Sequences
Lesson 2.2.3 How can I solve it?
Solving by Rewriting
Lesson Objective: Students will solve equations by first rewriting them as simpler equivalent equations.
Length of Activity: One day (approximately 50 minutes)
Core Problems: Problems 2-143 and 2-144
Ways of Thinking: Justifying, generalizing, choosing a strategy
Materials: Lesson 2.2.3 Resource Page, one for every four teams (optional)
Suggested Lesson Activity:
After a brief introduction of the lesson objective, direct teams to problems 2-143 and 2-144.
Optional: As teams finish problem 2-143 and move on to problem 2-144, distribute the solutions to these equations (Lesson 2.2.3 Resource Page). Having the solutions from the beginning can provide students motivation to focus on sound solving strategies and give them a sense of a goal. Ask teams to present strategies for simplifying and solving each equation or system, and have the teams field any questions from the class. Use this as an opportunity to encourage your students to be independent of you, checking the logic of each other’s ideas, rather than asking for your validation.
Then move teams on to problems 2-145 and 2-146, in which they will use their ideas of equivalence to solve two-variable equations for one of the variables.
If time permits, direct teams to problems 2-147 and 2-148, which will provide them further opportunity to think about and apply equivalence to solving problems.
Suggested Study Team Collaboration Strategies: for problem 2-144, Pairs Check, Round Table or Send a Spy
Closure: (10 minutes)
Ask teams to share their conclusions and results for problems 2-147 and 2-148. Problem 2-147 challenges students to distinguish quadratic equations in one variable from their corresponding parabolas. They should conclude that two quadratic equations in one variable that have the same solutions do not necessarily have the same graphs.
Problem 2-148 provides the opportunity to discuss multiple ways of seeing the tile pattern and the resulting different equivalent expressions that can represent the number of tiles in Figure x.
2.2.3 How can I solve it?
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Solving by Rewriting
Student pages for this lesson are 101 – 104.
In the past few lessons, you have worked on recognizing and finding equivalent expressions. In this lesson, you will apply these ideas to solve equations. As you work, use the questions below to keep your team’s discussion productive and focused.
How can we make it simpler? Does anyone see another way?
How can we be sure that our answer is correct?
2-143. Graciella was trying to solve the quadratic equation x2+2.5x!1.5=0. “I think I
need to use the Quadratic Formula because of the decimals,” she told Grover. Grover replied, “I’m sure there’s another way! Can’t we rewrite this equation so the decimals are gone?” [ a: Possibilities include 2x2+5x!3=0 and
10x2+25x!15=0; b: Students should factor the equation from part (a), which,
in the case of the answers given above, would result in (2x!1)(x+3)=0
or 5(2x!1)(x+3)=0; c: x=0.5 or x=!3 ]
a. What is Grover talking about? Rewrite the equation so that it has no decimals. b. Use your ideas from Lesson 2.2.2 to rewrite your equation again, expressing it
as a product.
c. Now solve your new equation. Be sure to check your solution(s) using Graciella’s original equation.
2-144. SOLVING BY REWRITING
Rewriting x2+2.5x!1.5=0 in problem 2-143 gave you a new, equivalent equation that was much easier to solve. How can each equation or system of equations below be rewritten so that it is easier to solve? With your team, find an equivalent equation or system for each part below. Be sure your new equations have no fractions or decimals and have numbers that are reasonably small. Then solve your new equation or system and check your answer(s) using the original equations. [ a:x=!5,!4; b: x=!2,!1 2; c: x=1; d: x=2; e: x=2,!y=!5; f: x=!5,!y=3 ] a. 100x2+100x=2000 b. 21 x2+ 43x! 12=0 c. 25x+ 35 =1 d. x!x3+ x2!1= 5!xx e. 15x+10y=!20 f. 3x+1+ 2y =!5
?
=
2-145. Is 6x+3y=12 equivalent to y=!2x+4? How can you tell using each
representation (table, graph, equation)? [ Yes, they are equivalent. They have the same solutions, so the values in the tables should be the same. The first equation can be solved for y to get the second equation. They have identical graphs. ]
2-146. Rewrite each of the following equations so that it is in y-form. Check to be sure your
new equation is equivalent to the original equation. [ a: y= 5
2 x!4; b: y= 2x !3 ]
a. 5x!2y=8 b. xy+3x=2
2-147. Angelica and D’Lee were working on finding roots of two quadratic equations:
y=(x!3)(x!5) and y=2(x!3)(x!5). Angelica made an interesting claim: “Look,” she said, “When I solve each of them for y=0, I get the same solutions. So these equations must be equivalent!”
D’Lee is not so sure. “How can they be equivalent if one of the equations has a factor of 2 that the other equation doesn’t?” she asked. [ a: No, they are not equivalent, as the values in the table would be different and the graph of the second equation is a vertical stretch of the first (although students will not have this vocabulary at this point); b: Yes, they are equivalent, as these equations have the same solutions, and if the first equation is multiplied by 2, the result is the second. ]
a. Who is correct? Is y=(x!3)(x!5) equivalent to y=2(x!3)(x!5)? How can you justify your ideas using tables and graphs?
b. Is 0=(x!3)(x!5) equivalent to 0=2(x!3)(x!5)? Again, how can you
justify your ideas?
2-148. Consider the tile pattern at right. [ a: The 100th
figure has 10,506 tiles; expressions vary but are equivalent to (x+2)(x+3); b: Figure 6 has 72 tiles; different expressions make no difference. ]
a. Work with your team to describe what the 100th
figure would look like. Then find as
many different expressions as you can for the area (the number of tiles) in Figure x. Use algebra to justify that all of your expressions are equivalent.
b. How can you use your expressions to find out more information about this pattern? Write and solve an equation to determine which figure number has 72 tiles. Do you get different results depending upon which expression you choose to use? Explain.
2-149. Rewrite each equation below. Then solve your new equation. Be sure to check your solution using the original equation. [ a: n=!2; b: x=!4,!1 ]
a. (n+4)+n(n+2)+n=0 b. 4x =x+3
2-150. Decide whether each of the following pairs of expressions or equations are equivalent.
If they are, show how you can be sure. If they are not, justify your reasoning completely. [ a: equivalent, b: equivalent, c: equivalent, d: not equivalent, e: not equivalent, f: not equivalent ]
a. (ab)2 and a2b2 b. 3x!4y=12 and y= 43x!3
c. y=2(x!1)+3 and y=2x+1 d. (a+b)2 and a2 +b2 e. x6
x2 and x
3 f. y=3(x!5)+2 and y=2x!8
2-151. Look back at the expressions in problem 2-150 that are not equivalent. Are there any
values of the variables that would make them equal? Justify your reasoning.
[ d: equal if a=0 or b=0, e: equal if x=1, f: equal if x=5 and y=2 ]
2-152. This problem is a checkpoint for multiplying polynomials. It will be referred to as Checkpoint 3.
Multiply and simplify each expression below. [ a: 2x3+2x2 !3x!3,
b: x3 !x2 +x+3, c: 2x2 +12x+18, d: 4x3!8x2 !3x+9 ]
a. (x+1)(2x2!3) b. (x+1)(x2!2x+3)
c. 2(x+3)2 d. (x+1)(2x!3)2
e. Check your answers to parts (a) through (d) by referring to the Checkpoint 3 materials located at the back of your book.
If you needed help multiplying and simplifying these expressions correctly, then you need more practice with problems like these. Review the Checkpoint 3 materials and try the practice problems. Also, consider getting help outside of class time. From this
2-153. Find the formula for t(n) for the arithmetic sequence in which t(15) = 10 and
t(63) = 106. [ 10=15m+b and 106=63m+b; m=2, b=!20, t(n)=2n!20 ]
2-154. Jillian’s parents bought a house for $450,000, and the value of the house has been increasing steadily by 3% each year. [ a: t(n)=450000(1.03)n; b: They will make $154,762.37 or 34.39% profit. ]
a. Find the formula t(n) that represents the value of the house each year.
b. If Jillian’s parents sell their house 10 years after they bought it, how much profit will they make? (That is, how much more are they selling it for than they
bought it for?) Express your answer as both a dollar amount and a percent of the original purchase price.
2-155. Factor 5x3y+35x2y+50xy completely. Show every step and explain what you did.
[ 5xy(x+2)(x+5) ]
2-156. Consider the sequence 10, 2, … [ a: –6, –14, –22, –30, t(n)=18!8n; b: 2
5 , 252 ,
2
125, 6252 , t(n)=50(15)n; c: Sequences and rules vary. ]
a. Assuming that the sequence is arithmetic with t(1) as the first term, write 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, write the next four terms of the sequence and then write a rule for t(n).
c. Create a totally different sequence that begins 10, 2, … For your sequence, write the next four terms and a rule for t(n).