Problem
2.1
Topic 2: Understanding Integers
for use after
Bits and Pieces I
(Investigation 3
)
Negative numbers are needed when the quantities are less than 0, such as
very cold temperatures. Temperatures in winter can easily go below 0ºF in
some locations. An altitude of 0 feet is referred to as sea level, but there are
places in the world that are below sea level.
The counting numbers and zero are called
whole numbers
.The first six
whole numbers are 0, 1, 2, 3, 4, and 5. You can extend a number line to the
left past zero.
The opposite of a positive number is a
negative number
.For example, the
number
–
2 is the
opposite
of
+
2. The set of whole numbers and their
opposites are called
integers
.
Emily, Sarah, Michael, Jacob, and Austin play a Question and Answer
game. A player steps forward for a correct answer, but steps backward
for an incorrect answer. During the first round, Michael takes five steps
backward. Jacob takes three steps forward. Emily takes three steps
backward. Austin does not move. Sarah takes two steps forward.
A. 1. Which integer describes Austin’s position in the game?
2. Draw a number line. Represent each player’s position on the
number line.
3. Who is in last place?
4. Which players are represented by opposites?
B. 1. In the next round each player moves two steps forward. Place all
five players on a new number line.
2. Are any players that were opposites before still opposites now?
Why or why not?
3. What does it mean when you read the numbers on the number line
from the left to the right?
C. In the final round, Emily stays in the same place, and Michael is at her
opposite. How many steps did Michael take in the final round?
?
5
?
4
?
3
?
2
?
1
0
?
1
?
2
?
3
?
4
?
5
Exercises
For Exercises 1–4, place each integer on a number line.
Then identify any opposites.
1.
–
1, 4, 2,
–
4, 3, 1
2. 2, 0,
–
3, 4,
–
1, 3
3.
–
5, 10,
–
2, 4, 0,
–
10
4.
–
5, 8,
–
7,
–
10, 5, 10
5. Use an integer to represent each play in a football game.
a. The fullback carries the ball for a gain of 6 yards.
b. The quarterback is sacked for a loss of 3 yards.
c. The play stops at the line of scrimmage for no gain.
6. Use an integer to represent each change to a bank account.
a. A deposit of $20 is made on Monday.
b. A check for $4 is written on Tuesday.
c. A check for $6 is written on Wednesday.
d. No transactions are made on Thursday.
7. Use an integer to represent each position of an elevator.
a. The elevator leaves the ground floor and arrives at the 12th floor.
b. The elevator leaves the ground floor and arrives at the second
basement level.
c. The elevator leaves the ground floor, arrives at the 7th floor, and
then travels down 3 floors.
8. Use an integer to represent time in seconds for a space ship launch.
a. Lift off.
b. The countdown begins with 10 seconds before lift off.
c. The space ship has been in the air for one minute.
d. Why do you think a launch countdown starts at
T-minus ten
seconds?
9. Use the number line below.
a. If
a
and
e
are opposites, what integer would you use to represent
c
?
b. Assign integer values to each point in part (
a
).
c. If
a
and
d
are opposites, is
c
positive or negative? Explain.
ab
cd
e
Topic 2: Understanding Integers
Guided Instruction
Mathematical Goals
• Compare and order positive and negative integers.
Vocabulary
•
whole numbers
•
negative number
•
opposites
•
integer
At a Glance
Explore extending the number line. Display a number line from
?
5 to
+
5
that can be copied by each student. Have students supply the natural
numbers of 1, 2, 3, 4, and 5. Add the zero as you mention whole numbers.
Use the number line to locate the negative integers as the opposites of the
natural numbers. Draw connector arrows between each pair of opposites.
•
What is the sum of opposite integers?
(zero)
•
What is another way that you could define opposites?
(Two numbers
that, when added together, have a sum of zero.)
Assign students to represent each of the students of Problem 2.1. These
students move forward or backward according to their role.
•
Which integer describes Austin’s position in the game?
(0)
Based on this answer, have students assign integer values to each of the
other students and answer Question A.
Read Question B, and have all students take two steps forward. Students
are now able to order the five integers represented by the players in the
game.
Read Question C. Give the students a little time to think about their
answer before the student representing Michael counts aloud the steps
needed to become Emily’s opposite.
Suppose the winner is the first person to reach
+
10.
•
Who has the best chance to win on the next question?
(Jacob)
•
What point value for the question is needed by that player?
(5)
•
Where would that player be located if they got the question wrong?
(0)
•
Who is the new leader?
(Sarah)
You will find additional work on integers in the grade 7 unit
Accentate
the Negative.
PACING
1 day
ACE Assignment Guide
for Topic 2
Core
1–9
Answers to Topic 2
Problem 2.1
A. 1.
0
2.
3.
Michael
4.
Jacob and Emily
B. 1.
2.
No, when Jacob and Emily both moved
forward two steps, their relationship to the
starting point (0) changed. The new
positions of these players are
?
1 and 5.
3.
You are reading the numbers in order from
the least to the greatest.
C.
4 steps; Emily started at
?
3, then moved
forward 2 steps to
?
1. For Michael to be her
opposite, Michael needs to be on
?
1.
Michael’s last move brought him to
?
3, so he
needs to move forward 4 steps to be Emily’s
opposite.
Exercises
1.
?
4 and 4
2.
?
3 and 3
3.
?
10 and 10
4.
?
10 and 10,
?
5 and 5
5. a.
?
6
b.
?
3
c.
0
6. a.
?
20
b.
?
4
c.
?
6
d.
0
7. a.
?
12
b.
?
2
c.
?
7,
?
4
8. a.
0
b.
?
10
c.
?
60
d.
The time before lift-off is negative ten
seconds
9. a.
0
b.
a
?
?
5,
c
? 0,
e
? 5
c.
Positive; point
b
would be zero because it is
midway between
a
and
d
. Point
c
is to the
right of point
b
.
?
10
?
7
?
5
0
5810
?
10
?
5
?
2 0
4
10
0
234
?
3
?
1
?
4
?
1 0 1 2 3 4
01
S
35
J
?
5
?
3
?
1
ME
A
01
S
35
J
?
5
?
3
?
1
ME
A