Positive and Negative Numbers

Positive and Negative Numbers  Objective To guide children as they investigate positive and negative numbers. www.everydaymathonline.com ePresent...
Author: Gerald Parsons
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Positive and Negative Numbers



Objective To guide children as they investigate positive and negative numbers.

www.everydaymathonline.com

ePresentations

eToolkit

Algorithms Practice

EM Facts Workshop Game™

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

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.

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

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Unit 9

Multiplication and Division

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Getting Started 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

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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



977 meters below sea level -977 m



4,240 meters above sea level +4,240 m or 4,240 m

Unit 9 Multiplication and Division

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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: ● ●





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,

6-pound loss: -6 lb; 6-pound gain: +6 lb

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 L LE LEM EM E M SOLVING SO S 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.

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Student Page Date

Time

LESSON 



5 units width = 9 units area = 45 square length =

1.

2 Ongoing Learning & Practice

Math Boxes

9 13

2.

Use the partial-products algorithm to solve. ×

units 2 factors of 45 and are

5

9

×

 Making a Line Plot

408 8

1,800 3,200 150 + 64 + 6 3,264 1,956

. 37 154 156

3.

652 3

4.

There are 54 candles. A box holds 12 candles. How many full boxes of candles are there?

(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?

When most children have completed their line plots, have them identify the median, mode, maximum, minimum, and range of their data.

6

Range:

A circle

INDEPENDENT ACTIVITY

118

79 80

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 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

190 180

thermometer could show? °F

104

b.

160

(Math Masters, p. 312)

Water boils

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

150 140

20 degrees warmer than -10°C?

10°C

4. How much colder is

100 90

5. Would 30°C be a good

temperature for swimming outside?

Yes

Explain.

50

Yes

For in–line skating?

Explain.

No

20

Body Temperature Room Temperature

10

40 30

temperature for ice-skating?

30

60

It would be very warm outside. 30° C is about the same as 86° F. 6. Would -6°C be a good

40

80 70

No

50

110

18° colder

-9°C than 9°C?

60

130 120

INDEPENDENT ACTIVITY

90

170

°C

3. What temperature is

For sledding?

100

200

2. What is the warmest temperature this

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

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

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Unit 9 Multiplication and Division

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Teaching Master Name

(Math Masters, p. 314)

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.

0 20

Try This

10

4. Do your own. Start at

0 10 20

5. Describe the relationships you see between the three numbers in each problem.

50 40 30 20 10

. Where did you land? . Count

10

Answers vary.

20

30

30 20

10

0 10 20

10 0

10

7

3

10

3. Start at 40. Count back 50. Where did you land?

Number Stories

20

15–30 Min

1. Start at 10. Count back 13. Where did you land?

 Solving Subtraction

INDEPENDENT ACTIVITY

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)

ENRICHMENT

40

40

5–15 Min 30

 Counting on a Number Line

INDEPENDENT ACTIVITY

2. Start at 15. Count up 22. Where did you land?

READINESS

50

50



50

9 13

Time

Negative Numbers on a Number Line

40

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

9 13 Exploring Order in Subtraction LESSON

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 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

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