Positive and
Negative Numbers
Objective To introduce addition involving negative integers.
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Practice
EM Facts
Workshop
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Teaching the Lesson
Key Concepts and Skills
• Compare and order integers.
[Number and Numeration Goal 6]
• Add signed numbers.
[Operations and Computation Goal 2]
• Identify a line of reflection.
[Geometry Goal 3]
Key Activities
Students review positive and negative
numbers on the number line, thinking of
them as reflected across the zero point.
They discuss and practice addition of
positive and negative numbers as accounting
problems, keeping track of “credits” and
“debits.” They play the Credits/Debits Game.
Ongoing Assessment:
Informing Instruction See page 825.
Family
Letters
Assessment
Management
Common
Core State
Standards
Curriculum
Focal Points
Ongoing Learning & Practice
Solving Fraction, Decimal,
and Percent Problems
Math Journal 2, p. 283
Students solve problems involving
fractions, decimals, and percents.
Math Boxes 10 6
Math Journal 2, p. 284
Students practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Differentiation Options
READINESS
Exploring Skip Counts on a Calculator
calculator
Students skip count on a calculator to
explore patterns in negative numbers.
READINESS
Using a Number Line to Add Positive
and Negative Numbers
masking tape
Students use a number line to add
positive and negative integers.
Use Math Boxes, Problem 1.
[Data and Chance Goal 4]
Study Link 10 6
Math Masters, p. 322
Students practice and maintain skills
through Study Link activities.
Key Vocabulary
opposite (of a number) credit debit
Materials
Student Reference Book, pp. 60 and 238
Study Link 10 5
Math Masters, pp. 320 and 468
transparencies of Math Masters, pp. 318
and 321 (optional) per partnership:
1 transparent mirror, deck of number cards
(the Everything Math Deck, if available) calculator (optional)
Advance Preparation
For Part 1, make and cut apart copies of Math Masters, page 320. Place them near the Math Message.
For the second optional Readiness activity in Part 3, use masking tape to create a life-size number line
(–10 to 10) on the floor.
Teacher’s Reference Manual, Grades 4–6 pp. 71–74, 100–102
822
Unit 10
Reflections and Symmetry
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP2, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
4.NF.6, 4.MD.1
Mental Math and Reflexes
Math Message
Pose problems involving comparisons of integers.
Suggestions:
Take a copy of Math Masters, page 320. Follow
the directions and answer the questions. Share
1 transparent mirror with a partner.
Are you better off if you have $3 or owe $10? Have $3
Owe $4 or owe $9? Owe $4
Owe $20 or owe $100? Owe $20
Which is greater?
–10 or 8? 8
5 or –1? 5
10 or –10? 10
Which is colder?
–3°C or 10°C? –3°C
–9°C or 19°C? –9°C
–7°C or –11°C? –11°C
Study Link 10 5 Follow-Up
Review answers. Have students share some of
the patterns they created on their own. An
overhead transparency of Study Link 10 5
(Math Masters, page 318) may be helpful.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Student Reference Book, p. 60; Math Masters, p. 320)
One way to think about a number line is to imagine the whole
numbers reflected across the zero point. Each of these positive
numbers picks up a negative sign as it crosses to the other side
of zero. The opposite of a positive number is a negative number.
Conversely, imagine the negative numbers reflected across the
zero point. The sign of each number changes from negative to
positive as it crosses to the other side of zero. The opposite of a
negative number is a positive number.
NOTE In this “flipping” of the number line, the zero point stays motionless,
like the fulcrum of a lever. Zero is the only number that equals its opposite.
When students place the transparent mirror on the line passing
through the zero point on Math Masters, page 320, the negative
numbers appear (reversed) across from the corresponding positive
numbers.
Name
LESSON
10 6
䉬
Date
Time
Positive and Negative Numbers
60
–10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 –1
0
1 2 34
1 2 3 4
1
2
3
4
5
6
7
8
9
10
Place your transparent mirror on the dashed line that passes through 0
on the number line above. Look through the mirror. What do you see?
What negative number image do you see . . .
1
above 2?
2
above 8?
8
4321
0
above 1?
109 8 7654321
0
1 2 3 4 5 6 7 8 9 10
12345678901
Math Masters, page 320
Lesson 10 6
823
Student Page
Read and discuss page 60 of the Student Reference Book with the
class. The diagram on the page is another way of showing that the
opposite of every positive number is a negative number, and the
opposite of every negative number is a positive number.
Fractions
Negative Numbers and Rational Numbers
People have used counting numbers (1, 2, 3, and so on) for
thousands of years. Long ago people found that the counting
numbers did not meet all of their needs. They needed numbers
for in-between measures such as 2 _12 inches and 6 _56 hours.
Fractions were invented to meet these needs. Fractions can
also be renamed as decimals and percents. Most of the numbers
you have seen are fractions or can be renamed as fractions.
Rename as fractions: 0, 12, 15.3, 3.75, and 25%.
0 = _01
12
12 = _
1
153
15.3 = _
10
375
3.75 = _
100
25
25% = _
100
However, even fractions did not meet every need. For example,
problems such as 5 - 7 and 2 _34 - 5 _41 have answers that are less
than 0 and cannot be named as fractions. (Fractions, by the way
they are defined, can never be less than 0.) This led to the
invention of negative numbers. Negative numbers are numbers
that are less than 0. The numbers - _12 , -2.75, and -100 are
negative numbers. The number -2 is read “negative 2.”
Negative numbers serve several purposes:
♦ To express locations such as temperatures below zero on
a thermometer and depths below sea level
Note
Every whole number
(0, 1, 2, and so on) can
be renamed as a fraction.
For example, 0 can be
written as _01 . And 8 can
be written as _81 .
Note
Numbers like -2.75 and
-100 may not look like
negative fractions, but
they can be renamed as
negative fractions.
♦ To show changes such as yards lost in a football game
11
-2.75 = -_
, and
4
♦ To extend the number line to the left of zero
100
-100 = -_
1
♦ To calculate answers to many subtraction problems
The opposite of every positive number is a negative number,
and the opposite of every negative number is a positive
number. The number 0 is neither positive nor negative;
0 is also its own opposite.
The diagram at the right shows this relationship.
The rational numbers are all the
numbers that can be written or renamed
as fractions or as negative fractions.
Using Credits and Debits
WHOLE-CLASS
ACTIVITY
to Practice Addition of
Positive and Negative Numbers
ELL
(Math Masters, p. 321)
Display a transparency of Math Masters, page 321. Tell students
that in this lesson they pretend that they are accountants for a
new business. They figure out the “bottom line” as you post
transactions.
Discuss credits (money received for sales, interest earned, and
other income) as positive additions to the bottom line, and debits
(cost of making goods, salaries, and other expenses) as negative
additions to the bottom line. Explain that you will label credits
with a “+” and debits with a “–” to keep track of them as positive
and negative numbers.
Student Reference Book, p. 60
To support English language learners, clarify any misconceptions
about the use of the words credits, debits, and bottom line in this
lesson as compared with students’ observations of the use of credit
and debit cards at stores.
Links to the Future
Students explore subtraction of positive and
negative integers in Lesson 11-6. Addition
and subtraction of signed numbers is a
Grade 5 Goal.
Adjusting the Activity
Have students experiment with their
calculators to find out how to enter negative
numbers and expressions with negative
numbers. On the TI-15 students use the
(–) key, and on the Casio fx-55, students
use the
key.
AUDITORY
824
KINESTHETIC
TACTILE
VISUAL
Unit 10 Reflections and Symmetry
Be consistent throughout this lesson in “adding” credits and debits
as positive and negative numbers, because Lesson 11-6 uses the
same format to show “subtraction” of positive and negative
numbers—the effect on the bottom line of “taking away” what
were thought to be credits or debits.
Following is a suggested series of transactions. Entries in black
are reported to the class; entries in color are appropriate student
responses. To support English language learners, discuss the
meaning of the words transaction and change.
Transaction
Start
Change
End/Start of
Next
Transaction
New business,
start at $0
$0
$0
$0
Credit (payment)
of $5 comes in
$0
add +$5
+$5
Credit of $3
+$5
add +$3
+$8
Debit of $6
+$8
add -$6
+$2
Debit of $8 (Be
sure to share
strategies.)
+$2
add -$8
-$6
Debit of $3
-$6
add -$3
-$9
Credit of $5
(At last!)
-$9
add +$5
-$4
Credit of $6
-$4
add +$6
+$2
Student Page
Playing the Credits/Debits Game
(Student Reference Book, p. 238; Math Masters, p. 468)
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VIN
V
IIN
NG
Students play the Credits/Debits Game to practice adding
positive and negative numbers. They record their work on
Math Masters, page 468.
Games
Credits/Debits Game
Materials □ 1 complete deck of number cards
□ 1 Credits/Debits Game Record Sheet
for each player (Math Masters, p. 468)
Players
2
Skill
Addition of positive and negative numbers
Object of the game To have more money after
adjusting for credits and debits.
Directions
You are an accountant for a business. Your job is to
keep track of the company’s current balance. The current
balance is also called the “bottom line.” As credits and debits are
reported, you will record them and then adjust the bottom line.
Each player uses one
Record Sheet.
1. Shuffle the deck and lay it number-side down between
the players.
Ongoing Assessment: Informing Instruction
2. The black-numbered cards are the “credits,” and the
blue- or red-numbered cards are the “debits.”
3. Each player begins with a bottom line of +$10.
Note
4. Players take turns. On your turn, do the following:
As students play, watch for those who are beginning to devise shortcuts for
finding answers. For example, most students will probably count up and back on
a number line. Some students may notice that when two positive numbers are
added, the result is “more positive”; when two negative numbers are added, the
result is “more negative”; and when a positive and a negative number are added,
the result is the difference of the two (ignoring the signs) and has the sign of the
number that is “bigger” in the sense of being farther from 0.
Do not try too hard to get explanations; these will evolve over time as students
have more experience with positive and negative numbers.
♦ Draw a card. The card tells you the dollar amount
If both players have
negative dollar amounts
at the end of the round,
the player whose
amount is closer
to 0 wins.
and whether it is a credit or debit to the bottom line.
Record the credit or debit in your “Change” column.
♦ Add the credit or debit to adjust your bottom line.
♦ Record the result in your table.
5. At the end of 10 draws each, the player with more money is
the winner of the round.
Beth has a “Start” balance of +$20. She draws a black 4.
This is a credit of $4, so she records +$4 in the “Change” column.
She adds $4 to the bottom line: $20 + $4 = $24. She records +$24 in
the “End” column, and +$24 in the “Start” column on the next line.
Alex has a “Start” balance of +$10. He draws a red 12. This is a debit of
$12, so he records -$12 in the “Change” column. He adds -$12 to the
bottom line: $10+(-$12) = -$2. Alex records -$2 in the “End” column.
He also records -$2 in the “Start” column on the next line.
Student Reference Book, p. 238
2 Ongoing Learning & Practice
Solving Fraction, Decimal,
INDEPENDENT
ACTIVITY
and Percent Problems
(Math Journal 2, p. 283)
Students solve problems involving equivalent fractions, decimals,
percents, and discounts.
Math Boxes 10 6
Game Master
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 284)
Name
Date
Time
Credits/Debits Record Sheets
4
3
6
5
4
3
+$10
22212019181716151413121110 9 8 7 6 5 4 3 2 1
5
7
Game 1
6
8
Change
Record Sheet
7
9
Start
8
10
1
9
2
10
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 10-3. The skill in Problem 5
previews Unit 11 content.
1 2
4 3
238
End, and
next start
0
1
Game 2
2
1
2
3
4
5
6
Start
7
+$10
8
End, and
next start
9 10 11 12 13 14 15 16 17 18 19 20 21 22
Change
Record Sheet
Math Masters, p. 468
Lesson 10 6
825
Student Page
Date
10 6
Review: Fractions, Decimals, and Percents
1. Fill in the missing numbers in
the table of equivalent fractions,
decimals, and percents.
2.
3.
4.
5.
Writing/Reasoning Have students write a response to the
following: The weights in Problem 5 are expressed in pounds. Make
a table to show equivalent weights in ounces for 50; 150; 500; and
1,000 pounds. Then explain how you converted the weights.
Time
LESSON
Fraction
Decimal
Percent
4
_
10
0.4
6
10
75
100
40%
60%
0.75
0.6
61 62
75%
Kendra set a goal of saving $50 in 8 weeks. During the first
2 weeks, she was able to save $10.
Pounds
50
150
500
1,000
10
_
a.
What fraction of the $50 did she save in the first 2 weeks?
b.
What percent of the $50 did she save?
c.
At this rate, how long will it take her to reach her goal?
50
20%
10
weeks
80
_
Shade 80% of the square.
a.
What fraction of the square did you shade?
b.
Write this fraction as a decimal.
c.
What percent of the square is nott shaded?
100
0.8
20%
Tanara’s new skirt was on sale at 15% off the original price.
The original price of the skirt was $60.
a.
How much money did Tanara save with the discount?
b.
How much did she pay for the skirt?
Sample answer: I know that there are 16 ounces in a pound, so I
multiplied each weight by 16 to get the number of ounces.
$9
$51
Star Video and Vic’s Video Mart sell videos at about the same regular prices. Both
1
stores are having sales. Star Video is selling its videos at _
off the regular price.
3
Vic’s Video Mart is selling its videos at 25% off the regular price. Which store has
the better sale? Explain your answer.
Star Video has the better sale since 13 = 33 31 %, which
_
Ounces
800
2,400
8,000
16,000
Ongoing Assessment:
Recognizing Student Achievement
_
Math Boxes
Problem 1
is more than 25%. So they’re taking more off their
regular prices.
Math Journal 2, p. 283
274-285_EMCS_S_MJ2_G4_U10_576426.indd 283
2/15/11 6:15 PM
Use Math Boxes, Problem 1 to assess students’ ability to express the
probability of an event as a fraction. Students are making adequate progress
3
if they design a spinner that is _14 red and _4 blue. Many students will design
a spinner that has 3 consecutive parts red and 9 consecutive parts blue.
Some students will explore other possibilities—for example, 2 consecutive
red parts, followed by 4 blue parts, 1 red part, and 5 blue parts.
[Data and Chance Goal 4]
Study Link 10 6
INDEPENDENT
ACTIVITY
(Math Masters, p. 322)
Home Connection Students compare and order
positive and negative numbers and add positive
and negative integers.
Student Page
Date
Time
LESSON
10 6
1. a.
3 Differentiation Options
Math Boxes
Make a spinner.
Color it so that
if you spin it
36 times, you
would expect it
to land on blue
27 times and
red 9 times.
Sample answer:
b.
Explain how you designed your spinner.
Sample answer: If it lands on
blue 27 out of the 36 spins,
27
_3
then __
36 , or 4 , of the board
should be blue. Likewise,
9
__
_1
36 , or 4 , of the board should
be red.
red
blue
READINESS
Exploring Skip Counts
82–86
2.
Complete.
Rule:
1
-_
4
Solve each open sentence.
in
out
8
_
16
4
_
16
a.
67.3 + p = 75.22
p=
_6
b.
6.86 - a = 2.94
a=
c.
x + 5.69 = 7.91
x=
d.
4.6 - n = 0.32
n=
8
_
8
3.
8
3
_
4
2
_
4
10
_
7
_
12
12
To explore patterns in negative numbers, have students skip count
on the calculator. Ask students to start with 30 and count back by
10s on their calculator as they say the numbers aloud. Stop at –10
and ask the following questions:
20
55 57
acute
Angle RUG is an
(acute or obtuse) angle.
34–37
5.
Sebastian and Joshua estimated the
weight of their mother. What is the most
reasonable estimate? Fill in the circle
next to the best answer.
R
U
Measure of ∠RUG =
G
20
A.
50 pounds
B.
150 pounds
C.
500 pounds
D.
1,000 pounds
°
93 142
143
140
Math Journal 2, p. 284
274-285_EMCS_S_MJ2_G4_U10_576426.indd 284
826
Unit 10 Reflections and Symmetry
5–15 Min
on a Calculator
10
_
15
_
20
4.
7.92
3.92
2.22
4.28
SMALL-GROUP
ACTIVITY
2/18/11 9:16 AM
Study Link Master
●
What does the calculator display show after zero? –10
Name
Date
STUDY LINK
10 6
䉬
●
●
How do you read this number? negative ten
Time
Positive and Negative Numbers
Write or to make a true number sentence.
1.
Can you predict what number will come next? –20
3
14
2.
7
7
3.
19
20
8
4.
60
10
List the numbers in order from least to greatest.
Have students continue counting back, stopping at –50.
Repeat the routine counting back with other numbers such as
2, 5, 25, and 100. Remind students to clear their calculators
after each count.
SMALL-GROUP
ACTIVITY
READINESS
Using a Number Line to
5.
1
2
1
4
5, 8, , , 1.7, 3.4
8
least
6.
14
7
43, 22, , 5, 3, 0
43
1
3
least
1
2
4
3.4
1.7
5
greatest
14
0
5
7
Sample answers:
1
4
7.
Name four positive numbers
less than 2.
8.
Name four negative numbers
greater than 3.
2
22
greatest
1
2
3
4
1
1
1
2
4
1
Use the number line to help you solve Problems 9–11.
5–15 Min
Add Positive and
Negative Numbers
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
9. a.
49
10. a.
53
11. a.
15
13
8
2 13
b.
4 (9) b.
(5) 3 11
b.
1
2
3
4
5
2
(2) 13
5
6
c.
(4) (9) c.
(5) (3) c.
7
8
9 10 11 12 13 14 15
15
13
8
(2) (13)
Practice
To explore addition of positive and negative integers using a
number line model, have students act out addition problems
by walking on a life-size number line from –10 to 10.
12.
14.
13.90
13.
7.26 1.94 5.84 8.75
15.
3.38 1.02 12.88 2.91
5.32
0.76 2.62
Math Masters, p. 322
The first number tells students where to start.
The operation sign (+ or –) tells which way to face:
+ means face the positive end of the number line.
– means face the negative end of the number line.
If the second number is negative, then walk backward.
Otherwise, walk forward.
The second number (ignoring its sign) tells how many steps
to walk.
The number where the student stops is the answer.
Example: –4 + 3
Start at –4.
Face the positive end of the number line.
Walk forward 3 steps.
–5
You are now at –1. So –4 + 3 = –1.
–2
–1
0
1
2
–6 + –3 = ? (Start at –6. Face in the positive direction.
Walk backward 3 steps. End up at –9.)
–10 –9
●
–3
-4 + 3 = -1
Suggestions:
●
–4
–8
–7
–6
–5
–4
–3
4 + –6 = ? (Start at 4. Face in the positive direction.
Walk backward 6 steps. End up at –2.)
–2
–1
0
1
2
3
4
5
Lesson 10 6
827