Lesson 3.2.3

HW: 3-59 to 3-63

Learning Target: Scholars will multiply positive and negative integers.

In this lesson, you will work with your team to continue thinking about what happens when you remove +and –tiles from a collection of tiles representing a number. You will extend your thinking to find ways of making your calculations more efficient when the same number of tiles are removed multiple times. Consider these questions as you work today:

 Is there a more efficient way to do this?

 How do these ideas compare with what we learned about
adding and multiplying integers in Chapter 2?

3-51. For each expression below, predict what you know about the result without actually calculating it. Can you tell if the result will be positive or negative? Can you tell if it will be larger or smaller than the number you started with? Be ready to explain your ideas.

  1. −1 − (−6.5)
  2. 2.2 − (−2.2)
  3. 12 − 22
  4. −100 − (−98)
  5. −10 − (−2) − (−2) − (−2)

3-52. Now draw (or describe) a diagram for each of the expressions in problem 351 and calculate the number that each of them represents.

3-53. Troy and Avery are working with the expression−10 −(−2) − (−2) − (−2)from part (e) of problem 3-51.

  1. Help them find a shorter way to write this expression.
  2. Imagine that their expression does not include the –10. How could they write the new expression? What number would this new expression represent? If you were to describe what this expression represents using + and– tiles, what would you say?

3-54.How could you evaluate the product −7(−11) ? Work with your team to make sense of −7(−11) . Prepare a brief presentation explaining why your result must be correct.

3-55. WHAT DOES IT MEAN?

Your task: Work with your team to create a poster that shows what it means to multiply a negative number by another negative number or to multiply a negative number by a positive number. To demonstrate your ideas, include:

  • Examples (from the list below or create your own).
  • Pictures or diagrams.
  • Any words necessary to explain your thinking.
  • Numerical sentences to represent each of your examples.

4(−3) / −4(−3) / −4(3)
2(−7) / −2(−7) / −2(7)

3-57. What does −18 ÷ 9equal? How do you know? Explain why your answer makes sense. Then complete the division problems below.

  1. 45 ÷ (−3)
  2. −32 ÷ (−8)
  3. −54 ÷ 6

3-59. Copy and simplify each expression below.

  1. 6 + (−18)
  2. −9 + (−9)
  3. −12.2 + 6.1 + 15.8

3-60. Find each of the following products or quotients without using a calculator. Draw a diagram or use words to explain how you know your product makes sense.

  1. 6(−3)
  2. −6(3)
  3. −8(−3)
  4. −8(0)
  5. −20 ÷ 5
  6. −36 ÷ (−9)

3-61. Simplify each of the following expressions without using a calculator. Draw diagrams or use words to show your thinking.

  1. −8 + −2(3)
  2. 4 + −5(−3) + (−7)
  3. (−4.25)(2) + (−4.25)(−2)

3-62.Simplify the following fraction expressions. Show all of your work.

3-63. Nathanwants you to solve this puzzle: “I am thinking of a number. If you divide my number by 3 and add –3, you will get 4. What is my number?” Show all of your work.

Lesson 3.2.3

  • 3-51.Answers vary.
  • 3-52.Diagrams vary.
  • 5.5
  • 4.4
  • –10
  • –2
  • –4
  • 3-53. See below:
  • –10– 3(–2)
  • –3(–2). The result is 6. "Negative three times negative two," or "take away 3 groups of negative 2".
  • 3-54.Students are likely to think about this as removing 7 groups of 11 negatives, so they would reason that one would have to start with many zeros, and after 77negativetiles were removed, there would be 77 + tiles remaining.
  • 3-55. Students are likely to describe the product of two negative numbers as removing groups of negatives, and the product of a positive and a negative as adding groups of negatives or as removing groups of positives.

 3-57.–2. Students are likely to think about an associated multiplication problem, such as ; see the “Suggested Lesson Activity” for further information.

  1. –15
  2. 4
  3. –9
  • 3-59. See below:
  • –12
  • –12
  • –18
  • 9.7
  • 3-60. See below:
  • –18
  • –18
  • 24
  • 0
  • –4
  • 4
  • 3-61. See below:
  • –2
  • –14
  • 12
  • 0
  • 3-62. See below:
  • or 1
  • or
  • or
  • 3-63. 21