Lesson 10.6 Positive and Negative Numbers

Lesson 10.6 Positive and Negative Numbers
Objective
1
To introduce addition involving negative integers.
materials
Teaching the Lesson
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.
Key Concepts and Skills
• Compare and order integers.
[Number and Numeration Goal 6]
• Add signed numbers.
[Operations and Computation Goal 2]
ⵧ Student Reference Book, pp. 60
and 238
ⵧ Study Link 10 5
䉬
ⵧ Teaching Master (Math Masters,
p. 320)
ⵧ Game Master (Math Masters,
p. 468)
ⵧ Transparencies (Math Masters,
pp. 318 and 321; optional)
ⵧ 1 transparent mirror per partnership
ⵧ deck of number cards (Everything
Math Deck, if available)
• Identify a line of reflection.
[Geometry Goal 3]
See Advance Preparation
Key Vocabulary
opposite (of a number) • credit • debit
Ongoing Assessment: Informing Instruction See page 825.
2
Ongoing Learning & Practice
Students solve problems involving fractions, decimals, and percents.
Students practice and maintain skills through Math Boxes and Study Link activities.
materials
ⵧ Math Journal 2, pp. 283 and 284
ⵧ Study Link Master (Math Masters,
p. 322)
Ongoing Assessment: Recognizing Student Achievement Use journal page 284.
[Data and Chance Goal 4]
3
materials
Differentiation Options
READINESS
Students skip count on a calculator to
explore patterns in negative numbers.
READINESS
Students use a number line to add positive
and negative integers.
ⵧ calculator
ⵧ number line
ⵧ masking tape
See Advance Preparation
Additional Information
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.
822
Unit 10 Reflections and Symmetry
Technology
Assessment Management System
Math Boxes, Problem 1
See the iTLG.
Getting Started
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
Study Link 10 5 Follow-Up
䉬
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
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.
䉬
1 Teaching the Lesson
䉴 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
8
above 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
䉬
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
1
5
for in-between measures such as 2 2 inches and 6 6 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.
Note
Examples Rename as fractions: 0, 12, 15.3, 3.75, and 25%.
0
0 1
12
12 1
153
15.3 10
375
3.75 100
Every whole number
(0, 1, 2, and so on) can
be renamed as a fraction.
For example, 0 can be
0
written as 1. And 8 can
8
be written as 1.
25
25% 100
However, even fractions did not meet every need. For example,
3
1
problems such as 5 7 and 2 4 5 4 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
1
that are less than 0. The numbers 2, 2.75, and 100 are
negative numbers. The number 2 is read “negative 2.”
Note
Numbers like 2.75 and
100 may not look like
negative fractions, but
they can be renamed as
negative fractions.
Negative numbers serve several purposes:
♦ To express locations such as temperatures below zero on
a thermometer and depths below sea level
䉴 Using Credits and Debits
WHOLE-CLASS
ACTIVITY
to Practice Addition of
Positive and Negative Numbers
(Math Masters, p. 321)
11
♦ To show changes such as yards lost in a football game
2.75 4, and
♦ To extend the number line to the left of zero
100 1
100
Links to the Future
♦ 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.
Students explore subtraction of positive and negative integers in Lesson 11-6.
Addition and subtraction of signed numbers is a Grade 5 Goal.
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.
–7 –6 –5 –4 –3 –2 –1
0
1
2
3
4
5
6
7
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.
Student Reference Book, p. 60
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.
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.
Teaching Master
Name
LESSON
10 6
䉬
Date
Time
Ledger
Transaction
Start
End/Start of
Next Transaction
Change
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.
–10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 –1
0
1
2
3
4
5
6
7
8
9
10
Math Masters, p. 321
824
Unit 10 Reflections and Symmetry
Student Page
Transaction
Start
Change
End/Start of
Next
Transaction
New business,
start at $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
$0
$0
Games
Credits/Debits Game
Materials 䊐 1 complete deck of number cards
䊐 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.
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:
If both players have
negative dollar amounts
at the end of the round,
the player whose
amount is closer
to 0 wins.
♦ Draw a card. The card tells you the dollar amount
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.
Examples 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.
Continue until most students respond with ease.
Student Reference Book, p. 238
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.
A U D I T O R Y
䉬
K I N E S T H E T I C
䉬
T A C T I L E
䉬
V I S U A L
䉴 Playing the Credits/Debits Game
PARTNER
ACTIVITY
(Student Reference Book, p. 238; Math Masters, p. 468)
Students play the Credits/Debits Game to practice adding
positive and negative numbers. They record their work on
Math Masters, page 468.
Game Master
Name
Date
Time
Credits/Debits Record Sheets
3
5
4
3
22212019181716151413121110 9 8 7 6 5 4 3 2 1
4
6
Game 1
5
7
Change
Record Sheet
6
8
Start
7
9
+$10
8
10
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.
End, and
next start
0
1
Game 2
2
1
2
Do not try too hard to get explanations; these will evolve over time as students
have more experience with positive and negative numbers.
1
9
2
10
Ongoing Assessment: Informing Instruction
1 2
4 3
238
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
䉬
825
Student Page
Date
Time
LESSON
10 6
䉬
2 Ongoing Learning & Practice
Review: Fractions, Decimals, and Percents
1. Fill in the missing numbers in
Fraction
the table of equivalent fractions,
decimals, and percents.
Decimal
4
10
Percent
0.4
6
10
75
100
61 62
40%
60%
0.6
0.75
䉴 Solving Fraction, Decimal,
75%
and Percent Problems
2. Kendra set a goal of saving $50 in 8 weeks. During the first
2 weeks, she was able to save $10.
10
50
a. What fraction of the $50 did she save in the first 2 weeks?
20%
10
b. What percent of the $50 did she save?
c. At this rate, how long will it take her to reach her goal?
3. Shade 80% of the square.
(Math Journal 2, p. 283)
weeks
80
100
a. What fraction of the square did you shade?
Students solve problems involving equivalent fractions, decimals,
percents, and discounts.
0.8
20%
b. Write this fraction as a decimal.
c. What percent of the square is not shaded?
INDEPENDENT
ACTIVITY
4. 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?
䉴 Math Boxes 10 6
$9
$51
INDEPENDENT
ACTIVITY
䉬
(Math Journal 2, p. 284)
5. Star Video and Vic’s Video Mart sell videos at about the same regular prices. Both
stores are having sales. Star Video is selling its videos at 1 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 13%, which
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.
is more than 25%. So they’re taking more off their
regular prices.
283
Math Journal 2, p. 283
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 1
夹
Use Math Boxes, Problem 1 to assess students’ ability to express the
probability of an event as a fraction. Students are making adequate progress
1
3
if they design a spinner that is 4 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]
䉴 Study Link 10 6
䉬
Student Page
Date
INDEPENDENT
ACTIVITY
(Math Masters, p. 322)
Time
LESSON
10 6
䉬
夹
Math Boxes
1. a. 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.
red
blue
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,
1
9
, or , of the board should
4
36
be red.
Home Connection Students compare and order
positive and negative numbers and add positive
and negative integers.
82–86
2. Complete.
Rule:
14
3. Solve each open sentence.
in
out
8
16
4
16
a. 67.3 p 75.22
p
8
8
6
8
b. 6.86 a 2.94
a
c.
x 5.69 7.91
x
3
4
2
4
10
12
7
12
15
20
10
20
d. 4.6 n 0.32
n
7.92
3.92
2.22
4.28
READINESS
55 57
acute
RUG is an
(acute or obtuse) angle.
4. Angle
3 Differentiation Options
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.
䉴 Exploring Skip Counts
SMALL-GROUP
ACTIVITY
5–15 Min
on a Calculator
R
A. 50 pounds
U
B. 150 pounds
G
C. 500 pounds
Measure of ⬔RUG 20
°
D. 1,000 pounds
93 142
143
284
Math Journal 2, p. 284
826
Unit 10 Reflections and Symmetry
140
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:
Study Link Master
●
What does the calculator display show after zero? –10
●
How do you read this number? negative ten
●
Can you predict what number will come next? –20
Name
Date
STUDY LINK
Time
Positive and Negative Numbers
10 6
䉬
Write or to make a true number sentence.
1.
3
14
2.
7
7
19
3.
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
䉴 Using a Number Line to
5.
1
4
8
1
1
2
4
3.4
1.7
5
least
6.
greatest
14
7
43, 22, , 5, 3, 0
43
3
least
14
0
5
7
Sample answers:
7.
Name four positive numbers
less than 2.
1
4
1
2
3
4
8.
Name four negative numbers
greater than 3.
2
1
2
22
greatest
1
1
1
4
Use the number line to help you solve Problems 9–11.
5–15 Min
Add Positive and
Negative Numbers
1
2
5, 8, , , 1.7, 3.4
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
9. a.
49
13
b.
4 (9) 10. a.
53
8
b.
(5) 3 11. a.
15
2 13
b.
11
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
䉯 The first number tells students where to start.
䉯 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.
䉯 If the second number is negative, then walk backward.
Otherwise, walk forward.
䉯 The second number (ignoring its sign) tells how many steps
to walk.
䉯 The number where the student stops is the answer.
Example: –4 3
䉯 Start at –4.
䉯 Face the positive end of the number line.
䉯 Walk forward 3 steps.
–5
䉯 You are now at –1. So –4 3 –1.
–4
–3
–2
–1
0
1
2
4 3 1
Suggestions:
●
–6 –3 ? (Start at –6. Face in the positive direction.
Walk backward 3 steps. End up at –9.)
–10 –9
●
–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
䉬
827
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