Learning Target: Scholars Will Develop an Understanding of Multiplication As Repeated Addition

Lesson 2.2.4

HW: 2-79 to 2-84

Learning Target: Scholars will develop an understanding of multiplication as repeated addition.

In previous lessons, you created expressions to help Cecil cross a tightrope. For example, when the tightrope was 16 feet long,you could have written an expression such as 7 + 7 + 7 + 7 +7 + 7 + 7 + 7 + (−5) + (−5) +(−5) + (−5) + (−5) + (−5) + (−5) + (−5). When a calculation involves adding the same number over and over, as this one does, is there a shorter way to write the expression? Today, you will consider this as you continue to write and simplify expressions.

2-67.Cecil, the tightrope walker introduced in problem 231, still needs your help. Hewants to cross a rope that is 6 feet long. Using only the lengths of 5 and 8feet, find at least two ways Cecil can move to reach the end of the rope at the ladder. For each solution, draw a diagram and write an expression.

2-68. Cecil moved across the rope as shown at right. Show two ways to represent his moves with number expressions. Do both ways get him to the same endpoint?

2-69. Cecil is now so good at crossing the tightrope that he can make a leap of 7 feet at a time. He crossed the rope in these leaps as shown below. Record his moves two ways. Which way is easier to

record?

2-70.When adding the same number several times, multiplication can help. For example, if the tightrope walker moved to the right 3 feet, 3 feet, 3 feet, and then 3feet, it is shorter to write 4 (3)instead of 3 + 3 + 3 + 3. Note that parentheses are another way to show multiplication. Use multiplication to write 2 + 2 + 2 + 2 + 2.

2-71.How did Cecil, the tightrope walker, move if he started at point A and his moves were recorded as the expression 3(6.2) + 2? Draw a diagram and record how far along his rope he was when he finished.

2-72.Imagine that Cecil, the tightrope walker, starts at point B and walks on the rope toward point A as shown at right.

1.  How should this be written? Is there more than one way?

2.  Where does he end up?

2-73.To represent 2(3), Chad drew the diagram at right. Howdo you predict Chad would draw 5(−3)? What is the value of this expression?

2-74.The two equal expressions 2(3)and 6 can be represented with the diagram at right. Draw similar diagrams for each of the expressions in parts (a) through (c) below.

1.  What does 2(3 + 5)mean? What is the value of this expression?
Describe it with words and a diagram.

2.  What does 2(3) + 5mean? What is the value of this expression?
Describe it with words and a diagram.

3.  Compare 2(3 + 5)with 2(3) + 5. How are these movements the same or different? Explain your thinking and draw a diagram.

2-75.Draw a diagram to represent the expression 3(−2.5) + (−4). Is this the same as 3(−2.5 + (−4))? Use diagrams to justifyyour decision.

2-79. Find the value of each of the following expressions. Use a tile diagram or a number line to help you, if you need it.

1.  3(4)

2.  4 + 11 + (–4)

3.  3.2(2)

4. 

5. 

6. 

2-80.Use the Distributive Property to rewrite each of the following products as sums, and then calculate the value, as shown in the example below.

Example: 4(307) = 4(300) + 4(7) = 1200 + 28 = 1228

7.  9(410)

8.  6(592)

2-81.Copy and complete each of the Diamond Problems below. The pattern used in the Diamond Problems is shown at right.

2-82. Copy and complete each of the portions webs below.

1. 

2. 

2-83.Camille had a very fun birthday party with lots of friends and family attending. The party lasted for 3 hours. She and her friends played games for of the time, ate pizza and cake for 50% of the time, and spent the remainder of the time opening presents. Draw a diagram and make calculations to show the amount of time spent opening presents.

2-84. Jahna measured the heights of the sunflowers growing in her backyard. Here are the heights that she found (in inches): 34, 48, 52, 61, 76, 76, 61, 84, 61, 39, 83, 61, 79, 81, 56, and 88.Find the mean and median of the heights. 2-84 HW eTool (CPM).

3.  Find the mean and median of the heights.

4.  Create a histogram to represent this data. Your histogram should have four bins, each with a width of 15.

Lesson 2.2.4

2-67. Answers will vary: 8 + 8 + (−5) + (−5), 5 + 5 + 5 + 5 + 5 + 5 + (−8) + (−8) + (−8)

2-68. Answers will vary: 6 + 6 + 6 + 6, 4(6), 2(6) + 2(6). Yes

2-69. Answers will vary: 7 + 7 + 7 + 7 + 7, 5(7)

2-70. 5(2)

2-71. 6.2 + 6.2 + 6.2 + 2, or moved to the right 6.2 feet three times and then another 2 feet to the right. 20.6 feet to the right of point A.

2-72. See below:

1.  5(−3), −15, −3 + (−3) + (−3) + (−3) + (−3)

2.  He ends up 15 feet to the left of point B.

2-73. He could draw five groups of three negatives. The value is −15 .

2-74. See below:

1.  Moving to the right 3 feet and 5 feet twice. 16. See diagram below.

2.  Moving to the right 3 feet twice and then to the right 5 feet. 11. See diagram below.

3.  They both have Cecil do two sets of moves, but what they repeat is different (both 3 and 5 feet repeat in the first case, versus just 3 feet in the second statement).

2-75. No, they are not the same; 3(−2.5) + (−4) = −7.5 + (−4) = −11.5; 3(−2.5 + (−4)) = 3(−6.5) = −19.5.

2-79. See below:

1.  12

2.  11

3.  6.4

4.  6

5.  15

6.  −51

2-80. See below:

1.  9(400 +10) = 3600 + 90 = 3690

2.  6(500 + 90 + 2) = 3000 + 540 + 12 = 3552 or 6(600 + (−8)) = 3600 + (−48) = 3552

2-81. See answers in bold in the diamonds below:

2-82. See below:

1.  , 45%

2.  one and twenty-seven hunderedths, 1.27

2-83. Diagrams vary; they should show that + + is equivalent to 3hours. The “present” portion should show 22.5 minutes or 0.375 hours.

2-84. See below:

1.  mean = 65, median = 61

2.  See histogram below.