Download How do we show integer addition with red and black cards? Another

Lesson
6 Another Way of Looking at Integer Addition
Problem Solving:
More Dot Graphs
Another Way of Looking at Integer Addition
How do we show integer addition with red and
black cards?
We have used the number line to show how to add positive and negative
numbers. Another way to help us understand adding positive and
negative numbers is to use colored cards.
A couple of common phrases are in the red and in the black.
The word “red” is sometimes used when we talk about losing money.
It is also used with negative numbers. The word “black” is sometimes
used when we talk about making money or positive numbers.
We use red cards to show negative numbers and black cards to show
positive numbers. Here’s a simple problem.
Example 1
Show 4 + 3 using cards.
4+3
Start with four cards. Because 4 is positive, these four cards are black.
Next, add three cards. Because 3 is positive, these three cards are
also black.
+
The answer is 4 + 3 = 7.
564 Unit 8 • Lesson 6
Lesson 6
Adding positive numbers using cards is nothing new, but adding positive
and negative numbers with cards is different. Let’s look at a problem
that uses negative numbers.
Example 2
Show 5 + −2 using cards.
5 + −2
Start with five cards. These cards are black because 5 is positive.
Then add −2. Use two red cards because 2 is negative.
That leaves three black cards.
The answer is 3.
Unit 8 • Lesson 6 565
Lesson 6
To help understand what happened, let’s look at the number line again.
Example 3
Show 5 + −2 using a number line.
0
1
2
3
4
5
6
7
8
9
10
Start at 0 and move 5 in a positive direction. Then add −2. Show this by
moving 2 in a negative direction. Again, as was shown using the cards,
the answer is 3.
In Example 2, the two red cards and the two black cards cancel each
other out.
What that really means is when they are added together, the sum is 0:
2 + −2 = 0. We can see this on a number line.
Example 4
The sum of opposites is zero. Show this on a number line.
0
1
2
3
4
We start at 0 and move 2 places in a positive direction. Then we add
−2 by shifting gears and moving 2 places in a negative direction. That
takes us right back to 0. When we add opposites, we get 0.
Or if we’re using cards, one red card cancels out one black card because
their sum is zero. Likewise, two red cards cancel out two black cards
because their sum is zero, and so on. This simple concept is one of the
main properties about numbers that we use over and over again in
higher-level math.
Apply Skills
Turn to Interactive Text,
page 292.
566 Unit 8 • Lesson 6
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
5
Lesson 6
Problem Solving: More Dot Graphs
How do we plot negative numbers on a dot graph?
We can show negative numbers on a dot graph just like on a bar graph.
We will review this concept by looking at a game in which people answer
trivia questions. For every correct answer we earn 10 points and for
every incorrect answer we lose 10 points. Here’s an example of the game.
Tyrone and Carson are playing the trivia game in their social studies
class. The table shows Tyrone’s questions, his answers, and his score.
Question
Answer
1. What is the longest river in the
United States?
2. Where is the tallest mountain in
the United States?
3. Which city is farther west—Reno,
Nevada, or San Diego, California?
4. What is the capital of New York?
5. What lake is next to Chicago?
Right or Wrong Points
Missouri
River
Right
10
Colorado
Wrong
0
Reno,
Nevada
Right
10
Albany
Right
20
Lake
Michigan
Right
30
Here’s a dot graph of Tyrone’s points.
35
30
Points
25
20
15
10
5
0
1
2
3
4
5
Question
The graph shows that Tyrone got the first answer correct. He missed
only one question—Question 2—and then correctly answered the rest
of the questions.
Unit 8 • Lesson 6 567
Lesson 6
The table summarizes Carson’s questions, answers, and his score.
Question
Answer
Right or Wrong Points
1. What state is east of Mississippi?
Alabama
Right
10
Appalachians
Right
20
Miami
Wrong
10
Montana
Wrong
0
Oklahoma
Wrong
−10
2. What is the mountain range
on the east coast of the
United States?
3. What is the capital of Florida?
4. What state has a golden bear
on its flag?
5. The Alamo is in what state?
Here’s a graph of Carson’s points.
30
Points
20
10
0
10
20
1
2
3
4
5
Question
We see that Carson missed enough questions to end with −10 points.
The graph also shows that Carson answered the first two questions
correctly and then missed all the remaining questions.
As with other graphs, dot graphs help us see the change from one point
to the next. They also help us see negative values easily.
Problem-Solving Activity
Turn to Interactive Text,
page 293.
568 Unit 8 • Lesson 6
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Lesson 6
Homework
Activity 1
Add the integers.
1. 2 + −2 ​
2. 2 + −5 ​
3. 5 + −5 ​
4. −3 + −4 ​
5. −10 + 10 ​
6. −1 + 9 ​
7. 8 + −2 ​
8. −10 + 9 ​
9. 3 + 2 ​
10. −6 + 6 ​
Activity 2
Tell what equation is represented by the cards in each of the problems, then
solve the equation. Remember, red cards cancel out black cards because their
sum is zero.
Model
Answer: 5 + −1 = 4
Notice we drew a line through one black card and one red card. They
cancel each other out because their sum is zero. We are left with four
black cards. That represents the number 4. The answer to the problem
is +4, or just 4.
1. Write a number sentence
using integers.
2. Write a number sentence
using integers.
3. Write a number sentence
using integers.
4. Write a number sentence
using integers.
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.
Unit 8 • Lesson 6 569
Lesson 6
Homework
Activity 3
Tell which number is bigger by writing > or <.
1. −10
−9
2. 0
−5
−1
3. −8
−9
4. 5
5. −9
1
6. −7
−6
Activity 4 • Distributed Practice
Solve.
1. What is 10% of 100? ​
3
2. Convert 5 to a decimal number. ​
3. Convert 0.08 to a percent. ​
4. 2.3 · 8 ​
5. 18.6 · 0.3 ​
4
3
3
6. 5 · 4 ​
5
6
4
7. 4 ÷ 3 ​
1
8
8. 117.98 + 207.45 + 316.05 + 269.71 ​
570 Unit 8 • Lesson 6
Copyright 2010 by Cambium Learning Sopris West®. All rights reserved. Permission is granted to reproduce this page for student use.