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 2-31, still needs your
help. He wants to cross a rope that is 6 feet long. Using only the lengths of 5 and 8 feet,
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 3 feet, 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.
A. How should this be written? Is there more than one way?
B. Where does he end up?
2-73. To represent 2(3), Chad drew the diagram at
right. How do 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.
A. What does 2(3 + 5) mean? What is the value of this expression?
Describe it with words and a diagram.
B. What does 2(3) + 5 mean? What is the value of this expression?
Describe it with words and a diagram.
C. 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.
2-76. MENTAL MATH
Work with your team to find strategies for figuring out mentally (without using a
calculator or writing anything down) how far Cecil moved in parts (a) and (b)
below. Then write a number expression for each set of moves along with its result.
A. Cecil traveled 105 feet to the left 7 times. How far did he end up from his starting
point?
B. Cecil repeated the following pattern 12 times: He traveled to the right 198 feet
and then to the left 198 feet. How far did he end up from his starting point?
2-77. When you need to multiply mentally, it is often useful to use the Distributive
Property. The Distributive Property states that when you multiply a sum by a number,
you must multiply each part of the sum by that number. For example, part (a) of problem
2-76 can be seen as 7(−105) = 7(−100) + (−5)). The Distributive Property tells us that
this is equal to 7(−100) + 7(−5) or −700 + (−35). You will explore the Distributive
Property further in Chapter 4.
Use the Distributive Property to rewrite and calculate each product shown below.
A.
B.
C.
D.
6(12 + (−4))
7(300 + (−10))
4 · 302
5(871)
2-78. LEARNING LOG
In your Learning Log, describe what you have learned about adding and multiplying
positive and negative integers. Be sure to include examples and diagrams or other
representations that explain your thinking. Title this entry “Adding and Multiplying
Integers” and label it with today’s date.
Addition of Integers
Recall that integers are positive and negative whole numbers and zero, that is, … –3, –2,
–1, 0, 1, 2, 3, … .
You have been introduced to two ways to think about addition. Both of them involve
figuring out which, if any, parts of the
numbers combine to form zero. One
way to think about this concept is to think about the tightrope walker from problem
2-31. If Cecil travels one foot to the right (+1) and one foot to the left (–1), he will end
up where he started, so the sum of (+1) and (–1) is zero.
Another useful strategy for finding zero is to use + and – tiles. The diagram at right
can be represented by the equation –1 + 1 = 0. You can use this same idea for adding
any two integers. Use + and – tiles to build the first integer, add the tiles for the
second integer, and then eliminate zeros. Study the examples shown in the diagrams
below.
Example 1: 5 + (–3)
5 + (–3) = 2
Example 2: –5 + (2)
–5 + (2) = –3
Example 3: –6 + (–2)
–6 + (–2) = –8
With practice, zeros can be visualized. This helps you determine how many remaining
positive or negative tiles show the simplified expression.