Positive and Negative
Numbers
Objective To guide children as they investigate positive and
negative numbers.
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Teaching the Lesson
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Making a Line Plot
• Compare and order positive and
negative numbers. Student Reference Book, pp. 79–81,
89A, and 89B
Class Data Pad (optional)
Children measure their pencils and
show the data by making a line plot.
[Number and Numeration Goal 6]
• Solve number stories involving the
addition and subtraction of positive and
negative numbers. [Operations and Computation Goal 2]
Key Activities
Children consider two uses of positive and
negative numbers: relating numbers to a
zero point and recording changes. They
solve number stories about positive and
negative numbers.
Math Boxes 9 13
Math Journal 2, p. 238
Children practice and maintain skills
through Math Box problems.
Ongoing Assessment:
Recognizing Student Achievement
Use Math Boxes, Problem 1. [Measurement and Reference Frames
Goal 2]
Key Vocabulary
Fahrenheit scale degrees Fahrenheit Celsius scale degrees Celsius
Materials
Home Link 9 13
Math Masters, p. 312
Children practice and maintain skills
through Home Link activities.
Math Journal 2, p. 237
Student Reference Book, pp. 170 and 171
Home Link 9 12
calculator slate display thermometer
(optional)
Advance Preparation
Teacher’s Reference Manual, Grades 1– 3 pp. 64–66
784
Unit 9
Multiplication and Division
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Counting on a Number Line
Math Masters, p. 313
Children count on a number line with
positive and negative numbers.
ENRICHMENT
Solving Subtraction Number Stories
Math Masters, p. 314
2 each of number cards 0 –10 (from the
Everything Math Deck, if available)
Children explore the importance of order
in solving subtraction problems.
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6
Getting Started
Content Standards
3.NBT.2, 3.MD.4, 3.MD.6, 3.MD.7b
Mental Math and Reflexes
Math Message
Children skip count on their calculators. They say the
Look at the thermometer on page 171 in your
Student Reference Book.
Which temperature is colder, -5°C or -10°C? -10°C +1°C or
-14°C? -14°C How do you know?
=
counts as they press
on their calculators.
Suggestions:
Start at 12; count down by 4s. 12, 8, 4, 0, -4, -8, -12, ...
Start at –20; count up by 5s. -20, -15, -10, -5, 0, 5, 10,
15, 20, ...
Start at –18; count up by 4s. -18, -14, -10, -6, -2, 2,
6, 10, 14, 18, ...
Home Link 9 12 Follow-Up
Have children share solution strategies for
Problems 4 and 5 with a partner.
Briefly review answers.
1 Teaching the Lesson
Math Message Follow-Up
NOTE See pages 265 and 266 in the
WHOLE-CLASS
DISCUSSION
(Student Reference Book, pp. 170 and 171)
Discuss any disagreements about answers. Colder is further down
on the temperature scale. If you have a display thermometer in
the classroom, ask children to show the pairs of temperatures.
Review the two thermometer scales on page 171 in the Student
Reference Book. The Fahrenheit scale, marked in degrees
Fahrenheit, is standard in the United States for everyday use.
For example, most U.S. newspapers report temperatures only in
degrees Fahrenheit, so Fahrenheit is used in the class high/low
temperature record. The Celsius scale, marked in degrees
Celsius, is standard throughout the rest of the world for everyday
use and everywhere, including the United States, for scientific
work.
Student Reference Book for directions to skip
counting on a calculator. When skip counting
beginning with a negative number, it is
necessary to enter the start number and then
+/–
press the
key to make the numbers
negative. For the
Mental Math and
Reflexes problem, the following key sequences
are needed:
=
TI 20: 20,
+/–
,
+
=
, 5,
,
,…
=
Casio: 5,
+
,
+
, 20,
+/–
,
=
,
,…
Student Page
Reference Frames
Note the different temperatures on each scale for the same
natural phenomena: the freezing and boiling points of water,
the freezing point of salt solutions, and room temperature and
body temperature.
Most thermometers have marks that are spaced
2 degrees apart.
Fahrenheit
thermometer
Celsius
thermometer
NOTE For additional practice with Fahrenheit and Celsius thermometer scales,
go to www.everydaymathonline.com.
Student Reference Book, p. 171
Lesson 9 13
785
NOTE Locations on a reference scale that
are expressed without a negative symbol are
usually assumed to be positive. However, it is
sometimes helpful to be more explicit by using
the + symbol.
Writing Temperatures above
WHOLE-CLASS
ACTIVITY
and below Zero
Help children make connections between temperatures expressed
in words and temperatures expressed with numbers and units.
Dictate temperatures above and below zero for children to write on
their slates. For this exercise, specify that all temperatures are on
the Celsius scale. Examples:
●
Write the number and unit for 5 degrees below zero. -5°C
●
Write the number and unit for 8 degrees above zero. 8°C or +8°C
Explain that thermometers can be thought of as number lines or
reference frames. Zero is a reference or beginning point from
which positive and negative numbers go in opposite directions.
Temperatures are expressed in words as above or below zero, or
with symbols as positive or negative numbers.
Adjusting the Activity
ELL
Have children think in terms of a number line. Ask questions like the
following: What is the distance (how many jumps) between +2 and -2? 4
What is the distance (how many jumps) between +100 and -1,000? 1,100
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
Using Sea Level as
V I S U A L
WHOLE-CLASS
ACTIVITY
a Zero Point
Discuss what children know about sea level. Sea level is an
agreed-upon zero point from which elevations, such as land
elevations and depths of oceans, are measured. Just as with the
discussion on temperatures, ask higher/lower questions. Examples:
●
Which is lower, 100 meters above sea level or 1,000 meters
below sea level? 1,000 meters below sea level
●
Which is lower, 2 meters above sea level or 2 meters below sea
level? 2 meters below sea level
Dictate elevations, and have children write them on their slates as
positive or negative numbers. Check that children write a negative
sign when needed and the appropriate unit. Examples:
786
Unit 9 Multiplication and Division
●
977 meters below sea level -977 m
●
4,240 meters above sea level +4,240 m or 4,240 m
Student Page
Expressing Changes with
WHOLE-CLASS
ACTIVITY
Positive and Negative Numbers
Date
LESSON
9 13
䉬
Time
Number Stories with Positive & Negative Numbers
Solve the following problems. Use the thermometer scale, the class
number line, or other tools to help.
90
1. The largest change in temperature in a single day took
Changes are often expressed with positive and negative numbers.
Discuss changes like the following:
●
●
●
●
6-pound loss: -6 lb; 6-pound gain: +6 lb
80
place in January 1916 in Browning, Montana. The
temperature dropped 100°F that day. The temperature
was 44°F when it started dropping.
How low did it go?
70
60
56°F
50
2. The largest temperature rise in 12 hours took place in Granville,
What was the low temperature?
20
33°F
10
0
3. On January 12, 1911, the temperature in Rapid City,
South Dakota, fell from 49°F at 6
A.M.
to 13°F at 8
40
30
North Dakota, on February 21, 1918. The temperature rose
83°F that day. The high temperature was 50°F.
10°C temperature drop: -10°C; 10°C temperature rise:
+10°C
°F
100
–10
A.M.
–20
in football, gain of 6 yards: +6 yards; 15-yard penalty:
-15 yards
By how many degrees did the
temperature drop in those 2 hours?
62°F
–30
–40
4. The highest temperature ever recorded in Verkhoyansk,
–50
Siberia, was 98°F. The lowest temperature ever recorded
there was 94°F.
lose $10: -$10; find $10: +$10
Have children suggest other change situations. Record them on
the board with positive and negative numbers.
You and the children then make up number stories involving
comparisons. Examples:
●
Is it better to owe $5 or to owe $10? Owe $5
●
Is it easier to carry my backpack if I put in 3 lb or take out
5 lb? Take out 5 lb
●
In football, is it worse for the team to get a 10-yard penalty or a
15-yard penalty? 15-yard penalty
What is the difference between
those two temperatures?
–60
–70
192°
–80
–90
5. Write your own number story using positive and
–100
negative numbers.
Sample answer: My cat weighed 16 lb. Then
she got sick and now weighs 13 lb. What was
her weight change? 3 lb.
Math Journal 2, p. 237
Solving Number Stories with
PARTNER
ACTIVITY
Positive and Negative Numbers
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
E
M
SO
S
SOLVING
OL
O
LV
LV
VING
VIN
IIN
NG
(Math Journal 2, p. 237)
Children use thermometer scales, number lines, or other tools to
find answers to the problems on journal page 237. Bring the class
together to share solutions and strategies.
Adjusting the Activity
Encourage children to look for patterns on the completed journal page.
For example, how do you find the difference between a positive temperature and
a negative temperature? Add the two numbers as though both were positive.
A U D I T O R Y
K I N E S T H E T I C
T A C T I L E
V I S U A L
Links to the Future
This lesson is an early exposure to adding and subtracting positive and negative
numbers. This early exposure provides a background for work with positive and
negative numbers that will continue through Grades 4, 5, and 6.
Solving problems and number stories involving positive and negative numbers is
a Grade 5 Goal.
Lesson 9 13
787
Student Page
Date
Time
LESSON
2 Ongoing Learning & Practice
Math Boxes
9 13
5 units
width = 9 units
area = 45 square
length =
1.
2.
Use the partial-products algorithm
to solve.
×
units
2 factors of 45
and
are
5
9
652
3
×
1,800 3,200
150 + 64
+ 6 3,264
1,956
.
37
154 156
3. There are 54 candles. A box holds
4.
12 candles. How many full boxes of
candles are there?
Making a Line Plot
408
8
(Student Reference Book, pp. 79–81, 89A, and 89B)
68 69
Children measure the lengths of their pencils to the nearest __12 inch
and record the measures on the board or Class Data Pad. Have
them copy the measures on paper and order them from shortest to
longest. Then they make a line plot with the horizontal scale
marked in __12 -inch units and show the data. Remind children that
since no pencil is 0 inches long, the scale can begin with the
shortest pencil length. Circulate to make sure that each child
labels the horizontal and vertical axes and gives the line plot a
title.
Fill in the oval for the best answer.
The degree measure of the angle is
180°.
54 ÷ 12 = ?
or ? × 12 = 54
There are 4 boxes of candles.
There are 6 candles
Number model:
less than 90°.
less than 270°.
more than 270°.
left over.
74
259 260
5.
167 168
6.
What 3-D shape is this a picture of?
Fill in the oval for the best answer.
Number of pets children have:
0, 4, 0, 1, 1, 3, 6, 2, 5
sphere
2
Median:
cylinder
6
0
Maximum:
pyramid
Minimum:
What is the shape of the base?
Range:
A circle
INDEPENDENT
ACTIVITY
When most children have completed their line plots, have them
identify the median, mode, maximum, minimum, and range of
their data.
6
118
79 80
Math Journal 2, p. 238
204-239_EMCS_S_MJ2_G3_U09_576418.indd 238
3/11/11 1:45 PM
Math Boxes 9 13
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 238)
Mixed Practice Math Boxes in this lesson are linked with
Math Boxes in Lessons 9-9 and 9-11. The skill in Problem
6 previews Unit 10 content.
Ongoing Assessment:
Recognizing Student Achievement
Date
HOME LINK
Time
Family
Note
Encourage your child to use the thermometer pictured here to answer questions about
thermometer scales, temperature changes, and temperature comparisons. If you have
a real thermometer, try to show your child how the mercury moves up and down.
170–173
000 000
Please return this Home Link to school tomorrow.
°F
thermometer could show?
-40
210
°C
2. What is the warmest temperature this
°F
104
b.
190
160
140
10°C
Home Connection Children answer questions about
thermometer scales, temperature changes, and
appropriate temperatures for various activities. Point
out the two scales on the thermometer (°F and °C).
80
70
4. How much colder is
18° colder
-9°C than 9°C?
temperature for swimming outside?
100
Yes
Explain.
It would be very warm outside.
30° C is about the same as 86° F.
temperature for ice-skating?
No
Yes
Explain.
30
20
Body
Temperature
Room
Temperature
60
50
10
40
30
6. Would -6°C be a good
40
80
70
No
50
110
90
5. Would 30°C be a good
60
130
120
For in–line skating?
(Math Masters, p. 312)
Water
boils
150
20 degrees warmer
than -10°C?
0
20
10
0
Water freezes at 0° C so there
would be ice to skate on. This would
–10
be dangerous for in–line skating.
–40
–20
Water
freezes
–10
–20
Salt
solution
freezes
–30
–30
–40
Math Masters, p. 312
267-318_EMCS_B_MM_G3_U09_576957.indd 312
788
INDEPENDENT
ACTIVITY
90
170
°C
3. What temperature is
For sledding?
100
200
180
thermometer could show?
220
°C
220
-40
b.
°F
Home Link 9 13
1. What is the coldest temperature this
a.
[Measurement and Reference Frames Goal 2]
Positive and Negative Temperatures
9 13
a.
Use Math Boxes, Problem 1 to assess children’s progress toward finding the
areas of rectangular shapes. Children are making adequate progress if they
find the area of the rectangle. Some children may identify two factors of 45.
Home Link Master
Name
Math Boxes
Problem 1
Unit 9 Multiplication and Division
2/18/11 7:37 PM
Teaching Master
Name
(Math Masters, p. 314)
0
10
20
Try This
When the
distances from 0 to the 2 numbers on the number line are added together, the
sum is the count up or count back number.
30
20
10
. Where did you land?
. Count
10
0
10
20
5. Describe the relationships you see between the three numbers in each problem.
50
40
50
30
20
30
20
10
0
10
20
10
0
10
7
3
10
4. Do your own. Start at
Number Stories
3. Start at 40. Count back 50. Where did you land?
15–30 Min
20
Solving Subtraction
INDEPENDENT
ACTIVITY
1. Start at 10. Count back 13. Where did you land?
ENRICHMENT
Show the jumps on the number lines.
To provide experience with finding distances between positive
and negative numbers on a number line, have children practice
counting on a number line. Children record their work on
Math Masters, page 313. Have children describe the relationships
they found.
20
(Math Masters, p. 313)
Answers vary.
40
5–15 Min
30
Counting on a Number Line
INDEPENDENT
ACTIVITY
2. Start at 15. Count up 22. Where did you land?
READINESS
40
50
䉬
40
9 13
Time
Negative Numbers on a Number Line
50
3 Differentiation Options
Date
LESSON
Math Masters, p. 313
To explore solving subtraction number stories have children follow
the steps on Math Masters, page 314. When children have
completed the page, discuss the patterns. Sample answers:
When the numbers on the cards are the same, the difference
between them is always 0. When the numbers are not the same,
the differences are opposites. For example, 6 - 2 = 4 and
2 - 6 = -4. 4 and -4 are opposites.
Planning Ahead
Teaching Master
Name
Date
Time
LESSON
9 13 Exploring Order in Subtraction
10
9
8
7
6
5
4
3
2
1
2
__ __ __
__ __ __
4
3
__ __ __
__ __ __
3
4
__ __ __
__ __ __
2
10 9
Example
8
7
4, 6
6
5
__ __ __
__ __ __
1
6-4=2
4 - 6 = -2
1
0
number.
number, the difference is a negative
number is subtracted from the smaller
is a positive number. When the larger
from the larger number, the difference
Sample answer:
When the smaller number is subtracted
matter?
2. How do you know?
Yes
order of numbers in a subtraction number sentence
1. Look at the number sentences you wrote. Does the
Number
Sentences
Numbers on
Cards
䉬 Follow the steps 3 more times.
䉬 Choose 2 cards. Record the numbers in the chart below.
䉬 Place the cards number-side down.
You will need number cards 0–10 (2 of each).
Answers vary.
For the optional Readiness activity in Lesson 10-2, you will need a
few rectangular prisms.
and one in which the smaller number is written first. Use the number line at the bottom of this page
to help you figure out the differences.
䉬
䉬 Write 2 subtraction number sentences in the table, one in which the larger number is written first
For Lesson 10-1, you will need four objects of various shapes,
volumes, and weights that can fit into an empty 2- or 3-pound
coffee can. A good mix of objects might be a baseball, a tennis ball
(or a rubber ball about the size of a tennis ball), a base-10 big
cube, and an unopened 16-ounce can of food. Place a label with
the name of each object on each item so that children can use
consistent names when they list the items.
314
Math Masters, p. 314
Lesson 9 13
789