Lesson 3.1.2

HW: Day 1: Problems 3-23 through 3-29, any 5

Day 2: Problems 3-30 through 3-36, any 5

Learning Target: Scholars will use an area model to multiply expressions. They will factor expressions and demonstrate equivalence.

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?

3-13. 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 show it.

  1. Help Graham find as many ways as possible to convince Jonah that he is incorrect. How can he rewrite (x + y)2 correctly?
  2. Are there any values for x and y for which (x + y)2 = x2 + y2? In other words, is (x + y)2 = x2 + y2 sometimes true? Justify your answer.

3-14. Do you think that an area model can help rewrite expressions that involve multiplication?

  1. The area model at right relates the expressions (2x − 3)(3x + 1) and 6x2 − 7x − 3. With your team, discuss how it can be used to show that these expressions are equivalent. Be prepared to explain your ideas.
  2. Use an area model to find an expression equivalent to (5k − 3)(2k − 1).
  3. Use an area model to write a product that is equivalent to x2 − 3x − 4.

3-15. Rewrite each of the following products as a sum and each sum as a product, drawing an area model when appropriate.

  1. 2x2 + 5x + 2
  2. (3x − 1)(x + 2y − 4)
  3. (x − 3)(x + 3)
  4. 4x2− 49
  5. (p2 + 3p + 9)(2p − 1)
  6. (4 − x)(x2 + 1) + (3x − 5)

3-16. 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.

  1. p(p + 3)(2p − 1)
  2. x(x + 1) + (3x − 5)

3-17. Copy each area model below and fill in the missing parts. Then write the two equivalent expressions represented by each model. Be prepared to share your reasoning with the class.

3-18.Shinna noticed a similarity in parts (c) and (d) of problem 3-15.

  1. Look back at those two problems and their rewritten form. What might Shinna have noticed? Discuss this with your team and be prepared to share your ideas with the class.
  2. Shinna thinks she has found a shortcut that will allow her to rewrite expressions such as those written below without drawing a diagram. What do you think she has figured out? Try your ideas on the expressions shown below.
  3. w2 − 81
  4. 4m2 − 1
  5. x2 − 16y2

3-19.Shinna has noticed that differences of squares can be factored easily.

  1. Decide which of the expressions below can be seen as a difference of squares and can therefore be factored using Shinna’s shortcut. For each difference of squares, show the squares clearly and then write the product. For example, 16x2 − 9y2 can be rewritten as (4x)2 − (3y)2 and then as (4x − 3y)(4x + 3y).
  2. a2 − 4b2
  3. 2x2 − 16
  4. −x2 + y4
  5. 4a2 + 9b2
  6. Write two more expressions of your own that are differences of squares and show each in factored form.

3-20.Shinna wants to factor 9x2y4 − z6. “Wait!” she says. “I think I can see a way to use my shortcut!”

  1. Discuss this with your team. Is Shinna’s expression a difference of squares? If so, what are the squares? If not, explain why. Be ready to share your ideas with the class.
  2. Shinna decided to rewrite her expression so that its structure was simpler to see. She wrote 9x2y4 − z6 as U2 − V2. What was she using U to represent? What about V?
  3. George is confused! “Shinna,” he says, “There was no U or V in your problem! What are you doing?” Explain to George what is going on.
  4. Help Shinna finish factoring the expression 9x2y4 − z6 by factoring U2 − V2 and then substituting the original expressions for U and V.

3-21. How can you use this method of substitution to make use of what you know about other expressions? Work with your team to describe the structure of each of the expressions in parts (a) through (d) below. Use substitution, when appropriate, to make the structure clear. For example, 25x2 − 100y4 is a difference of squares and can be rewritten as U2 − V2 with U = 5x and V = 10y2.

  • The following questions might be useful:
  • What do all of these expressions have in common?
  • How might we substitute U and V to make rewriting simpler?
  • a2 + 2ab + b2
  • x2 − 6x + 9
  • 9x2 + 30xy + 25y2

3-23. Decide whether each of the following pairs of expressions are equivalent for all values of x (or a andb). If they are equivalent, show how you can be sure. If they are not, justify your reasoning completely.

  1. (x + 3)2 and x2 + 9
  2. (x + 4)2 and x2 + 8x + 16
  3. (x + 1)(2x − 3) and 2x2 − x − 3
  4. 3(x − 4)2 + 2 and 3x2 − 24x + 50
  5. (x3)4 and x7
  6. ab2 and a2b2

3-24. Look back at the expressions in problem 3-23 that are not equivalent. For each pair of expressions, are there any values of the variable(s) that would make the two expressions equal? Justify your reasoning. 3-24 HW eTool

3-25. Jenna wants to solve the equation 2000x − 4000 = 8000.

  1. 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?)
  2. Justify that your equation in part (a) is equivalent to 2000x − 4000 = 8000 by showing that they have the same solution.
  3. Now Jenna wants to solve . Write and solve an equivalent equation with easier numbers that would give her the same answer.

3-26. Find an equation for each sequence below. Then describe its graph. 3-26 HW eTool (Desmos).

3-27. For the function h(x) = −3x2 − 11x + 4, find the value of h(x) for each value of x given below.

  1. h(0)
  2. h(2)
  3. h(−1)
  4. h()
  5. For what value(s) of x does h(x) = 0?

3-28. Find the x-intercepts for the graph of y − x2 = 6x.

3-29. Multiply each pair of polynomial functions below to find an expression for f(x) · g(x). 3-29 HW eTool (Desmos).

  1. f(x) = 2x, g(x) = (x + 3)
  2. f(x) = (x + 3), g(x) = (x − 5)
  3. f(x) = (2x + 1), g(x) = (x − 3)
  4. f(x) = (x + 3), g(x) = (x + 3)

3-30. Describe how the graph of y + 3 = −2(x + 1)2 is different from y = x2.

3-31. Given the parabola f(x) = x2 − 2x − 3, complete parts (a) through (c) below.

  1. Find the vertex by averaging the x-intercepts.
  2. Find the vertex by completing the square.
  3. Find the vertex of f(x) = x2 + 5x + 2 using your method of choice.
  4. What are the domain and range for f(x) = x2 + 5x + 2?

3-32. Simplify each of the following expressions, leaving only positive exponents in your answer.

  1. (x3y−2)−4
  2. −3x2(6xy − 2x3y2z)

3-33. Determine if each of the following functions are odd, even or neither.

  1. y = 3x3
  2. y = x2 + 16

3-34. You decide to park your car in a parking garage that charges $3.00 for the first hour and $1.00 for each hour (or any part of an hour) after that.

  1. How much will it cost to park your car for 90 minutes?
  2. How much will it cost to park your car for 118 minutes? 119 minutes?
  3. How much will it cost to park your car for 120 minutes? 121 minutes?
  4. Graph the cost in relation to the length of time your car is parked.
  5. Is this function continuous?
  6. Describe how the graph of this function will change if the parking garage raises their parking rate so that the first hour is now $5.00.

3-35. Give the equation of each circle below in graphing form. 3-35 HW eTool (Desmos).

  1. A circle with radius of 12 centered at the point (−2, 13).
  2. A circle with center (−1, −4) and radius 1.
  3. A circle with equation x2 + y2 − 6x +16y + 57 = 0 . (Hint: Complete the square for both x and y.)

3-36. Giuseppe decides that he really wants some ice cream, so he leaves the house at 3:00 p.m. and walks to the ice cream parlor. He arrives at 3:15 (the ice cream parlor is 6 blocks away). He buys an ice cream cone and sits down to eat it. At 3:45 he heads back home, arriving at 4:05. Find Giuseppe’s average walking rate in blocks per hour for each of the following situations.

  1. His trip to the ice cream parlor.
  2. His trip back home.
  3. The entire trip including the time spent eating.

Lesson 3.1.2

  • 3-13.See below:
  • Explainations vary; (x + y)2 = x2 + 2xy + y2.
  • (x + y)2 = x2 + y2 only when x = 0 or y = 0; justifications vary.
  • 3-14.See below:
  • Explanations vary, but should contain the idea that area of a rectangle is the same as the product of its dimensions.
  • 10k2– 11k + 3
  • (x– 4)(x + 1)
  • 3-15. See below:
  • (2x+ 1)(x + 2)
  • 3x2– 13x + 6xy– 2y + 4
  • x2– 9
  • (2x– 7)(2x + 7)
  • 2p3 + 5p2 + 15p– 9
  • –x3 + 4x2 + 2x– 1
  • 3-16.See below:
  • 2p3+ p2– 3p
  • x2 + 4x– 5
  • 3-17.See below:
  • y(x+ 3 +y) =xy+ 3y +y2
  • (x+ 8)(x+ 3) =x2+ 11x+ 24
  • (5x− 3)(2x− 4y+ 5) = 10x2− 20xy+ 19x+ 12y− 15
  • Possible answers include (x+ 12)(x+ 1) =x2+ 13x+ 12, (x+ 6)(x+ 2) =x2+ 8x+ 12,and (x+ 4)(x+ 3) =x2+ 7x+ 12
  • 3-18.See below:
  • Students should describe the difference of squares, although they may use different words.
  • She has figured out that the difference of squares can be written as a product of the square roots as follows:i: (w + 9)(w− 9), ii: (2m + 1)(2m− 1),iii: (x + 4y)(x− 4y)
  • 3-19.See below:
  • Parts (i) and (iii) are differences of squares. i: a2− 4b2 = (a)2− (2b)2;iii:−x2 + y4 = (y2)2− (x)2.
  • Expressions vary. Examples include: 16x2 − 9y2 = (4x + 3y)(4x− 3y) and x2− 81 = (x + 9)(x− 9)
  • 3-20.See below:
  • Yes; it is the difference of the square of 3xy2 and the square of z3.
  • Urepresents 3xy2and V represents z3.
  • Explanations vary.
  • U2− V2 = (U + V)(U− V); (3xy2 + z3)(3xy2− z3).
  • 3-21. These can all be expressed in the form U2 + 2UV + V2 = (U + V)2.
  • (a + b)2
  • (x− 3)2
  • (3x + 5y)2
  • ((a + 7)− 5)2
  • 3-22.See below:
  • Answers will vary.
  • 3-23. See below:
  • not equivalent
  • equivalent
  • equivalent
  • equivalent
  • not equivalent
  • not equivalent
  • 3-24.See below:
  • equal if x = 0
  • equal
  • equal
  • equal
  • equal if x = 0 or x = 1
  • equal if a = 1 or a = 0
  • 3-25.See below:
  • Possibilities include:x– 2 = 4 or 2x– 4 = 8
  • They have the solution x = 6
  • 3– x = 7, x =–4
  • 3-26. See below:
  • t(n) =–3n + 17, points along a line with y-intercept (0, 17) and slope–3
  • t(n) = 50(0.8)n, points along a decreasing exponential curve with y-intercept (0, 50)
  • 3-27.See below:
  • 4
  • –30
  • 12
  • –2
  • x =–4,
  • 3-28.(0, 0) and (–6, 0)
  • 3-29.See below:
  • 2x2 + 6x
  • x2– 2x– 15
  • 2x2– 5x– 3
  • x2 + 6x +9
  • 3-30.The first graph opens downward, is stretched, and has its vertex at (–1, –3). The second is the parent graph.
  • 3-31.See below:
  • (1,–4)
  • (1,–4)
  • (–2.5,–4.25)
  • Domain: –∞ < x < ∞, Range: y ≥–4.25
  • 3-32.See below:
  • –18x3y + 6x5y2z
  • 3-33.See below:
  • odd
  • even
  • even
  • 3-34. See below:
  • $4.00
  • $4.00
  • $4.00. $5.00
  • See graph below.
  • No, it is a step function.
  • The graph will shift (translate) upward by $2.00.
  • 3-35. See below:
  • (x + 2)2 + (y– 13)2 = 144
  • (x + 1)2 + (y + 4)2 = 1
  • (x– 3)2 + (y + 8)2 = 16
  • 3-36.See below:
  • 24 blocks per hour
  • 18 blocks per hour
  • 11.08 blocks per hour