Table of Contents - Rowland Blogs

Table of Contents  CHAPTER 3: EXPRESSIONS AND EQUATIONS PART 1 (4‐5 WEEKS) ................................................................................ 3 3.0 Anchor Problem: Tiling a Garden Border .............................................................................................................................................. 6 SECTION 3.1: COMMUNICATE NUMERIC IDEAS AND CONTEXTS USING MATHEMATICAL EXPRESSIONS AND EQUATIONS ................. 8 3.1a Class Activity: Matching Numerical Expressions to Stories ........................................................................................................ 9 3.1b Class Activity: Numeric and Algebraic Expressions ..................................................................................................................... 13 3.1b Homework: Matching and Writing Expressions for Stories ..................................................................................................... 16 3.1b Additional Practice ...................................................................................................................................................................................... 18 3.1c Class Activity: Algebra Tile Exploration ............................................................................................................................................. 20 3.1c Homework: Algebra Tile Exploration ................................................................................................................................................. 23 3.1d Class Activity: More Algebra Tile Exploration ................................................................................................................................ 26 3.1d Homework: More Algebra Tile Exploration ..................................................................................................................................... 29 3.1e Class Activity: Vocabulary for Simplifying Expressions .............................................................................................................. 31 3.1e Homework: Solidifying Expressions ..................................................................................................................................................... 34 3.1f Class Activity: Iterating Groups .............................................................................................................................................................. 35 3.1f Homework: Iterating Groups ................................................................................................................................................................... 42 3.1g Class Activity: More Simplifying ............................................................................................................................................................ 43 3.1g Homework: More Simplifying ................................................................................................................................................................. 47 3.1g Additional Practice: Iterating Groups ................................................................................................................................................ 48 3.1 g Additional Practice: Simplifying .......................................................................................................................................................... 49 3.1h Class Activity: Modeling Context with Algebraic Expressions ................................................................................................. 51 3.1h Homework: Modeling Context with Algebraic Expressions ...................................................................................................... 53 3.1i Class Activities: Properties. ....................................................................................................................................................................... 54 3.1i Homework: Properties ................................................................................................................................................................................ 59 3.1j Class Activity: Using Properties to Compare Expressions ........................................................................................................... 61 3.1k Classwork: Modeling Backwards Distribution ................................................................................................................................ 64 3.1k Homework: Modeling Backwards Distribution .............................................................................................................................. 69 3.1l Self‐Assessment: Section 3.1 ..................................................................................................................................................................... 70 SECTION 3.2 SOLVE MULTI‐STEP EQUATIONS ................................................................................................................................................ 71 3.2a Classroom Activity: Model Equations ................................................................................................................................................. 72 3.2a Homework: Model and Solve Equations ............................................................................................................................................ 77 3.2b Class Activity: More Model and Solve One‐ and Two‐Step Equations .................................................................................. 81 3.2b Homework: More Model and Solve One‐ and Two‐Step Equations ....................................................................................... 84 3.2c Class Activity: Model and Solve Equations, Practice and Extend to Distributive Property ........................................ 87 3.2c Homework: Model and Solve Equations, Practice and Extend to Distributive Property ............................................. 90 3.2d Class Activity: Error Analysis .................................................................................................................................................................. 93 3.2d Homework: Practice Solving Equations (select homework problems) ............................................................................... 95 3.2e Homework: Solve One‐ andTwo‐ Step Equations (practice with rational numbers) ................................................... 98 3.2e Extra Practice: Equations with Fractions and Decimals ........................................................................................................ 100 3.2f Class Activity: Create Equations for Word Problems and Solve ........................................................................................... 102 3.2f Homework: Create Equations for Word Problems and Solve ................................................................................................ 104 3.2g **Class Challenge: Multi‐Step Equations ........................................................................................................................................ 106 3.2h Extra Practice: Solve Multi‐Step Equation Review .................................................................................................................... 107 3.2i Self‐Assessment: Section 3.2 .................................................................................................................................................................. 113 SECTION 3.3: SOLVE MULTI‐STEP REAL‐WORLD PROBLEMS INVOLVING EQUATIONS AND PERCENTAGES .................................... 114 3.3a Classroom Activity: Percents with Models and Equations ..................................................................................................... 115 3.3a Homework: Percents with Models and Equations ..................................................................................................................... 117 3.3b Class Activity: Percent Problems ........................................................................................................................................................ 119 3.3b Homework: Percent Problems ............................................................................................................................................................. 122 3.3c Class Activity: More Practice with Percent Equations .............................................................................................................. 124

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3.3c Homework: More Practice with Percent Equations .................................................................................................................. 126 3.3d Self‐Assessment: Section 3.3 ................................................................................................................................................................. 127

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CHAPTER 3: Expressions and Equations Part 1 (4-5 weeks) UTAH CORE Standard(s): Expressions and Equations 1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.1 2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.EE.2 3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 7.EE.3 4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations to solve problems by reasoning about the quantities. 7.EE.4 a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. 7.EE.4a

CHAPTER OVERVIEW: The goal of chapter 3 is to facilitate students’ transition from concrete representations and manipulations of arithmetic and algebraic thinking to abstract representations. Each section supports this transition by asking students to model problem situations, construct arguments, look for and make sense of structure, and reason abstractly as they explore various representations of situations. In Chapter 3, students work with fairly simple expressions and equations to build a strong intuitive understanding of structure (for example, students should understand the difference between 2x and x2 or why 3(2x  1) is equivalent to 6x  3.) They will continue to practice skills manipulating algebraic expressions and equations throughout Chapters 4 and 5. In Chapter 6 students will revisit ideas in this chapter to extend to more complicated contexts and manipulate with less reliance on concrete models. Section 3.1 reviews and builds on students’ skills with arithmetic from previous courses to write basic numerical and algebraic expressions in various ways. In this section students should understand the difference between an expression and an equation. Further, they should understand how to represent an unknown in either an expression or equation. Students will connect manipulations with numeric expressions to manipulations with algebraic expressions. They will then come to understand that the rules of arithmetic are naturally followed when working with algebraic expressions and equations. Lastly, students will name the properties of arithmetic. By the end of this section students should be proficient at simplifying expressions and justifying their work with properties of arithmetic. Section 3.2 uses the skills developed in the previous section to solve equations. Students will need to distribute and combine like terms to solve equation. This section will rely heavily on the use of models to solve equations. Section 3.3 ends the chapter with applications of solving equations involving percent including ones with percent of increase and percent of decrease. Problems in this section should be solved using models. There will be similar exercises in Chapter 4 where students will used an algebraic equation approach.

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VOCABULARY: expression, equation, simplify, rational number, integer, term, like terms, constant, variable, factor, product, coefficient, unknown

CONNECTIONS TO CONTENT: Prior Knowledge Students will extend the skills they learned with manipulatives in previous grades with addition/subtraction and multiplication/division of whole numbers/integers to algebraic expressions in a variety of ways. For example, in elementary school students modeled 4  5 as four “jumps” of five on a number line. They should connect this thinking to the meaning of “4x”. Students also modeled multiplication of whole numbers using arrays in earlier grades,; in this chapter they will use that logic to multiply using variables. Additionally, in previous grades, students explored and solidified the idea that when adding/subtracting one must have “like units.” Thus, when adding 123 + 14, we add the “ones” with the “ones,” the “tens” with the “tens” and the “hundreds” with the “hundreds;” or we cannot add ½ and 1/3 without a common denominator because the a unit of ½ is not the same as a unit of 1/3. Students should extend this idea to adding variables. Hence, 2x + 3x is 5x because the unit is x, but 3x + 2y cannot be simplified further because the units are not the same (three units of x and two units of y.) In 6th grade students solved one step equations. Students will use those skills to solve equations with more than one-step in this chapter. Earlier in this course, students developed skills with rational number operations. In this chapter, students will be using those skills to solve equations that include rational numbers. Future Knowledge As students move on in this course, they will continue to use their skills in working with expressions and equations. Those same skills will be used to solve more real-life applications that use equations and well as solving inequalities. In future courses, students will be able to solve equations of all forms by extending properties.

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MATHEMATICAL PRACTICE STANDARDS (emphasized): Make sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments and critique the reasoning of others Model with mathematics

Attend to precision

Students will make sense of expressions and equations by creating models and connecting intuitive ideas to properties of arithmetic. Properties of arithmetic should be understood beyond memorization of rules. Students will, for example, note that x + x + x + x + x is the same as 5x. Students should extend this type of understanding to 5(x + 1) meaning five groups of (x + 1) added together, thus simplifying to 5 x + 5. For each of the properties of arithmetic, students should connect concrete understanding to abstract representations. Students should be able to explain and justify any step in simplifying or solving an expression or equation in words and/or pictures. Further, students should be able to evaluate the work of others to determine the accuracy of that work and then construct a logical argument for their thinking. Students should be able to model all expression and equations throughout this chapter. Further, they should be able to interchange models with abstract representations. Students demonstrate precision by using correct terminology and symbols when working with expressions and equations. Students use precision in calculation by checking the reasonableness of their answers and making adjustments accordingly.

Using models students develop an understanding of algebraic structures. For example, in section 3.2 students should understand the structure of an equation like 3x + 4 = x + 5 as meaning the same thing as 2x = 1 or x = ½ Look for and when it is “reduced.” Another example, in section 3.3 students will use a make use of model to show that a 20% increase is the original amount plus 0.2 of the structure original amount or 1.2 of the original amount (though they will not write equations until Chapter 4.) Students demonstrate their ability to select and use the most appropriate tool Use appropriate (paper/pencil, manipulatives, and calculators) while solving problems. tools Students should recognize that the most powerful tool they possess is their strategically ability to reason and make sense of problems. Students will study patterns throughout this chapter and connect them to both Look for and their intuitive understanding and properties of arithmetic. express regularity in repeated reasoning

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3.0 Anchor Problem: Tiling a Garden Border Imagine that you are putting 1-foot square tiles around the edge of a square garden. Without counting directly, how could you figure out how many tiles go around the garden that is 4 feet by 4 feet? Write down four ways that you could quickly “add up” the tiles. 4 feet across

Method 1: Method 2: Method 3: Method 4:

Without counting directly, how could you figure out how many tiles go around the garden that is 5 feet by 5 feet? Show how you could adapt the methods that you used above to “add up” these tiles. 5 feet across

Method 1: Method 2: Method 3: Method 4:

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10 feet across

Show how you could adapt the methods that you used above to “add up” the number of tiles for a 10 foot by 10 foot square garden. Method 1: Method 2: Method 3: Method 4:

What if the garden were 20 feet by 20 feet? Demonstrate how each method would work now. 20 feet across

Method 1: Method 2: Method 3: Method 4:

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Section 3.1: Communicate Numeric Ideas and Contexts Using Mathematical Expressions and Equations Section Overview: This section contains a brief review of numerical expressions. Students will recognize that a variety of expressions can represent the same situation. Models are encouraged to help students connect properties of arithmetic in working with numeric expressions to working with algebraic expressions. These models, particularly algebra tiles, aid students in the transition to the abstract thinking and representation. Students extend knowledge of mathematical properties (commutative property, associative property, etc.) from purely numerical problems to expressions and equations. The distributive property is emphasized and factoring, “backwards distribution,” is introduced. Work on naming and formally defining properties is at the end of the section. Through the section students should be encouraged to explain their logic and critique the logic of others.

Concepts and Skills to be Mastered (from standards) By the end of this section, students should be able to: 1. Use the Distributive Property to expand and factor linear expressions with rational numbers 2. Combine like terms with rational coefficients 3. Recognize and explain the meaning of a given expression and its component parts. 4. Recognize that different forms of an expression may reveal different attributes of the context.

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3.1a Class Activity: Matching Numerical Expressions to Stories Four students, Aaron, Brianna, Chip, and Dinah, wrote a numerical expression for each story problem. Look at each student’s expression and determine whether or not it is appropriate for the given story problem. Explain why the expression “works” or “doesn’t work.” 1. Josh made five 3-pointers and four 2-pointers at his basketball game. How many points did he score? Expression Evaluate Does it work? Why or Why Not? a. 3 + 3 + 3 + 3 + 3 + 2 + 2 +2 + 2 b. 5 + 3 + 4 + 2 c. (5 + 3)(4 + 2) d. 5(3) + 4(2)

2. I bought two apples for $0.30 each and three pounds of cherries for $1.75 a pound. How much did I spend? Expression Evaluate Does it work? Why or Why Not? a. 2(0.30) + 3 (1.75) b. 2(1.75) + 3 (0.30) c. 0.30 + 0.30 + 1.75 + 1.75 + 1.75 d. (2+ 3)(0.30 + 1.75)

3. I bought two apples for $0.30 each and three oranges for $0.30 each. How much money did I spend? Expression Evaluate Does it work? Why or Why Not? a. (0.30 + 0.30 + 0.30) + (0.30 + 0.30) b. 3(0.30) + 2(0.30) c. 5(0.30) d. 0.60 + 0.90

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4. Aunt Nancy gave her favorite niece 3 dollars, 3 dimes, and 3 pennies. How much money did her niece receive? Expression a. 3(1.00) + 3(0.10) + 3(0.01)

Evaluate Does it work?

Why or Why Not?

b. 3(1.00 + 0.10 + 0.01) c. 3 + 1.00 + 0.10 + 0.01 d. 3.00 + 0.30 + 0.03

5. Aunt Nancy gave each of her two nephews the same amount of money. Each nephew received one dollar, one quarter, and one dime. How much did the two nephews receive altogether? Expression a. 1 + 1 + 0.25 + 0.25 + 0.10 + 0.10

Evaluate

Does it work?

Why or Why Not?

b. 1 + 0.25 + 0.10 c. 2 (1 + 0.25 + 0.10) d. 2(1) + 0.25 + 0.10

6. Aunt Nancy gave her favorite niece two dollars, 1 dime, and 3 pennies. How much money did her niece receive?

7. Uncle Aaron gave 8 dimes, 2 nickels, and 20 pennies to his nephew. How much money did he give away?

8. I bought 2 toy cars for $1 each and 3 toy trucks for $1.50 each. How much did I spend? 9. The football team scored 1 touchdown, 3 field goals, and no extra points. How many points did they score in all? Hint: a touchdown is worth 6 points, a field goal worth 3, and an extra point worth one.

10. I had $12. Then I spent $2 a day for 5 days in a row. How much money do I have now? 11. I earned $6. Then I bought 4 candy bars for $0.75 each. How much money do I have left?

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3.1a Homework: Matching Numerical Expressions to Stories Four students, Aaron, Brianna, Chip, and Dinah, wrote a numerical expression for each story problem. Look at each student’s expression and determine whether or not it is appropriate for the given story problem. Explain why the expression “works” or “doesn’t work.” 1. I bought two toy cars for $5 each and three toy trucks for $7 each. How much did I spend? Expression

Evaluate Does it work?

Why or Why Not?

a. 2(5) + 3(7) b. 2(3) + 5(7) c. (2 + 3)(5 + 7) d. (2 + 3) + (5 + 7)

2. The football team scored three touchdowns, two field goals, and two extra points. How many points did they score in all? (Hint: a touchdown is 6 points, a field goal is 3 points, and an extra point is just 1 point) Expression a. 3(6) + 2(3) + 2(1)

Evaluate Does it work?

Why or Why Not?

b. 6 + 6 + 6 + 3 + 3 + 1 + 1 c. (6 + 6 + 6) + (3 + 3) + (1 + 1) d. 18 + 6 + 2

3. I earned $6. Then I bought 4 candy bars for $0.50 each. How much money do I have left? Expression a. 6 – 0.50 – 0.50 – 0.50 – 0.50

Evaluate Does it work?

b. 6 – 4(0.50) c. 6 – (0.50 – 0.50 – 0.50 – 0.50) d. 6 – (0.50 + 0.50 + 0.50 + 0.50)

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Why or Why Not?

4. I earned $5. Then I spent $1 a day for 2 days in a row. How much money do I have now? Expression a. 5 − 1 + 1

Evaluate

Does it work?

Why or Why Not?

b. 5 – 1 – 1 c. 5 – (1 – 1) d. 5 – (1 + 1)

5. Uncle Aaron gave 2 dimes, 3 nickels, and 2 pennies to his nephew. How much money did he give away? Expression a. 2(0.10) + 3(0.05) + 2(0.01)

Evaluate

Does it work?

Why or Why Not?

b. 2(0.10 + 0.01) + 3(0.05) c. 0.20 + 0.15 + 0.02 d. 2 + 0.10 + 3 + 0.05 + 2 + 0.01

Write an expression of your own for each problem. Then evaluate the expression to solve the problem. 6. Josh made ten 3-pointers and a 2-pointer at his basketball game. How many points did he score?

7. I bought three apples for $0.25 each and 3 pounds of cherries for $1.75 a pound. How much money did I spend? 8. I bought five apples for $0.30 each and 5 oranges for $0.30 each. How much money did I spend?

9. I bought two t-shirts at $12 each and 3 sweaters for $20 each. How much did I spend altogether?

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3.1b Class Activity: Numeric and Algebraic Expressions Read each story problem. Determine if you think the expression is correct. Evaluating the expression for the given value. Explain why the expression did or didn’t work for the given problem. 1. Ryan bought 3 CDs for x dollars each and a DVD for $15. How much money did he spend? Expression

Correct expression?

Evaluate x=7

Did it work?

Why or why not?

a. 3 + x + 15 b. 15x + 3 c. 15 + x + x + x d. 3x + 15

2. I started with 12 jellybeans. Sam ate 3 jellybeans and then Cyle ate y jellybeans. How many jellybeans were left? Expression a.

12 – 3 – y

b.

12 – (3 – y)

c.

12 – (3 + y)

d.

9–y

Correct expression?

Evaluate y=6

Did it work?

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Why or why not?

3. Kim bought a binder for $5, colored pencils for $2, and 4 notebooks for n dollars each. How much did she spend? Expression

Do you think it will work?

Evaluate (use n = 3)

Did it work?

Why or why not?

a. 5 + 2 + 4 + n b. 4n + 5 + 2 c. 7 + 4n d. n + n + n + n + 7

For each context below, draw a model for the situation, label all parts and then write an expression that answers the question. The first exercise is done for you. 4. Jill bought 12 apples. Jan bought x more apples than Jill. How many apples did Jan buy? Jan bought 12 + x apples.

5. Josh won 12 tickets. Evan won p tickets less than Josh. How many tickets did Evan win?

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6. Tim is 3 years younger than his brother. If his brother is y years old, how old is Tim?

7. I washed w windows less than Carol, who washed 8 windows. If I get paid $2 for each window I wash, how much did I earn?

8. Jan bought a more apples than Jill. Jill bought 4 apples. Each apple costs $0.10. How much money did Jan spend on apples?

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3.1b Homework: Matching and Writing Expressions for Stories Read each story problem. Determine which expressions will work for the story problem given. Try evaluating the expression for a given value. Explain why the expression did or didn’t work for the given problem. 1. Bob bought 5 books for x dollars each and a DVD for $12. How much money did he spend? Expression

Do you think it will work?

Evaluate (use x = 5)

Did it work?

Why or why not?

a. 5 + x + 12 b. 5(x)12 c. x + x + x + x + x + 12 d. 5x + 12

2. Jim won 30 tickets. Evan won y tickets less than Jim did. How many tickets did Evan win? Expression

Do you think it will work?

Evaluate (use y = 6)

Did it work?

Why or why not?

a. 30 – y b. y  30 c. y + 30 d. 30 ÷ y

Draw a model and then write an expression for each problem. 3. I did 4 more problems than Minnie. If I did p problems, how many did Minnie do?

4. I bought x pairs of shoes for $25 each and 2 pairs of socks for $3 each. How much did I spend?

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5. I bought m gallons of milk for $2.59 each and a carton of eggs for $1.24. How much did I spend?

6. Paul bought s sodas for $1.25 each and chips for $1.75. How much did he spend?

7. Bob and Fred went to the basketball game. Each bought a drink for d dollars and nachos for n dollars. How much did they spend on two drinks and two orders of nachos?

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3.1b Additional Practice Draw a model and write an expression for each problem. 1. Marina has $12 more than Brandon. Represent how much money Marina has.

2. Conner is three times as old as Jackson. Represent Conner’s age.

3. Diane earned $23 less than Chris. Represent how much Diane earned.

4. Juan worked 8 hours for a certain amount of money per hour. Represent how much Juan earned.

5. Martin spent 2/3 of the money in his savings account on a new car. Represent the amount of money Martin spent on a new car.

6. Brianne had $47 dollars. She spent $15 on a new necklace and some money on a bracelet. Represent the amount of money Brianne has now.

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7. For 5 days Lydia studied math for a certain amount of time and read for 15 minutes each day. Represent the total amount of time Lydia studied and read over the 5 day period.

8. Carlos spent $8 on lunch, some money on a drink and $4 on ice cream. Represent how much money Carlos spent.

9. Nalini has $26 dollars less than Hugo. Represent the amount of money Nalini has.

10. Bruno ran four times as far as Milo. Represent the distance Bruno ran.

11. Christina earned $420. She spent some of her earnings on her phone bill and spent $100 on new clothes. Represent the amount of money Christina now has.

12. Camille has 4 bags of candy. Each bag has 3 snicker bars and some hard candy. Represent the amount of candy Camille has.

13. Heather spent ¼ of the money in her savings account on a new cell phone. Represent the amount of money Heather spent on the new cell phone.

14. Miguel is 8 years older than Cristo. Represent Miguel’s age.

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3.1c Class Activity: Algebra Tile Exploration In using Algebra tiles, every variable is represented by a rectangle, positive or negative and every integer is represented by a square, positive or negative.

Key for Tiles: 1

=1

1 = –1

x

=x

x

= –x

Write an expression for what you see and then write the expression in simplest form. 1. 2. 3. 1

1

1

1

1

1

1

1

1

x

1

1

x

x

x

1

1

x

x

x

4.

1

5.

x

x

x

x

x x

1

6.

x

1

x

1

x

1

x 1

x x

1

x 1

20

x

1

x

Use the algebra tile key above to model each expression on your desk. Sketch a picture in the space below. 7. x+6

8. –3 x + –2

9. 4x–1

Is there more than one way to model each of the expressions above? Justify.

Draw a model for each of the expressions below using the key from previous page. Simplify if you can. 10. 2x + 1 + x

11. 3x +4 + (–2)

12. 2x + x – 2 + 3

13. –3x + 1 + –x

14. 2x + –3 + –2x

15. –x + 3 + 4x

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16. –2x + 4 + x  7

17. 4x – 3 + 2 – 2x

18. –4x – 1 + 3x + 2 – x

19. x+x+x+x

20. x–x–x–x

21. –x – x – x – x

22. 3 – 2x + x  5

23. 4 – 2 – 4x – 2

24. 2x – x – 3 + 5

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3.1c Homework: Algebra Tile Exploration Use the key below to interpret or draw the algebraic expressions in your homework. Key for Tiles: 1

=1

1 = –1

x

=x

x

= x

Write an expression for each model below. 1. 2. 1

1

1

1

1

4.

x

x

1

1

x

x

1

1

x

x

x

x x

1

1

x

x

x

x

x

x

x

1

1

x

x

1

1

5.

x x 1

3.

6. 1

x

x

x

1

x

x

x

1

x

1

1

1

x

1

x

x

x

x

23

1

1

1

1

7. 2x + 4

8. x–5

9. 2x – 3 + 5

10. 3x + 2 – 2x

11. 2x + 1 + 3 – 5

12. –2x + 4 – 3

13. –2x + 3 + 5x – 2

14. 5x – 3 – 4 + x

15. –3x + 1 + 2x – 3

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16. x + 4 + –3x – 7

17. –x – 3 + 2x – 2

18. 4x – 3 – 7x + 4

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3.1d Class Activity: More Algebra Tile Exploration Miguel saw the following two expressions: 17 + 4 + 3 + 16

43 – 8 – 3 + 28

He immediately knew the sum of the first group is 40 and the sum of the second set is 50. How do you think he quickly simplified the expressions in his head?

Key for Tiles: 1

=1

1 = –1

x

=x

x

= –x

For 1-16 model each expression using Algebra Tiles. Then simplify each expression. 1. 8x + 2x + 7

2. 3 + 2x + x

3. 5x – 9x

4. –6 + 4x + 9 – 2x

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5. –3 + 3x + 11 – 5x

6. 9x – 12 + 12

7. 1 + x + 5x – 2

8. –4x – 5x

Your friend is struggling to understand what it means when the directions say “simplify the expression.” What can you tell your friend to help him?

Your friend is also having trouble with expressions like problems #5 and #8. He’s unsure what to do about the “ – “. What might you say to help him?

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For 9 – 16 use the key below. 1 1

9.

=1

x

=x

= –1

x

= –x

x+x+y+x+2

y

y

10.

2x + y + x + 3 + 3y + 2

11. 3x + –x + y + –y + 2

12. x + y – x – y + 2

13. –2x + 2y – y – y – x

14. –2 + 3x – 4 + 2x – y + 2y

15. 5x – 2y + 4 – 3x + y + x – 2

16. –5 + x – y – 2y + 3x – 7

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=y = y

3.1d Homework: More Algebra Tile Exploration

Key for Tiles: 1

=1

1 = –1

x

=x

x

= –x

For 1-16 model each expression using Algebra Tiles. Then simplify each expression, combining like terms. .

1. 5x + 3x + 5

2. 1 + 3x + x

3. 3x – 5x

4. 7 – 2x – 9 + 4x

5. –4x – 5 + 2x

6. 5 – 4x + 5x

7. –4x – 5 + x + 7x

8. x – 6x

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For 9 – 16 use the key below. 1 1

9.

=1

x

=x

= –1

x

= –x

y

y

2x + x + 2y + x + 1

10. x + 3y + x + 2 + y + 1

11. x + –2x + 3y + y + 3

12. x + 3y – 1x + 2y + 1

13. –2x + y – 3y + y + x

14. 5 + x – 4 + x – 2y + y

15. 5x – y + 2 – 4x + 2y + x + 2

16. –2 + 2x – 2y + x – 3

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=y = y

3.1e Class Activity: Vocabulary for Simplifying Expressions 1. In groups of 2 or 3 students, consider the following expressions: a) 2x + 5 + 3y, b) 2x + 5 + 3x, and c) 2x + 5x + 3x. How are these expressions similar? How are they different?

Parts of an Algebraic Expression: Use the diagrams to create definitions for the following vocabulary words. Be prepared to discuss your definitions with the class. Terms Constants

x42y35y

x42y35y There are five terms in this expression. The terms are x, –4, 2y, 3, and –5y

The constants are –4 and 3. (Recall that subtracting is like adding a negative number.)

Coefficients

Like Terms

x42y35y

x42y35y

The coefficients are 1, 2, and –5

2y and –5y are like terms. –4 and 3 are also like terms.

Terms: Constant: Coefficient: Like Terms: 2. Identify the terms, constants, coefficients, and like terms in each algebraic expression. Expression Terms Constants Coefficients

4x, –x, 2y, –3

–3

4, –1, 2

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Like Terms

4x, –x

3. Simplifying Algebraic Expressions. Use the vocabulary words “constant,” “coefficient,” and “like terms” to explain in writing how to simplify each algebraic expression. 3x + 4x

8n + 4 + 4n

6x + 4 – 5x + 7

Simplify each expression. 9. 2x – y + 5x

10. 5y + x + y

11. 2y + 8x + 5y – 1

12. 10b + 2 – 2b

13. 8y + x – 5x – y

14. 9x + 2 – 2

15. –2x – 6 + 3y + 2x – 3y

16. 6m + 2n + 10m

17. 7b – 5 + 2b – 3

18. 2a – 3 + 5a + 2

19. 8x + 5 – 7y + 2x

20. 4y + 3 – 5y – 7

21. 6x + 4 – 7x

22. 2x + 3y – 3x – 9y + 2

32

23. –2b + a + 3b

24. m – 5 + 2 – 3n

25. 4 + 2r + q

26. 5h – 3 + 2k – h + k

27. 5t – 3 – t + 2

28. c + 2d – 10c + 4

33

3.1e Homework: Solidifying Expressions Matching: Write the letter of the equivalent expression on the line. 1. ____ 3x – 5x

a) –8a

2. ____ 4a – 12a

b) 2x – 6

3. ____ 3x + 5x

c) –x + y

4. ____ 16a + a

d) 9x – 4y

5. ____ 2x – 2y + y

e) –2x

6. ____ 2x – 2 – 4

f) 2x – y

7. ____ x – y + 2x

g) x + y

8. ____ –y + 2y – x

h) 3x – y

9. ____ 5x + 4y – 3x – x + 3y – 6y

i) 17a

10. ____ 4x + 3y + 5x – 7y

j) 8x

Simplify each expression by combining like terms. 11.

21.

12.

22.

13.

23.

14.

24.

15.

25.

16.

26.

17.

27.

18.

28.

19.

29. 30.

20. 34

3.1f Class Activity: Iterating Groups Review: Show different ways to expand 5(10).

Draw different array and number line models to show 5(10) is 50; use your answers above to come up with different representations.

35

36

1 1

=1

x

=x

= –1

x

= –x

y

y

=y = y

Use the key above for the following: 1. Model: 2x + 2y

Can you write 2x + 2y in a different way? How?

2. Suppose for the expression 2x + 2y that y = 5. In the space below, create a new model for the expression and write the expression in different ways.

37

3. Model: 4x + 12

Can you write 4x + 12 in different ways? How?

4. Model 6x + 12

Can you write 6x + 12 in different ways?

How are problems # 3 and # 4 related? How are they related to # 1?

38

5. Model 5(2x + 1) and then simplify.

Write 5(2x +1) in two different ways.

6. Model 4(3x – 2) and simplify.

Write 4(3x – 2) in three different ways.

What does the number in front of the parentheses tell you about the grouping?

39

7. Model 3(2x – 5) and simplify.

Write 3(2x – 5) in three different ways. In 6th grade you learned that expressions like 6(2 + 3) could be written as 6(5) or 6(2) + 6(3). We extended that thinking to expressions like 5(2x + 1) and found it could be written as 5(2x) + 5(1) or 10x + 5. In exercises 6 and 7 we saw that expressions like 4(3x – 2) can be written as 12x – 8; 12 + (–8); 4(3x) – 4(2); 4(3x) + (–4(2)). How can you use what we’ve learned about integers and what we know about writing expressions with parentheses to re-write expressions that have “ – “ in the groupings?

Use what you have learned to rewrite each numeric expression. The first one is done for you: 9. 3(1 + 6) 10. 5 (4 – 1) 8. 4(2 + 3) = 4(2) + 4(3) =8 + 12 = 4(5) = 20

11. 2(7 – 2)

12. 4(3 – 5)

13. 5(2 + 3)

40

Draw a model for each expression, then rewrite the expression in an equivalent form. 14. 3(x + 2)

15. 2(3x + 5)

16. 3(x + 1)

17. 4(3x – 1)

18. 2(3 + x)

19. 3(3x – 2)

In sixth grade you talked about order of operations, what is the order of operations and how is it related to what’s you’ve been doing above?

41

3.1f Homework: Iterating Groups Simplify each of the following. Draw a model to justify your answer. 1. 3(x + 1)

2. 2(3x + 2)

3. 4(x + 3)

4. 2(3x – 1)

5. 3(2x – 3)

6. 5(x – 1)

The expressions 2(5x – 3) can be written and 10x – 6 OR 10x + (–6). Write the following expressions in two different ways as the example shows: 7. 4(3x – 5)

8. 2(7x – 3)

42

3.1g Class Activity: More Simplifying Review: Discuss the following questions in groups of 2 to 3: What is the opposite of “forward 3 steps”?

What is the opposite of “turn right”

What is the opposite of “forward three steps then turn right”?

Using that logic above, what do you think each of the following means? –x

–(x)

(x+1)

(1  x)

What does “ – “ in front of a set of parentheses tell us?

43

Review: Draw a model of 3(x + 1) then simplify:

What do you think –3(x + 1) means?

In groups of 2 or 3 students, simplify each of the following. Be ready to justify your answer. 2. – (x + 2) 3. – (3x + 2) 1. 4(x + 1)

4. – 2(x + 3)

5. – 3(x – 2)

6. 2(3 – x)

7. – 2(5 – 3x)

8. – (4 – 3x)

9. – 4(2x + 3)

44

Combining ideas: Review and combine ideas: Simplify the following expression: 3x + 5 – x 3(x + 2) – 3(x + 2) + x – 4

In groups of 2 or 3 students, model and simplifying the following exercises. Be prepared to justify your answer. 10. 3x + 5 – x + 3(x + 2) 11. 3x + 5 – x – 3(x + 2)

12. 2(x – 1) + 4x – 6 + 2x

13.

Explain your strategy for simplifying problems 10 through13.

45

– 2(x – 1) + 4x – 6 + 2x

Practice: 13. 5 + 2(x – 3)

14. 7x – 2(3x +1)

15. 6x – 3 + 2x – 2(3x + 5)

16. –9x + 3(2x  5) + 10

17.

18. –(4x – 3) – 5x + 2

(5 – 3x) – 7x + 4

19. 9 – 8x – (x + 2)

20. 15 – 2x – (7 – x)

46

3.1g Homework: More Simplifying Simplify the following expressions: 1. 3(2x + 1)

2. – 3(2x + 1)

3. – 3(2x – 1)

4. – (x + 4)

5. – (x – 4)

6. – (4 – x)

7. – 2(4x – 3)

8. – 5(3x + 2)

9. – 7(2x – 5)

10. 5x + 2(x + 3)

11. 5x + 2(x – 3)

12. 5x – 2(x + 3)

13. 5x – 2(x – 3)

14. 3x + 2 – 4x + 2(3x + 1)

15. –7x + 3 + 2x – 3(x +2)

16. 10x – 4 – 7x – 4(2x  3)

17. 4x – 5(2x  5) – 3x + 4

18. x – 7 – 2(5x – 3) + 4x

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3.1g Additional Practice: Iterating Groups Matching: Write the letter of the equivalent expression on the line 1. ____

a)

2. ____

b)

3. ____

c) d)

4. ____

e)

5. ____

f)

6. ____

g)

7. ____

h)

8. ____

i)

9. ____

j)

10. ____

Practice: Simplify each expression. 11.

20.

29.

12.

21.

30.

13.

22.

31.

23.

32.

14.

15.

33.

24.

34.

16.

25.

17.

26.

35.

18.

27.

36.

19.

28.

37. 48

3.1 g Additional Practice: Simplifying

1.

5x  10y  2x  4y  3x

2.

5x  10y  2x  4y  3x

3.

2x  5y  2x  4y  2z

4.

7w  3q  5  8q  6  10w

5.

3p  2q  4 p  4q  6  4

6.

17v  2  12v 12 15v

7.

3c  6c  5c  2d  4d  3d

8.

10y  10y  3

9.

31y  5x  4  12  13x  23y

10.

3y  2x  5y  5x  10x

11.

4(2x  5)

12.

6(4  2y)

13.

(8  6v)(2)

14.

(3  5h)(3)

16.

1(x  2y)

18.

(3k  5)

15.

17.

6

8

(3x  6)

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19.

(7h  2k)

20.

7(x  5q)

21.

3  4(2x  5)

22.

12  6(4  2y)

23.

5  (8  6v)

24.

4  (3  5h)

25.

2(6h  8)  10

26.

5y  1(x  2y)  6

27.

5x  (3x  6)  6

28.

5k  (3k  5)  8

29.

4h  (7h  2k)  5

30.

7x  7(x  5q)

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3.1h Class Activity: Modeling Context with Algebraic Expressions Look back at the anchor problem. In particular, look back at your work for the two situations below. Recall you were putting 1-foot square tiles around the edge of a square garden and you were trying to figure out how many tiles you’d need for different gardens. Your task was to express the number of tiles you’d need in four different ways.

What if the garden were 100 feet by 100 feet? Demonstrate how each method would work now. 100 feet across

Method 1: Method 2: Method 3: Method 4:

What if the garden were n feet by n feet? Demonstrate how each method would work now. Simplify each method. n feet across

Method 1: Method 2: Method 3: Method 4:

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In the context above you wrote several expressions for each situation; often there is more than one equivalent way to algebraically model a context. Below are contexts, write two equivalent expressions for each situation. It may be helpful to draw a model. 1. Marty and Mac went to the hockey game. Each boy bought a program for 3 dollars and nachos for n dollars. Write two different expressions that could be used to represent how much money the boys spent altogether. Expression 1:

Expression 2:

2. The cooking club would like to learn how to make peach ice cream. There are 14 people in the club. Each member will need to buy 3 peaches and 1 pint of cream to make the ice cream. Peaches cost x cents each, and a pint of cream costs 45 cents. Write two different expressions that could be used to represent the total cost of ingredients for all 14 members of the club. Simplify each expression.

3. Leo and Kyle are training for a marathon. Kyle runs 10 mile per week less than Leo. Write two expressions to represent the distance Kyle ran over 12 weeks if L equals the distance Leo ran every week.

4. Harry is five years younger than Sue. Bridger is half as old as Harry. Write two different expressions that could be used to represent Bridger’s age in terms of Sue’s age. Simplify each expression. (Hint: use a variable to represent Sue’s age.)

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3.1h Homework: Modeling Context with Algebraic Expressions Below are contexts. Write two different expressions for situations 1 and 2 and then simplify. For situations 3 and 4, write an algebraic expression. Draw a model for each. 1. Marie would like to buy lunch for her three nieces. She would like each lunch to include a sandwich, a piece of fruit, and a cookie. A sandwich costs $3, a piece of fruit costs $0.50, and a cookie costs $1. Write two different expressions that could be used to represent the total price of all three lunches. Then simplify each expression that you wrote.

2. Boris is setting up an exercise schedule. For five days each week, he would like to play a sport for 30 minutes, stretch for 5 minutes, and lift weights for 10 minutes. Write two different expressions that could be used to represent the total number of minutes he will exercise in five days. Then simplify each expression that you wrote.

3. Five girls on the tennis team want to wear matching uniforms. They know skirts will costs $24 but are not sure about the price of the top. Write two different expressions that could be used to represent the total cost of all five skirts and tops if x represents the price of one top. Simplify each expression.

4. Drake, Mike, and Vinnie are making plans to go to a concert. The tickets will cost $30 each, and each boy plans to buy a t-shirt for t dollars. Write two different expressions that could be used to represent the total cost for all three boys. Simplify each expression.

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3.1i Class Activities: Properties. In mathematics, there are things called “properties;” you may think of them as “rules.” Properties are the rules for a set of numbers. In today’s lesson, we are going to formally define the properties of arithmetic that you’ve used all along in math. There is nothing new in the properties discussed in this section. Everything you expect to work will work. We’re just giving vocabulary to what you’ve been doing so that when you construct an argument for an answer, you’ll be able to use language with precision. By the end of this section, you should be able to define and explain the properties in pictures, words, and symbols. Commutative Property The word “commute” means “to travel” or “change.” It’s most often used in association with a location. For example, we say people commute to work. For each of the following pairs of expressions, the operation is the same, but the constants have been commuted. Determine if the pairs are equivalent, be able to justify your answer. From these pairs, we are going to try to define the Commutative Property. 1.

2.

12 + 4

9.8 – 3.4

4 + 12

3.4 – 9.8

3.

4.

12 – 4

5·4

4 – 12

4·5

5.

6.

3 · 0.9

18 ÷ 6

0.9 · 3

6 ÷ 18

What pattern are you noticing?

Commutative Property:

54

Associative Property The word “associate” means “partner” or “connect.” Most often we use the word to describe groups. For example, if a person goes to Eastmont Middle School and not Indian Hills Middle School, we would say that person is associated with Eastmont Middle School. For each of the following pairs of expressions, the operations are the same, but the constants have been associated in different ways. Determine if the pairs are equivalent; be able to justify your answer. From these pairs, we’re going to try to define the Associative Property. (12 + 4) + 6

(12 – 4) – 3

12 + (4 + 6)

12 – (4 – 3)

(3 + 5) + 7.4

(20.9 – 8) – 2

3 + (5 + 7.4)

20.9 – (8 – 2)

(5 · 4) ·

(18 ÷ 6) ÷ 3

5 · (4 ·

)

18 ÷ (6 ÷ 3)

(6 · 2) · 5

(24 ÷ 12) ÷ 3

6 · (2 · 5)

24 ÷ (12 ÷ 3)

What patterns do you notice about the problems that were given?

Associative Property:

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Identity Property The word “identity” has to do with “sameness.” We use this word when we recognize the sameness between things. For example, you might say that a Halloween costume cannot really hide a person’s true identity. Above we defined the Associative and Commutative Properties for both addition and multiplication. We need to do the same thing for the Identity Property. What do you think the Identity Property for Addition should mean? Give examples of what you mean:

Identity Property of Addition:

What do you think the Identity Property for Multiplication should mean?

Give examples of what you mean:

Identity Property of Multiplication:

Multiplicative Property of Zero What do you think this property tells us?

Multiplicative Property of Zero

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Distributive Property of addition over multiplication Like all the other properties above, we’ve used this property throughout section 3.1. Below, first show the property to show 2(3 + 4) and then show it for a(b + c).

Inverse Properties The word “inverse” means “opposite” or “reverse.” You might say, forward is the inverse of backward. There is an inverse for both addition and multiplication.

What do you think should be the additive inverse of 3?

What do you think would be the additive inverse of –3?

What do you think would be the multiplicative inverse of 3?

What do you think would be the multiplicative inverse of 1/3?

Inverse Property of Addition: Inverse Property of Multiplication:

57

Properties of Mathematics: Name Property Algebraic Statement Identity Property of Addition

Meaning

Identity Property of Multiplication

Multiplicative Property of Zero

Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition Associative Property of Multiplication Distributive Property of Addition over Multiplication Additive Inverse

Multiplicative Inverse

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Examples

3.1i Homework: Properties Complete the table below: 1. Identity Property of Addition:

Show the Identity Property of Addition with 2

Show the Identity Property of Addition with 3

2. Identity Property of Multiplication:

Show the Identity Property of Multiplication with 23

Show the Identity Property of Multiplication with 3b

3. Multiplicative Property of Zero:

Show Multiplicative Property of Zero with 43.581

Show the Multiplicative Property of Zero with 4xy

4. Commutative Property of Addition: 4.38 + 2.01 is the same as: x + z is the same as: 5. Commutative Property of Multiplication: ab = ba

∙ 6k

6. Associative Property of Addition: (a + b) + c = a + (b + c)

is the same as: is the same as:

(1.8 + 3.2) + 9.5 is the same as: (x + 1) + 9 is the same as:

7. Associative Property of Multiplication: (2.6 · 5.4) · 3.7 is the same as: (wh)l is the same as: Use the listed property to fill in the blank. 8. Multiplicative Inverse:

1

3 (____) = 1

¼ (____) = 1

9. Additive Inverse: a + (–a) = 0 + ____ = 0

59

____ + –x = 0

Name the property demonstrated by each statement.

10. 11. 12. 13.

9∙7 3 6

7∙9 3∙6

25 + (–25) = 0 5

1 5

1

14.

(x + 3) + y = x + (3 + y)

15.

1mp = mp

16.

9 + (5+35) = (9+5) + 35

17.

0 + 6b = 6b

18.

7x  0 = 0

19.

4(3z)=(43)z

20.

x+4=4+x

21.

3(x + 2) = 3x + 6

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3.1j Class Activity: Using Properties to Compare Expressions Evaluate the following pairs of expressions. Write whether or not the two expressions are equivalent. If the expressions are not equivalent, correct expression 2 to make it equivalent. Use properties to explain how you can know that the expressions are equivalent or not without evaluating them. Expression 1 Equivalent? Expression 2 Explanation  or ≠ 3 25 ∙ 4 5 88

47 14

3 ∙ 25 4 63

47

133 2

5

63

88 + 133(2) + 14

25 + 4(3 + 1)

4(3 + 1) + 25

Using Properties to Justify Steps for Simplify Expressions Example: Jane wants to find the sum: 3 + 12 + 17 + 28. She uses the following logic, “3 and 17 are 20, and 12 and 28 are 40. The sum of 20 and 40 is 60.” Why is this okay? The table below shows how to justify her thinking using properties to justify each step. Statement

Step

Justification

3 + 12 + 17 + 28

No change, this is where she started.

This expression was given.

3 + 17 + 12 + 28

Commutative Property of Addition

20 + 40

The 17 and the 12 traded places. Jane chose to add the numbers in pairs first, which is like inserting parentheses. Jane found the sums in the parentheses first.

60

And so . . . 3 + 12 + 17 + 28 = 60

(3 + 17) + (12 + 28)

61

Associative Property of Addition Jane is now following the Order of Operations.

1) The expression 3(x – 4) – 12 has been written in three different ways. State the property that allows each change. Expression Step Justification This expression was given, only rewritten 3(x + (–4)) + (–12) No change using the idea that a – b = a + (–b) (–12) + 3(x + (–4)) (–12) + 3x + 3(–4) 2) The expression 2(3x + 1) + –6 x + –2 has been written in four different ways. State the property that allows each change. Expression Step Justification 2(3x + 1) + –6x + –2

No change

6x + 2 + –6x + –2 6x + –6x + 2 + –2 0+0 Review: Look back at Chapter 2 and review addition/subtraction with the chip model. 3) Model 3 + (–5) to find the sum 3 + (–5) Step 1 Step 2

Justify Step 1:

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Step 2:

Step 3:

Step 3 1 1

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4) In chapter 2 you learned that a negative times a negative produces a positive product. We used models to discover why this is true. In groups of 2-3, write a more formal proof.

Here is a proof: we start with –1(0) = 0

Statement –1(0) = 0

Step Given

Justification Multiplicative Property of Zero

–1(–1 + 1) = 0

0 was replaced with (–1 + 1)

Additive Inverse Property

–1(–1) + –1(1) = 0 –1(–1) + 1(–1) = 0 –1(–1) + –1 = 0 –1(–1) must equal 1 because if we get 0 when we add it to –1, it must be the additive inverse of –1

63

3.1k Classwork: Modeling Backwards Distribution Review: below is a review of modeling multiplication with an array. 13

Factors: 1, 3 There is one group of 3 Product/Area: 3

23

33

Factors:

Factors:

Product/Area:

Product/Area:

Use the Key below to practice using a multiplication model. 1

x2

22

44

x

xx

Factors:

Factors:

Factors:

Product/Area:

Product/Area:

Product/Area:

Look at the three models above. Why do you think 22 is called “two squared”? 32 is called “3 squared”? and 42 is called “four squared”?

64

1. Build the factors for 3(x + 2) on your desk. Then build the area model. Draw and label each block below. What are the factors of the multiplication problem? What is the area ? What is the product of the multiplication problem?

2. Build the factors for 3(2x +1) on your desk. Then build the area model. Draw and label each block below. What are the factors of the multiplication problem? What is the area? What is the product of the multiplication problem?

3. Build the factors for 2(x + 4). Build the area and draw. What is the area or product? What are the factors?

4. Build the factors for x(x + 3). Build the area and draw. What is the area or product? What are the factors?

65

5. Build the factors for x(2x + 5). Build the area and draw. What is the area or product? What are dimensions or factors?

Review concepts: 6. Write each as the sum of two whole number and the product of two integers. Model your expression: a) 15 b) 9 c) 35

7. Simplify each; use a model and words to explain the difference between the two expressions: a) x + x b) x  x

66

Example: Model the expression 2x + 4 on your desk. Find the factors and write 2x + 4 factored form. What are the dimensions (factors) of your rectangle? Draw them. Length:

Width:

What is the area (product) of the rectangle? Write 2x + 4 in factored form:

8) 6x + 3

9) 3x + 12

10) 5x + 10

8) 6x + 3

9) 3x +12

10) 5x + 10

Factors:

Factors:

Factors:

6x + 3 in factored form:

3x + 12 in factored form:

5x +10 in factored form:

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13. x2 + 3x

11. 6x + 2

12. x + 4

6x + 2

x+4

x2 + 3 x

Factors:

Factors:

Factors:

6x + 2 in reverse distributed form:

x + 4 in reverse distributed form:

2x 2 + 4x in reverse distributed form:

Practice: Write each in reverse distributed form. Use a model if you’d like. 14. 30x + 6 15. 4b + 28 16. 3m – 15

17. 4n – 2

18.

25b – 5

19. 4x – 8

Look at problems 14, 15, and 19. How else might these expressions be factored?

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3.1k Homework: Modeling Backwards Distribution Write each in reverse distributed from. Use a model to justify your answer. 1. 2x + 4

2. 3x + 12

3. 2x + 10

4. 3x + 18

5. x2 + 2x

6. x2 + 5x

Simplify each expression. Draw a model to justify your answer. 7. 2x + 3x

8. (2x)(3x)

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3.1l Self-Assessment: Section 3.1 Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that best describes your progress in mastering each skill/concept.

Skill/Concept

Beginning Understanding

1. Use the Distributive Property to expand and factor linear expressions with rational numbers. 2. Combine like terms with rational coefficients. 3. Recognize and explain the meaning of a given expression and its component parts. 4. Recognize that different forms of an expression may reveal different attributes of the context.

70

Developing Skill and Understanding

Deep Understanding, Skill Mastery

Section 3.2 Solve Multi-Step Equations Section Overview: This section begins by reviewing and modeling one- and two-step equations with integers. Students then learn to apply these skills of modeling and solving to equations that involve the distributive property and combining like terms. Students learn to extend these skills to solve equations with rational numbers. Next, students will write and solve word problems that lead to equations like those that they have learned to solve. Students will continue to practice all the skills that they have learned, including critiquing another’s work to find the error.

Concepts and Skills to be Mastered (from standards ) By the end of this section, students should be able to: 1. Solve multi-step equations fluently including ones involving calculations with positive and negative rational numbers in a variety of forms. 2. Convert between forms of a rational number to solve equations. 3. Use variables to create equations that model word problems. 4. Solve word problems leading to linear equations. 5. Connect arithmetic solution processes that do not use variables to algebraic solution processes that use equations. 6. Critique the reasoning of others.

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3.2a Classroom Activity: Model Equations Use any method you’d like to solve each of the following. Draw a model to justify your answer: 3. 3n = 18 1. m + 3 = 7 2. 8 = k – 2

4. 17 = 2j + 1

5. j/2 = 6

6. y + 3 = –5

It may have been easy to solve some (or all) of the above in your head, that’s good; that means you’re making sense of the problem. In this section, we are going to focus on the structure of equations and how properties of arithmetic allow us to manipulate equations. So, even though the “answer” is important, more important right now is that you understand the underpinnings of algebraic thinking. Evaluate the expressions 2x + 1 for each of the given values: 7. Evaluate 2x + 1 for x = 3 8. Evaluate 2x + 1 for x = –2

Solve each equation in any way you want: 8. 2x + 1 = 5 9. 2x + 1 = 9

9. Evaluate 2x + 1 for x = –3

10. 2x + 1 = –5

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What is the difference between an equation and an expression?

For this activity use the following Key to represent variables and integers. Note: “x” or “–x” can be any variable.

x = 1=

–x = –1 =

11. Use a model to solve x – 1 = 6. Write the algebraic procedure you followed to solve.

x–1=6

=

73

12. Use a model to solve x – 3 = 5. Write the algebraic procedure you followed to solve.

x–3=5

= = =

13. Use a model to solve 8 = 7 + m. Write the algebraic procedure you followed to solve.

=

8=7+m

= = 74

14. Use a model to solve 6 = 3x. Write the algebraic procedure you followed to solve. 6 = 3x

= = = 15. Use a model to solve 8 = –2m. Write the algebraic procedure you followed to solve.

8 = –2m

= = = 75

16. Use a model to solve – 5 + 3n = 7. Write the algebraic procedure you followed to solve.

= = =

76

3.2a Homework: Model and Solve Equations Model and solve each equation below. Draw algebra tiles to model. Use the Key below to model your equations.

x = 1=

–x = –1 =

1. x – 6 = –9

= = = 77

2. –15 = x – 14 –15 = x – 14

= = = 3. m + 2 = –11

=

m + 2 = –11

= = 78

4. 4n = –12 4n = –12

= = = 5. –15 = –3m

–15 = –3m

= = = 79

6. 3t + 5 = 2

=

3t + 5 = 2

= = 7. 8 = 2p – 4 8 = 2p – 4

= = = 80

3.2b Class Activity: More Model and Solve One- and Two-Step Equations Now let’s formalize the solving equation process. The answers are obvious in these first few equations. We use basic equations to think about solving more complicated ones. Example 1 is done for you. What are the solving action? Record the steps using Algebra

Model/Draw the Equation 1. x + 5 = 8

Add –5 to both sides. x58

- 5 -5 x =3

=

2. 5 = x + 8

=

Look at # 1 and # 2, why are the answers different?

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Check solution in the equation.

x58 (3)  5  8

True, so the solution is correct.

3. 3x = –6

=

Explain the logic above in #3.

How might you use related logic to model x/3 = –6?

4. x/3 = –6

=

5.

(1/2)x = 3

=

In problems # 3 and # 4, what happened to the terms on both sides of the equation?

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6. 2x – 5 = –9

=

7. 7 = 3x – 2

=

8. –5 = –3 + 2x

=

9.

7 + x/2 = –3

=

10. –3 = x/2 + 2

=

83

3.2b Homework: More Model and Solve One- and Two-Step Equations What are the solving action? Record the steps using Algebra

Model/Draw the Equation 1. 2 = x + 5

=

2. –12 = 3x

=

3.

–(x/4) = –8

=

4.

–2 = (1/3) x

=

84

Check solution in the equation.

5. –9 = x/2 – 5

=

6. –3x + 2 = –13

=

7. –11 = –4x – 3

=

85

8. –4 + n/3 = –2

=

9.

x/3 – 5 = –2

=

10. 2 + 5x = –8

=

11.

(1/2) x – 5 = – 3

=

86

3.2c Class Activity: Model and Solve Equations, Practice and Extend to Distributive Property Practice: Use a model to solve each. Show your algebraic manipulations on the right. 1. –16 = 6a – 4

=

2.

7 = 6 – n/7

=

3. –10 = –10 – 3x

=

4. Review: Create a model and then use the model to simplify each of the following expressions a) 3(2x + 1) b) –2(3x + 2) c) –4(2x – 3)

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Use a model to solve each. Show your algebraic manipulations on the right. 5. 2(x + 1) = –8

=

6. 6 = –3(x – 4)

=

7. –12 = –3(5x – 1)

=

88

8. –4(3 – 2m) = –12

=

9. –(1/2)(4x + 2) = –5

=

10. 3(2x – 4) + 6 = 12

=

89

3.2c Homework: Model and Solve Equations, Practice and Extend to Distributive Property Use a model to solve each. Show your algebraic manipulations on the right. 1. 9 = 15 + 2p

=

2. –7 = 2h – 3

=

3.

–5x – 12 = 13

=

90

4. 6 = 1 – 2n + 5

=

5.

8x – 2 – 7x = –9

=

6. 2(n – 5) = –4

=

91

7. –3(g – 3) = 6

=

8. –12 = 3(4c + 5)

=

92

3.2d Class Activity: Error Analysis Students in Mrs. Jones’ class were making frequent errors in solving equations. Help analyze their errors. Examine the problems below. When you find the mistake, circle it, explain the mistake and solve the equation correctly. Be prepared to present your thinking. Student Work Explanation of Mistake Correct Solution Process 1. 6t  30

2.

3.

8  5c  37

4.

5.

4x  3  17

93

6. 3(2x  4)  8

7. 3x  2x  6  24

8.

9. 2(x  2)  14

10. 3(2x  1)  4  10

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3.2d Homework: Practice Solving Equations (select homework problems) Solve each equation, use Algebra Tiles if that will help you. 1. 8  t  25

2. 2n  5  21

3. 3 y  13

4. 12  5k  8

5. 5  b  8

6. 5  6a  5

7. 8 

10.

n 5 7

t 42 3

8. 8 

x 5 7

11. 9 

9.

n 6 8

12.

y  2  10 3

y  4  12 5

13. 8  6   p  8

14. 7  8x  4x  9

15. 8x  6  8  2x  4

16. 6(m  2)  12

17. 5(2c  7)  80

18. 5(2d  4)  35

19. 3(x 1)  21

20. 7(2c  5)  7

21. 6(3d  5)  75

22. 4 14  8m  2m

23. 1  5 p  3p  8  p

24. 5 p  8 p  4  14

25. 2 p  4  3p  9

26. 8  x  5  1

27. 12  20x  3  4x

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3.2e Class Activity: Solve One- and Two-Step Equations with Rational Numbers (use algebra to find solutions) Before we begin… …how can we find the solution for this problem? …do you expect the value for x to be larger or smaller than 4 for these problems? Explain.

.25x  4

…how can you figure out the solutions in your head?

Solve the equations for the variable. Show all solving steps. Check the solution in the equation (example #1 check: –13(3) = 39, true.). Be prepared to explain your work. 1.  13m  39 3. y  25  34 4. 2y  24 2.

Check: 5.

6. 13  25  y

7.

Check:

96

8.

9.

11.

10.

12.

Check: 13.

14.

15.

17.

18. 8.38v 10.71  131.382

20.

21.

Check: 16. 9.2r  5.514  158.234

Check: 19.

Check:

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3.2e Homework: Solve One- andTwo- Step Equations (practice with rational numbers) Solve the equations for the variable in the following problems. Use models if desired. Show all solving steps. Check the solution in the equation. 1. 22  11k 3. x 15  21 2.

Check: 4. 3y  36

6. 54  16  y

5.

Check: 7.

8.

9.

Check: 10.

11.

12.

Check: 98

13. 5b  0.2

15.

14.

Check: 16. 3.8  13.4 p  460.606

17. 0.4 x  3.9  5.78

18.

Check: 19.

20.

21.

Check:

99

3.2e Extra Practice: Equations with Fractions and Decimals 1.

2.

3.

5.

6.

Check: 4.

Check: 7.

9.

8.

Check:

100

10.

11.

12.

14.

15.

17.

18.

Check: 13.

Check: 16.

Check:

101

3.2f Class Activity: Create Equations for Word Problems and Solve

For each, draw a model to represent the context and then determine which of the equations will work to answer the question. Explain your reasoning. 1. Brielle has 5 more cats than Annie. If Brielle has 8 cats, how many does Annie have?

What is the unknown/variable? a = the number of cats that Annie has Equation

Will it work?

Why or Why Not?

a. 8 = 5 + a b. 5 = 8 + a c. 8 = 5a d. 5 = 8a 2. Three pounds of fruit snacks cost $4.25. How much does one pound of fruit snacks cost?

What is the unknown? (What does the variable represent?) y = Equation

Will it work?

Why or Why Not?

a. y + 3 = 4.25 b. 3y = 4.25 c. y + y + y = 4.25 d. 4.25 = 3 y

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3. Jim bought a tie for x dollars and a jacket for $37.50. Jack bought the same items as Jim. Together, they spent $80. How much did each tie cost?

What is the unknown? (What does the variable represent?) x = Equation

Will it work?

Why or Why Not?

A x + x + 37.50 + 37.50 = 80 B 2x + 2(37.50) = 80 C 2(x + 37.50) = 80 D 2x + 80 = 37.50 4. Bo bought some songs for $0.79 each, an album for $5.98, total price $8.35. How many songs did he buy?

What is the unknown? (What does the variable represent?) x = Equation

Will it work?

Why or Why Not?

A 0.79x + 5.98 = 8.35 B 5.98x + 0.79 = 8.35 C 8.35 + 5.98 + 0.79 = x D 5.98 + 0.79x = 8.35

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3.2f Homework: Create Equations for Word Problems and Solve

Draw a model for each context and then write the sentence as an algebraic equation. An example is given below. My mom’s height (h) is 8 inches more than my height (60 inches).

What is unknown? My mom’s height. Equation:

or

Solution: My mom’s height =

Write out what each unknown stands for, write an equation to model the problem, then (if possible) find the solution. Note: A solution can only occur if enough information is given. Make sure your equation matches your model. 1. The blue jar has 27 more coins than the red jar.

2. My age is twice my cousin’s age.

3. A car has two more wheels than a bicycle.

4. Art’s jump of 18 inches was 3 inches higher than Bill’s.

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5. The sum of a number and its double is eighteen.

6. The $5 bill was $3 more than the cost of the notebook

7. The visiting team’s score was five points less than our score (50 points).

8. The number of minutes divided by sixty gives us 3 hours.

9. Bill has twice as much money as I do. Our money together is $9.

10. A large popcorn and a drink together cost the same as the movie ticket. I spent $10 on all three.

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3.2g **Class Challenge: Multi-Step Equations

1. The following problems will involve all 5 steps below. Discuss these steps as a class—make sure everyone agrees and understands the five steps. 1. 2. 3. 4. 5.

Distribute Collect like terms Variables on one side, constants on other side Divide (or multiply by fraction) to get a variable coefficient of 1 Check solution.

2. Using the 8 problems below, do one of the following.  

Put one at a time on the chalkboard and have groups “relay race” to complete. Then compare and correct the steps and solutions Have groups be responsible for one problem to present to the class.

a. 2(4x + 1) − 11 x = –1 b. 3 = 2(x + 3) + x + 2 x + 2 c. 3(x – 6) − 4(x + 2) = – 21 d. 7(5x – 2) − 6(6x – 1)= – 4 e. 2a + (5a – 13) + 2a – 3 = 47 f. 3a + 5(a – 2) – 6a = 24 g. 13 = 2(c + 2) + 3c + 2c + 2 h. 3(y + 7) – y = 18

3. Can you write and solve an equation for this problem? You are playing a board game. You land on a railroad and lose half your money. Then you must pay $1000 in taxes. Finally you pay half the money you have left to get out of jail. If you now have $100, how much money did you start with?

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3.2h Extra Practice: Solve Multi-Step Equation Review 1. 6.2d  124

2.

3.

5.

6.

8.

9.

Check: 4. g 12.23  10.6

Check: 7.

Check:

107



10. 28  8x  4

11.

12.

Check: 13.

14.

15.

Check: 16.

18. 3(x  1)  2(x  3)  0

17.

Check:

108

19.

21.

20.

Check: Solve Multi-Step Equations (distribution, rational numbers) 22. 4(x  2x)  24 23. 5(5  x)  65

24. 6(4  6x)  24

Check: 25. 3(2x  3)  57

26. 2(4x  1)  42

Check: 109

27. 7(2x  7)  105

28. 3(7x  8)  150

29. 5(7x  5)  305

30. 5(1 7x)  320

Check: 31. 3(5x  6)  78

32. 5(2x  3)  96

33.

Check: 34.

36.

35.

Check:

110

37.

38.

39.

41. 7(5x  8)  91

42. 2(4x  2)  76

44. 4(9  x)  12

45. 5(7  6x)  175

Check:

40. 2(4x  8)  32

Check: 43. 7(3x  7)  175

Check: 111

46. 2(1x  4)  18

47. 3(7  4x)  33

48. 3(10  6x)  84

Check: 49. 5(1x  7)  40

50. 8  3(5  2x) 1

51.

Check: 52.

53.

54.

56.

57.

Check: 55.

Check: 112

3.2i Self-Assessment: Section 3.2

Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that best describes your progress in mastering each skill/concept.

Skill/Concept

Beginning Understanding

1. Solve multi-step equations fluently including ones involving calculations with positive and negative rational numbers in a variety of forms. 2. Convert between forms of a rational number to solve equations. 3. Use variables to create equations that model word problems. 4. Solve word problems leading to linear equations. 5. Connect arithmetic solution processes that do not use variables to algebraic solution processes that use equations. 6. Critique the reasoning of others.

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Developing Skill and Understanding

Deep Understanding, Skill Mastery

Section 3.3: Solve Multi-Step Real-World Problems Involving Equations and Percentages Section Overview: Students will learn how to solve percent problems using equations. They will begin by modeling percent problems using a drawn model. Then they will translate that model into an equation which they will then solve. Students will use similar reasoning to move to problems of percent of increase and percent of decrease. Finally, students will put all of their knowledge together to solve percent problems of all types.

Concepts and Skills to be Mastered (from standards ) By the end of this section, students should be able to: 1. Recognize and explain the meaning of a given expression and its component parts when using percents. 2. Solve multi-step real-life percent problems involving calculations with positive and negative rational numbers in a variety of forms. 3. Convert between forms of a rational number to simplify calculations or communicate solutions meaningfully.

114

3.3a Classroom Activity: Percents with Models and Equations Draw a model to help you solve the problems below. Then choose the algebraic equation(s) that represents the situation. Solve that equation. 1.

To pass a test you need to get at least 70% correct. There are 50 questions on the test. How many do you need to answer correctly in order to pass? a. Model:

b. Choose the appropriate equation(s). Justify your choice.

2. 25% of a box of cookies is oatmeal. If there are 50 oatmeal cookies in a box, how many cookies are there total? a. Model:

b. Choose the appropriate equation(s). Justify your choice. (1/4)(50) = c

c(1/4) = 50

c = 50(4)

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1/4 = c/50

Draw a model for each and then write an algebraic equation to solve each percent problem.

3. 30% of my books are science fiction. If I have 60 science fiction books, how many books do I have?

5. 15% of the day was spent cleaning the house. If there are 24 hours in the day, how many of them were spent cleaning the house?

6. I got 52 out of 60 questions right on the last History test. What percent correct did I get?

7. A pair of jeans are on sale for $35. Originally they were $45. What percent of the original price is the sale price?

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3.3a Homework: Percents with Models and Equations Draw a model to help you solve the problems below. Then choose the algebraic equation(s) that would represent the problem. Solve that equation. 1. To get an A in math class, I need to get a 90% on the test. If the test has 40 questions, how many do I need to get right in order to get an A? a. Model:

b. Choose the appropriate equation(s). Justify your choice.

2. 25% of the club came to the meeting. 3 people were at the meeting. How many people are in the club? a. Model:

b. Choose the appropriate equation(s). Justify your choice.

3. 32 of the 48 people at the gym are wearing blue. What percent are wearing blue? a. Model:

b. Choose the appropriate equation(s). Justify your choice.

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Draw a model for each and then write an algebraic equation to solve each percent problem.

4. 65% of the population needs to vote for the new law in order for it to pass. There are 800 voters. How many need to vote for the new law in order for it to pass?

5. 5% of the apples have worms in them. 10 apples had worms in them. How many apples are there total?

6. 250 students dressed up for Spirit Day. There are 800 students. What percent dressed up for Spirit Day?

7. 55% of my shirts are purple. If I have 20 shirts, how many of them are purple?

8. 5 out of the 7 ducklings have yellow feathers. What percent of the ducklings have yellow feathers?

9. 30% of the sodas are grape. There are 24 grape sodas. How many sodas are there total?

10. I made 36 cupcakes, that’s 80% of what I need. How many cupcakes will I be making all together?

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3.3b Class Activity: Percent Problems

For each context below: a) draw a model, b) select the expression that represents the situation, and then c) justify your answer: 1. Last week, Dirk jumped y inches in the long jump. This week, he increased the length of his jump by 10%. a. Draw a model to represent this situation.

b. Which of the following algebraic expressions represents the length of his jump now? (Be prepared to explain your answers and how you know they are correct!) 0.10y

0.90y

1.10y

y + 0.10y

c.

2. Hallie wants to buy a pair of jeans for h dollars. She knows she will have to pay 6% sales tax along with the price of the jeans. a. Draw a model to represent this situation.

b. Which of the following algebraic expressions represents the price of the jeans with tax? 0.06h

0.94h

0.6h

1.6h

1.06h

c. If the original price of the jeans was $38, what is the price with tax? Show at least three different ways to get your answer. 119

3. Jamie has a box with x chocolates in it. Her little brother ate 25% of the chocolates. a. Draw a model to represent this situation.

b. Which of the following algebraic expressions represents the amount of chocolate in her box now? (Be prepared to explain your answers and how you know they are correct!) 0.25x

0.75x

x – 0.25x

(1 – 0.25)x

4. Drake wants to buy a new skateboard with original price of s dollars. The skateboard is on sale for 20% off the regular price. a. Draw a model to represent this situation.

b. Which of the following algebraic expressions represents the sale price of the skateboard? s – 0.80

0.2s

0.8s

s – 0.20

1s – 0.20 s

c. If the original price of the skateboard was $64, what is the sale price? Show at least three different ways to get your answer.

5. Alayna makes delicious cupcakes. She estimates that one-dozen cupcakes cost $7.50 to make. She wants a 50% mark up on her cupcakes. How much should she sell one-dozen cupcakes for? a. Draw a model to represent this situation:

b. Write an expression to represent what she should charge.

c. How much should she charge?

120

6. There were 850 students at Fort Herriman Middle School last year. The student population is expected to increase by 20% next year. What will the new population be? Draw a model to represent this situation.

Write an expression to represent the new population.

What will the new population be?

7. A refrigerator at Canyon View Appliances costs $2200. This price is a 25% mark up from the whole sale price. What was the wholesale price? Draw a model to represent this situation.

Write an expression to represent the whole sale price

What was the whole sale price?

8. Carlos goes the ski shop to buy a $450 snowboard that’s on sale for 30% off. When he gets to the store, he gets a coupon for an addition 20% the sale price. What will he pay for the snow board? Write an expression to represent the problem Draw a model to represent this situation. situation.

What will Carlos pay for the snowboard?

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3.3b Homework: Percent Problems 1. Dean took his friend to lunch last week. His total bill was b dollars. He wants to tip the waitress 20%.

How much will Dean pay, including the 20% tip? (Don’t worry about tax in this problem.) b. Draw a model to represent this situation.

c. Which of the following expressions represent the amount that Dean will pay?

0.8b

b + 0.2b

1.2b

1.8b

1 + 0.2b

d. If Dean’s bill was $22.50, how much will Dean pay including the 20% tip? Show at least three different

ways to get your answer.

5. Philip took a vocabulary test and missed 38% of the problems. There were q problems on the test. a. Draw a model to represent this situation.

b. Which of the following expressions represent the number of problems that Philip got correct? 0.38q

q – 0.38

q – 0.38q

0.62q

(1 – 0.38)q

c. Which of the following expressions represent the number of problems that Philip missed? 0.38q

1q – 0.62q

q – 0.38q

0.62q

(1 – 0.38)q

d. If there were 150 problems on the test, how many did Philip get correct? Show at least three different ways to get your answer.

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6. Chris would like to buy a picture frame for her brother’s birthday. She has a lot of coupons but is not sure which one to use. Her first coupon is for 50% off of the original price of one item. Normally, she would use this coupon. However, there is a promotion this week and the frame is selling for 30% off, and she has a coupon for an additional 20% any frame at regular or sale price. Which coupon will get her the lower price? She is not allowed to combine the 50% off coupon with the 20% off coupon. a. Draw a model to show the two different options. 50% off coupon:

30% off sale with additional 20% off coupon:

b. Let x represent the original price of the picture frame. Write two different expressions for each option. 50% off coupon:

30% off sale with additional 20% off coupon:

c. Which coupon will get her the lowest price? Explain how you know your answer is correct.

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3.3c Class Activity: More Practice with Percent Equations 1. Selina gave the waiter a $2.25 tip at the restaurant. If her meal cost $12.50, what percent tip did she give? a) Let t represent the percent of the tip. Which of the following equations are true?

b) What is percent tip did she give?

Write an equation with a variable for each problem below, then solve the equation. Justify your work. 2. Beatrice bought a new sweater. She paid $3 in sales tax. If sales tax is 6%, what was the original price of the sweater?

3. Jill bought a bracelet that cost $5. Sales tax came out to be $0.24. What is the sales tax rate?

4. Kaylee is training for a marathon. Her training regiment is to run 12 miles on Monday, increase that distance by 25% on Wednesday, and then on Saturday increase the Wednesday distance by 25%. How far will she run on Saturday?

124

5. Juan is trying to fit a screen shot into a report. It’s too big, so he reduces is first by 30%. It still doesn’t fit, so he reduces that image by 20%. What percent of the original image did he paste into his report? If the original image was 8 inches wide, how wide is the twice reduced image?

6. The size of Mrs. Garcia’s class increased 20% from the beginning of the year. If there are 36 students in her class now, how many students were in her class at the beginning of the year?

7. Mika and her friend Anna want to give 20% of the money they make at a craft fair to charity. If Mika makes $500 and they want to give a total of $150. How much will Anna have to make?

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3.3c Homework: More Practice with Percent Equations

Write an equation with a variable for each problem. Then solve the equation. Justify your answer. 1. John paid $3.45 in sales tax on his last purchase. What was the original price if the tax rate is 3%?

2. Newt paid $45 in sales tax for his new television. If tax is 6%, what was the original price of the television?

3. Carter gave the waitress a tip of $8.75. If the original price of his meal was $24.95, what percent of the price was the tip?

4. Louie paid a total of $12.55, with tax, for his new frying pan. If the original price of the pan was $11.95, what was the tax rate?

5. Robert paid $14.41, with tax, for his model airplane kit. If tax was 6%, what was the original price of the kit?

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3.3d Self-Assessment: Section 3.3

Consider the following skills/concepts. Rate your comfort level with each skill/concept by checking the box that best describes your progress in mastering each skill/concept.

Skill/Concept

Beginning Understanding

1. Recognize and explain the meaning of a given expression and its component parts when solving problems with percents. 2. Solve multi-step real-life percent problems involving calculations with positive and negative rational numbers in a variety of forms. 3. Convert between forms of a rational number to simplify calculations or communicate solutions meaningfully.

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Developing Skill and Understanding

Deep Understanding, Skill Mastery