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1 Table of Contents CHAPTER 3: EXPRESSIONS AND EQUATIONS PART 1 (4 5 WEEKS) Anchor Problem: Tiling a Garden Border... 6 SECTION 3.1: COMMUNICATE NUMERIC IDEAS AND CONTEXTS USING MATHEMATICAL EXPRESSIONS AND EQUATIONS a Class Activity: Matching Numerical Expressions to Stories b Class Activity: Numeric and Algebraic Expressions b Homework: Matching and Writing Expressions for Stories b Additional Practice c Class Activity: Algebra Tile Exploration c Homework: Algebra Tile Exploration d Class Activity: More Algebra Tile Exploration d Homework: More Algebra Tile Exploration e Class Activity: Vocabulary for Simplifying Expressions e Homework: Solidifying Expressions f Class Activity: Iterating Groups f Homework: Iterating Groups g Class Activity: More Simplifying g Homework: More Simplifying g Additional Practice: Iterating Groups g Additional Practice: Simplifying h Class Activity: Modeling Context with Algebraic Expressions h Homework: Modeling Context with Algebraic Expressions i Class Activities: Properties i Homework: Properties j Class Activity: Using Properties to Compare Expressions k Classwork: Modeling Backwards Distribution k Homework: Modeling Backwards Distribution l Self Assessment: Section SECTION 3.2 SOLVE MULTI STEP EQUATIONS a Classroom Activity: Model Equations a Homework: Model and Solve Equations b Class Activity: More Model and Solve One and Two Step Equations b Homework: More Model and Solve One and Two Step Equations c Class Activity: Model and Solve Equations, Practice and Extend to Distributive Property c Homework: Model and Solve Equations, Practice and Extend to Distributive Property d Class Activity: Error Analysis d Homework: Practice Solving Equations (select homework problems) e Homework: Solve One andtwo Step Equations (practice with rational numbers) e Extra Practice: Equations with Fractions and Decimals f Class Activity: Create Equations for Word Problems and Solve f Homework: Create Equations for Word Problems and Solve g **Class Challenge: Multi Step Equations i Self Assessment: Section SECTION 3.3: SOLVE MULTI STEP REAL WORLD PROBLEMS INVOLVING EQUATIONS AND PERCENTAGES a Classroom Activity: Percents with Models and Equations a Homework: Percents with Models and Equations b Class Activity: Percent Problems b Homework: Percent Problems c Class Activity: More Practice with Percent Equations c Homework: More Practice with Percent Equations

2 3.3d Self Assessment: Section

3 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 x 2 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. 3

4 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 , 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 6 th 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. 4

5 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 Look for and make use of structure Use appropriate tools strategically Look for and express regularity in repeated reasoning 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 when it is reduced. Another example, in section 3.3 students will use a model to show that a 20% increase is the original amount plus 0.2 of the 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 (paper/pencil, manipulatives, and calculators) while solving problems. Students should recognize that the most powerful tool they possess is their ability to reason and make sense of problems. Students will study patterns throughout this chapter and connect them to both their intuitive understanding and properties of arithmetic. 5

6 3.0 Anchor Problem: Tiling a Garden Border Imagine that you are putting 1-foot square tiles around the edge of a square garden. Look at the answers below this will help you guide the discussion. Students may use a variety of methods to write each. Have students work in groups of four and present their answers to the class. REVISIT during 3.1i. 4 feet across 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. Various methods are reasonable. Method 1: 4(4) + 4, 4 sides of 4 tiles, then 4 corners Method 2: , 6 on top, 6 on bottom, 4 on the left, 4 on the right Method 3: , outer square minus inner square Method 4: 4 6 4, 4 sides of 6, subtract the tiles counted twice Invest time in talking about the first representation many possible approaches, but all simplify to the same answer. 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: 4(5) + 4, 4 sides of 5 tiles, then 4 corners Method 2: , 7 on top, 7 on bottom, 5 on left, 5 on right Method 3: , outer square minus inner square Method 4: 4 7 4, 4 sides of 7, subtract the tiles counted twice Students will begin to be fluent in the different ways of expressing the area of the border. 6

7 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. 20 feet across Method 1: 4(10) + 4, 4 sides of 10 tiles, then 4 corners Method 2: , 12 on top, 12 on bottom, 10 on left, 10 on right Method 3: , outer square minus inner square Method 4: , 4 sides of 12, subtract the tiles counted twice What if the garden were 20 feet by 20 feet? Demonstrate how each method would work now. Method 1: 4(20) + 4, 4 sides of 20 tiles, then 4 corners Method 2: , 22 on top, 22 on bottom, 20 on left, 20 on right Method 3: , outer square minus inner square Method 4: , 4 sides of 22, subtract the tiles counted twice In the discussion, help students connect the different representations and notice that all represent the same amount/value. When they present their solutions, have them justify the logic for their representation and/or how it is connected to other representations. Record at least one of the solutions so that you can return to it during later lessons. Make sense of the problems Reason Abstractly Look for and express regularity in repeated reasoning 7

8 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. 8

9 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. Teachers may want to review order of operations before beginning. See 3.1b for where this lesson is leading. 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 Yes Each basket is added together, giving the total. b No 5+3 is not an accurate interpretation of five 3-pointers c. (5 + 3)(4 + 2) 48 No 5+3 is not an accurate interpretation of five 3-pointers d. 5(3) + 4(2) 23 Yes It is the same as A. 5(3) means 5 groups of 3 as above. 4(2) means 4 groups of 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) $5.85 Yes The algebra describes the situation 2 groups of (0.30) etc b. 2(1.75) + 3 (0.30) $4.40 No This describes two pounds of cherries and 3 apples c $5.85 Yes The algebra describes the situation as in A but now using repeated addition d. (2+ 3)( ) $10.25 No We cannot combine groups and amounts and then multiply. 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. ( ) + ( ) $1.50 Yes Added each purchase individually. b. 3(0.30) + 2(0.30) $1.50 Yes This uses multiplication instead of repeated addition c. 5(0.30) $1.50 Yes This expression works because the cost is always **compare to 2D d $1.50 Yes This sums the total cost of apples and the total cost of oranges 9

10 4. Aunt Nancy gave her favorite niece 3 dollars, 3 dimes, and 3 pennies. How much money did her niece receive? Expression Evaluate Does it work? Why or Why Not? a. 3(1.00) + 3(0.10) + 3(0.01) $3.33 Yes This takes three of each unit of money, then adds them together. b. 3( ) $3.33 Yes Since all the money came in groups of three, this sums one set, then multiplies. **compare to 2D and 3C. c $4.11 No This took three as its own value, rather than operating on the other values. d $3.33 Yes This is like A after multiplication. 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 Evaluate Does it work? Why or Why Not? a $2.70 Yes This adds each piece of money in succession. b $1.35 No This is the amount given to one nephew, not the two together. c. 2 ( ) $2.70 Yes Each nephew received the same amount of money, thus two times the amount one received will work. d. 2(1) $2.35 No This does not account for one of the quarters and one of the dimes. 6. Aunt Nancy gave her favorite niece two dollars, 1 dime, and 3 pennies. How much money did her niece receive? 2(1) + 1(0.10) + 3(0.01) $ Uncle Aaron gave 8 dimes, 2 nickels, and 20 pennies to his nephew. How much money did he give away? 8(0.10) + 2(0.05) + 20(0.01) $ I bought 2 toy cars for $1 each and 3 toy trucks for $1.50 each. How much did I spend? 2(1) + 3(1.50) $ 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. 1(6) + 3(3) 15 points 10. I had $12. Then I spent $2 a day for 5 days in a row. How much money do I have now? 12 5(2) $2 11. I earned $6. Then I bought 4 candy bars for $0.75 each. How much money do I have left? 6 4(0.75) $3 10

11 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) $31 Yes This expression describes the situation b. 2(3) + 5(7) $41 No This expression pairs the wrong quantities together. c. (2 + 3)(5 + 7) $60 No Cannot combine quantities and multiply by value if the values are different. d. (2 + 3) + (5 + 7) $17 No This expression does not compensate for repeated purchases at a price. 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 Evaluate Does it work? Why or Why Not? a. 3(6) + 2(3) + 2(1) 26 Yes This expression multiplies the correct points the correct number of times. b Yes This is the same as above only using repeated addition instead of multiplication. c. ( ) + (3 + 3) + (1 + 1) 26 Yes This is the same as above only grouping the sets of addition d Yes This is the same as A or C after the first step of simplification. 3. I earned $6. Then I bought 4 candy bars for $0.50 each. How much money do I have left? Expression Evaluate Does it work? Why or Why Not? a $4 Yes It gives the total, then subtracts each expenditure. b. 6 4(0.50) $4 Yes It is like A only multiplying instead of repeated subtraction. c. 6 ( ) $7 No We need to subtract the sum of 0.50 four times. d. 6 ( ) $4 Yes This subtracts the sum of 0.50 four times. 11

12 4. I earned $5. Then I spent $1 a day for 2 days in a row. How much money do I have now? Expression Evaluate Does it work? Why or Why Not? a $5 No This shows spending a dollar and then getting a dollar. b $3 Yes This expression accurately describes the situation c. 5 (1 1) $5 No We need to subtract the sum of the two dollars. d. 5 (1 + 1) $3 Yes We are subtracting the sum of the two dollars. 5. Uncle Aaron gave 2 dimes, 3 nickels, and 2 pennies to his nephew. How much money did he give away? Expression Evaluate Does it work? Why or Why Not? a. 2(0.10) + 3(0.05) + 2(0.01) $0.37 Yes This accounts for each of the coins. b. 2( ) + 3(0.05) $0.37 Yes This grouped the dimes and pennies, which is accurate. c $0.37 Yes This is like A after multiplication d $7.16 No This misrepresents the idea of 2 dimes as Write an expression of your own for each problem. Then evaluate the expression to solve the problem. There are various accurate expressions for these problems. They should each result in the value given. 6. Josh made ten 3-pointers and a 2-pointer at his basketball game. How many points did he score? 10(3) + 1(2) 32 points 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? 3(0.25) + 3(1.75) $ I bought five apples for $0.30 each and 5 oranges for $0.30 each. How much money did I spend? 5(0.30) + 5(0.30) $ I bought two t-shirts at $12 each and 3 sweaters for $20 each. How much did I spend altogether? 2(12) + 3(20) $84 12

13 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. Through this work, discuss how ideas are related to the previous day s work. Note changes in order and grouping and issues with Ryan bought 3 CDs for x dollars each and a DVD for $15. How much money did he spend? Expression Correct Evaluate expression? x 7 Did it work? Why or why not? a. 3 + x $25 No 3 + x does not translate 3 CDs for x dollars. b. 15x $108 No 15x does not represent a DVD for $15 c x + x + x --- $36 Yes There is 15 for the DVD, then three CDs, each represented with an x. d. 3x $36 Yes This uses multiplication to make expression C more concise. 2. I started with 12 jellybeans. Sam ate 3 jellybeans and then Cyle ate y jellybeans. How many jellybeans were left? Expression Correct Evaluate expression? y 6 Did it work? Why or why not? a y Yes This accurately represents losing 3 and then y. b. 12 (3 y) No I lost the sum of 3 and y. c. 12 (3 + y) Yes This expression groups all the jellybeans that were eaten and subtracts them from the original amount. d. 9 y Yes Yes, this expression begins with the amount I had after Sam ate Jellybeans. 13

14 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 Evaluate Did it it will work? (use n 3) work? Why or why not? a n --- $14 No 4 + n does not accurately represent 4 notebooks for n dollars each. b. 4n $19 Yes This sums 4 times the cost of a notebook with the other two costs. c n --- $19 Yes This expression has already summed the other two costs. d. n + n + n + n $19 Yes This expression uses repeated addition for the notebooks. 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. a. Jan bought x more apples than Jill. How many apples did Jan buy? Jan bought 12 + x apples. Help students connect the model with the expression. Ask students if they could draw a model of the context differently. Jill s 12 apples Jill s 12 apples Jill bought 12 apples, Jan bought x more Jan bought x more 5. Josh won 12 tickets. Evan won p tickets less than Josh. How many tickets did Evan win? 12 p x 14

15 6. Tim is 3 years younger than his brother. If his brother is y years old, how old is Tim? y 3 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? $2(8 w) 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? $0.10(4 + a) 15

16 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 Evaluate Did it it will work? (use x 5) work? Why or why not? a. 5 + x $22 No 5 + x does not accurately represent 5 books for x dollars each. b. 5(x) $300 No The DVD is in addition to the books. c. x + x + x + x + x $37 Yes This expression uses repeated addition to account for the cost of the books. d. 5x $37 Yes This expression uses multiplication to account for the cost of the books. 2. Jim won 30 tickets. Evan won y tickets less than Jim did. How many tickets did Evan win? Expression Do you think Evaluate Did it it will work? (use y 6) work? Why or why not? a. 30 y tickets Yes This is an accurate representation of the expression b. y tickets No Jim won more tickets than Evan, thus we must subtract from Jim s amount. c. y tickets No Evan won less tickets than Jim, thus we need to subtract. d. 30 y tickets No The difference between their amounts is absolute; we must subtract. 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? p 4 4. I bought x pairs of shoes for $25 each and 2 pairs of socks for $3 each. How much did I spend? $25x + 2($3) 16

17 5. I bought m gallons of milk for $2.59 each and a carton of eggs for $1.24. How much did I spend? m($2.59) + $ Paul bought s sodas for $1.25 each and chips for $1.75. How much did he spend? s($1.25) + $ 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? 2d + 2n 17

18 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. b is the amount of money Brandon has. Then Marina has b + 12 dollars. 2. Conner is three times as old as Jackson. Represent Conner s age. j is Jackson s age. Then Conner is 3 j or 3j years old. 3. Diane earned $23 less than Chris. Represent how much Diane earned. c is amount Chris earned. Diane earned $23 less than Chris or c Juan worked 8 hours for a certain amount of money per hour. Represent how much Juan earned. x is amount of money earned per hour. Juan worked at this rate for 8 hours so 8x. 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. x is amount in Martin s account. He spent two-thirds of this amount: (2/3) x or (2/3)x. 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. y is amount of money spent on bracelet. Brianne now has $47 $15 y or $32 y. 18

19 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. m is amount of time she studied math each day. This was repeated for 5 days so 5( m + 15 ). 8. Carlos spent $8 on lunch, some money on a drink and $4 on ice cream. Represent how much money Carlos spent. d is amount Carlos spent on a drink. He spent $8 + d + $4 or $12 + d. 9. Nalini has $26 dollars less than Hugo. Represent the amount of money Nalini has. h is the amount that Hugo has. Nalini has $26 less than him or h 26 dollars. 10. Bruno ran four times as far as Milo. Represent the distance Bruno ran. m is the distance Milo ran. Bruno ran a distance 4 times as long: 4m. 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. p is the amount spent on the phone bill. $420 p $100 or $320 p. 12. Camille has 4 bags of candy. Each bag has 3 snicker bars and some hard candy. Represent the amount of candy Camille has. h is the amount of hard candy in each bag. Camille has 4(3 + h) pieces of candy. 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. a is the amount in her account. She spent ¼ of this or (¼) a or (1/4)a or a/4 on the cell phone. Discuss these various representations with the class. 14. Miguel is 8 years older than Cristo. Represent Miguel s age. c is Cristo s age. Miguel is 8 years older or c

20 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 x x 1 x Algebra tiles can be easily made with card stock. It is best to start with manipulatives and then move to drawing only. Have students draw all their models. Attention will have to be paid to: a ( b) a + b and a b a + ( b). For example, #4 can be expressed as 4x 1 or 4x + 1; take time to discuss this. For problems 10 through 18 have students create the model in more than one way (e.g. 2x+3 can be modeled with 5x + 3x ). Model, Structure Write an expression for what you see and then write the expression in simplest form x x x x x x x x x x x x x x x + ( 2) or x x 2 or x + ( 2) talk about these two different ways to write the expression 6. 1 x x x 1 1 x + ( 1) or x 1 x x x x x x x x 20

21 Use the algebra tile key above to model each expression on your desk. Sketch a picture in the space below. 7. x x x 1 Is there more than one way to model each of the expressions above? Justify. Discuss other possible representations (zero pairs in the representations.) Draw a model for each of the expressions below using the key from previous page. Simplify if you can x x 11. 3x +4 + ( 2) 3x x + x x x + 1 x x x 13. 3x x 4x x x x x 3x

22 16. 2x x 7 x 3 or x + ( 3) 17. 4x x 2x x 1 + 3x + 2 x 2x x + x + x + x 4x x + x 5 x 2 or x + ( 2) 20. x x x x -2x x 2-4x 21. x x x x 4x Look at 20 and 21 together 24. 2x x x

23 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 x x 1 x Write an expression for each model below x x 1 or x + ( 1) x x x x x x x x x x 3x 3 or 3x + ( 3) 1 1 x x x x x x x x x x 1 1 Algebra tiles can be easily made with card stock. It is best to start with manipulatives and then move to drawing only. Have students draw all their models throughout this chapter. Attention will have to be paid to: a ( b) a + b and a b a + ( b). For example, #4 can be expressed as 4x 1 or 4x + 1; take time to discuss this. For problems 10 through 18 have students create the model in more than one way (e.g. 2x+3 can be modeled with 5x + 3x ) x x x x x x x x x x x 2 or x + ( 2) x x x 1 2x

24 Model each integer or expression on your desk. Sketch a model in the space below. 7. 2x x x x + 2 2x 13. 2x x x x x 12. 2x x x 3 24

25 16. x x x 3 + 2x x 3 7x

26 3.1d Class Activity: More Algebra Tile Exploration Miguel saw the following two expressions: 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? In this lesson you will focus on commuting. You are not teaching the property explicitly, rather you should highlight that you are changing the order of addition to simplify. Students will have done this in previous grades. In this section you are adding the idea that a b a + ( b) allows us to change order when dealing with. Further, you are extending the notion to algebraic expressions. Key for Tiles: x 1 x x x For 1-16 model each expression using Algebra Tiles. Then simplify each expression. 1. 8x + 2x x x + x 3 + 3x 3. 5x 9x 4x x + 9 2x 3 + 2x Reason Abstractly Structure 26

27 x x 2x x x x + 5x 2 6x x 5x 9x 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? Answers will vary. Discuss simplify v evaluate v solve and expression v equation. Also why we simplify when does it help and when is it easier to not simplify? 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? Discuss a b a + ( b); changing all subtract to add the opposite. 27

28 For 9 16 use the key below. 1 1 x x y y 1 x 1 x y y 9. x + x + y + x x + y + x y + 2 3x + y + 2 3x + 4y x + x + y + y x + y x y + 2 2x x + 2y y y x x 4 + 2x y + 2y 5x + y 6 3x 15. 5x 2y + 4 3x + y + x x y 2y + 3x 7 2 3x y + 2 4x 3y 12 28

29 3.1d Homework: More Algebra Tile Exploration Key for Tiles: 1 1 x x 1 x 1 x For 1-16 model each expression using Algebra Tiles. Then simplify each expression, combining like terms x + 3x + 5 8x x + x 1 + 4x 3. 3x 5x 2x x 9 + 4x 2 + 2x 5. 4x 5 + 2x 2x 5 or 2x + ( 5) x + 5x x x 5 + x + 7x 4x 5 or 4x + ( 5) 8. x 6x 5x 29

30 For 9 16 use the key below. 1 1 x x y y 1 x 1 x y y 9. 2x + x + 2y + x x + 3y + x y + 1 2x + 4y + 3 4x + 2y x + 2x + 3y + y x + 3y 1x + 2y + 1 x + 4y + 3 5y x + y 3y + y + x x 4 + x 2y + y 2x y + 1 x y or x + (-y) 15. 5x y + 2 4x + 2y + x x 2y + x 3 2x + y + 4 3x 2y 5 or 3x + ( 2y) + ( 5) 30

31 3.1e Class Activity: Vocabulary for Simplifying Expressions In groups of 2 or 3 students, consider the following expressions: a) 2x y, b) 2x x, and c) 2x + 5x + 3x. How are these expressions similar? How are they different? Students will note that a) cannot be simplified, b) can be simplified to two terms and c) can be simplified to one term. They may also note that the three are algebraic expressions (none are strictly numeric.) Have them explain why some of the expressions can be simplified but others not. Help student note that terms can be combined if they are the same. Note, too, that they likely do not have the vocabulary to attend to precision when talking about the expression. Introduce the need for terminology. 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 There are five terms in this expression. The terms are x, 4, 2y, 3, and 5y Coefficients x 4 2y 3 5y The constants are 4 and 3. (Recall that subtracting is like adding a negative number.) Like Terms The coefficients are 1, 2, and 5 2y and 5y are like terms. 4 and 3 are also like terms. Students will create various acceptable definitions. Here are some referenced definitions, however, it is important to attend to precision. Terms: a part of an algebraic expression, either a product of numbers and variable(s) or simply a number Constant: a fixed value, not a variable Coefficient: a factor of the term Like Terms: terms with the same variable(s) x 4 2y 3 5y x 4 2y 3 5y 2. Identify the terms, constants, coefficients, and like terms in each algebraic expression. Expression Terms Constants Coefficients Like Terms 4x, x, 2y, 3 3 4, 1, 2 4x, x 3z, 2z, 4z, 1 1 3, 2, 4 3z, 2z, 4z 2, 3b, 5a, b 2 3, 5, 1 3b, b a, b, c, d none 1, 1, 1, 1 None 31

32 3. Simplifying Algebraic Expressions. Use the vocabulary words constant, coefficient, and like terms to explain in writing how to simplify each algebraic expression. Exercises 9-28 are review. 3x + 4x Various correct answers. i.e: "3x and 4x are like terms. The coefficient 3 for the first term tells us there are 3 x terms, the coefficient 4 for the second term tells us there are 4 x terms. Because they are like terms we can combine them to say there are 7 x terms or 7x. 8n n 6x + 4 5x + 7 Simplify each expression. 9. 2x y + 5x 7x y 11. 2y + 8x + 5y 1 7y + 8x y + x 5x y 4x + 7y 15. 2x 6 + 3y + 2x 3y b 5 + 2b 3 9b x + 5 7y + 2x 10x 7y y + x + y 6y + x b + 2 2b 8b x x 16. 6m + 2n + 10m 16m + 2n 18. 2a 3 + 5a + 2 7a y + 3 5y 7 y x + 4 7x 22. 2x + 3y 3x 9y + 2 x x 6y + 2

33 23. 2b + a + 3b b + a 24. m n m 3 3n r + q 26. 5h 3 + 2k h + k 4 + 2r + q 4h + 3k t 3 t + 2 4t c + 2d 10c + 4 9c + 2d

34 3.1e Homework: Solidifying Expressions Matching: Write the letter of the equivalent expression on the line. 1. _e 3x 5x 2. _a 4a 12a 3. _ j 3x + 5x 4. _i 16a + a 5. _f 2x 2y + y 6. _b 2x _h x y + 2x 8. _c y + 2y x 9. _g 5x + 4y 3x x + 3y 6y 10. _d 4x + 3y + 5x 7y Simplify each expression by combining like terms b x y a 2b b x 3.4y or x y a) 8a b) 2x 6 c) x + y d) 9x 4y e) 2x f) 2x y g) x + y h) 3x y i) 17a j) 8x 22. 6m z y x u b y z w

35 3.1f Class Activity: Iterating Groups Review: Show different ways to expand 5(10). Have students come up with as many ways as possible to expand 5(10). Record student responses, have then justify that their expression is equivalent to 5(10). Possible expansions: (0+10), 5(1+9), 5(2+8), 5(0)+5(10), 5(1)+5(9), 5(2)+5(8), 0+50, 5+45, 10+40, Students explored the distributive property in 6 th grade, so these are not new concepts. Draw different array and number line models to show 5(10) is 50; use your answers above to come up with different representations. Students should have done a similar exploration in elementary school as early as 3 rd grade. Two ideas should be highlighted: 1) Area representation and its relationship to the distributive property: Help students note that all models for 50 are rectangles with areas of 50 some are single rectangles (as in 1x50 or 5x10) while others are the sum of rectangles (as in 5(3+7) is a larger rectangle of two smaller rectangles of area 15 and 35.) Sample models are shown below student pages do not include the illustrations below (an example of 1x50 did not fit on the page.) Look specifically at all the rectangles when one side is 5. For all of those, the other side is always 10, even if the 10 is split into two parts. Later in this section, time will be spent on the distinction between 5(2) and 5 2, it would be useful to note the representation of 5(5+5) below. 2) Iteration and its relationship to the distributive property: For example, one can see two iterations of twentyfive for 2 x 25 or five iterations of ten for 5 x 10. Another way to see the iterations is on a number line model (the last model example shown.)

36 (3+7) If no students draw this model, suggest it to students and then ask them if they can reorder the jumps. i.e. help them see they can pair a jump of 3 and 7 to make jumps of

37 1 1 x x y y 1 x 1 x y y Use the key above for the following: 1. Model: 2x + 2y Can you write 2x + 2y in a different way? How? x + x + y + y 2(x + y) Student should note a) there are TWO groups of one x and one y, i.e. two groups of (x + y). b) They are reordering terms. Also, have students look carefully at the rectangle they create with each. Note that the rectangle for 2(x + y) is the most square. This idea will be tied to the greatest common factor later in this section. 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. In prior grades, students worked with expressions like 5(3 + 7) and learned that this could be written as either 5(10) or 5(3) + 5(7). This is the first time they will be extending the concept to algebraic expressions this is often a very difficult transition for students. Help students see that it doesn t matter whether they are working with constants or unknowns, everything works the same. Help them see that numeric expressions like 5(3 + 7) can be written as EITHER 5(10) or 5(3) + 5(7), but an expression like 5 (x + 7) can only also be written as 5x + 35 distributing is the only way to eliminate the grouping. You are developing intuitive understanding in this discussion. Properties will be formalized later in this section. 37

38 3. Model: 4x + 12 Can you write 4x + 12 in different ways? How? x + x + x + x x + 2x (2x + 6) 4(x + 3) There are FOUR groups of (x + 3). There are also TWO groups of 2x + 6. Have students look at the two models. They should note that 4(x + 3) is the most square representation. This will not matter in this section. But later it will matter, so it s good to note it now. 4. Model 6x + 12 Can you write 6x + 12 in different ways? x + x + x + x + x + x x + 2x + 2x (2x + 4) THREE groups of (2x + 4) 6(x + 2) SIX groups of (x + 2) Note: 6(x + 2) is the most square representation. How are problems # 3 and # 4 related? How are they related to # 1? Answers will vary. Note that in both 3 and 4 there are a couple choices for writing the expression in written form (later n the section this will be discussed as reverse distribution. ) They should note that all are equivalent forms of the same expression. 38

39 5. Model 5(2x + 1) and then simplify. 10x + 5 Have students draw their models and then present how they arrived at their simplified versions. Ask students to critique the arguments presented. Students should explain saying something like: There are FIVE groups of (2x + 1). Then you reorder the terms so that you are combining all the x terms and all the constant terms. Write 5(2x +1) in two different ways. 10x + 5; 5(2x) + 5(1) 6. Model 4(3x 2) and simplify. 12x 8 Take time to discuss the different ways that this expression can be written (below.) On the next page you will push to formalize a(b c) a(b + ( c)) ab + ac Write 4(3x 2) in three different ways. 12x 8; 12x + ( 8); 4(3x) 4(2); 4(3x) + ( 4(2)) What does the number in front of the parentheses tell you about the grouping? The number tells you how many groups of the amount in the ( ) there are. Discuss equivalence here. 39

40 7. Model 3(2x 5) and simplify. Write 3(2x 5) in three different ways. e.g. 3(2x) 3(5); 6x 15; 6x + ( 15) In 6 th 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? Have students work in pairs to discuss this and present their thinking. a(b c) a(b + ( c)) ab + ac Explanations will vary: There are a groups of (b + c) which gives you a groups of b and a groups of c. Use what you have learned to rewrite each numeric expression. The first one is done for you: 8. 4(2 + 3) 9. 3(1 + 6) (4 1) 4(2) + 4(3) (5) (7 2) 12. 4(3 5) 13. 5(2 + 3) 40

41 Draw a model for each expression, then rewrite the expression in an equivalent form (x + 2) 3x (3x + 5) 6x (x + 1) 3x (3x 1) 12x + ( 4) or 12x (3 + x) 6 + 2x 19. 3(3x 2) 9x + ( 6) or 9x 6 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? PEMDAS two things should be discussed here: 1) we simplify groupings first if we can, as in problems # 8 through # 12. When we cannot, we use our understanding of the expressions to simplify. In other words, for expressions like 3(x + 2), we understand that to mean there are 3 groups of (x + 2). This allows us to reorder our terms and combine like terms. 2) Using a b a + ( b) allows us to change order when there is subtraction. 41

42 3.1f Homework: Iterating Groups Simplify each of the following. Draw a model to justify your answer. 1. 3(x + 1) 3x (3x + 2) 6x (x + 3) 4x (3x 1) 6x 2 or 6x + ( 2) 5. 3(2x 3) 6x + ( 9) 6. 5(x 1) 5x + ( 5) 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

43 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? You are moving students to thinking very ABSTRACTLY in this section. They will need to MAKE SENSE of the STRUCTURE to interpret and USE A MODEL. Using that logic above, what do you think each of the following means? x the opposite of x (x+1) the opposite of x + 1. Talk about taking the opposite of ALL of x + 1. Relate to above. Thus it means x 1 or x + ( 1). If students build this, they may note that it is also the same as 1 x or 1 + ( x) (x) the opposite of x note that these first two mean exactly the same thing. (1 x) the opposite of the entire quantity 1 x; 1 + x or x + ( 1) or x 1. All should be discussed. What does in front of a set of parentheses tell us? Take the opposite of the stuff on the inside the grouping. It would be a good idea to have student generate their own examples here. 43

44 Review: Draw a model of 3(x + 1) then simplify: What do you think 3(x + 1) means? Students need to make sense of the expression. This means the OPPOSITE of 3 groups of (x + 1) [3(x + 1)]. You want students to reason that they can take 3 groups of (x + 1) and then take the OPPOSITE of that expression. Take time to explore this expression. In groups of 2 or 3 students, simplify each of the following. Be ready to justify your answer. 1. 4(x + 1) 2. (x + 2) 3. (3x + 2) 4x (x + 3) 2x (5 3x) x x (x 2) 3x (4 3x) x 3x (3 x) 6 2x 9. 4(2x + 3) 8x 12 44

45 Combining ideas: Review and combine ideas: Simplify the following expression: 3x + 5 x 2x + 5 3(x + 2) 3x + 6 3(x + 2) + x 4 2x + ( 10) In groups of 2 or 3 students, model and simplifying the following exercises. Be prepared to justify your answer x + 5 x + 3(x + 2) 11. 3x + 5 x 3(x + 2) 5x + 11 x 1 Compare problems 10 and (x 1) + 4x 6 + 2x 8x 8 Explain your strategy for simplifying problems 10 through (x 1) + 4x 6 + 2x 4x 4 Compare problems 12 and 13 45

46 Practice: (x 3) 2x x 2(3x +1) x x 3 + 2x 2(3x + 5) 2x (5 3x) 7x x x (x + 2) 16. 9x + 3(2x 5) x (4x 3) 5x + 2 9x + 5 9x + 7 x x (7 x) 46

47 3.1g Homework: More Simplifying Simplify the following expressions: 1. 3(2x + 1) 6x (2x + 1) 6x (2x 1) 6x (x + 4) x (4x 3) 8x x + 2(x + 3) 7x x 2(x 3) 3x x 4 7x 4(2x 3) 5x (x 4) x (3x + 2) 15x x + 2(x 3) 7x x + 2 4x + 2(3x + 1) 5x x 5(2x 5) 3x + 4 9x (4 x) 4 + x or x (2x 5) 14x x 2(x + 3) 3x x x 3(x +2) 8x x 7 2(5x 3) + 4x 5x 1 47

48 3.1g Additional Practice: Iterating Groups Matching: Write the letter of the equivalent expression on the line 1. b 2. i 3. j 4. a 5. g 6. h 7. e 8. f 9. c 10. d Practice: Simplify each expression x y x n a) b) c) d) e) f) g) h) i) j) 29. 6x x 13. 3k p p 14. 2b n n m 24. t y 7z 16. rs + rt k x x k v x n c j t 48

49 3.1 g Additional Practice: Simplifying 1. 5x 10y 2x 4y 3x 2. 5x 10y 2x 4y 3x c 6c 5c 2d 4d 3d 8. 10y 10y x v x 5y 2x 4y 2z 3p 2q 4 p 4q y 5x x 23y x y w 3q 5 8q 6 10w 17v 2 12v 12 15v 4(2x 5) 6(4 2y) ( 8 6v)( 2) (3 5h)( 3) 3y 2x 5y 5x 10x 15h 9 1(x 2y) x + 2y (3x 6) ( 3k 5) 18. 3k + 5 (7h 2k) 7( x 5q) h 2k 7x + 35q 49

50 (2x 5) (4 2y) 8x 17 12y ( 8 6v) 4 (3 5h) 6v 3 5h + 7 2(6h 8) 10 5y 1(x 2y) 6 12h 6 x + 7y + 6 5x (3x 6) 6 5k ( 3k 5) 8 2x 8k h (7h 2k) 5 7x 7( x 5q) h 2k 5 14x 35q 50

51 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. This should be a class discussion. Ask students how their understanding has evolved since doing this problem at the beginning of the section. Review the work students did originally. Focus on equivalence of expressions. 100 feet across n feet across What if the garden were 100 feet by 100 feet? Demonstrate how each method would work now. Method 1: 4(100) + 4, 4 sides of 100 tiles, then 4 corners Method 2: , 102 on top, 102 on bottom, 100 on left, 100 on right Method 3: , outer square minus inner square Method 4: , 4 sides of 102, subtract the tiles counted twice What if the garden were n feet by n feet? Demonstrate how each method would work now. Simplify each method. Method 1: 4(n) + 4, 4 sides of n tiles, then 4 corners Method 2: (n+2) + (n+2)+ n + n, n+2 on top, n+2 on bottom, n on left, n on right Method 3: (n+2) 2 n 2, outer square minus inner square Method 4: 4(n+2) 4, 4 sides of (n+2), subtract the tiles counted twice 51

52 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: 2(3 + n) 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. 14(3x ) 14 3 x x cents 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. 12(L 10) 12 L + 12( 10) 12L 120 miles 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.) ( s 5) ( 5) s s s

53 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. 3( ) $ 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. 5( ) 5(45) minutes 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. 5(24 + x) x 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. 3(30 + t) t t t 90 3t dollars 53

54 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. equivalent not equivalent; however 12 + ( 4) ( 4) Ask students if there is a way to commute with subtraction. 5. equivalent What pattern are you noticing? 2. not equivalent equivalent not equivalent Commutative Property: a + b b + a; ab ba; Addition and multiplication are commutative (you can change the order of addition or multiplication.) Also discuss that subtraction expressions can be changed to addition expressions with integers and how doing this allows order to change. After you ve talked about multiplicative inverse (below), come back here and discuss how division expression could be changed to multiplication expressions to change order. 54

55 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 equivalent 12 + (4 + 6) (3 + 5) equivalent 3 + ( ) (5 4) equivalent 5 (4 ) (6 2) 5 equivalent 6 (2 5) (12 4) 3 not equivalent 12 (4 3) Ask how both could be written to make them equivalent 12 + ( 4 + 3) (12 + 4) + 3 (20.9 8) 2 not equivalent 20.9 (8 2) (18 6) 3 not equivalent 18 (6 3) Ask how both could be written to make them equivalent (18 1/6) (1/3) 18 (6 (1/3)) (24 12) 3 not equivalent 24 (12 3) What patterns do you notice about the problems that were given? Associative Property: (a + b) + c a + (b + c); (ab)c a(bc); Addition and multiplication are associative (you can change the grouping), but subtraction and division are not. As above, discuss how one might work with subtraction expressions to change grouping. Discussion of division should happen in conjunction with multiplicative inverse. 55

56 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? Have students brainstorm. Answers will vary. Look for something like doesn t change the identity of the expression. Give examples of what you mean: Identity Property of Addition: a + 0 a; You can add 0 to anything and it won t change the expression. Discuss zero pairs from Chapter 2 here. What do you think the Identity Property for Multiplication should mean? Have students brainstorm. Answers will vary. Look for something like doesn t change the identity of the expression. Give examples of what you mean: Identity Property of Multiplication: a(1) a; You can multiply anything by 1 and it won t change the expression. Multiplicative Property of Zero What do you think this property tells us? Multiplicative Property of Zero a(0) 0; 0 times anything results in 0. For advanced students, a discussion of fields or groups would be good in this section. For example you may want to talk about properties with different Isometries 56

57 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: a + ( a) 0; discuss how this property is related to the additive identity Inverse Property of Multiplication: a(1/a) 1 for a 0; discuss how this property is related to the multiplicative identity. 57

58 Properties of Mathematics: Name Property Algebraic Statement Meaning Examples Identity Property of Addition 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 Adding zero to a number does not change the number. Multiplying a number by one does not change the number. Multiplying a number by zero results in zero. Reversing the order of addition does not change the result. Reversing the order of multiplication does not change the result. Changing the grouping of addition does not change the result. Changing the grouping of multiplication does not change the result. a groups of (b + c) 0 Multiplying 1 for a 0 A number added to its opposite will result in zero. a number by its inverse will result in one (2 + 5) 3(2) + 3(5) 3(2 5) 3(2) + 3( 5)

59 3.1i Homework: Properties Complete the table below: 1. Identity Property of Addition: 0 2. Identity Property of Multiplication: 1 3. Multiplicative Property of Zero: 0 0 Show the Identity Property of Addition with Show the Identity Property of Addition with Show the Identity Property Show the Identity Property of Multiplication with 23 of Multiplication with 3b b Show Multiplicative Show the Multiplicative Property of Zero with Property of Zero with 4xy Commutative Property of Addition: is the same as: Commutative Property of Multiplication: ab ba x + z is the same as: is the same as: 6k is the same as: 6 6. Associative Property of Addition: (a + b) + c a + (b + c) ( ) is the same as: ( ) (x + 1) + 9 is the same as: x + (1 + 9) 7. Associative Property of Multiplication: ( ) 3.7 is the same as: 2.6 ( ) (wh)l is the same as: Use the listed property to fill in the blank. 8. Multiplicative Inverse: ¼ ( 4 ) 1 9. Additive Inverse: a + ( a) x + x 0 59

60 Name the property demonstrated by each statement Commutative Property of Multiplication Associative Property of Multiplication ( 25) 0 Additive Inverse Property Multiplicative Inverse 14. (x + 3) + y x + (3 + y) Associative Property of Addition mp mp Identity Property of Multiplication (5+35) (9+5) + 35 Associative Property of Addition b 6b Identity Property of Addition 18. 7x 0 0 Multiplicative Property of Zero 19. 4(3 z)(4 3)z Associative Property of Multiplication 20. x x Commutative Property of Addition 21. 3(x + 2) 3x + 6 Distributive Property 60

61 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 Multiplication is Associative, I can group multiplication in different ways and still get the same answer Subtraction is not Commutative, I cannot change the order of subtraction. If I change all subtractions to the addition of the opposite term, then I can make changes. Correct: ( 63) (2) + 14 Addition is Commutative, I can change the order of addition and still get the same answer (3 + 1) 4(3 + 1) + 25 Addition is Commutative, I can change the order of addition and still get the same answer. Using Properties to Justify Steps for Simplify Expressions Example: Jane wants to find the sum: 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 No change, this is where she started. This expression was given The 17 and the 12 traded places. Commutative Property of Addition (3 + 17) + ( ) Jane chose to add the numbers in pairs first, which is like inserting parentheses. Associative Property of Addition Jane found the sums in the parentheses first. Jane is now following the Order of Operations. 60 And so

62 1) The expression 3(x 4) 12 has been written in three different ways. State the property that allows each change. Expression Step Justification 3(x + ( 4)) + ( 12) No change This expression was given, only rewritten using the idea that a b a + ( b) ( 12) + 3(x + ( 4)) Addition order changed Commutative Property of Addition/Addition is Commutative ( 12) + 3x + 3( 4) Three groups of (x + ( 4)) written out Distributive Property 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 Given expression 6x x + 2 Multiplied 2 by both 3x and 1 Distributive Property Commutative Property of 6x + 6x Changed the order of the terms Addition/Addition is Commutative x + ( 6x) and 2 + ( 2) both sum to 0 Additive Inverse 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 Step Justify Step 1: this is a representation of the expression Step 2: 1 + ( 1) 0, so each of these pairs are zero: 1 is the additive inverse of 1. Step 3: this is a representation of what remains when the zero pairs are removed. 62

63 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. Answers will vary. Look for logic and the use of properties to justify steps. Students may use models. Have students present their proof to the class. Here is a proof: we start with 1(0) 0 Statement Step Justification 1(0) 0 Given Multiplicative Property of Zero 1( 1 + 1) 0 0 was replaced with ( 1 + 1) Additive Inverse Property 1( 1) + 1(1) 0 1 was multiplied by each term in Distributive Property parentheses 1( 1) + 1( 1) 0 The 1(1) got switched to 1( 1) changed Commutative Property of multiplication the order of multiplication 1( 1) ( 1)1 was replaced with 1 Identity Property of Multiplication 1( 1) must equal 1 because if we get 0 when we add it to 1, it must be the additive inverse of 1 Additive Inverse Property 63

64 3.1k Classwork: Modeling Backwards Distribution Review: below is a review of modeling multiplication with an array Factors: 1, 3 Factors: 2, 3 Factors: 3, 3 There is one group of 3 There are two groups of 3 There are three groups of 3 Product/Area: 3 Product/Area: 6 Product/Area: 9 Use the Key below to practice using a multiplication model. 1 x 2 x x x Factors: 2, 2 Factors: 4, 4 Factors: x, x Product/Area: 4 Product/Area: 16 Product/Area: x 2 Look at the three models above. Why do you think 2 2 is called two squared? 3 2 is called 3 squared? and 4 2 is called four squared? The difference between 4 4 (4 2 ) and (2(4)) is difficult for many students. Discuss the geometric representations. 64

65 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? 3, x + 2 What is the area? 3x + 6 What is the product of the multiplication problem? 3x 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? 3, 2x +1 What is the area? 6x + 3 What is the product of the multiplication problem? 6x Build the factors for 2(x + 4). Build the area and draw. What is the area or product? 2x + 8 What are the factors? 2 and x Build the factors for x(x + 3). Build the area and draw. What is the area or product? x 2 + 3x What are the factors? x and x

66 5. Build the factors for x(2x + 5). Build the area and draw. What is the area or product? 2x 2 + 5x What are dimensions or factors? x and 2x + 5 Review concepts: 6. Write each as the sum of two whole number and the product of two integers. Model your expression: a) 15 answers will vary: etc. b) 9 c) Discuss: sometimes we break numbers apart using addition, other times we break them apart with multiplication. Why? 7. Simplify each; use a model and words to explain the difference between the two expressions: a) x + x 2x b) x x x 2 Discuss the geometric representation of each. 66

67 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: x + 2 Width: 2 What is the area (product) of the rectangle? 2x + 4 Write 2x + 4 in factored form: 2(x + 2) 8) 6x + 3 9) 3x ) 5x ) 6x + 3 Factors: 3 and 2x + 1 6x + 3 in factored form: 3(2x + 1) 9) 3x +12 Factors: 3 and x + 4 3x + 12 in factored form: 3(x + 4) 10) 5x + 10 Factors: 5 and x + 2 5x +10 in factored form: 5(x + 2) 67

68 11. 6x x x 2 + 3x 6x + 2 Factors: 2 and 3x + 1 6x + 2 in reverse distributed form: 2(3x + 1) x + 4 Factors: 1 and x + 4 x + 4 in reverse distributed form: x + 4 or 1(x + 4) x x Factors: x and x + 3 2x 2 + 4x in reverse distributed form: x(x + 3) Practice: Write each in reverse distributed form. Use a model if you d like x + 6 6(5x + 1) 15. 4b (b + 7) 16. 3m 15 3(m 5) 17. 4n 2 2(2n 1) b 5 5(5b 1) 19. 4x 8 4(x 2) Look at problems 14, 15, and 19. How else might these expressions be factored? e.g. 14 could be written and 2(15x + 3). Discuss most square representation of each this should not be a lesson on factoring. 68

69 3.1k Homework: Modeling Backwards Distribution Write each in reverse distributed from. Use a model to justify your answer. 1. 2x + 4 2(x + 2) 2. 3x (x + 4) 3. 2x (x + 5) 4. 3x (x + 6) 5. x 2 + 2x x(x + 2) Simplify each expression. Draw a model to justify your answer. 7. 2x + 3x 5x 8. (2x)(3x) 6x 2 6. x 2 + 5x x(x + 5) 69

70 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 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. Beginning Understanding Developing Skill and Understanding Deep Understanding, Skill Mastery 70

71 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. 71

72 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: student workbook does not have models as shown in teacher edition below. 1. m k n 18 To find k, we must take away 2 from both k 2 and j j/ y To find y, we must take away 3 from both the y +3 and the 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 7 Discuss how the unknown in an expression can represent any number. Also, discuss evaluate v. solve. Solve each equation in any way you want: 8. 2x x x 2-3 Discuss how/why 7-8 simplify to different values. 9. Evaluate 2x + 1 for x 3 x x x -3 72

73 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 x Use a model to solve x 1 6. Write the algebraic procedure you followed to solve. x 1 6 Throughout these exercises, students should discuss properties of arithmetic. x additive inverse x 7 73

74 12. Use a model to solve x 3 5. Write the algebraic procedure you followed to solve. Talk about equality; adding 3 to both sides of the equation maintains equality. 3 and (-3) are additive inverses. x 3 5 x additive inverse x Use a model to solve m. Write the algebraic procedure you followed to solve m m +( 7) +( 7) additive inverse 1 m 74

75 14. Use a model to solve 6 3x. Write the algebraic procedure you followed to solve. 15. Use a model to solve 8 2m. Write the algebraic procedure you followed to solve. Students might take the opposite of both sides: -8 2m. Allow students time to Make Sense of the Problem 6 3x (1/3) 6 (1/3) 3x or 6/3 3/3 x multiplicative inverse 2 x 8 2m 75

76 16. Use a model to solve 5 + 3n 7. Write the algebraic procedure you followed to solve. 76

77 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. 1. x 6 9 x x 1 1 x

78 2. 15 x small negative squares 14 small negative squares and a positive x (rectangle) 15 x m rectangle, 2 small squares 11 negative small squares m

79 4. 4n m students may take the opposite of both sides to model m 79

80 6. 3t rectangles, 5 small squares 2 small squares p 4 80

81 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. 1. x x + 8 Model/Draw the Equation Look at # 1 and # 2, why are the answers different? Discuss structure. What are the solving action? Record the steps using Algebra Add 5 to both sides. x x 3 Show work x 3 Check solution in the equation. x 5 8 (3) 5 8 True, so the solution is correct. Show check 81

82 3. 3x 6 Show work Show check x 2 Explain the logic above in #3. How might you use related logic to model x/3 6? 4. x/3 6 Show work Show check 1/3 of an x is -6, so if we triple both sides, a whole x is -18. x (1/2)x 3 Show work Show check x 6 In problems # 3 and # 4, what happened to the terms on both sides of the equation? Both sides of the equation were multiplied by the multiplicative inverse of the coefficient of x. 82

83 6. 2x 5 9 Show work Show check x x 2 Show work x 3 Show check x Show work Show check x x/2 3 Show work Show check x x/2 + 2 Show work Show check x 10 83

84 3.2b Homework: More Model and Solve One- and Two-Step Equations 1. 2 x + 5 Model/Draw the Equation What are the solving action? Record the steps using Algebra Show work Check solution in the equation. Show check x 3. (x/4) 8 x 3 Show work Show check x 4 Show work x Show work 2 (1/3) x x 6 Show check Show check 84

85 5. 9 x/2 5 Show work Show check x x x x 3 x 2 85

86 n/3 2 Show work Show check x 6 9. x/3 5 2 Show work Show check x x 8 Show work Show check x (1/2) x 5 3 Show work Show check x 4 86

87 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. Additional practice is available at 3.2h a n/ x 7 **This will cause questions: Students will be unsure what to do with 0 3x. Discuss x 0 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) 87

88 Use a model to solve each. Show your algebraic manipulations on the right. 5. 2(x + 1) 8 Students may distribute first OR may chose to divide first. Discuss both approaches. x (x 4) (5x 1) x 2 x 1 88

89 8. 4(3 2m) (1/2)(4x + 2) (2x 4) **Again, they may be unsure what to do when nothing is left on side of the equation. i.e. they will get 8m 0. Look back at Review #3. m 0 **Look for different strategies: doubling both sides OR distributing the ½. Talk about both. x 2 x 3 89

90 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 p h x h 2 x 5 90

91 n + 5 n x 2 7x (n 5) 4 x 7 n 3 91

92 7. 3(g 3) 6 g (4c + 5) **This is the first equation that has a fraction answer. c 9/4 92

93 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 c 37 4x 3 17 The student forgot the negative. Multiplicative inverse is ( 1/6) The Multiplicative invers is 4/3 On the second line the student forgot to keep the negative. a b a + ( b) The student subtracted when division was necessary. Student applied additive inverse rather than multiplicative inverse

94 6. 3(2x 4) x 2x (x 2) 14 3(2x 1) 4 10 The student did not distribute accurately. There are 3 groups, of (2x 4). So there will be 6x and 8. The student subtracted 2x twice from the same side of the equation. The equality was not maintained. The student failed to note a ( b) a + b. Taking away 2x is the same as adding 2x. The student did not distribute correctly. Likely not accounting for the negatives. The student added terms that were not like terms. 94

95 3.2d Homework: Practice Solving Equations (select homework problems) Solve each equation, use Algebra Tiles if that will help you. Assign a portion of the problems: divide the problems among groups of students. Each student should do 6-10 problems (for example, a column if in groups of three), then student pairs check each other s homework the next day t n y t 13 n 10 y k 8 4 k 5. 5 b 8 13 b a 5 0 a 7. 8 n x 7 5 y n 6 t 21 x 120 n 24 y t n 8 6 y y p x 4x x 6 8 2x 4 6 p 4 x 3 x 16. 6(m 2) (2c 7) (2d 4) 35 4 m 6 x 4.5 c 1.5 d 19. 3(x 1) (2c 5) (3d 5) c 2.5 d m 2m p 3p 8 p p 8 p m 1 p 6 p p 4 3p x x 3 4x -1 p 12 x x 15/24 5/8 95

96 3.2e Class Activity: Solve One- and Two-Step Equations with Rational Numbers (use algebra to find solutions) Before we begin Additional practice is available at 3.2h 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. Throughout this assignment, students should make sense of the equation. Drawing models is important throughout. As students become more proficient, discuss multiplicative inverse. Structure Model how can you figure out the solutions in your head? Review equivalent forms of rational numbers from Chp 1 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 m y y 24 m 3 2. m 32 y 59 y 12 Check: 5. x ¼ y 12 y.25x 4 Compare 4 and 7. In 4) two y s are -24, in 7 half an x is 6. In both we want to know what one unknown is. 7. x x 8 96

97 Check: a d Check: a 2.5 Check: r v r 16.6 x 40 v 14.4 Check: n x 2 Check: 97

98 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 k 3. x k x 49 x 36 Check: 4. 3y 36 y 12 Check: 7. x x 5 8. x 12 Check: m j y 70 y x 8 Check: 98

99 13. 5b 0.2 b Check: p x p Check: 19. x 20 Check: 20. x 4.7 x m x 25 99

100 3.2e Extra Practice: Equations with Fractions and Decimals Discuss clearing fractions conceptually first. Reprint. Check: Not Do 6. Only. Purposes Check: n x 3.5 Instructional for Draft: Check: 100

101 x 6 a 3.75 Check: 13. Check: 16. Check: q 15.5 w k h

102 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 a Yes This accurately describes the situation. b a No This has swapped the value that Brielle has with its increase over Annie's. c. 8 5a No This says Brielle has 5 times as many cats as Annie. d. 5 8a No This is a few steps off: wrong operator, wrong place. 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 the cost of one pound of fruit snacks Equation Will it work? Why or Why Not? a. y No This says one pound costs three dollars less than $4.25. b. 3y 4.25 Yes This shows three pounds by multiplying one pound by three. c. y + y + y 4.25 Yes This shows three pounds by repeated addition of one pound. d y Yes This is the same as b, but order has changed. 102

103 3. Jim bought a tie for x dollars and a jacket for $ 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 the cost of a tie Equation Will it work? Why or Why Not? A x + x Yes This equation lists all items and sums them. B 2x + 2(37.50) 80 Yes This equation counts each item twice, then sums them. C 2(x ) 80 Yes This equation sums each boy's cost, then multiplies by two. D 2x No This equation has mixed the total and the jacket cost. 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 the number of songs that Bo bought Equation Will it work? Why or Why Not? A 0.79x Yes This equation multiplies the song number by the song price. B 5.98x No This equation is multiplying the album cost by the song number. C x No This equation shows x as a sum of prices. D x 8.35 Yes This is the same as equation via the commutative property. 103

104 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: h 8 60 or h Solution: My mom s height 68 inches 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. b is the number of coins in blue jar r is the number of coins in red jar b r A car has two more wheels than a bicycle. c is the number of wheels on a car 2. My age is twice my cousin s age. m is the my age c is the cousin s age m 2c 4. Art s jump of 18 inches was 3 inches higher than Bill s. b is the height of Bill s jump b is the number of wheels on a bicycle c b is the b + 3 b 15

105 5. The sum of a number and its double is eighteen. 6. The $5 bill was $3 more than the cost of the notebook n is the unknown number n is the cost of the notebook n + 2n 18 n 6 The unknown number was The visiting team s score was five points less than our score (50 points). v is the visiting team s score 50 5 v The visiting team's score (v) was 45 points. 9. Bill has twice as much money as I do. Our money together is $9. m is the my money m + 2m 9 My money (m) is $3 5 n + 3 n 2 The notebook costs $2. 8. The number of minutes divided by sixty gives us 3 hours. m is the the number of minutes 3 Solution: The number of minutes (m) is A large popcorn and a drink together cost the same as the movie ticket. I spent $10 on all three. t is the cost of a movie ticket t + t 10 The cost of a movie ticket (t) is $5. 105

106 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. Distribute 2. Collect like terms 3. Variables on one side, constants on other side 4. Divide (or multiply by fraction) to get a variable coefficient of 1 5. 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 x 1 b. 3 2(x + 3) + x + 2 x + 2 x 1 c. 3(x 6) 4(x + 2) 21 x 5 d. 7(5x 2) 6(6x 1) 4 x 4 e. 2a + (5a 13) + 2a 3 47 a 7 f. 3a + 5(a 2) 6a 24 a 17 g. 13 2(c + 2) + 3c + 2c + 2 c 1 h. 3(y + 7) y 18 y 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? Various equation may work. This is one example 100 You started with $2,

107 3.2h Extra Practice: Solve Multi-Step Equation Review d 124 d k a 100 Check: 4. g g h 15 Check: w Check: 107

108 x 4 x x s 26 Check: x 1 w 21 Check: (x 1) 2(x 3) 0 v 9 x Check: 108

109 v 46 w 1.2 Check: Solve Multi-Step Equations (distribution, rational numbers) 22. 4(x 2x) (5 x) ( 4 6x) 24 x 2 Check: x 8 x 8 x ( 2x 3) (4x 1) ( 2x 7) 105 x 5.5 x 11 Check: 109

110 28. 3(7x 8) (7x 5) (1 7x) 320 x 6 x 8 x 9 Check: 31. 3(5x 6) (2x 3) 96 x 4 Check: 34. x x 8.1 x a x 10 Check: 110

111 m b n Check: 40. 2(4x 8) (5x 8) ( 4x 2) 76 x 6 Check: x 1 x (3x 7) ( 9 x) ( 7 6x) 175 x 6 x 7 Check: 111

112 46. 2(1x 4) (7 4x) (10 6x) 84 x 5 x 1 x 3 Check: 49. 5(1x 7) (5 2x) x 1 Check: 52. x 3 x Check: k k r x 2 r Check: 112

113 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 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. Beginning Understanding Developing Skill and Understanding Deep Understanding, Skill Mastery 113

114 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. Teachers note: In this section, students will solve percent problems using equations. Percent problems will be revisited in Chapter 4 and solved using proportional thinking. You will want to relate the concepts in that chapter. In this chapter, students may still prefer models to solve problems involving percent, that s fine. Help students connect models to equations throughout. 114

115 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. Access background knowledge regarding drawing models for percentage problems (chapter 1) and fluency between fractions, decimals and percent. Students have not experienced equations and percentages simultaneously. Observe their reasoning about which equation works to find the solution. Help them reason through what each equation says/means, then practice on their own. There are several ways students may write equations. 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. c 35 questions correct 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) 1/4 c/50 c 200 total cookies 115

116 Draw a model for each and then write an algebraic equation to solve each percent problem % of my books are science fiction. If I have 60 science fiction books, how many books do I have? 0.30x 60 so x 200 books 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? 0.15(24) h h 3.6 hours 6. I got 52 out of 60 questions right on the last History test. What percent correct did I get? 60x 52 so x 86 2/3% 7. A pair of jeans are on sale for $35. Originally they were $45. What percent of the original price is the sale price? 45x % 116

117 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. r 36 questions correct 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. p 12 total members 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. b 0.66 so 66 2/3% are wearing blue 117

118 Draw a model for each and then write an algebraic equation to solve each percent problem % 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? 800(0.65) p p 520 voters 5. 5% of the apples have worms in them. 10 apples had worms in them. How many apples are there total? w(0.05) 10 w 200 total apples students dressed up for Spirit Day. There are 800 students. What percent dressed up for Spirit Day? p 31.25% dressed for Spirit Day 7. 55% of my shirts are purple. If I have 20 shirts, how many of them are purple? 20(0.55) p p 11 purple shirts 8. 5 out of the 7 ducklings have yellow feathers. What percent of the ducklings have yellow feathers? y 71.4 % have yellow feathers 9. 30% of the sodas are grape. There are 24 grape sodas. How many sodas are there total? s(0.30) 24 s 80 total sodas 10. I made 36 cupcakes, that s 80% of what I need. How many cupcakes will I be making all together? 0.8x 36. I need 45 cupcakes total 118

119 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 y c. I need all of y and then 0.1 of y. Discuss why the other two are common mistakes. 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. $ (38) p 38+38(.06) p 38(1+.06) p 119

120 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. x 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. s b. Which of the following algebraic expressions represents the sale price of the skateboard? s s 0.8s s s 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. $ (0.8) p 64 64(0.20) p 64(1 0.20) p 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. 1.5($7.50) c. How much should she charge? $ 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? 120

121 Draw a model to represent this situation. Write an expression to represent the new population. 1.20(850) What will the new population be? 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. What was the whole sale price? $ Write an expression to represent the whole sale price $2200/ 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? Draw a model to represent this situation. Write an expression to represent the problem situation. What will Carlos pay for the snowboard? $ b Homework: Percent Problems (0.80)(0.70)($450) 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.)

122 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 b 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. $ (1.2) p (0.2) p 22.50(1+.2) p 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. 93 correct (150) c 0.62(150) c 150(1 0.38) c 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 122

123 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: x(0.5) x x(0.5) 30% off sale with additional 20% off coupon: (0.7x)0.8 (x 0.3x)0.8 c. Which coupon will get her the lowest price? Explain how you know your answer is correct. The 50% off coupon will get her the lowest price. 123

124 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? 0.18 t so the tip was 18% Write an equation with a variable for each problem below, then solve the equation. Justify your work. Have students do these in groups of 2-3 and then present their answers to the class 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? 0.06x 3 x 50 The original price of the sweater was $50 dollars. 3. Jill bought a bracelet that cost $5. Sales tax came out to be $0.24. What is the sales tax rate? 5t 0.24 t Sales tax is 4.8%. 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? d (1.25)*(1.25)(12) On Saturday, she will run miles. 124

125 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? (0.8)(0.7)(8 inches) The image is now 4.48 inches wide. 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? x the number of students at the beginning of the year. x x 36, 30 students 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 $150. How much will Anna have to make? Let A be the amount Anna will need to make; 0.2(500 + A) 150 A$