common Core Standards in the classroom. Each TMS Skills Sheet addresses a Common Core cluster within each domain of the Common Core State Standards for Math.

TMS Sheets are not intended to be used only once. Many of them cover many standards and can and should be used until the teacher feels that those standards have been addressed to sufficiently meet the needs of the students s/he is teaching. Although the Standards for Mathematical Practice are not outlined within these lesson suggestions, teachers should continue to provide opportunities for these practice for students.

An example for the top of a TMS Skills Sheet for Algebra follows. Note the following: • The objective which is the cluster, is indicated.

• The pre-requisite standards for the skill indicate what students are expected to know at the end of the level prior to the current cluster.

• Standards for the current grade-level skills are indicated.

> If the current cluster is considered a Modeling Standard, it will be indicated with a ∆ following the objective number.

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• Students that show mastery of the current cluster can be challenged to explore the growth cluster.

Teaching Math Skills

Seeing Structure in Expressions (SSE.A)

Current

Growth

Write and interpret numerical expressions

Interpret the structure of expressions

Write expressions in equivalent forms to solve problems

(5.OA.A.1, 5.OA.A.2)

(A.SSE.A.1∆, A.SSE.A.2)

(A.SSE.B.3, A.SSE.4)

Performance Task: Dream Car

Performance Task: Dream Car

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Pre-Requisite

represent the following situations and define the variable used: The skills sheets will eachWrite be an setexpression up withtothe following sections:

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Review & Pre-Assessment

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Objective: Interpret the structure of expressions

1. A phone bill cost $25 a month and $0.05 for each picture message sent. Write an expression to

• Review & Pre-Assessment represent the cost of the phone bill. (25 + 0.05x; where x represents the number of picture messages

• Instruction

• Scaffolding

• Evidence of Learning for every incorrect answer. Write an expression to represent your score on the test. (5c – 3w; where

sent) 2. A taxi cab costs a $5 flat rate and $1.50 per mile. Write an expression to represent the total cost of the taxi ride. (5 + 1.50x; where x represents total miles) 3. On your math test your teacher gives you 5 points for every correct answer and takes away 3 points

4. Tickets to the movies cost $5 for children and $9 for adults. Write an expression to represent the total cost paid to see a movie. (5c + 9a; where c represents the number of children attending the movie, and a represents the number of adults)

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c represents the number or problems correct, and w represents the number of problems incorrect)

5. A sweater at the store is 25% off. Write an expression to represent the sale price of the sweater. (x – 0.25x or 0.75x; where x represents the original cost of the sweater) 6–10. Challenge students to evaluate each above expression for a given value of your choice. (Answers

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CoreConnects: Mathematics will vary.)

CoreConnects: Mathematics – How to Use the Teaching Math Skills Sheets Review and Pre-Assessment This section provides a set of review and pre-assessment questions. As noted above, they will align to the pre-requisite skills required for the current cluster. Students should have sufficient knowledge of these review items in order to continue with instruction at this level. If they do not, the teacher should refer to the pre-requisite cluster, and start instruction with that skills sheet.

Instruction This section presents several instructional activities that pertain to the current cluster. They are bulleted activities that follow a Directed Math Activity Format. Within the instruction, specific Practice Standards have been indicated.

Teachers should look for student engagement in these areas. Each instructional activity is designed to cover one or two class sessions, depending on session length.

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Evidence of Learning

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It is rare that all students will be working at the exact same pace and at the exact same level. This section provides a few options for either additional support or additional challenge for those students who may need it.

Once an adequate amount of time has been spent on the standards and cluster, the teacher should be looking for evidence of learning. At the end of each Skills Sheet, there is a bulleted list of what is expected of students upon mastery of this cluster.

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These skills sheets have been provided as a guide for instruction. Teachers are encouraged to supplement them with additional activities that align to the standards indicated.

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Teaching Math Skills

Seeing Structure in Expressions (SSE.B)

Pre-Requisite

Current

Growth

Use properties of operations to generate equivalent expressions

Write expressions in equivalent forms to solve problems

Rewrite rational expressions

(7.EE.A.1, 7.EE.A.2)

(A.SSE.B.3∆, A.SSE.B.4∆) Performance Task: Dream Car

(A.APR.D.6, A.APR.D.7) Performance Task: Ancient Algebra

2. 6s + 30

(6(s + 5))

3. 4(y + 7) – 9

(4y + 19)

4. (b + 6) – 5 + 9 5. 3(2 + 2p)

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(7(x + 4))

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Have students write an equivalent expression for each of the following (answers may vary):

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Review & Pre-Assessment 1. 7x + 28

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Objective: Write expressions in equivalent forms to solve problems

(b + 10) (6 + 6p)

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6. A rectangle is four times as long as it is wide. One expression that represents the perimeter would be w + 4w + w + 4w. Write two other expressions to represent the perimeter of the rectangle.

(10w; 2(4w) + 2(w))

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Have students read the problems carefully and answer each question:

7. A rectangle is three times as long as its width. One possible way to write an expression to find the perimeter would be 3x + 3x + x + x. Write the expression in two other ways. (8x; 2(3x) + 2(x)) 8. Peter says the two expressions 7z – 4 + 3z – 6 and 4z – 10 are equivalent. Is he correct? (Yes)

10. An equilateral triangle has a perimeter of 18x + 30. What is the length of each side of the triangle?

(6x + 10)

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9. An equilateral triangle has a perimeter of 12x + 9. What is the length of each side of the triangle?

(4x + 3)

Instruction FINDING THE ZEROS OF A QUADRATIC FUNCTION:

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• Have students graph a linear equation. As a class, discuss the different ways for graphing this equation: creating a chart, using the slope and y-intercept, graphing the zeros. If multiple methods were used, have students share their method with the class. Encourage students to graph the equation using another method. (PS.3)

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Seeing Structure in Expressions (SSE.B)

Teaching Math Skills

• Provide students with a graph of a quadratic function. As a class, compare and contrast the graph of the linear equation with the graph of the quadratic equation (e.g., one is curved, one is straight, etc.).

Discuss the graphing techniques previously used. Will they work when graphing quadratic functions?

Why or why not? Which method would be the most effective? Guide students to see that a table of values can be used if students identify the zeros and the maximum value. (PS.3, PS.5, PS.6)

• Provide students with a quadratic equation and graph that can represent one of the discussed scenarios.

As a class, create a table of values to identify the zeros of the function (the point where the x coordinate or y coordinate are zero). Compare the zeros with the graph. How do they compare? (PS.3, PS.5, PS.6) • Provide pairs of students with one to three additional functions and a graph. Challenge students to identify the zeros of the function using both a table and then by graphing the function and locating the zeros on the graph. Challenge students to compare the results. Students should realize that the points where the line crosses the x- and y-axis correlates with the zeros in the chart. (PS.4, PS.5, PS.6)

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• As a class, discuss issues that may arise from finding the zeros of a quadratic function using a table of values. Guide students to see that for some functions it may be difficult to find the exact values that will yield a zero. Explain that another way to solve these problems is to factor the function, set each factor equal to zero, and solve. Guide students to see that this process works because when one factor is equal to zero, the product of the factors is also equal to zero. As a class, explore this process for two to three of the quadratic functions previously explored. (PS.3, PS.5, PS.6) • Have students work with a partner to solve three to five additional problems by factoring to find the zeros of a quadratic function. Discuss the results as a class. If students are ready, have them complete an additional two to three problems independently. (PS.1, PS.2, PS.5)

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• Discuss what types of real-life scenarios may result in a curved graph (e.g., throwing a ball, shooting a basket, kicking a ball, launching a rocket, etc.). (PS.3, PS.4)

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COMPLETING THE SQUARE TO CONVEY THE VERTEX FORM AND FINDING THE MAXIMUM OR MINIMUM VALUE OF A QUADRATIC FUNCTION: • As a class, discuss that there are many ways to write the same function. Why might we choose one way over another? Guide students to see that different forms of an equation make it easier to find out certain information, e.g., in slope-intercept form you can quickly identify the slope and y-intercept, which can be useful for graphing or finding the rate of change. (PS.3)

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• Discuss what a vertex is. Guide students to see that it is the middle of a parabola. Have students create several parabolas and identify the vertex of each. Share graphs and discuss observations. Make sure students observe that it represents either the maximum or minimum point on the graph. Discuss why this might be useful information. Guide students to see that equations are used to represent all the solutions represented by the function. The maximum or minimum value is used in finding the maximum or minimum solutions. Ask students to provide real-life situations in which they may want to find the maximum or minimum values (e.g., maximizing profits). (PS.3, PS.4, PS.5)

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Teaching Math Skills

Seeing Structure in Expressions (SSE.B) Algebra

• Explain that much like slope-intercept form there is a vertex form. Explain that vertex form is f(x) = – a(x – h)2 + k where the vertex is (h, k). Provide students with five to seven problems in vertex form and challenge them to identify the vertex. Note: make sure students are properly identifying the signs of the coordinates. (PS.1, PS.2, PS.3) • Provide students with an expression NOT in slope-intercept form. As a class, discuss how to rearrange the expression so that it is in slope-intercept form. Guide students to see that if they complete the square this will provide the needed form. (PS.1, PS.2, PS.5)

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• If students are not familiar with completing the square, introduce the concept using flats to represent x2, rods to represent x, and units to represent the constants. Build a square to represent the expression.

Begin with the flat representing the x2. Next divide the rods into two groups and line the first half up vertically along the right side of the flat, and the other half horizontally along the bottom of the flat. This will create most of the square with the bottom right corner missing. Have students arrange the units they have to fill in the vacant area. Students will see that they need some units to fill in the gaps.

(PS.4, PS.5, PS.6)

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• Relate the method used with the blocks to the algebraic method. Challenge students to factor five to seven polynomials with a partner by competing the square. Once students have completed the square and discussed their results, challenge them to find the vertex of each expression. (PS.1, PS.2, PS.3)

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• Write the factored equation for the new completed square. Make sure that students understand that the units added in have to be subtracted from the final equations so as now to change the value of the expression. For example, if students are factoring the equation x2 + 6x = 0, they can model the equation by placing one flat (x2) and 3 rods on each side of the flat (6x). Students can write the factored equation by looking at the edge of the square. If each edge have one rod and 3 units, but 4 units to be added to complete the square. The factored equation would be (x + 3)2 – 9. Help students to see that they need to subtract the units added because they where needed to complete the square but are not part of the original equation. Confirm that this works by checking the results against the original expression. (PS.4, PS.5, PS.6)

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Seeing Structure in Expressions (SSE.B)

Teaching Math Skills

• Provide students with the graph of each expression that they just factored. Make sure that half have maximums and half have minimums. Have students work with a partner and compare the graph to the equation. Challenge them to discover a connection between the equations and if the graph is a maximum or minimum. Guide students to see that if a > 0 then the vertex is a minimum. If a < 0 then the vertex is a maximum. (PS.5, PS.6, PS.7)

WRITING AN EQUIVALENT FORM OF AN EXPONENTIAL FUNCTION AND IDENTIFY RATE OF GROWTH OR DECAY: • Introduce the concept of exponential functions and as a class discuss real-life situations that may be exponential functions (e.g., the value of a car). Discuss the form of an exponential function: y = bx.

(PS.3, PS.4) • Provide students with five to ten graphs and their corresponding functions. Make sure to include positive and decimal values for b. Challenge students to work with a partner to identify whether the graphs are showing growth or decay, and to identify any patterns in the base that can help them identify if it is a growth or decay function based on the base. Guide students to see if the base is greater than one then it is a growth function. If the base is between zero and one then it is a decay function. If the technology is available have students confirm their theories by graphing four to seven additional functions on a graphing calculator or computer. (PS.3, PS.4, PS.5)

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• As a class, explore the parts of an exponential function. Select one function previously explored and add a constant on front of the base, e.g., 3(2)x. Challenge students to create a table of vales for x = 0, 1, 2, 3, 4, and 5. (PS.3, PS.5, PS.6)

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• Provide students with three to five problems. Challenge them to find the vertex by completing the square and determine whether it is a maximum or minimum. If time allows, explore one to three real-life applications to find maximums and minimums of an expression. (PS.1, PS.2, PS.3)

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• Have students graph the function by plotting the values from the table. Discuss that the constant is the initial value. Because the base is 2, the function doubles each time (if it is was 3 it would triple, if it was 4 it would quadruple, etc.), and the exponent represents the number of times (factors) it has changed.

(PS.3, PS.5, PS.6) • Explain that when a function changes by a fixed amount each time it can be represented by the expression A = a(1 + r)t; where a is the initial value, r is the percent of change, and t is time. As a class, rearrange the previous expression to fit this form: 3 (1 + 1.00)x noting that this function is a 100% change. (PS.3) • Introduce the exponential model. Explain that it is an expression or equation used to represent real quantities or relationships. Provide students with an example of an exponential model. For instance: I paid $19,000 for my new car. It loses 8% of its value each year. The expression that represents this scenario would be 19,000(1+ (-0.08))t. Discuss that 8% is subtracted because the value is decreasing each year making this an exponential decay problem. The simplified expression would be 19,000(0.92)t.

Discuss how this expression compares to the original. Guide students to see that this means that the car is maintaining 92% of its value each year. Note: if the percent of change is not given it can be calculated

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Teaching Math Skills

Seeing Structure in Expressions (SSE.B)

• As a class, discuss how you might find the rate of decay for one day. Guide students to see that because there are 365 days in a year the exponent representing time can be multiplied by 365, and because we do not want to change the value of the expression we must also multiply by 1/365. The property of exponents can then be used to distribute the exponent (1/365) to the base resulting in the expression: 19,000(0.9998)365t. Working from the equation a(1 + r)t, this results in a decrease in value of 0.02% per day. (PS.4, PS.5, PS.6, PS.7)

Scaffolding Additional Support

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• Challenge students to identify a real-life situation in which there is exponential decay or growth and create an expediential model for the situation by identifying the initial value, and percent of change for a given time period. Extend the activity by challenging students to identify the percent change for a smaller increment of time. (PS.1, PS.2, PS.3, PS.4)

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• Provide students with two to four additional scenarios. Have them work with a partner to find the exponential model and rate of change. Once this is complete challenge them to find the rate of change for a smaller increment of time (e.g., day, second, minute, week, etc.). (PS.1, PS.2, PS.3)

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by using the formula for rate of change (difference of the original and final amount divided by the original amount). (PS.4, PS.5, PS.6)

Additional Challenge or Rigor

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• Allow students to use manipulatives such as counters or base ten blocks when solving problems • Guide students with questioning

• Review the properties and graphs of linear equations • Have students work with single-digit numbers

• Pair students with a struggling learner for peer tutoring

• Have students create their own problems and answers to trade with a classmate

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• Allow struggling learners to work with a partner

• Have students identify and explain a real-life application of quadratic functions • Challenge students to write two equivalent forms for a quadratic or linear equation

• Students will be able to write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the meaning of the zeros

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Evidence of Learning

» Given a quadratic function, students will be able to explain the meaning of the zeros of the function; that is if f(x) = (x – c) (x – a) then f(a) = 0 and f(c) = 0 » Given a quadratic expression, students will be able to explain the meaning of the zeros graphically; that is for an expression (x – a) (x – c), a and c correspond to the x-intercepts (if a and c are real)

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Algebra

• Students will be able to write expressions in equivalent forms by completing the square to convey the vertex form, to find the maximum or minimum value of a quadratic function, and to explain the meaning of the vertex

Teaching Math Skills

• Students will be able to use properties of exponents (such as power of a power, product of powers, power of a product, and rational exponents, etc.) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay

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Seeing Structure in Expressions (SSE.B)

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Teaching Math Skills

Congruence (CO.B)

Pre-Requisite

Current

Growth

Understand congruence and similarity using physical models, transparencies, or geometry software

Understand congruence in terms of rigid motions

Prove geometric theorems

(G.CO.B.6, G.CO.B.7, G.CO.B.8)

Geometry

Objective: Understand congruence in terms of rigid motions

(G.CO.C.9, G.CO.C.10, G.CO.C.11)

Performance Task: Investment Property Geometry

(8.G.A.1, 8.G.A.2, 8.G.A.3, 8.G.A.4, 8.G.A.5)

Review & Pre-Assessment Determine if the two figures are congruent by using transformations. Explain your reasoning.

G

C

F

H

E

(These rectangles are congruent. Rectangle EFGH is reflected and translated to the right of ABCD.) H

SA

2.

Geometry

D

B

E

A

M PL

1.

F

G

Geometry

N

P

M (These two triangles are not congruent. The angles and sides are not the same.)

CoreConnects: Mathematics

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Geometry

Geometry

Congruence (CO.B)

Teaching Math Skills

3.

X

Q Z

Y

S

(These two figures are congruent. Triangle QRS is reflected to the right and translated down from triangle XYZ.) Determine whether the two images are congruent. If they are, explain in detail why they are congruent.

Also, use the congruent symbol to match up the vertices.

4.

6

y Figure 2

E

Geometry

R

E

M PL

4 2

B

Geometry

-6

-4

-2 M

0

x

Q

2

4

6

-2

Figure 1

-4

U

L

SA

-6

Geometry

(These two figures are congruent. Figure 2 was produced by translating 4 units to the right and 5 units up from Figure 1. U is congruent to B; M is congruent to E; L is congruent to Q.) 5.

6

y

K

4 F

2

Figure 2

-4

-2

0

2

-2

Geometry

-4

x

Y

X -6

H

4

6

Figure 1 Z

-6

(These two figures are not congruent. The length of Figure 1 is not the same as Figure 2.)

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Teaching Math Skills

Congruence (CO.B)

F

C

6

y

4 Figure 1

Figure 2

D

Geometry

6.

A

2

x

P -6

-4

-2

0 -2

2

4

6

R

Geometry

-4 -6

E

(These two figures are congruent. Figure 2 was produced by reflecting Figure 1 and translating

2 units down. P is congruent to R; F is congruent to A; C is congruent to D.)

M PL

Instruction

PREDICT THE EFFECTS RIGID MOTION HAS ON FIGURES IN THE COORDINATE PLANE: Geometry

• Have students draw a trapezoid on sheet of graph paper, cut it out, place the trapezoid in quadrant 1 on a coordinate plane, and trace it. Ask students what would happen if we rotated the trapezoid 90 degrees counterclockwise. As a class, model this transformation by placing the cutout of the trapezoid on its outline and rotating it 90 degrees counterclockwise. Students should see that the trapezoid has moved into quadrant 2, thus changing the trapezoid’s orientation and location. (PS.5, PS.6)

SA

• Discuss how rotating the same trapezoid 90 degrees clockwise would differ from 90 degrees counterclockwise. Model this transformation by placing the cutout of the trapezoid on its outline and rotating it 90 degrees clockwise. Students will see that the trapezoid rotates into quadrant 4 instead of into quadrant 2. (PS.3, PS.5, PS.6)

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Geometry

• Have students trace the trapezoid overlapping quadrants 3 and 4. Ask students to predict what would happen if the trapezoid was translated 6 units up and 3 units to the left. Help students to understand that translating a figure is similar to sliding or shifting a figure across a coordinate plane. Have students use the cutout to model the transformation. Students should recognize that a translation 6 units up and

Geometry

• Have students trace their trapezoid in quadrant 3 of a coordinate plane. As a class, discuss what the effect reflecting the trapezoid over the y-axis would have. Discuss the fact that the y-axis would act like a mirror, resulting in a reflected image of the trapezoid in quadrant 4. After students have agreed upon a result, have students model the transformation by placing the cutout over the outline and reflecting it over the y-axis. Challenge students to model the reflection over the x-axis and compare the reflection of the trapezoid to a y-axis reflection of the trapezoid. Guide students to understand that the x-axis reflection will also result in a mirror image, however, the x-axis will act like the mirror instead of the y-axis. Students should recognize that the reflected trapezoid will be in quadrant 2 instead of quadrant 4. (PS.3, PS.5, PS.6, PS.7)

Geometry

Congruence (CO.B)

Teaching Math Skills

3 units to the left will result in a trapezoid in which every point has shifted 6 units up and 3 units to the left. (PS.5, PS.6, PS.7) • Challenge students to brainstorm some real-life situations in which it would be necessary to predict the outcome of a rigid transformation. Guide students to make connections to topics such as architecture, engineering, fashion, etc. (PS.3, PS.7, PS.8) • Have students work with a partner to perform three to five additional rigid transformations. Have students make and model the transformations as needed. (PS.1, PS.2, PS.4, PS.6)

• Have students cut out a triangle and trace it on a coordinate plane. Ask students to rotate, reflect, and translate the triangle on the same coordinate plane. As a class, discuss the results. (PS.3, PS.5, PS.6) • Challenge students to explore what the triangles that have undergone rigid transformations (translated figures, rotated figures, and reflected figures) have in common. Through discussion, guide students to recognize that while their placement or orientation will differ, all the triangles are still the same shape and size as the original triangle. (PS.7, PS.8)

E

Geometry

DEVELOPING THE DEFINITION OF CONGRUENCE:

M PL

Geometry

• Ask students what mathematical vocabulary relates to the concept of figures that are the same shape and size. Help students to understand that the word congruence relates to this concept because it means that two objects are the same shape and size, regardless of placement or orientation. (PS.6) • Have students explore three to five shapes that have been transformed (at least one should not be congruent to the original shape). Challenge students to determine if the transformed shape is congruent to the original shape. Students may choose to cut out the original shape so that they can physically manipulate it to determine congruence. (PS.4, PS.5, PS.6)

SA

• Provide students with two triangles that have side lengths and angle measurements that are equivalent.

Have students measure each side of the triangles and label the lengths. Ask students to determine if the triangles are congruent. Students may do this visually or may cut out the triangles and manipulate them to determine congruence. Guide students to understand that the triangles are congruent because they are the same shape and size. (PS.5, PS.6) • Ask students to measure and label the angles in both triangles with a protractor. Guide students to recognize that both triangles have angles that have exactly the same measurements. Ask students to conjecture whether angle measurements alone are an accurate predictor of congruence. Challenge students to construct a triangle that has the same angle measurements as one of the original triangle but has one or more different side lengths. (PS.4, PS.5, PS.6)

Geometry

Geometry

SHOWING TRIANGLE CONGRUENCE USING CONGRUENCE OF CORRESPONDING SIDES AND CORRESPONDING ANGLES:

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Teaching Math Skills

Congruence (CO.B)

Geometry

• Ask students to measure and label the angles in both triangles with a protractor. Guide students to recognize that both triangles have angles that have exactly the same measurements. Ask students if matching angle measurements alone are an accurate predictor of congruence. Guide students to understand that matching angle measures alone are not a predictor of congruence because figures can be similar when they have the same angle measurements. If students were able to previously create a triangle with the same angle measures and different lengths, revisit the models in the discussion to further support the concept. (PS.3, PS.5, PS.6)

Geometry

• Provide students with two triangles that are different sizes but are a similar triangle. Ask students to measure and label the lengths of the sides of the triangles and determine if the triangles are congruent. Students may do this visually or may cut out the triangles and manipulate them to determine congruence. Guide students to understand that the triangles are not congruent because they are different sizes. (PS.5, PS.6)

E

• Discuss whether student conjectures were correct or incorrect. Ask students to develop a rule for determining congruence. Guide students to develop a rule, including the fact that all sides and angles in a triangle must be equal for two triangles to be congruent. (PS.7, PS.8)

M PL

• Provide students with three to five sets of triangles. Challenge students to determine congruence by measuring the sides and angles. (PS.1, PS.2, PS.5) DEVELOPING AND EXPLAINING THE TRIANGLE CONGRUENCE CRITERIA (ASA, SSS, AND SAS):

SA

Geometry

• Provide students with two congruent triangles, including the side measures. Ask students if the measurements for all three sides of each triangle would be enough information to determine if the triangles were congruent. Challenge students to create a triangle with the same side measurements but with different angle measurements. This can be done with by having students cut a pipe cleaner to the length of each side, and arrange the pipe cleaners to create triangles with different angle measures.

Angle measures can be verified by using a protractor. Students should recognize that it is impossible to create triangles with the same side measures and different angle measures. Guide students to see that if three sides of each triangle are congruent, then the triangles are congruent. (PS.6, PS.7, PS.8)

a

Geometry

• Have students draw a triangle as shown below. Ask students if there is a relationship between the angles of the triangles and their corresponding sides, for example angle A and side a. (PS.5, PS.6)

A

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Geometry

• Ask students how side a would be affected if angle A was either larger or smaller. Have students create a triangle where angle A is lager, and one where angle A is smaller and measure the corresponding side a for each. Students should recognize the larger the angle, the longer the side opposite the angle is.

(PS.5, PS.6, PS.7)

Congruence (CO.B)

Teaching Math Skills

Geometry

• Provide students with two congruent triangles. Have students measure two sides and the angle between as shown below. Have students apply the idea of corresponding sides and angles in a triangle to consider whether two sides and the corresponding angle of the missing side of each triangle would be enough to determine congruence. (PS.6, PS.7)

40° 12

Geometry

40°

14

14

12

E

• Guide students to understand that the 40 degree angle corresponds with the length of the missing side, and that because we have congruent given side lengths, the corresponding sides will be equal, thus all three sides in each triangle must be equal, yielding congruent triangles. Guide students to understand that when a side, followed by an angle, followed by a side are congruent in two triangles, the result will always be two congruent triangles by SAS, or Side Angle Side. (PS.5, PS.6, PS.7)

Geometry

M PL

• Challenge students to use two of their pipe cleaners for the first activity to create a triangle that has those side lengths and the same angle measure in between them as the original triangle, but is not congruent. Students should recognize that this is not possible because the angle locks the sides in place. Have students change the angle that is congruent to the original. They should see that it WILL be possible to create a different triangle. Guide students to see that the theorem only works then the congruent angle is between the two congruent sides. (PS.4, PS.5, PS.6)

Geometry

SA

• Ask students to apply the idea of corresponding sides and angles in a triangle to consider whether two angles and the length of the side that does not correspond to either of the given angles (the side in between the angle) would be enough to determine congruence of two triangles (pictured below).

(PS.3, PS.6, PS.7)

40°

40°

12

12 90°

Geometry

• Ask students what sides the given angles correspond with. Guide students to understand that the 40 degree angle and 90 degree angle correspond with the lengths of the missing sides. (PS.3, PS.6) • Challenge students to determine if the given side plus the angles of the two corresponding sides will be enough information to determine congruence. Guide students to understand that the given side plus the two missing sides’ angles will prove congruence because the lengths of all three sides are accounted for, making an angle followed by a side, followed by an angle—or ASA—a valid way to prove congruence of two triangles. (PS.3, PS.6, PS.7)

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Teaching Math Skills

Congruence (CO.B)

Scaffolding Additional Challenge or Rigor

• Allow struggling learners to work with a partner

• Pair students with a struggling learner for peer tutoring

E

Additional Support

Geometry

• Provide students with five to seven pairs of triangles. Each pair should have select measurements marked. Challenge students to use these measurements and the theorems reviewed to determine congruence. (PS.1, PS.2, PS.5)

Geometry

• Challenge students to use one of their pipe cleaners for the first activity to create a triangle that has that side length and the same two adjacent angle measures, but that is not congruent to the original triangle. Students should recognize that this is not possible because the angles lock the sides in place.

Have students change one of the congruent angles to the angle across from the congruent side, not adjacent to it. They should see that it WILL be possible to create a different triangle. Guide students to see that the theorem only works then the congruent side is between the two congruent angles. (PS.4, PS.5, PS.6)

• Challenge students to think of examples of congruence in real-world situations

• Guide students with questioning

• Provide students with geometric solids to count model congruence

SA

• Have students work with geometry software to model transformations and congruence

• Have students create their own problems and answers to trade with a classmate • Have students identify and explain a real-life application of congruence

Geometry

M PL

• Allow students to use manipulatives such as: counters or tokens, base ten blocks, and multiplication flash cards when solving problems

• Challenge students to explore congruence of three-dimensional objects

Evidence of Learning

• Knowing that rigid transformations preserve size and shape or distance and angle, students will be able to use this fact to connect the idea of congruency and develop the definition of congruent

Geometry

• Students will be able to use descriptions of rigid motion and transformed geometric figures to predict the effects rigid motion has on figures in the coordinate plane

• Students will be able to use the definition of congruence, based on rigid motion, to show two triangles are congruent if and only if their corresponding sides and corresponding angles are congruent • Students will be able to use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence criteria: ASA, SSS, and SAS

Geometry

CoreConnects: Mathematics

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Teaching Math Skills

M PL SA

Geometry

Geometry

Geometry

E

Geometry

Geometry

Congruence (CO.B)

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Algebra – Dream Car Algebra

You would like to purchase your dream car. It is time to do your research and make sure that your dream car is worth the cost.

Part A Research your dream car. How much does the car cost?

What other factors do you need to consider that may impact the price of the car?

Algebra

E

What would your monthly payment be, including the interest for a five-year loan and for a three-year loan?

Part B

Algebra

M PL

How much will you ACTUALLY pay for the car after you have completed your payments on a five-year loan?

On a three-year loan?

SA

Cars decrease in value as soon as they are driven off of the lot. Therefore your dream car’s value has a rate of decay. Why do you think this is?

What do you think the graph of the value of your car would look like? Why do you think this?

Algebra

Confirm your results by looking up the value of your car in “good” condition by examining the value of your dream car after time has passed, and graphing the results on a sheet of graph paper. Remember to label your axes.

Age of Car: Value:

New

1 yr. old

2 yrs. old

3 yrs. old

4 yrs. old

5 yrs. old

6 yrs. old

8 yrs. old

10 yrs. old

20 yrs. old

Algebra

CoreConnects: Mathematics

25

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Algebra

CoreConnects: Mathematics – Algebra – Dream Car

Part C This relationship can be represented by the equation A = a(1 + r)t where A represents the actual value, t represents time in years, and a represents the initial value. Explain why this equation is representative of the rate of decay of a car. Why is there a 1 in the equation?

Algebra

Use the equation to explore the projected value of your dream car after time has passed. Use a different color and add this to the graph above.

Age of Car: Projected Value:

New

1 yr. old

2 yrs. old

3 yrs. old

4 yrs. old

5 yrs. old

6 yrs. old

8 yrs. old

10 yrs. old

20 yrs. old

M PL

SA

If you were to sell your car after four years would you make enough to pay off the balance of your loan?

Why or why not?

Algebra

Algebra

Algebra

E

How do the two graphs compare? What factors could explain the similarities and differences in the graphs?

CoreConnects: Mathematics

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CoreConnects: Mathematics – Algebra – Dream Car

Common Core Standard Assessed Skill A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression, such as terms, factors, and coefficients. 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. A.SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

Part A

Algebra

You would like to purchase your dream car. It is time to do your research and make sure that your dream car is worth the cost.

Algebra

Answer Key/Teacher Guide

Research your dream car. How much does the car cost?

M PL

E

Teacher: Have students research car costs online, or in ads. If resources are not available then discuss the average cost of a car and guide students to an appropriate price.

What other factors do you need to consider that may impact the price of the car?

Algebra

Teacher: Help students to understand that interest affects the amount that you end up paying after the loan is paid off. The length of the loan also makes a difference. Sales tax, title, and transfer taxes may also apply depending on the state. Students may also choose to add extra features and warranties to their car. If students make a down payment or a trade-in this will help to reduce their monthly payment.

SA

What would your monthly payment be including the interest for a five-year loan and for a three-year loan?

Teacher: Discuss that the interest rate is the yearly interest rate and compounds each year with respect to their payment. Their monthly payment will be made up of interest and principal. A monthly auto loan calculator can be used to explore payment.

Teacher: Students should see that although their monthly payment is more on a three-year loan they are paying less interest and therefore paying less.

Algebra

How much will you ACTUALLY pay for the car in after you have completed your payments on a five-year loan?

On a three-year loan?

Part B Cars decrease in value as soon as they are driven off of the lot. Therefore your dream car’s value has a rate of decay. Why do you think this is?

CoreConnects: Mathematics

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Algebra

Teacher: Help students to see that factors such as age, lack of warranty, wear and tear, millage, and new technology pay a role in the deprecation.

Algebra

CoreConnects: Mathematics – Algebra – Dream Car What do you think the graph of the value of your car would look like? Why do you think this?

Teacher: Help students to understand that the graph will decrease slowly at first and then sharply as the car starts to age.

Confirm your results by looking up the value of your car in “good” condition by examining the value of your dream car after time has passed, and graphing the results on a sheet of graph paper. Remember to label your axis.

Algebra

Teacher: Values will differ depending on the cars selected. If technology is not available to research prices, students can make predictions and graph the results. Discuss the reasonableness of the predictions and allow students time to make changes as needed. Samples values are listed below for a $20,000 car. Also, discuss an appropriate scale for the graph.

Age of Car: Value:

New

1 yr.

2 yrs.

3 yrs.

4 yrs.

5 yrs.

6 yrs.

8 yrs. 10 yrs. 20 yrs. old old old old old old old old old $20,000 $18,000 $17,000 $15,000 $14,000 $13,000 $12,000 $10,000 $8,500 $4,000

E

Dream Car Deprecia4on

M PL

25000

Value

Algebra

20000 15000 10000 5000

SA

0

0

5

10

15

20

Projected Actual

25

Algebra

Years Owned

Part C

This relationship can be represented by the equation A = a(1 + r)t where A represents the actual value, t represents time in years, and a represents the initial value. Explain why this equation is representative of the rate of decay of a car. Why is there a 1 in the equation?

Algebra

Teacher: Help students to understand that decay would be a negative rate of change. If a car is decreasing in value by 8 percent each year that means that it would maintain 92% of its value each year. (1 – 1.08 = 0.92)

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CoreConnects: Mathematics – Algebra – Dream Car

Teacher: The sample data was generated for a $20,000 car depreciating at a rate of 8% a year and rounded to the nearest dollar.

Algebra

Use the equation to explore the projected value of your dream car after time has passed. Use a different color and add this to the graph above.

Age of New 1 yr.

2 yrs.

3 yrs.

4 yrs.

5 yrs.

6 yrs.

8 yrs. 10 yrs. 20 yrs.

Car: old old old old old old old old old Projected $20,000 $18,400 $16,928 $15,574 $14,328 $13,182 $12,127 $10,264 $8,688 $3,774 Value: How do the two graphs compare? What factors could explain the similarities and differences in the graphs?

If you were to sell your car after four years would you make enough to pay off the balance of your loan? Why or why not?

Algebra

Teacher: The data on the actual graph will vary more than the data on the projected graph.

SA

Algebra

M PL

E

Teacher: Students should refer to part A to calculate how much they would owe, and part C to determine the value of their car after four years. If time allows explore other options.

Algebra Algebra

CoreConnects: Mathematics

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M PL SA

Algebra

Algebra

Algebra

E

Algebra

Algebra

CoreConnects: Mathematics – Algebra – Dream Car

CoreConnects: Mathematics

30

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Geometry – Investment Property Geometry

You are investing in a new studio condo development. As part of the investment process you are involved with the design of the condos.

Part A A studio condo is a condo where there is no bedroom. What areas need to be included in the condo?

(Note: Areas do not always need to be separated by a wall.)

E

Geometry

What appliances and features need to be included in the condo? Research the approximate dimensions of each item (round dimensions to the nearest foot).

Part B

Front of Condo B

SA

Front of Condo A

Geometry

M PL

Below is the blueprint for four condos. They are built to touch each other on two sides. Each block on the blueprint represents two square feet. Help design Condo A. Make sure to include the designated areas, appliances, and features identified in Part A. Don’t forget the front door! You may add walls for the closest and bathroom as needed.

Geometry

Front of Condo C

Front of Condo D

Geometry

CoreConnects: Mathematics

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Geometry

CoreConnects: Mathematics – Geometry – Investment Property

Part C When developments are constructed the same layouts, materials, and features are often used in each unit to maximize efficiency and value. Condo B will be the mirror image of Condo A. Complete the blueprint of Condo B. How does the layout of Condo B compare to the layout of Condo A?

M PL SA

Geometry

Geometry

Geometry

E

Geometry

Condos C and D will have the same blueprints as Condos A and B respectively, but will be rotated 90 degrees.

Complete the blueprint for condos C and D. Discuss how Condo C and Condo D compare to Condo A.

CoreConnects: Mathematics

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CoreConnects: Mathematics – Geometry – Investment Property

Geometry

Common Core Standard Assessed Skill G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.

Specify a sequence of transformations that will carry a given figure onto another.

G.CO.A.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Geometry

Answer Key/Teacher Guide

E

You are investing in a new studio condo development. As part of the investment process you are involved with the design of the condos.

Part A

Teacher: Students should identify a sleeping/living area, kitchen, and bath at minimum. They may choose to add other features, but try to limit additional features to three.

What appliances and features need to be included in the condo? Research the approximate dimensions of each item (round dimensions to the nearest foot).

Geometry

M PL

A studio condo is a condo where there is no bedroom. What areas need to be included in the condo? (Note: areas do not always need to be separated by a wall.)

Below is the blueprint for four condos. They are built to touch each other on two sides. Each block on the blueprint represents two square feet. Help design Condo A. Make sure to include the designated areas, appliances, and features identified in Part A. Don’t forget the front door! You may add walls for the closest and bathroom as needed.

Geometry

Part B

SA

Teacher: Help students to identify features that are standard in all homes no matter the size, and some items that may be common. E.g. bathroom: sink, tub, toilet; kitchen: stove, cabinets/counter, refrigerator, dishwasher (optional); living area: fireplace; bedroom: closet; other features: washer/ dryer, coat closet. Dimensions can be researched online, looked up in an ad, measured in the classroom, or estimated.

Teacher: Discuss the placement of each object. Students may choose to eliminate some luxury items. Students can also make paper models of each item in their home, and trace them when they like their placement.

Geometry

CoreConnects: Mathematics

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CoreConnects: Mathematics – Geometry – Investment Property Geometry

Front of Condo A

Bathroom

Front of Condo B

Kitchen

E

Front of Condo D

M PL

Geometry

Part C

Kitchen

Front of Condo C

Bathroom

Living Room/Sleeping

Geometry

Living Room/Sleeping

When developments are constructed the same layouts, materials, and features are often used in each unit to maximize efficiency and value. Condo B will be the mirror image of Condo A. Complete the blueprint of Condo B. How does the layout of Condo B compare to the layout of Condo A?

SA

Teacher: Help students to understand that Condo B is a reflection of Condo A, when the wall separating them is similar to the Y-axis.

Teacher: The original condo and a rotation is depicted above in Condo D. Have students predict what the layouts will look like and make changes as needed. If students made paper models of the appliances and features they can use them to model the rotations and reflections. As students move from condo to condo ask them to identify and explain any patterns in their rotations of the appliances. Condo D is a reflection and a rotation of Condo A; Condo B is a reflection of Condo A; and Condo D is a rotation of Condo A. Condo C is a reflection of Condo D.

Geometry

Geometry

Condos C and D will have the same blueprints as Condos A and B respectively, but will be rotated 90 degrees.

Complete the blueprints for condos C and D. Discuss how Condo C and Condo D compare to Condo A.

CoreConnects: Mathematics

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E

M PL

SA

E

M PL

SA