Download Unit Review Integers Finding Points on a Graph Lesson 15

Lesson
15 Unit Review
Integers
Problem Solving:
Finding Points on a Graph
Integers
What do we need to know about positive and
negative numbers?
When we work with integers, we extend our number line to the left
of zero. Numbers on the left of zero are called negative numbers. The
numbers to the right of zero are positive numbers.
Review 1 shows that if we draw a vertical line at zero on the number
line, we have a line of symmetry that divides positive and negative
numbers. Everything around a line of symmetry is an exact reflection of
the image on the other side.
Review 1
How does zero make a line of symmetry on the number line?
line of symmetry
−5
−4
−3
−2
negative numbers
−1
0
1
2
3
4
positive numbers
5
Positive and negative numbers continue on the number line infinitely,
and each number has an opposite. For instance, 5 is the opposite of
−5. A number and its opposite are always an equal distance from zero.
Review 2 shows that −3 is 3 units away from zero, just as 3 is 3 units
away from zero.
Review 2
What do opposites look like on the number line?
3 units
−5
618 Unit 8 • Lesson 15
−4
−3
−2
−1
3 units
0
1
2
3
4
5
Lesson 15
When we compare positive and negative numbers, we only need to
remember one simple rule.
Rule:
Any number to the left of another number on the number line
is smaller.
For example, −9 is smaller than 5. Let’s look at these numbers on the
number line in Review 3. We see that −9 is also smaller than −7. We use
the inequality symbol for less than (<) to show that −9 is smaller.
Review 3
How do we compare integers using the number line?
The number −9 is smaller than 5 because it’s to the left of it.
left
right
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
1
2
3
4
5
6
7
8
9 10
−9 < 5
We see on the number line that −9 is to the left of 5. That means it’s
smaller. We use the inequality symbol for less than (<) to show −9
is smaller.
The number −9 is smaller than −7 because it’s to the left of it.
left
right
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
1
2
3
4
5
6
7
8
9 10
We also see that −9 is smaller than −7.
−9 < −7
Unit 8 • Lesson 15 619
Lesson 15
How do we add and subtract positive and
negative numbers?
When we add positive and negative numbers, it’s important to know
which way to move on the number line. When we add a positive number,
we move in a positive direction. When we add a negative number, we
move in a negative direction. Review 1 shows what this looks like on the
number line.
Review 1
Move in a positive
direction (right) when
we add a positive number.
What does movement on the number line look like?
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
1
2
3
4
5
6
7
8
9 10
1
2
3
4
5
6
7
8
9 10
Move in a negative
direction (left) when we
add a negative number.
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
620 Unit 8 • Lesson 15
Lesson 15
Now let’s look at some problems involving the addition of positive
and negative numbers.
Review 2 starts with the problem 6 + 2 and then shows what it looks like
to calculate the answer to the problem 6 + −2. To make things easier to
understand, we are going to use arrows to show positive and negative
movement on the number line. In both cases, we start at zero and move
6 places in the positive direction.
In the first problem, we move 2 places more in the positive direction. In
the problem 6 + −2, we move 2 places in the negative direction.
Review 2
How do we add positive and negative numbers?
6+2
+6
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
1
2
3
+2
4
5
6
7
8
9 10
6
7
8
9 10
6+2=8
−2
6 + –2
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
+6
1
2
3
4
5
6 + –2 = 4
Unit 8 • Lesson 15 621
Lesson 15
Another way to think about addition of positive and negative numbers is
through the concept of cancellation. Review 3 uses red and black cards
to represent the problem 5 + −2. The black cards are positive and the
red cards are negative. We see in the example how 2 red cards cancel
out 2 black cards, leaving 3 black cards. That is the same thing that
happens when we solve the problem 5 + −2 = 3.
Review 3
How can cards help us understand the way positive and negative
numbers cancel each other out?
5 + −2 =
3
Cancellation is just like adding opposites. In the example above, we
cancelled by adding 2 + −2 = 0. The numbers 2 and −2 are an equal
distance from the number line, so the answer is always zero.
622 Unit 8 • Lesson 15
Lesson 15
This last rule keeps us from being confused when we see operations like
addition or subtraction and the negative sign for a number.
Rule:
Subtraction is the same as adding the opposite.
Review 4 shows us how the problem 4 − 3 is the same as 4 + −3.
Review 4
How are subtraction and adding the opposite the same?
4−3=1
−3
0
1
2
3
4
5
6
7
1
2
3
4
5
6
7
4 + −3 = 1
0
The same rule can be used in problems like 4 − −3. This problem is the
same as 4 + 3. We are adding the opposite, so that means we add the
opposite of −3, which is 3. So, 4 − −3 = 7 and 4 + 3 = 7.
Unit 8 • Lesson 15 623
Lesson 15
What about adding and subtracting
big numbers?
Number lines and cards are good tools for thinking about adding and
subtracting small integers. Working with bigger numbers requires good
number sense and a calculator.
Review 1 shows the problem −78 − 110. We begin by placing −78 on the
number line. We know that it is to the left of zero, and we approximate
its location with a tick mark. Then we use the rule: subtraction is the
same as adding the opposite. That lets us change the problem to
−78 + −110. Adding −110 means that we are moving in a negative
direction on the number line.
Now that we have an idea where the answer is, we can solve the
problem using any method and have a way of checking our answer
using good number sense.
Review 1
How do we work with large integers?
−78 − 110
Start by locating −78 on the number line.
−78
0
Next use the “add the opposite” rule to change the problem.
We know that −78 − 110 is the same as −78 + −110.
Here is how we would use number sense to answer the problem. The
number −110 is the same as −100 + −10. We move −100 to the left and
then move −10 more.
−10
−100
−178
−188
−78 − 110 = −188
624 Unit 8 • Lesson 15
−78
0
Lesson 15
When do we work with integers on a
daily basis?
Integers help us understand things we do every day. They are especially
useful when we talk about money. They can help us understand how
much money we make and how much we spend.
• Income is the money we make. This can come from different
sources, depending on what we do.
• Expenses are the money we spend. The money can also go
to different places, depending on what we need to buy.
Whether we are talking about our personal bank account or running
a business, we want our income to be greater than our expenses. This
means that we are “in the black.” If our expenses are greater than our
income, then we say that we are “in the red.”
Review 1
How do we use integers to talk about money?
Problem:
A store owner makes $1,385.27 on Monday. On that same day, the
owner spent $1,527.38 on a new computer for his business. What is
his profit for that day?
We subtract to find the difference between what the owner made and
what he spent.
$1,385.27 – $1,527.38 = –$142.11
The store owner is in the red, or has a negative profit, for Monday.
Apply Skills
Turn to Interactive Text,
page 315.
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
Unit 8 • Lesson 15 625
Lesson 15
Problem Solving: Finding Points on a Graph
What is important about dot graphs?
A dot graph is another way to represent data. Review 1 shows how
we use a dot graph to represent data from call-in requests to a radio
station. The graph shows how many callers requested the song “Your
World” at various times throughout the day. If we want to find out how
many requests there were at a particular time of day, we would track
along the bottom, locate that time, and move up to the dot. The dot tells
us how many callers requested the song at that time.
Review 1
How do we represent data on a dot graph?
Number of Callers
Telephone Requests for “Your World”
135
130
125
120
115
110
105
100
95
90
85
80
10 AM–
11 AM
11 AM–
12 PM
12 PM–
1 PM
1 PM–
2 PM
2 PM–
3 PM
3 PM–
4 PM
Time of Day
The most important idea about dot graphs is that we cannot connect
the dots with lines. Why? We can only use lines if we know that the
change between the dots follows the line.
Connecting the dots in Review 1 would mean that there were 130 calls
between 12 pm and 1 pm. We do not know this, and there is no reason
to believe that the change from one dot to the next follows a straight
line. Dots on this graph just tell us how many calls we have during an
hour’s time.
626 Unit 8 • Lesson 15
Lesson 15
Why are coordinate graphs important?
Coordinate graphs appear all the time in late middle school and high
school math. They are an important way of showing what equations
look like. In this unit, we have learned how to find the x and y
coordinates using the axes of the coordinate graph. Review 1 presents
some of the key vocabulary that we need to know in order to work with
coordinate graphs.
Review 1
What are key vocabulary words for coordinate graphs?
y-axis
II
y
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3–2 –1
–1
–2
–3
–4
–5
–6
–7
–8
III
Quadrant
I
Origin (0, 0)
1 2 3 4 5 6 7 8
x
x-axis
IV
Unit 8 • Lesson 15 627
Lesson 15
We learned how to plot points on a coordinate graph using the x- and
y-axes. Review 2 shows two graphs. In the first graph, Points A and B
are symmetrical on the y-axis. That means they are the same distance
from the y-axis. The second graph shows Points C and D. They are
symmetrical on the x-axis. Symmetry is one of many ways of thinking
about points on a coordinate graph.
Review 2
How can points be symmetrical on a coordinate graph?
y
B
(–3, 5)
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3–2 –1
–1
–2
–3
–4
–5
–6
–7
–8
y
8
7
6
5
4
3
2
1
A
(3, 5)
1 2 3 4 5 6 7 8
x
–8 –7 –6 –5 –4 –3–2 –1
–1
–2
–3
–4
–5
–6
–7
–8
The points on each graph are symmetrical. Point A and Point B are the
same distance from the y-axis, and their y-coordinates are the same.
Point C and Point D are the same distance from the x-axis, and their
x-coordinates are the same.
Problem-Solving Activity
Turn to Interactive Text,
page 317.
628 Unit 8 • Lesson 15
Reinforce Understanding
Use the mBook Study Guide
to review lesson concepts.
C
(4, 2)
1 2 3 4 5 6 7 8
D
(4, –2)
x