Are they equivalent? - CHAPTER 2. Sequences and Equivalence

Sequences

Lesson 2.2.1 Are they equivalent?

Equivalent Expressions

Lesson Objective: Students will identify equivalent expressions and develop and share algebraic strategies for demonstrating equivalence.

Length of Activity: One day (approximately 50 minutes)

Core Problems: Problems 2-118 through 2-120

Ways of Thinking: Justifying, generalizing, choosing a strategy

Materials: None

Section Overview: Section 2.2 begins an explicit focus on algebraic manipulation skills that continues throughout the course. In this lesson, students focus on equivalence and strategies for rewriting equations and expressions. The understanding and skills that students gain in this lesson will be essential

H T A

Suggested Lesson Activity:

As teams start on problem 2-118, encourage them to find as many ways as they can to see the number of tiles in the pool border and to create expressions that represent each of these ways of picturing the tiles. After teams have had time to generate ideas, bring the class together and have teams share their expressions and reasoning. Challenge the class to think of additional ways to write an expression to represent the situation. Then pose the question, “How can we tell that they are all equivalent?” Students may decide that they can prove equivalence by evaluating the expressions for a given value of x. If so, ask if one value of x is enough. If they claim that it is, give a counterexample, such as the expressions

3x+4 and x+8 evaluated for x=2. Students may also decide to show equivalence by simplifying or manipulating expressions algebraically to show that they are the same. In this case, ask for justification for the steps they take. This in an opportunity to remind students that they are using algebraic properties such as the

Commutative Property, the Associative Property, the Distributive Property, and the properties of zero and one (if these are important requirements of your curriculum). Encourage all relevant ideas. Then move teams on to problems 2-119 and 2-120. Notice that in problem 2-120, students are asked to write at least three equivalent expressions for each given expression. So, rather than just simplifying using memorized rules, students will have to think about what they can do to the expressions without changing their value. This would be a good place to do a “Pairs Check.”

Suggested Study Team Collaboration Strategy: for problem 2-120, Pairs Check

Closure: (10 minutes)

Give teams the opportunity to share any particularly interesting ideas that came up in problem 2-120. Then have them complete the Learning Log entry in problem 2-121 to explain their understanding of equivalence.

2.2.1 Are they equivalent?

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Equivalent Expressions

Student pages for this lesson are 94 – 97.

In Chapter 1, you looked at ways to organize your algebraic thinking using multiple

representations. In the first part of this chapter, you used multiple representations to analyze arithmetic and geometric sequences. In this section, you will focus on equations and expressions while experimenting with equivalent expressions and rewriting equations to solve them more easily.

2-118. Kharim is designing a tile border to go around his new square

swimming pool. He is not yet sure how big his pool will be, so he is calling the number of tiles that will fit on each side x, as shown in the diagram at right. [ a: Answers vary but could include 4(x!1), 2(x!2)+2x, 4(x!2)+4, 4x!4, etc; b: All expressions should simplify to 4x!4;

c: Explanations vary. ]

a. How can you write an algebraic expression to represent the total number of tiles Kharim will need for his

border? Is there more than one expression you could write? With your team, find as many different expressions as you can to represent the total number of tiles Kharim will need for the border of his pool. Be prepared to share your

strategies with the class.

b. Find a way to demonstrate algebraically that all of your expressions are

equivalent, that is, that they have the same value.

c. Explain how you used the Distributive, Associative, and Commutative Properties in part (b).

x tiles

?

=

2-119. Jill and Jerrell were looking back at their work on problem 1-54 (“Analyzing Data from a Geometric Relationship”) in Lesson 1.2.1. They had come up with two different expressions for the volume of a paper box made from cutting out squares of dimensions x centimeters by x centimeters. Jill’s expression was (15!2x)(20!2x)x, and Jerrell’s

expression was 4x3!70x2 +300x. [ a:They are equivalent; b:Methods vary, but see the notes in the “Suggested Lesson Activity” section for a description of some possible methods; c: They are all equivalent, but methods vary—for example, students may show that (20!2x)x=(10!x)2x or that

(15!2x)(10!x)2x multiplies out to 4x3!_70x2 +_300x_{using the Distributive,}

Commutative, and Associative Properties; d: Not necessarily—this describes the volume of a box with dimensions 2x by (20–2x) by (10–x), while Jeremy’s box is only the same when x=5. ]

a. Are Jill’s and Jerrell’s expressions equivalent? Justify your answer.

b. If you have not done so already, find an algebraic method to decide whether their expressions are equivalent. What properties did you use? Be ready to share your strategy.

c. Jeremy, who was also in their team, joined in on their conversation. He had yet another expression: (15!2x)(10!x)2x. Use a strategy from part (b) to decide whether his expression is equivalent to Jill’s and/or Jerrell’s. Be prepared to share your ideas with the class.

d. Would Jeremy’s expression represent the dimensions of the same paper box as Jill’s and Jerrell’s? Explain.

2-120. For each of the following expressions, find at least three equivalent expressions. Be

sure to justify how you know they are equivalent. [ Possible responses include a: x2 +₆_x+₅_,_x2 +₃_x+₃_x+₉!₄_,₍_x+₅₎₍_x+₁₎_{; b:}₈_a6_b9_,16a6b9 2 , (2a2_b3₎₍₂_a2_b3₎₍₂_a2_b3₎_{; c:}_m3_n9_,_m!_m!_n!_n!_n!_n!_n!_m!_n!_n!_n!_n_,₍_mn3₎3_; d: 9p4q2 q6 , 9p4 q4 , ( 3p2 q2 ) 2_] a. (x+3)2!4 b. (2a2b3)3 c. m2n5!mn4 d. (3p2q q3 ) 2

2-121. LEARNING LOG

What does it mean for two expressions to be equivalent? How can you tell if two expressions are equivalent? Answer these questions in your Learning Log. Be sure to include examples to illustrate your ideas. Title this entry “Equivalent Expressions” and label it with today’s date.

2-122. For each of the following expressions, find at least three equivalent expressions.

Which do you consider to be the simplest? [ Answers vary but are equivalent to: a: 4x2 !₁₂_x+₁₄_{, b:}81y4

x4 ]

a. (2x!3)2+5 b. (3x2y

x3 )

2-123. Match the expressions on the left with their equivalent expressions on the right.

Assume that all variables represent positive values. Be sure to justify how you know each pair is equivalent. [ a: 3, b: 4, c: 1, d: 5, e: 2 ]

a. 4x2y4 1. 2x y

b. 8x2y 2. 2y 2x

c. 4x2y 3. 2xy2

d. 16xy2 4. 2x 2y

e. 8xy2 5. 4y x

2-124. Donnie and Dylan were both working on simplifying the expression at right. The first step of each of their work is shown below.

Donnie: 8x15y12

512x3y9 Dylan: (

x4y

4 ) 3

Each of them is convinced that he has started the problem correctly. Has either of them made an error? If so, explain the error completely. If not, explain how they can both be correct and verify that they will get the same, correct solution. Which

student’s method do you prefer? Why? [ They are both correct: x12y3

64 .

Preferences vary. ]

(2x5y4

8xy3 ) 3

2-125. While Jenna was solving the equation 150x+300=600, she wondered if she could first change the equation to x+2=4. What do you think? [ a: They both have the solution x=2. b: She divided both sides of the equation by 150 and used the Distributive Property. c: Answers vary. One way to rewrite the equation is t!2=5.]

a. Solve both equations and verify that they have the same solution.

b. What did Jenna do to the equation 150x+300=600 to change it to x+2=4? c. In the same way, rewrite 60t!120=300.

2-126. Solve this system for m and b: 342=23m+b

147=10m+b

[!m=15,!b=!3!]

2-127. Tanika made this sequence of triangles:

[ a: (4, 8, 4 3), (5, 10, 5 3); b: The long leg is n 3 units long, and the hypotenuse is 2n units long. ]

a. If the pattern continues, what do you think the next two triangles in the sequence would be?

b. Write a sentence to explain how to find the long leg and hypotenuse if you know the short leg (i.e., if the base is n units long).

2-128. Consider the sequence 3, 9, … [ a: 15, 21, 27, 33, t(n)=6n!3; b: 27, 81, 243, 729, t(n)=3n_{; c: Sequences and rules vary. ]}

a. Assuming that the sequence is arithmetic with t(1) as the first term, find the next four terms of the sequence and then write a rule for t(n).

b. Assuming that the sequence is geometric with t(1) as the first term, find the next four terms of the sequence and then write a rule for t(n).

c. Create a sequence that begins with 3 that is neither arithmetic nor geometric. For your sequence, write the next four terms and, if you can, write a rule for t(n).

2-129. Classify the triangle with vertices A(3, 2), B(–2, 0), and C(–1, 4) by finding the length

of each side. Be sure to consider all possible triangle types. Include sufficient evidence to support your conclusion. [ This is a scalene triangle, because the sides

2 4 6 1 ₂ ₃ 3 2 3 3 3 30˚ 60˚ 30˚ 60˚ 30˚ 60˚

Lesson 2.2.2 Are they equivalent?

Area Models and Equivalent Expressions

Lesson Objective: Students will use an area model to multiply expressions. They will factor expressions and demonstrate equivalence.

Length of Activity: One day (approximately 50 minutes)

Core Problems: Problems 2-130 through 2-132

Ways of Thinking: Justifying, generalizing, choosing a strategy

Materials: None

Note: If you have determined (by assessing homework or some other method) that your students are very comfortable multiplying and factoring expressions, you may choose not to spend a whole class period on this lesson. Also note that problem 2-130 is designed to allow you to assess students’ understanding of multiplying binomials.

Suggested Lesson Activity:

As teams work on problem 2-130, encourage them to find multiple ways to prove that (x+y)2 ! x2+y2. As you circulate, take note of the strategies being developed. Some common strategies are substitution of numbers, diagrams using discrete

objects (as shown in the diagram at right), area models, and use of the Distributive Property or the FOIL acronym to show steps. Ask each

team to share a strategy until they have all been discussed.

Then move teams on to problem 2-131, in which they will use an area model to find equivalent expressions, or at least demonstrate

equivalence. Area models were used heavily in Algebra Connections, so for those of your students who used that text, this should be a reminder. For students who learned to multiply and factor expressions using another method, area models can still be a powerful tool for demonstrating equivalence and understanding multiplication and division of expressions using the Distributive Property. Ask teams to share strategies for part (c), as a way to remind students of factoring methods.

Move teams on to problem 2-132 and, if time permits, problems 2-133 and 2-134. As you circulate, encourage students to discuss and share methods in teams.

Stuggested Study Team Collaboration Strategy: for problem 2-132, Pairs Check

Closure: (10 minutes)

Have teams share strategies for finding equivalent expressions for parts (b) and (d) of problem 2-132.

( + )2 = ( )2 = ( )2 + ( )2 = +

2.2.2 Are they equivalent?

Area Models and Equivalent Expressions

Student pages for this lesson are 98 – 100.

In this lesson, you will continue to think about equivalent expressions. You will use an area model to demonstrate that two expressions are equivalent and to find new ways to write expressions. As you work with your team, use the following questions to help focus your discussion.

How can we be sure they are equivalent? How would this look in a diagram? Why is this representation convincing?

2-130. Jonah and Graham are working together. Jonah claims

that (x+y)2 =x2+y2. Graham is sure Jonah is wrong, but he cannot figure out how to prove it. Help Graham find as many ways as

possible to convince Jonah that he is incorrect. How can he rewrite (x+y)2 correctly? [ Proofs vary; (x+ y)2 =_x2 +_2xy+ _y2_{. ]}

2-131. How can an area model help relate the expressions (2x!3)(3x+1) and

6x2 !7x!3? [ a: (3x+1)(2x!3)=6x2+₂_x!₉_x!₃=₆_x2 !₇_x!₃_,

b: 10k2 !₁₁_k+₃_{, c:}₍_x!₄₎₍_x+₁₎_{, d: To find the value in each cell, each term in} one expression is multiplied by each term in the other, and this is the distributive property. ]

a. Copy the area diagram at right onto your own paper. With your team, discuss how it can be used to show that these expressions are equivalent.

b. Use an area model to find an expression equivalent to (5k!3)(2k!1).

c. Use an area model to find an expression equivalent to x2 !3x!4. d. How does the area model help use the distributive property?

?

=

2x + 1 3x – 3 6x2 !3 !9x 2x

2-132. Use an area model that shows an equivalent expression for each of the following expressions. [ a: 9m2 !₃₀_m+₂₅_{, b:}₍₂_x+₁₎₍_x+₂₎_{, c:}₃_x2 !₁₃_x+₆_xy!₂_y+₄_, d: (2x!5)(x+3), e: x2 !₉_{, f:}₍₂_x!₇₎₍₂_x+₇₎_] a. (3m!5)2 b. 2x2₊₅_x₊₂ c. (3x!1)(x+2y!4) d. 2x2+x!15 e. (x!3)(x+3) f. 4x2!49

2-133. With your team, decide whether the following expressions can be represented with a

model and rewrite each expression. Be prepared to share your strategies with the class. [ a: 2p3+₅_p2 !₃_p_{, b:}_x2 +₄_x!₅_]

a. p(p+3)(2p!1) b. x(x+1)+(3x!5)

2-134. Copy each area model below and fill in the missing parts. Then write the pairs of equivalent expressions represented by each model. Be prepared to share your reasoning with the class.

[ a: y(x+3+ y)=xy+3y+ y2_{, b:}₍_x+₈₎₍_x+₃₎= _x2+₁₁_x+₂₄_,

c: (5x!3)(2x!4y+5)=10x2 !₂₀_xy+₁₉_x+₁₂_y!₁₅_{, d: Likely answers include}

(x+12)(x+1)=x2+₁₃_x+₁₂_,₍_x+₆₎₍_x+₂₎=_x2 +₈_x+₁₂_{, and}

(x+4)(x+3)=x2 +₇_x+₁₂_{, although other answers are possible. ]}

a. b. c. d. 3 8x x2 –3 –6x –15 –4y –20xy 12 x x2 xy x y2 3

2-135. Decide whether each of the following pairs of expressions or equations is equivalent for all values of x (or a and b). If they are equivalent, show how you can be sure. If they are not, justify your reasoning completely. [ a: not equivalent, b: equivalent, c: equivalent, d: equivalent, e: not equivalent, f: not equivalent ]

a. (x+3)2 and x2+9 b. (x+4)2 and x2+8x+16

c. (x+1)(2x!3) and 2x2 !x!3 d. 3(x!4)2+2 and 3x2!24x+50 e. (x3)4 and x7 f. ab2 and a2b2

2-136. Look back at the expressions in problem 2-135 that are not equivalent. For each pair,

are there any values of the variable(s) that would make the two expressions equal?

Justify your reasoning. [ a: equal if x=0, e: equal if x=0 or x=1, f: equal if

a=1 or a=0 ]

2-137. Jenna wants to solve the equation 2000x!4000=8000. [ a: Possibilities include

x!2=4 or 2x!4=8; b: They have the solution x=6; c: 3!x=7, x=!4. ]

a. What easier equation could she solve instead that would give her the same solution? (In other words, what equivalent equation has easier numbers to work with?)

b. Justify that your equation in part (a) is equivalent to 2000x!4000=8000 by showing that they have the same solution.

c. Now Jenna wants to solve ₅₀3 !₅₀x = ₅₀7 . Write and solve an equivalent equation with easier numbers that would give her the same answer.

2-138. Find a rule for each sequence below. Then describe its graph. [ a: t(n)=!3n+17, points along a line with y-intercept (0, 17) and slope –3; b: t(n)=50(0.8)n_, points along a decreasing exponential curve with y-intercept (0, 50) ]

a. _n _t(n) b. 3 8 5 2 7 –4 n t(n) 1 40 2 32 3 25.6

2-139. Given that n is the length of the bottom edge of the backward L-shaped figures below, what sequence is generated by the total number of dots in each figure? What is the 46th

term, or t(46), of this sequence? The nth

term? [ odd numbers; 46th

term: 91;

nth

term: 2n!1 ]

n = 1 2 3 4 5

2-140. For the function h(x)=!3x2!11x+4, find the value of each expression below.

[ a: 4; b: –30; c: 12; d: !21

4 ; e: x=!4,!13 ]

a. h(0) b. h(2) c. h(!1) d. h(1₂)

e. For what value(s) of x does h(x)=0?

2-141. Find the x-intercepts for the graph of y!x2 =6x. [ (0, 0) and (–6, 0) ]

2-142. Multiply each pair of functions below to find an expression for f(x)!g(x).

[ a: 2x2+₆_x_{, b:}_x2!₂_x!₁₅_{, c:}₂_x2 !₅_x!₃_{, d:}_x2 +_6x+₉_]

a. f(x)=2x, g(x)=(x+3) b. f(x)=(x+3), g(x)=(x!5) c. f(x)=(2x+1), g(x)=(x!3) d. f(x)=(x+3), g(x)=(x+3)

In document CHAPTER 2. Sequences and Equivalence (Page 68-79)