Download Introduction to Integers Get the Point?

Introduction to Integers
Get the Point?
ACTIVITY
1.8
SUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/
Retell, Create Representations, Quickwrite, Self Revision/
Peer Revision
ACTIVITY 1.8 Investigative
Introduction to Integers
Activity Focus
My Notes
• Comparing and ordering
integers
• Absolute value
• Adding and subtracting integers
Ms. Martinez has a point system in her classroom. Students earn
points for participation, doing homework, using teamwork, and
so on. However, students lose points for talking or not completing
homework or class work.
Ms. Martinez tells the class that at the end of the year each
student in the group with the most points will receive a book or
DVD. She assigns a letter to each student so she can easily track
point totals. One student is A, the next is B, and so on.
Materials
• Two-color counters
• Class number line or tape to
make one
1. T his table shows each student’s total points at the end of the
week. Your teacher will assign you a letter.
A
−3
B
3
C
8
D
−1
E
0
F
−5
G
−6
H
10
I
7
J
−4
K
1
L
2
M
−3
N
−2
O
12
P
1
Q
−7
R
2
S
−1
T
6
U
−4
V
−1
W
9
X
3
Chunking the Activity
#1–2
#3
#4–7
a. Write the total points and the letter assigned to you on a
sticky note. Then post it on the class number line.
#8–11
#12–13
#14–15
#16–19
#20–22
Paragraph Summarize/
Paraphrase/Retell
b. Copy the letters from the class record on this number line.
1 Self Revision/Peer Revision,
Create Representations (a, b),
Quickwrite (c), Debriefing In
order to keep this activity simple,
students are identified by letters.
Assign each student in your class
a letter and give each a sticky
note. Draw a number line on the
board or wall ahead of time that
goes at least from -12 to 12.
Once students have posted their
numbers on the number line,
spend time discussing how to
order integers. Do not introduce
the term integer yet. A common
mistake is finding -2 on the
number line, and then putting -3
to the right of it.
Answers may vary depending on class size. Sample answer:
© 2010 College Board. All rights reserved.
S
UM V P R X
QG F J AN D E K L B
–6 –4 –2
0
2
T I C WH
4
6
8
O
10 12
c. In terms of points, what do the numbers to the right of zero
represent?
points earned by the students
d. What do the numbers to the lef t of zero represent?
points lost by the students
e. Describe how you knew where to place your number on the
number line.
Answers may vary. Sample answer for account F: I had −5.
Since it is negative, I started counting to the left of zero.
−5 is after −4 but before −6 so I put it right between them.
f. Student E was in class for only 2 days during the week. On
the first day, E was awarded points, and on the second day,
E lost points. Explain why E’s score is zero.
The number of points awarded and lost must have been the
same, resulting in a score of zero.
Unit 1 • Number Concepts
1/13/11 10:23:31 AM
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045-053_SB_MS1_1-8_SE.indd 45
45
Unit 1 • Number Concepts
45
2 Students should discuss
multiple real-life situations where
integers are used, such as bank
accounts, golf scores, elevations,
temperatures, football yardages,
weight changes, and so on.
Paragraph Marking the Text
3 Create Representations,
Think/Pair/Share This question
has been scaffolded for students.
The first two number lines have
been labeled for them; however,
they must label the second
two, and draw the final two on
their own.
ACTIVITY 1.8
continued
Introduction to Integers
Get the Point?
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Create Representations, Think/Pair/Share
My Notes
CONNECT TO HISTORY
The Common Era is the calendar
system now used throughout the
world. This system is like a number line because the year numbers
increase as time moves on. The
label CE can be used for these
years. For example, the first year
of the twenty-first century could
be written 2001 CE.
2. We have seen how negative numbers can be used to represent
points lost by the students. Name at least three other uses for
negative numbers in real life.
Answers may vary. Sample answers: overdrawn bank accounts,
golf scores, ocean depths, temperatures below zero, football
yardage losses, weight loss, stock market losses
Ms. Martinez sometimes assigns cooperative learning groups. She
assigns each group member a role based on his or her total points.
T he roles are reporter (lowest total), recorder (next to lowest total),
facilitator (next to highest total), and timekeeper (highest total).
3. Use the number lines.
a. Order the points for the members in each group from lowest
to highest.
Group 2: −6, −5, 0, 10
Group 1: −3, −1, 3, 8
A
−3
Differentiating Instruction
Use tape to create a number line
on the floor. Have students stand
at points on the number line to
represent each number earned by
the group.
–4 –2
B
3
0
C
8
2
4
D
−1
6
8
G
Giving students roles in
ccooperative learning
groups he
helps to maintain
individual accountability and
participation.
TEACHER TO
TEACHER
J
−4
K
1
L
2
R
2
S
−1
0
G
−6
2
4
H
10
6
8
10
M
−3
N
−2
O
12
P
1
Group 6: −4, −1, 3, 9
Group 5: −7, −1, 2, 6
Q
−7
–6 –4 –2
F
−5
Group 4: −3, −2, 1, 12
Group 3: −4, 1, 2, 7
I
7
E
0
© 2010 College Board. All rights reserved.
ACTIVITY 1.8 Continued
T
6
U
−4
V
−1
W
9
X
3
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045-053_SB_MS1_1-8_SE.indd 46
46 SpringBoard® Mathematics with Meaning™ Level 1
Introduction to Integers
ACTIVITY 1.8
Get the Point?
continued
3 Create Representations (b)
SUGGESTED LEARNING STRATEGIES: Create Representations,
Marking the Text, Summarize/Paraphrase/Retell
4 The purpose of Question 4 is
My Notes
for students to discover that -3
and 3 are the same distance from
zero. Be sure students do not
write -3 for Part b.
b. Use your work in Part a to determine who will have each
role. Record the letters for each student in the table.
Reporter
Recorder
Group 1
Group 2
A
D
B
C
G
F
E
H
Group 3
Group 4
J
K
L
I
M
N
P
O
Group 5
Q
S
R
T
Group 6
U
V
X
W
ACTIVITY 1.8 Continued
Facilitator Time Keeper
Paragraph Marking the Text
56 These questions give students
the opportunity to practice with
absolute value.
4. Look at Students A and B.
Paragraph Summarize/
Paraphrase/Retell
a. How many points does A need to earn to have a total of 0?
3 points
b. How many points does B have to lose to have a total of 0?
Differentiating Instruction
3 points
Some students may need to walk
a number line 3 steps to the right
and 3 steps to the left to see
that both are the same number
of steps, simply in different
directions.
c. What do you notice about the distances of their point totals
from zero?
Both are the same distance from zero.
© 2010 College Board. All rights reserved.
Numbers that are the same distance from zero and are on
dif ferent sides of zero on a number line, such as −3 and 3, are
called opposites. Absolute value is the distance from zero and
is represented with bars: |−3| = 3 and |3| = 3. Absolute value is
always positive because distance is always positive.
5. Now f ind C’s and D’s distances from zero.
C’s distance from zero is 8; D’s distance from zero is 1.
6. What is the absolute value of zero? 0
ACADEMIC VOCABULARY
The absolute value of a
number is the distance of
the number from zero on
a number line. Distance or
absolute value is always
positive. For example, the
absolute value of both –6
and 6 is 6.
7 Debriefing Students may
need reminders again about signs:
> versus <.
T
TEACHER
TO
TEACHER
FFor more practice, have
sstudents pair and play
War using a regular deck of cards.
Black cards are positive numbers
and red cards are negative
numbers. As an alternative to
using a standard deck of cards,
have students make integers cards
of their own.
Absolute value can be used to compare and order positive and
negative numbers. The negative number that is the greatest distance
from zero is the smallest. |−98| = 98 and the |−90| = 90. Therefore,
−98 is further left from zero than −90 is, so −98 is less than −90.
7. Use this method to compare each pair of negative numbers.
−15
> −21
−392 < −390
−2,840
> −2,841
Unit 1 • Number Concepts
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47
12/16/09 5:46:36 PM
MINI-LESSON: Absolute Value
Use a vertical number line to discuss temperature, as this is the context
typically used to introduce negative numbers in the elementary grades.
Give daily highs and lows that span from positive to negative and discuss
their distance from the zero (above zero and below zero temperatures).
Use a string to demonstrate these distances and show that 5 degrees
above zero (+5) and 5 degrees below zero (-5) are the same distance
from 0. Stress that distance cannot be negative, as the length of the
string they are using is a positive length.
If students need additional support, have them walk a number line on the
floor. They should start at zero and walk to +6. Next, students start back
at zero and walk to -6. Did they walk the same number of steps? Yes,
they are simply walking in different directions.
Suggested Assignment
CHECK YOUR UNDERSTANDING
p. 53, #1–3
UNIT 1 PRACTICE
p. 60, #53–57
Unit 1 • Number Concepts
47
Paragraph Summarize/
Paraphrase/Retell The number
lines in this part of the student
text are shown with plus signs to
stress that all numbers have signs
(+ or -), except for zero. This
helps students see the relationship
between opposites and to relate
their distances from zero. It is left
to the teacher’s discretion whether
students write the plus as well as
the minus when first working with
integers. As part of discussing
the vocabulary term integer, ask
students what the opposite of
zero is, and lead them to discover
that zero is its own opposite.
8 Identify a Subtask Before
students work on Question 8,
tape a number line on the floor,
and have a student volunteer walk
each step to show addition.
TThis is a good time for
sstudents to write in their
note
math notebooks,
do a vocabulary
organizer, and/or add to the
interactive word wall.
TEACHER TO
TEACHER
ACTIVITY 1.8
continued
Introduction to Integers
Get the Point?
SUGGESTED LEARNING STRATEGIES: Summarize/
Paraphrase/Retell, Identify a Subtask
My Notes
ACADEMIC VOCABULARY
Integers are the natural
numbers, their opposites,
and zero.
The opposite of 0 is 0.
The number lines below give visual representations of the integers.
Notice that zero is the only integer that is neither positive nor
negative.
Opposites
0 +1 +2 +3 +4 +5 +6
–6 –5 –4 –3 –2 –1
Opposite of Natural Numbers
Natural Numbers
WRITING MATH
0 +1 +2 +3 +4 +5 +6
–6 –5 –4 –3 –2 –1
Place a negative sign in front of a
number to indicate its opposite.
Positive Integers
Negative Integers
The opposite of 4 is -4.
The opposite of -4 is –(-4) = 4.
MATH TERMS
T he cooperative groups will f ind their totals to determine which
group has the most points at this time.
8. Group 1 uses a number line to f ind their total.
Most mathematicians use Z to
refer to the set of integers. This
is because in German, the word
Zahl means “number.”
A
−3
–3 –2 –1
WRITING MATH
To avoid confusion, use
parentheses around a negative number that follows an
operation symbol.
B
3
0
1
C
8
2
3
D
−1
4
5
6
7
8
To add with a number line, start at the f irst number. T hen move to
the right to add a positive number, or to the lef t to add a negative
number.
© 2010 College Board. All rights reserved.
ACTIVITY 1.8 Continued
a. Add A and B, or −3 + 3: Put your pencil at −3 and move it
to the right 3 places to add 3. 0
b. Add C and D, or 8 + (−1): Put your pencil on 8 and move it
to the lef t 1 place to add −1. 7
c. Combine the sums of Parts a and b.
0 +7 =7
48 SpringBoard® Mathematics with Meaning™ Level 1
TThink of the Cartesian coordinate system as a vertical
n
number
line and a horizontal number line that are
perpendic
perpendicular
to each other in a plane, with (0, 0) as the point of
intersection. Have students practice plotting points on a coordinate
grid. You can use coordinate grids or number lines to reinforce
absolute value and comparing and ordering integers.
TEACHER TO
TEACHER
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045-053_SB_MS1_1-8_SE.indd 48
Introduction to Integers
ACTIVITY 1.8
Get the Point?
continued
9a Quickwrite (9, 10a), Think/
Pair/Share (10), Create
Representations (11),
Debriefing Students need to
understand zero pairs in order to
add and subtract integers with
algebra tiles and counters. In
Question 11, they should start at
0, move left 5, left 6 more, then
right 10.
SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/
Share, Create Representations, Role Play, Use Manipulatives
My Notes
9. What happens when you add a number and its opposite, for
example, 3 and −3?
The sum is 0.
10. A number and its opposite are called additive inverses.
a. Why do you think they are also called a zero pair?
T hey are a pair of numbers that have a sum of zero.
b. Write 2 more zero pairs.
ACADEMIC VOCABULARY
A number and its opposite
are called additive inverses.
The sum of a number and its
additive inverse is zero.
Answers may vary. Sample answer: −7 and 7, 13 and −13.
Paragraph Role Play, Use
Manipulatives
11. Use one number line to f ind the total points for group 2. −1
E
0
F
G
−5 −6
–12 –10 –8 –6 –4 –2
0
2
H
10
4
Remember, zero pairs have a sum
of zero (-1 + 1 = 0), so they are
eliminated.
6
8
10 12
Group 3 decides to add their points using positive and negative
counters.
© 2010 College Board. All rights reserved.
I
7
A positive counter
J
−4
K
1
ACTIVITY 1.8 Continued
b Use Manipulatives Students
should role-play the guided
instruction on using counters
to add Group 3’s accounts, and
then apply the process to solving
Question 13.
J
2
is +1. A negative counter
is −1.
To add I and J, f irst take 7 positive counters and 4 negative
counters.
Next, combine the counters in zero pairs.
3 positive counters remain, so 7 + (−4) = 3
12. What is the point total of Group 3? 6
Unit 1 • Number Concepts
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Unit 1 • Number Concepts
49
ACTIVITY 1.8 Continued
c Use Manipulatives,
Debriefing Students will
probably need additional practice
with adding integers using
counters either after reading the
introduction or before moving
onto Question 14. Time for this
practice has been allowed in the
suggested pacing.
ACTIVITY 1.8
continued
Introduction to Integers
Get the Point?
My Notes
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Look for a Pattern, Group Discussion, Self Revision/Peer
Revision, Identify a Subtask
13. Use positive and negative counters to add the points of the
students in Group 4.
M
-3
N
-2
O
12
P
1
a. Draw counters to show -3 + (-2).
+
Suggested Assignment
CHECK YOUR UNDERSTANDING
p. 53, #4–5
= –5
b. What is the sum for students O and P?
12 + 1 = 13
UNIT 1 PRACTICE
p. 60, #58
c. Draw counters to add your answers to Parts a and b. 8
Paragraph Role Play, Use
Manipulatives
Group 5 looks at number relationships in order to find a way to add
integers without the help of number lines or counters.
First, they look at the sums of integers that have the same signs:
Discussion Group, Self Revision/
Peer Revision Students move
from the concrete counters and
pictorial representations to the
more abstract method of adding
integers. The term generalization
in Question 14 indicates that
students must find a pattern and
write a rule. Although students
may not be expected to memorize
these rules at this time, you may
want to encourage them to do so.
3+5=8
-2 + -3 = -5
-3 + -5 = -8
4 + 9 = 13
2+3=5
-4 + -9 = -13
© 2010 College Board. All rights reserved.
de Look for a Pattern,
14. Look for a pattern and write a generalization for adding
integers with the same signs.
Answers may vary. Sample answer: Add the numbers as if they
had no signs and use the common sign of the integers as the
sign of the sum.
Next, they consider sums of integers with different signs:
-2 + 5 = 3
7 + -11 = -4
4 + -3 = 1
-8 + 1 = -7
15. Look for a pattern and write a generalization for adding
integers with different signs.
Answers may vary. Sample answer: Ignore the signs; the
sum is the difference between the greater number and the
lesser number; use the sign of the greater number as the sign
of the sum.
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50 SpringBoard® Mathematics with Meaning™ Level 1
Introduction to Integers
ACTIVITY 1.8
Get the Point?
continued
f Identify a Subtask Students
now apply their generalizations
for adding integers.
SUGGESTED LEARNING STRATEGIES: : Identify a Subtask,
Create Representations, Role Play
My Notes
16. Use these generalizations to find the total points for the
students in Group 5.
Q
-7
R
2
S
-1
g Create Representations
hi Create Representations,
Debriefing Students compile
their results to answer the initial
question of who wins the 50% off
their class rings.
T
6
a. Q + R -7 + 2 = -5
b. (Q + R) + S -5 + (-1) = - 6
c. Find the sum for all four students. -6 + 6 = 0
Paragraph Role Play The
purpose is to give students
a need to subtract integers.
Students should again role-play
with counters as a strategy to
understand the guided instruction
that follows. Students discover
that subtracting integers is merely
adding their opposites.
17. Use your generalizations to find the total points for the
students in Group 6. 7
U
-4
V
-1
W
9
ACTIVITY 1.8 Continued
X
3
18. Compile the total group points in the table below.
G1
G2
G3
G4
G5
G6
7
-1
6
8
0
7
© 2010 College Board. All rights reserved.
19. Which group has the most points?
Group 4
You can evaluate the numerical expression 8 - (-1) using
positive and negative counters.
MATH TERMS
A numerical expression is a
number, or a combination of
numbers and operations.
• Begin with 8 positive counters (+8):
• You must subtract a negative counter (-1), but there are none.
So, add a zero pair, 1 positive and 1 negative.
+
• Now, subtract the negative counter, leaving 9 positive counters.
So, 8 - (-1) = 9.
+
Thus, subtracting a negative 1 is like adding a positive 1.
8 - (-1) = 8 + 1
Unit 1 • Number Concepts
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In this activity negative numbers are put in parentheses
w
when they are being subtracted to avoid the confusion of
subtra
two subtraction
symbols. You may want to make students aware of
another method; for example, -2 and -2 are the same number.
They will see negative numbers written in different ways. While
there is no right or wrong way to write a negative, many teachers
fear that by using the raised negative sign students will not
understand that subtracting 2 and -2 are the same. This is important
when doing integer operations.
TEACHER TO
TEACHER
Unit 1 • Number Concepts
51
ACTIVITY 1.8 Continued
ACTIVITY 1.8
continued
Introduction to Integers
Get the Point?
jl Identify a Subtask,
Use Manipulatives, Think/
Pair/Share These questions
provide scaffolded practice with
subtracting integers. Additional
practice is recommended after
these problems.
SUGGESTED LEARNING STRATEGIES: Use Manipulatives,
Identify a Subtask, Think/Pair/Share
My Notes
20. Draw counters to find the difference: 3 - (-4). Do your work
in the My Notes space.
a. What type of counters do you need to start, and how many
do you need?
3 positive counters
Suggested Assignment
b. What type of counters do you need to subtract, and
how many?
CHECK YOUR UNDERSTANDING
p. 53, #6–7
4 negative counters
c. Notice that you do not have the counters you need to
subtract. What can you add to give you the counters you
need without changing the starting value?
UNIT 1 PRACTICE
p. 61, #59–60
4 zero pairs
d. Cross out four negative counters (subtract - 4). Remaining
counters: = 3 - (-4) = 7
21. Complete this statement to show how to compute 3 - (-4)
by adding the opposite.
© 2010 College Board. All rights reserved.
Instead of subtracting -4, add its opposite, 4.
The addition expression is 3 + 4 , so 3 - (-4) = 7 .
22. Find the difference two ways.
a. -5 - 7
Use counters: -12
Add the opposite: -12
b. -2 - (-3)
Use counters: 1
Add the opposite: 1
c. 8 - (-6)
Use counters: 14
Add the opposite: 14
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52 SpringBoard® Mathematics with Meaning™ Level 1
Introduction to Integers
ACTIVITY 1.8
Get the Point?
continued
ACTIVITY 1.8 Continued
Answer Key
1a.-10
CHECK YOUR UNDERSTANDING
b.+25
Write your answers
answers on
on notebook
notebook paper.
paper.Show your work.
4. Evaluate each expression.
Show your work.
a. -3 + 9
b. -5 + (-7)
1. Write an integer to represent each situation.
c. -12 + 6
d. -24 - 11
a. 10 yard loss in football
e. -13 - (-8)
b. Earn $25 at work
d. Elevation of 850 feet above sea level
e. Change in score after an inning with
no runs
f. The opposite of losing 50 points in a game
2. Evaluate each expression.
1
c. - __
2
3. The following table shows the high and
low temperatures of 5 consecutive days in
February in North Pole, Alaska.
© 2010 College Board. All rights reserved.
High
Low
Mon
1
-13
b. |-11|
Tues
-29
-45
Wed
-27
-54
|
Thurs
5
-2
d.+850
f. 31 - (-10)
e. 0
5. During their possession, a football team
gained 5 yards, lost 8 yards, lost another
2 yards, then gained 45 yards. What were
the total yards gained or lost?
c. 2 degrees below zero
a. |54|
c. -2
|
Fri
7
1
a. Order the high temperatures from warmest
to coldest over this five-day period.
f. +50
2a. 54
6. In North America the highest elevation
is Denali in Alaska at 20,320 feet above
sea level and the lowest elevation is Death
Valley in California at 282 feet below sea
level. Write and evaluate an expression
with integers to find the difference between
the elevations.
7. On winter morning, the temperature fell
below -6°C. What does this temperature
mean in terms of 0˚C?
8. MATHEMATICAL Using a number line,
R E F L E C T I O N explain how you can
order integers.
b.11
c. _21_
3a.Friday: 7, Thursday: 5,
Monday: 1, Wednesday: -27,
Tuesday: -29
b.No, the three warmest
days are in the same order,
but the last two days are
switched. The order is
Friday: 1, Thursday: -2,
Monday: -13, Tuesday: -45,
Wednesday: -54.
4a.6
b. Is the order of days from warmest to coldest
daily low temperatures the same as for the
daily high temperatures? Explain.
b.-12
c. -6
d.-35
e. -5
f. 41
5. 40 yards gained
6. 20,320 - (-282); 20,602 ft
Unit 1 • Number Concepts
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1/13/11 12:29:11 PM
7. Answers may vary. Sample
answer: The temperature was
more than 6° below 0.
8. Answers may vary. Sample
answer: A number line helps
me see where each number
is compared to the other
numbers. It gives me a way
to put the numbers in order
one at a time. It is easier to
put a number between two
numbers on a number line
than between two numbers in
a list.
Unit 1 • Number Concepts
53