Download NS6-63 Opposite Values

NS6-63 Opposite Values
Pages 78–79
Goals
STANDARDS
6.NS.C.5
Students will understand that positive and negative numbers are used
together to describe quantities that have opposite values.
Vocabulary
PRIOR KNOWLEDGE REQUIRED
cancel
electric charge
electron
integer
negative
positive
proton
Can compare and order integers
Knows that positive and negative numbers are used together
to describe quantities that have opposite directions
Can locate integers on a number line
MATERIALS
BLM Game Board (p. M-24)
lots of counters
a small token for each student
a die for each student
lots of two-color counters
Using integers to describe gains and losses.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ACTIVITY
Give each student a copy of a game board (from BLM Game Board),
40 counters, a die, and a small token to use as a game piece. The
counters can be used to represent dollar bills, which the players will
gain or lose as they play the game. Players begin the game by owning
20 counters, and the goal is to exceed that amount by the end of the
game. On each turn, players roll the die and move forward on the
game board the given number of spaces. If they land on +3, for example,
they would add three counters to their stack. If they land on −3, they
would remove three counters from their stack. The game ends when
the player reaches or passes the “Finish” mark. Players win if they reach
the finish with more than 20 counters (or dollars), lose if they finish with
less than 20, and tie if they finish with exactly 20. Play until everyone
has finished at least one game.
When students finish playing, point out that they used integers to show
gains and losses. ASK: What else have we used integers to describe?
(sample answers: locations above or below sea level; temperatures above
and below zero; time zone ahead of or behind London, England; golf score
above/below par; +/− ratings in some sports)
(MP.4)
The Number System 6-63
Why integers are convenient to describe gains and losses. Tell students
that integers are convenient for describing things that are opposite in some
way. Locations above and below sea level are opposite directions from
M-1
sea level. Gains and losses are opposite actions or opposite values. Point
out that if they gain $3 on one move and lose $3 on the next move, they
end up with the same amount they started with. So these actions cancel
each other out.
(MP.2)
Exercises: Write the integer that describes the action.
a) a gain of $4
b) a loss of $6
c) a loss of $5
d) a gain of $2
Answers: a) +4, b) −6, c) −5, d) +2
Comparing gains and losses. Write on the board:
Day 1
Gain of $3
Day 2
Loss of $4
Tell students that on Day 1, you gained $3 and on Day 2, you lost $4.
ASK: Which day was the better day? (Students can signal their answer
by holding up one or two fingers.) Repeat for various situations.
Day 1
Day 2
a)
Gain of $4
Loss of $3
b)
Loss of $2
Loss of $1
c)
Gain of $3
Gain of $4
d)
Loss of $5
Gain of $4
Answers: a) Day 1, b) Day 2, c) Day 2, d) Day 2
Connecting better results to greater integers. Remind students that
integers can be shown on a number line. Give volunteers an index card
labeled with a specific gain or loss: gain of $5, loss of $3, gain of $1, loss
of $2. Have the volunteers tape their card to the board at the appropriate
place on the number line.
−4
−3
−2
−1012345
Then tell students that the cards are in order from least integer to greatest
integer, but you want the cards to be in order from worst result to best
result. Ask a volunteer to move any cards they need to. (They don’t have
to move any cards at all.) Point out that a better result, in this particular
context, means a greater integer. If one result is better than another, the
better result has a greater integer than the other result.
(MP.2)
Exercises: Say which result is better. Then write an integer inequality.
a) a gain of $3, a loss of $5 c) a loss of $3, a loss of $5
b) a gain of $2, a gain of $5
d) a loss of $6, a loss of $2
Answers: a) a gain of $3, +3 > −5; b) a gain of $5, +2 < +5;
c) a loss of $3, −3 > −5; d) a loss of $2, −6 < −2
M-2
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
−5
Using integers to describe positive and negative electric charges. Ask
students what they use electricity for. (sample answers: turning on lights,
watching television, and so on) Tell students that protons and electrons
are what create electricity. Protons have a positive electric charge and
electrons have a negative electric charge. Tell students that a proton has
a charge of +1 and an electron has a charge of −1. Write this on the board:
proton = +1electron = −1
Exercises: What is the electric charge?
a) 3 protons
b) 2 electrons
c) 4 protons
d) 5 electrons
Bonus: 1,700 electrons
Answers: a) +3, b) −2, c) +4, d) −5, Bonus: −1,700
proton electron
+
-
Representing protons and electrons with pictures. Show students how
you will represent a proton and an electron. Tell students that protons and
electrons actually look nothing like what you are drawing—they are just
symbols to represent what they mean, not what they look like.
Exercises: What is the electric charge?
a)
+ +
b)
- - -
c)
+ + +
+ + +
d)
- - -
Answers: a) +2, b) −3, c) +6, d) −4
Electrons and protons cancel each other out. Tell students that protons and
electrons cancel each other out when they are near each other. Draw two
protons on the board. SAY: This object has a charge of +2. Then draw
an electron with the two protons. SAY: The electron cancels out one of the
protons. It’s as though neither of them is there. Show their removal as below.
+ + COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: Now what is the electric charge? (+1) Point out that because there
are more protons than electrons, the electric charge is positive.
Exercises: What is the electric charge?
a)
+ - -
b)
+ - - - -
c)
+ - + + - +
- -
Answers: a) −1, b) −4, c) 0
Point out that when there are more electrons than protons, the charge is
negative. Then SAY: Protons and electrons have opposite electric charges
that cancel each other out. That’s why it is convenient to use integers to
represent them.
The Number System 6-63
M-3
Extension
(MP.1, MP.3)
Take 10 two-color counters and toss them. The counters that land on yellow
represent protons and the counters that land on red represent electrons.
Determine the resulting electric charge by pairing up protons with electrons
and seeing what’s left. Record the results of various tosses and investigate
the following question:
Can the resulting electric charge ever be an odd number? Explain.
(You can tell students that an integer is odd if the number part without the
plus or minus sign is odd.)
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: No, because you always pair up the counters. So what’s left is 10
minus an even number, which always leaves an even number of counters.
M-4
Teacher’s Guide for AP Book 6.2
NS6-64 Debits, Credits, and Debt
Pages 80–81
STANDARDS
6.NS.C.7d
Goals
Vocabulary
Students will understand the relationship between debits, credits,
and debt. Students will understand that a greater debt is worse than
a lesser debt.
balance
bank account
credit
debit
debt
integer
negative
positive
PRIOR KNOWLEDGE REQUIRED
Can compare and order integers
Understands that integers can be used for actions or values that cancel
each other out
Understands that, in the context of money, a better result corresponds
to a greater integer
MATERIALS
an index card
Introduce debits and credits. Tell students that when you put money into
your bank account, the bank calls it a credit, and when you take money
out of your account, the bank calls it a debit. All bank accounts start at $0.
Draw on the board:
Debit
Credit
Balance
$3
$3
$2
SAY: After I put $3 into my account, my balance is $3. Then I took out $2.
ASK: Now what is my bank account balance? ($1) Fill in the balance. Add
three more rows as shown:
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Debit
Credit
Balance
$3
$3
$2
$1
$8
$5
$2
Have volunteers tell you the balance after each action. ($9, $4, $2)
Introduce negative bank accounts. Tell students that a bank account can
be negative. It means that you owe money to the bank. It is not a good
position to be in, but it can happen. For example, if you put $2 into your
account, you can take out $3, but you would then owe $1 to the bank.
The Number System 6-64
M-5
Using a number line to determine the balance. Add three rows to
the table:
$2
$4
$5
$3
−4 −3 −2 −10
+1 +2 +3 +4
Draw a number line from −4 to +4, and place an index card at the +2
location to show the current balance, as in the margin. Tell students that you
had $2 in your account, then you took out $4. You want to move the index
card four places to show how much you have now. ASK: Should I move it
right or left? (left) PROMPT: After I take money out, does my bank account
have more money in it or less? (less) Demonstrate moving the card four
places left. Then write the balance (−$2) in the balance column. Have
volunteers move the card and write the balance for the next two actions.
(+$3 and $0)
Have students draw a number line on grid paper and use a small token to
represent the balance in the exercise below.
(MP.4)
Exercise: The account starts at $0. Write the balance after each action.
(Answers are in italics.)
Debit
Credit
Balance
$4
$4
$6
−$2
$1
−$1
$2
−$3
$3
−$6
$8
+$2
Debit
Credit
$300
$600
Balance
$300
$200
$400
−$300
−$100
−$500
$900
$200
+$400
+$200
Comparing bank account balances. Write on the board:
Ari: −$3
M-6
Bob: −$4
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Bonus: Draw a number line from −500 to +500 to find the balance.
(MP.3)
SAY: Ari owes $3 and Bob owes $4. ASK: Who is in a better position? (Ari)
Why? (because he owes less money) Repeat for Ari’s bank account having
+$4 and Bob’s having −$5. (Ari is in a better position because it is better
to have money than to owe money) SAY: It’s better to have more money in
your account and it’s worse to owe money to your bank.
(MP.4)
Exercises: Whose bank account is better?
a)Ari: −$2 b)Ari: +$2 c)Ari: −$3 d)Ari: +$4 e)Ari: +$2
Bob: +$3Bob: $0Bob: −$1 Bob: +$7 Bob: −$3
Answers: a) Bob’s, b) Ari’s, c) Bob’s, d) Bob’s, e) Ari’s
ASK: How can you tell from the integers whose bank account is better?
(the greater integer corresponds to the better account balance)
Introduce the word “debt.” Tell students that the word debt is used to
describe how much money you owe. Write on the board:
Ari owes $30 and Bob owes $40.
ASK: Who owes more money? (Bob) SAY: Bob owes more money so his
debt is greater. ASK: Who is in a better position? (Ari) Point out that Ari’s
bank account is in a better position because Bob’s debt is greater than
Ari’s. Have volunteers write the integers to show each person’s bank
account balance. (Ari’s is −$30 and Bob’s is −$40)
Exercises: Whose debt is greater? Whose bank account balance is greater?
Ari
Bob
Whose debt
is greater?
Whose bank account
balance is greater?
a)
−$2
−$3
Bob’s
Ari’s
b)
−$2
−$1
Ari’s
Bob’s
c)
−$3
−$4
Bob’s
Ari’s
d)
−$5
−$6
Bob’s
Ari’s
−$3.74
−$3.50
Ari’s
Bob’s
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Bonus
Have students draw a number line from −4 to +4 and place the bank
account balances on the number line.
A. a debt of $3
B. a balance of −$4
C. owing $2
Then have students put the balances in order from best to worst. ASK: How
is this order shown on the number line? (read the amounts from right to left)
Extension
(MP.8)
The balance starts at 0. Determine the balance after a credit of $85 and
a debit of $83, then determine the balance after a debit of $85 and a credit
of $83. How do the balances compare? (they are opposite numbers)
Use this observation to determine the balance after a credit of $283.50
and a debit of $285.47.
Answer: −$1.97
The Number System 6-64
M-7
NS6-65 The Meaning of Zero
Pages 82–83
Goals
STANDARDS
6.NS.C.5
Students will use positive and negative numbers to represent quantities
in real-world contexts and will explain the meaning of 0 in each situation.
Vocabulary
bank account balance
Celsius
electron
elevation
Fahrenheit
integer
negative
positive
proton
sea level
time zone
(MP.6)
PRIOR KNOWLEDGE REQUIRED
Is familiar with the Fahrenheit temperature scale
Can use integers to represent temperatures, elevations, bank account balances, and electric charges
The units need to be emphasized. Remind students that using integers
can be a mathematical way of saying two things are opposite directions from
a chosen point. For example, locations above sea level can be chosen as
positive and locations below sea level can be chosen as negative. Point
out that students have to decide on a unit of measurement before they can
know what the integers mean. For example, does +3 mean 3 meters above
sea level, 3 feet above sea level, or 3 miles above sea level? Allow a volunteer
to decide on a unit (say, feet). Tell students that elevation tells how high up
something is. Sea level is the “0” of elevation.
Exercises: What does each integer mean in the context of elevation?
a) +4 b) −2
c) 0
d) −1,000
Answers: a) 4 feet above sea level, b) 2 feet below sea level, c) at sea level,
d) 1,000 feet below sea level
(MP.6)
Exercises: What does the integer mean in the context? Remember to
write the units.
a) +5 (temperature)
b) −3 (time zone)
c) +4 (elevation)
d) 0 (temperature) e) 0 (time zone)
f) 0 (elevation)
Answers: a) 5 degrees warmer than 0°F, b) 3 hours behind London, England,
c) 4 feet above sea level, d) 0°F, e) the time in London, England, f ) sea level
When 0 is arbitrary. Point out that when integers are used to mean opposite
directions, the point chosen as 0 is arbitrary. Sea level didn’t have to be the
chosen level, nor did London, England, have to be chosen as the “0” time
M-8
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
ASK: What does −3 mean for temperature? Again, allow students to decide
on the unit. (say, degrees Fahrenheit) Point out that, whichever scale is
chosen, −3 means three degrees colder than zero, and +3 means three
degrees warmer than 0. Remind students that integers are also used for
time zones, with the time in London, England, being 0. ASK: Does anyone
know what units are used—minutes, hours, or days? (hours)
zone. ASK: Do you think 0°F has to be the 0 for temperature, or could a
different temperature be chosen as 0? PROMPT: Does anyone know of any
temperature scale that uses a different temperature as 0? (Celsius)
Celsius and Fahrenheit. Tell students that the two most common temperature scales are Fahrenheit and Celsius and the scales use different points
as 0. Draw on the board the two temperature scales and point out that both
0 and 100 were chosen based on water in the Celsius scale. SAY: 0 is the
temperature at which ice becomes water, and 100 is the temperature at which
water becomes steam. Draw on the board:
°C
Water boils at
this temperature.
°F
+100°
+100°
Ice melts at this
temperature.
0°
+32°
0°
−40°
−40°
ASK: Which is warmer, 0°C or 0°F? (0°C) How can you tell? (it is higher up on
the scale) 100°C or 100°F? (100°C) −40°C or −40°F? (they are the same) How
can you tell? (they are at the same height on the scale) At what temperature
in Fahrenheit does ice melt? (32°F)
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When 0 means two quantities are the same. SAY: When the gains and
losses are the same, they cancel each other out and you get $0. When
amounts that cancel are equal in other contexts, you get 0 too. For example,
when the number of protons equals the number of electrons, there is no
electric charge. ASK: If you put the same amount of money in your bank
account as you took out, what is your balance? (0) If your team scored
the same number of points as the other team while you were playing,
what is your +/− rating? (0)
Exercises: What integer represents the amount?
a)
b)
c)
d)
e)
f)
the same number of electrons as protons
4 protons and 4 electrons
4 protons and 5 electrons
2 more protons than electrons
the same amount put into the account as taken out
3 more points scored for the team than against the team
Answers: a) 0, b) 0, c) −1, d) +2, e) 0, f) +3
The Number System 6-65
M-9
Extensions
(MP.6)
1. Use the temperature scales on AP Book 6.2 p. 82 to estimate.
a) 0°F ≈ °C (MP.1, MP.3)
b) 100°F = °C
Bonus: Which temperature is warmer, −50°C or −50°F? How do
you know?
Answers: a) about −18 or −20, b) about +40, Bonus: −50°F will be
closer to −40°F than −50°C will be to −40°C. That’s because 0°F is
closer to −40°F than 0°C is to −40°C. So, −50°F is warmer than −50°C.
(MP.1)
2. Pretend your time zone is 0. What would the time zone be in
London, England?
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Answer: Answers will vary, but it will be the opposite integer to
whatever your integer time zone is.
M-10
Teacher’s Guide for AP Book 6.2
NS6-66 Opposite Integers Again
Page 84
Goals
STANDARDS
6.NS.C.6a
Students will understand that the opposite of the opposite of a number is
the number itself. Students will understand that amounts that cancel are
represented by opposite integers.
Vocabulary
cancel
credit
debit
electron
integer
negative
opposite number
positive
proton
PRIOR KNOWLEDGE REQUIRED
Knows that opposite integers are the same distance from 0
Knows that integers can be used to represent quantities that cancel
each other out
Knows that protons and electrons cancel each other out to produce
no electric charge
Knows that debits and credits are actions that cancel each other out
MATERIALS
an index card
Review opposite integers as integers that are the same distance from
0, in opposite directions. Draw on the board:
−6 −5 −4 −3 −2 −10123456
ASK: How far from 0 is 3? Place an index card at 0. ASK: How many
places would it have to move to get to 3? (three places) Demonstrate
how it moves three places to the right. Repeat for −3. (also three places,
but this time to the left)
Exercises: How far from 0 is the integer?
a) −4b)
+4
c) −1
d) +1
e) −6f)
+6
Answers: a) 4, b) 4, c) 1, d) 1, e) 6, f) 6
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Exercises: What other integer is the same distance from 0?
a) −7
b) +8
c) −19
d) +41
e) −6,000
Answers: a) +7, b) −8, c) +19, d) −41, e) +6,000
Remind students that the integer that is the same distance from 0, but in
the opposite direction, is called the opposite integer. It has the opposite
sign, + or −, but the same number part.
The opposite of the opposite is the original number. ASK: What is
the opposite of +11? (−11) What is the opposite of −11? (+11)
Write on the board:
+11 The Number System 6-66
−11 +11
M-11
ASK: What is the opposite of the opposite of −7? (−7) Repeat for 1,000,000
(1,000,000) and −800,000 (−800,000). Point out that the opposite of the
opposite of any number is the original number.
(MP.8)
Exercises: Write the opposite of the opposite integer.
a) +5b)
−12c)
−304
d)47
e)+618
Answers: a) +5, b) −12, c) −304, d) 47, e) +618
Remind students that fractions and decimals can have opposites too. ASK:
What is the opposite of 3.5? (−3.5) What is the opposite of −3.5? (+3.5)
What is the opposite of the opposite of −3/4? (−3/4)
(MP.8)
Exercises: Write the opposite of the opposite number.
1
7
a) −4.7
b) −3 c) +3.02d)
5
4
Answers: a) −4.7, b) −3 1/4, c) +3.02, d) 7/5
Amounts that cancel are represented by opposite integers. Remind
students that 0 can mean that two opposite quantities are equal. For example,
if the amount you put into your bank account equals the amount you took
out, then your bank account has exactly 0 dollars in it. Remind students
that the bank calls the amount you put into your account a credit. ASK:
What does the bank call the amount you take out? (a debit) What cancels
out a credit of $3? (a debit of $3) In electric charge, what cancels out
5 electrons? (5 protons)
Tell students to leave space between the parts in these exercises.
Exercises: Write what cancels out the amounts.
a) 4 protons
b) debit of $6
Bonus: e) 275 electrons
c) 7 electrons
d) credit of $5
f) 804 protons
Answers: a) 4 electrons, b) credit of $6, c) 7 protons, d) debit of $5,
Bonus: e) 275 protons, f) 804 electrons
4 protons
4 electrons
Ask volunteers to write the integers that represent each amount.
+4
(MP.7, MP.8)
−4
Exercises: The next exercise uses the same examples as the previous
exercise. Write the integer for the amount and for the amount that cancels.
What do you notice?
a) 4 protons
b) debit of $6
Bonus: e) 275 electrons
c) 7 electrons
d) credit of $5
f) 804 protons
Answers: a) +4, −4; b) −6, + 6; c) −7, +7; d) +5, −5;
Bonus: e) −275, +275; f) +804, −804; opposite integers represent
amounts that cancel each other out
M-12
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Write the answer to part a) on the board:
Extensions
1. a) Which number is equal to its own opposite?
b) How many electrons cancel out 0 protons?
c) What debit cancels out a credit of $0?
Answers: a) 0, b) 0 electrons, c) a debit of $0
(MP.1)
2. a) Name a context in which opposite integers do not represent
amounts that cancel.
b) Name a different context, not done in class (i.e., not electricity or
debits/credits), in which opposite integers do represent amounts
that cancel.
Sample Answers: a) temperature, time zone, elevation; b) gains and
losses, +/− ratings
(MP.8)
3. What is the opposite of the opposite of the opposite of −5? What is the
opposite of the opposite of the opposite of the opposite of −5?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: +5, −5
The Number System 6-66
M-13
NS6-67 Distance Apart
Pages 85–86
Goals
STANDARDS
6.NS.C.8
Students will find the distance apart between two integers.
PRIOR KNOWLEDGE REQUIRED
Vocabulary
Can recognize opposite signs of numbers as indicating locations
on opposite sides of 0 on the number line
integer
negative
opposite number
positive
sign (+ or −)
MATERIALS
masking tape
Finding the distance between integers that have opposite signs.
Mark a starting point in the middle of the classroom using masking tape,
for example, and mark equal-length steps in both directions.
Ask two volunteers to start at the , standing back-to-back, and walk in
opposite directions. Ask the first volunteer to take two steps and the second
volunteer to take three steps. The volunteers should now be facing away
from each other. Ask one of them to predict how many steps apart they are
(five), and then to turn around and verify that. Draw on the board:
−5
−4
−3
−2
−1012345
a) −2 and +4b)
−3 and +1c)
−5 and +2
Students should tell you they are: a) six steps apart, b) four steps apart,
c) seven steps apart. ASK: If you know how far from 0 each integer is, and
you know they are in opposite directions from 0, how can you tell how far
apart they are? (add the distances from 0)
(MP.8)
Exercises: How far apart are the integers?
a) −2 and +1b)
−1 and +4c)
−2 and +2
d) −20 and +30e)
−300 and +4f)
−8 and +70
Answers: a) 3, b) 5, c) 4, d) 50, e) 304, f) 78
M-14
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
SAY: The 0 is like the starting point. A negative sign tells you to walk left,
a positive sign to walk right, and the number tells you how many steps to
take. Now continue playing the same game with pairs of volunteers, but this
time only tell them their integer. (Make sure the whole class can hear you
telling both volunteers their integers.) Use integers with opposite signs:
(MP.4, MP.6)
Word problems practice. Point out that the distance between two integers
can be used to answer many questions in real-life. For example, how much
warmer is one temperature than another, or how many hours apart are two
time zones, or what the distance is between two elevation levels above or
below sea level.
a)The temperature on Monday was +4°F and the temperature on Tuesday
was −3°F. How much colder was it on Tuesday than on Monday?
b)The average temperature in January is −7°C and in July is 22°C. How
much warmer is July’s average than January’s?
c)A bird is 300 feet above sea level and a fish, directly below, is 400 feet
below sea level.
i) Write integers to represent their elevations.
ii) How far apart are the integers?
iii) How far apart are the bird and the fish?
d)Judy’s time zone is +2 and Tim’s time zone is −5. How many hours
apart are they?
Answers: a) 7 degrees colder, b) 29 degrees warmer,
c) i) bird: +300, fish: −400, ii) 700, iii) 700 feet, d) 7 hours
Finding the distance between integers that have the same sign. Ask
for two volunteers to start at the but, this time, they walk in the same
direction. Ask one volunteer to take five steps and ask the second volunteer
to take three steps in the same direction as the first volunteer. Ask the first
volunteer to guess how far apart the two volunteers are, then turn around
and check their guess. Now repeat the game using integers, asking
students with a positive integer to move right and students with a negative
integer to move left from the starting point. Use only pairs of integers that
have the same sign:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
a) +2 and + 5
b) −5 and −2c)
+4 and +3d)
−3 and −4
Students should tell you they are a) three steps apart, b) three steps apart,
c) one step apart, d) one step apart. ASK: If you know how far from 0 each
integer is, and you know they are in the same direction from 0, how can you
tell how far apart they are? (subtract the shorter distance from the longer
distance) Point out that they should subtract the shorter distance from the
longer distance. Write on the board:
−8 and −11
ASK: What is the longer distance? (11) What is the shorter distance? (8)
So they are 11 − 8 = 3 units apart.
(MP.8)
Exercises: How far apart are the integers?
a) −8 and −1b)
−5 and −6c)
+3 and +7
d) −2 and −8e)
+300 and +4f)
−8 and −70
Answers: a) 7, b) 1, c) 4, d) 6, e) 296, f) 62
The Number System 6-67
M-15
(MP.4, MP.6)
Word problems practice.
a)The temperature at the North Pole on Monday was −20°F and on
Tuesday was −30°F. Which day was colder? How much colder?
b)A fish is 200 meters below sea level and another fish is 500 meters
below sea level.
i) Use integers to represent their elevations.
ii) How far apart are the fish?
iii) How far apart are the integers?
c)Sam’s time zone is +2 and Anne’s is +5.
How many hours apart are they?
d)Cat’s time zone is −8 and Molly’s time zone is −3.
How many hours apart are they?
Answers: a) Tuesday, 10 degrees colder, b) i) −200, −500, ii) 300 feet,
iii) 300, c) 3 hours, d) 5 hours
AP Book Bonus. Provide the bonus question on AP Book 6.2 p. 86 as
a challenge. Students will need to decide whether to add or subtract the
distances from 0. Many students will struggle with this question, but will
be able to get it after doing Lesson NS6-67. Allow students to struggle with
it now, then come back to it after they complete the next lesson. You can
then use it to make the point that when they work hard, they will often find
something that they used to find difficult to be easy. The extensions can
also be attempted again after the next lesson.
Extensions
(MP.1)
1.The distance between a positive integer and a negative integer is 5.
What might the integers be?
Bonus: List all possible solutions.
Answers: +4 and −1, +3 and −2, +2 and −3, +1 and −4
2.The distance between two negative numbers is 5/7. Give one pair of
numbers that work.
Sample answer: −1/7 and −6/7
3. The distance between 62 and a negative integer is 100. What is the
negative integer?
Answer: −38
4.The temperature in the South Pole in January can reach as high as 9°F.
In July, it can get 100° lower. What is the temperature in July?
Answer: −91°F
5.Relate the ideas in this lesson to 0.
Answer: Explain to students that the distance between 0 and 3 is either
3 + 0 or 3 − 0, so both adding and subtracting work.
M-16
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
(MP.1)
NS6-68 Absolute Value
Pages 87–88
STANDARDS
6.NS.C.6a, 6.NS.C.8
Goals
Students will understand the absolute value of an integer as its distance
from 0. Students will add or subtract the absolute values to find the
distance between two integers, as appropriate.
Vocabulary
absolute value
integer
positive
negative
PRIOR KNOWLEDGE REQUIRED
Can find the distance between two integers on the same side of 0
Can find the distance between two integers on opposite sides of 0
Can find an integer’s distance from 0
Can use an integer’s sign (+ or −) to decide on which side of 0 it is
Knows that positive integers can be written with or without a sign
Introduce the word absolute value. Tell students that the absolute value of
a number is its distance from 0. Point out that the absolute value is easy to
find—it’s just the number part, no matter what the sign is.
Exercises: What is the absolute value of the number?
a) −3
b) +5 c) −4
d) 7
e) 6
f) −6
g) 0
5
Bonus: h) −31.4
i) 7.09
j) −
8
Answers: a) 3, b) 5, c) 4, d) 7, e) 6, f) 6, g) 0, Bonus: h) 31.4, i) 7.09, j ) 5/8
Using a picture to decide whether to add or subtract the absolute
values. Draw on the board:
−80
+3
−8
−3 0
ASK: How far apart are the integers −8 and +3? (11) Did you add or
subtract the absolute values? (add) Write on the board:
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
8 + 3 = 11
Repeat for −8 and −3. (subtract the absolute values to get 5 units apart)
8−3=5
(MP.7)
Exercises: Would you add or subtract the absolute values to find the
distance apart?
a)
−200
+11
?
The Number System 6-68
b)
−14 −8 0
?
M-17
c)
d) 0
+5
?
−290
+9
+60
?
Bonus
e) f)
−1.8 −0.70
?
−8.6
0
15.04
?
Answers: a) add, b) subtract, c) subtract, d) add, Bonus: e) subtract, f ) add
Knowing when to add or subtract without a picture. Tell students that you
want a way to find the distance between the integers without using the
picture. ASK: How can you tell from the picture whether to add or subtract
the absolute values? (if the integers are in the same direction from 0, subtract
the absolute values; if they are in opposite directions, add the absolute values)
ASK: Can you decide without a picture whether the integers are on the same
side of 0 or on opposite sides of 0? (yes) How? (by looking at the signs)
PROMPT: What part of the integer—the sign or the number—tells you which
side of 0 the integer is on? (the sign)
Remind students that a negative sign says move left from 0 and a positive
sign says move right from 0. ASK: What does no sign say? (move right from
0) Remind students that a positive integer can have a positive sign, but
it doesn’t have to—an integer with no sign is positive. Write on the board
several pairs of integers:
a) −3 and +4
d) 2 and −6
b) −2 and −8
e) 4 and 5
c) +3 and +1
f) −5 and −1
For each pair of integers, ASK: Are these integers on the same side of 0?
Students can signal thumbs up for yes and thumbs down for no.
Answers: a) no, b) yes, c) yes, d) no, e) yes, f) yes
(MP.7)
Opposite sides of 0 add the absolute values
Same side of 0 subtract the absolute values
Exercises: Would you add or subtract the absolute values to find the
distance apart?
a) −3 and +6
d) +7 and +84
b) −8 and −20
e) −32 and −5
c) −18 and +26
f) +17 and −6
Answers: a) add, b) subtract, c) add, d) subtract, e) subtract, f) add
Remind students that when they subtract the absolute values, they subtract
the smaller absolute value from the greater absolute value.
M-18
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Write on the board:
(MP.7)
Exercises: Write an addition or subtraction equation to find the distance
between the numbers.
a) −30 and +14b)
−18 and −6
c) 7 and −9
d) −3 and −15e)
+4 and +9f)
−8 and 12
Bonus: g) −5.8 and −3.2 h) −8.6 and 5.4 i) 6.02 and −3.7
Answers: a) 30 + 14 = 44, b) 18 − 6 = 12, c) 7 + 9 = 16, d) 15 − 3 = 12,
e) 9 − 4 = 5, f) 8 + 12 = 20, Bonus: g) 5.8 − 3.2 = 2.6, h) 8.6 + 5.4 = 14,
i) 6.02 + 3.7 = 9.72
(MP.4)
Day
Temperature
(°F)
Monday
+11
Tuesday
+8
Wednesday
+2
Thursday
Friday
0
−4
Using in-between points to find the total distance. Remind students
that the distance between two integers can be used to answer real-life
questions. Draw on the board the table in the margin. ASK: How much
warmer is Monday than Tuesday? (three degrees) How much warmer is
Tuesday than Wednesday? (six degrees) How much warmer is Monday
than Wednesday? (nine degrees) How did you get that? (11 − 2) Is there
another way? PROMPT: How can you use three degrees and six degrees?
(add them to get nine degrees)
Now repeat for comparing Wednesday to Friday by using Thursday. Point
out that this time, it’s the absolute values that are being added. ASK: Why is
that? (because Thursday’s temperature is 0) Emphasize that +2 and −4 are
on opposite sides of 0, so you can add their distances from 0 to get their
distance from each other.
Exercises: How much warmer is:
a) Monday than Friday?
b) Tuesday than Friday?
Answers: a) 11 + 4 = 15, b) 8 + 4 = 12
Point out that 0 being the in-between value is just a special case. It’s always
true that adding the distances from an in-between point is the total distance
between the numbers.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
NOTE: Students who had difficulty with the AP Book Bonus problem from
the previous lesson (on AP Book 6.2 p. 86) can attempt it now.
Extensions
1. Which integer, +7 or −10, is farther from −2?
Answer: +7
(MP.1)
2. Write in the chart whether you would add or subtract the absolute values
to find the distance between the integers.
The Number System 6-68
+
+
–
–
Answer:
+
+ subtract
–
add
–
add
subtract
M-19
NS6-69 Concepts in Absolute Value
Pages 89–91
STANDARDS
6.NS.C.7, 6.NS.C.8
Goals
Students will use absolute value notation. Students will interpret absolute
value as magnitude in real-world contexts. Students will distinguish
comparisons of absolute value from comparisons of actual value.
Vocabulary
absolute value
debt
distance apart
integer
negative
negative sign
opposite integer
positive
positive sign
time zone
PRIOR KNOWLEDGE REQUIRED
Understands the absolute value of an integer is its distance from 0
Can convert measurements in meters to measurements in centimeters
Can use integers to represent amounts in a variety of contexts (elevation, temperature, time zone, bank accounts)
Understands that opposite integers are the same distance from 0
Can add, subtract, or compare absolute values of integers to answer real-world questions
Can compare actual values of integers to answer real-world questions
Introduce absolute value notation. Remind students that the absolute
value of an integer is its distance from 0. Tell students that because absolute
value is used a lot, mathematicians have created a notation for it so that
you don’t have to keep writing out the words. Write on the board:
|−3| = 3
Read the equation as: The absolute value of negative three is three. Point
out that the absolute value of an integer is the number part without the sign.
Exercises: Write the number.
a) |−9| = b)|+10| = d) |+3.4| = e)
−3
1
= 2
c) |−12| = f) |−8.7| = Absolute value in context. Choose a student and ask other students to
decide how far, in steps, they are from the chosen student. Point out that
the distance is always positive, no matter what direction you come from.
Tell students that a fish is five meters below sea level. ASK: What integer
can we use to describe the fish’s location? (−5) How far is the fish from
sea level? (5 m) Point out that the distance from sea level is positive, even
when the location is negative. The distance from sea level is the absolute
value of the integer. Write on the board:
|−5| = 5, so the fish is 5 m from sea level
SAY: Writing the answer using absolute value notation tells me not only
what answer you got, but how you got it. It’s a way to justify your answer.
M-20
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answers: a) 9, b) 10, c) 12, d) 3.4, e) 3 1/2, f) 8.7
(MP.4, MP.6)
Exercises: Justify your answers using absolute value notation.
a) How far above sea level is +7 km?
b) How far below sea level is −7 km?
c) Jon’s time zone is −5. How many hours behind London is that?
d) Jade’s time zone is +2. How many hours ahead of London is that?
e) Lina’s bank account is −$100. How much debt does she have?
Answers: a) |+7| = 7, so 7 km, b) |−7| = 7, so 7 km, c) |−5| = 5,
so 5 hours, d) |+2| = 2, so 2 hours, e) |−100| = 100, so $100
(MP.2)
Comparing and ordering absolute values. Write several pairs of integers
on the board, and ask students to point to the number with the greater
absolute value. Point out that students can pretend the sign is not there
and just compare the numbers without the signs.
a) −7 +3
b) −6 +8
c) −15 −20
d) −20 +15
Answers: a) −7, b) +8, c) −20, d) −20
Exercises: Write the absolute values in order from least to greatest.
a)|−7|, |+4|, |−3|
b) |−8|, |+4|, |15|
c) |−9|, |+10|, |−5|
Bonus
|−5|, |3|, |−2|, |7|, |+9|, |−8|
Answers: a) |−3| < |+4| < |−7|, b) |+4| < |−8| < |15|,
c) |−5| < |−9| < |+10|, Bonus: |−2| < |3| < |−5| < |7| < |−8| < |+9|
Some students may need to write the absolute values of the integers first,
before putting the absolute values in order.
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Comparing absolute values in context. SAY: A bird is flying 400 meters
above sea level and a fish is swimming 200 meters below sea level. Let’s
mathematize the situation. ASK: Using sea level as 0, what is the bird’s
location? (+400) And the fish’s location? (−200) ASK: Which is closer to
sea level: the bird or the fish? (the fish) SAY: −200 is closer to 0 than +400
is, so its absolute value—its distance from 0—is shorter. Write on the board:
|−200| is less than |+400|
(MP.3, MP.4)
Exercises: Which is closer to sea level: the bird or the fish?
a) The bird is at +300 ft; the fish is at −500 ft.
b) The bird is at +20 m; the fish is at −30 m.
c) The bird is at +53 m; the fish is at −28 m.
Include questions with decimals.
d) The bird is at +6.4 m; the fish is at −5.8 m.
e) The bird is at +31.25 m; the fish is at −31.37 m.
Answers: a) the bird, b) the bird, c) the fish, d) the fish, e) the bird
The Number System 6-69
M-21
Comparing absolute values in different units. Tell students that now they
will have to be careful to look at the units when they compare the distances
from 0. Remind students that 1 meter is 100 centimeters. ASK: How many
centimeters is 5 meters? (500)
(MP.2, MP.3)
Exercises: Which is closer to sea level: the bird or the fish? Change both
measurements to centimeters to check.
a) The bird is at +300 cm; the fish is at −7 m.
b) The bird is at +20 cm; the fish is at −300 m.
c) The bird is at +53 cm; the fish is at −2 m.
d) The bird is at +542 cm; the fish is at −3 m.
Answers: a) the bird, b) the bird, c) the bird, d) the fish
(MP.3)
Adding, subtracting, or comparing absolute values. ASK: Would you
add, subtract, or compare absolute values to answer these questions?
Explain your answer.
a) How much higher is +5 m than −3 m?
b) What is closer to sea level: +5 m or −3 m?
c) How much warmer is −2°C than −5°C?
Answers: a) add, because +5 and −3 have opposite signs, b) compare
because closeness to 0 is the absolute value, c) subtract, because −2 and
−5 have the same sign
Comparing absolute values and comparing actual values in context.
Tell students that sometimes they need to compare the absolute values of
integers and sometimes they need to compare the actual values of integers.
It just depends on what the context is and what they need to know. Remind
students that your time zone tells you how many hours you are ahead of or
behind London, England. Write on the board:
Billy’s time zone: +1
ASK: Is Pam ahead of or behind London? (behind) How do you know? (the
negative sign) Is Billy ahead of or behind London? (ahead of) How do you
know? (the positive sign) ASK: Whose time is ahead of the other’s? (Billy’s)
Who will need to change their clock more if they visit London? (Pam) Point
out that by deciding whose time is ahead, they compared the actual values of
the integers: positive 1 is greater than negative 5, so Billy’s time is ahead of
Pam’s. By deciding who has to change their clock more, they are comparing
how far each time zone is from 0, so they are comparing the absolute values.
(MP.3, MP.4)
Exercises: Answer these questions. Did you compare the absolute values
or the actual values of the integers?
a) Who will celebrate the New Year first, Pam or Billy?
b) Who is more likely to have jet lag if they visit London, Pam or Billy?
M-22
Teacher’s Guide for AP Book 6.2
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Pam’s time zone: −5 c) Ahmed lives in time zone +3, Rosa in −2, and Nomi in −5.
i) Who will celebrate the New Year first?
ii) Who will need to change their clock more hours if they decide
to meet in London?
iii) Who will celebrate the New Year last?
Answers: a) Billy, actual values, b) Pam, absolute values, c) i) Ahmed,
actual values, ii) Nomi, absolute values, iii) Nomi, absolute values
NOTE: Questions 7 and 8 on AP Book 6.2 p.90 refer to years AD and BC.
There is not technically a year 0 so, in this sense, the calculation of how
far apart two years are will be off by 1. For example, 1 AD and 1 BC are
actually only 1 year apart, not two, as adding the absolute values would
suggest. However, since the years are rounded to the nearest hundred,
this will not make a difference.
Extensions
(MP.2)
1. Which is closer to sea level: a bird at +13 feet or a fish at −152 inches?
Answer: the fish
2. Point out that absolute values are just positive numbers, so you can
add, subtract, multiply, and divide them.
a) |−3| + |4| b) |−9| − |4| c) |−2| × |−3| d) |+8| ÷ |−2|
Answers: a) 7, b) 5, c) 6, d) 4
3. Which absolute value is greater?
a) |−3.6| or |+4|b) −4
1
or |−4.2|
2
c) |+2.7| or |−2.91|
Answers: a) |+4|, b) |−4 1/2|, c) |−2.91|
4. What is the opposite integer of |−3|?
COPYRIGHT © 2013 JUMP MATH: NOT TO BE COPIED. CC EDITION
Answer: −3
5. Make up your own question in which the absolute values need to be
compared. Make up your own question in which the actual values need
to be compared.
6. Tell students that the strength of an electric charge in an object is actually
given by the absolute value, not the actual value, of the electric charge.
So an object with a charge of −3 has a stronger electric charge than an
object with a charge of +2, even though −3 is less than +2. What matters
is that |−3| > |+2|. Which charge is stronger?
a) −4 or −5
b) +3 or −7
c) −8 or + 10 d) +8 or +9
Answers: a) −5, b) −7, c) +10, d) +9
The Number System 6-69
M-23