Subtracting Negative Integers
Notes:
Comparison of CST questions to the skill of subtracting negative integers.
5th Grade/65
Total # of
Benchmark
Questions
22
Total # of Concept Questions
0 (1 Subtraction w/- answer)
NS2.1 Add, subtract, multiply and
divide with decimals; add with
negative integers; subtract positive
integers from negative integers;
and verify the reasonableness of
the results.
6th Grade/65
17
0 (4 Adding +/- Integers)
NS2.3 Solve addition, subtraction,
multiplication and division
problems, including those that
arise in concrete situations, that
use positive and negative integers
and combinations of these
operations.
7th Grade/65
14
0 (3 Absolute Value)
NS1.2 Add, subtract, multiply and
divide rational numbers (integers,
fractions and terminating
decimals) and take positive
rational numbers to whole number
powers.
Although the skill itself is not emphasized in the test, experience tells us
that students still struggle when working with the concept of negative and
positive numbers and operations.
This is the order in which to introduce integers:
1. Using a number line to add integers (developing the language used for
working with integers)
2. Tile spacers (concrete)
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MDC@ACOE 10/17/10
3. Introduce the rules of adding integers (by now students have figured
it out, anyway)
4. Introduce subtracting integers: number line, tile spacers,
decomposing, rules
Once students have developed the number sense necessary to succeed in
adding and subtracting integers, they can transfer that skill into adding and
subtracting positive and negative decimals and fractions and combining like
terms.
Regarding this lesson:
This lesson assumes students have had little exposure to different
strategies, and a general weakness in working with integers, so there is a
lot of scaffolding embedded. Please be sure to begin the lesson where your
students’ needs are.
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Subtracting Negative Integers Warm-Up
Quadrant II: 3rd Grade #46, NS2.8 Quadrant I: 5th Grade #47, AF1.2 If n = 31, what is the value of 6 – n ?
Tony had $20. He paid $8 for a ticket to a baseball game. At the game, he bought a hotdog for $3. What amount of money did Tony have then? Show two ways to solve. Show at least two ways to solve. Quadrant III: 6th Grade #27, NS2.3 Quadrant IV: 7th Grade #59, AF4.1
Solve: One morning, the temperature was 5° below zero. By noon, the temperature rose 20° Fahrenheit (F) and then dropped 8°F by evening. What was the evening temperature? Solve: What is the value of x if -­‐3x + 2 = -­‐7 ? What are some errors students might make? What are some errors students might make? Today’s Objective: Develop a variety of strategies to solve addition and subtraction problems, while providing opportunities to develop number sense regarding adding and subtracting integers. Page 3 of 26
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Subtracting Negative Integers Warm-Up Solutions
Quadrant II: 3rd Grade #46, NS2.8 Quadrant I: 5th Grade #47, AF1.2 If n = 31, what is the value of 6 – n ?
Tony had $20. He paid $8 for a ticket to a baseball game. At the game, he bought a hotdog for $3. What amount of money did Tony have then? + + + + + + + + + + + + + + + 9 12 20 + + + + + = $9 6 – 31
+(-­‐3) + (-­‐8) 6 – 31
= 6 – 6 – 25
= 6 + (-31)
= 0 – 25
+ + + + + +
= -25
_ _ _ _ _ _
_ _ _ _ _ _
_ _ _ _ _ _
_ _ _ _ _ _
_ _ _ _ _ _
= $9 _
= 24 negatives
= -24
th
Quadrant III: 6 Grade #27, NS2.3 Quadrant IV: 7th Grade #59, AF4.1
Solve: What is the value of x if Solve: One morning, the temperature was 5° below zero. By noon, the temperature rose 20° Fahrenheit (F) and then dropped 8°F by evening. What was the evening temperature? -­‐3x + 2 = -­‐7 ? -­‐3x + 2 = -­‐7 -­‐5 + 20 + (-­‐8) -­‐3x +2 = -­‐7 + 2 -­‐2 = 20 + (-­‐5) + (-­‐8) -­‐3x = -­‐7 -­‐2 = 20 + (-­‐13) -­‐3x = -­‐9 = 7 + 13 + (-­‐13) -­‐3 ·∙ x = -­‐3 ·∙ 3 = 7 + 0 x = 3 = 7 Possible errors: adding positive integers Possible errors: adding 2 to both sides, +7 – 2 Page 4 of 26
MDC@ACOE 10/17/10
Standard: 7NS1.2: Add, subtract, multiply and divide rational numbers
(integers, fractions and terminating decimals) and take positive rational
numbers to whole number powers.
Objective: Students will be able to add and subtract integers.
Define Integers: Any member of the set I=
So, integers are the positive and negative whole numbers, including zero.
APK (Activate Prior Knowledge): “Where do we see integers in the real
world?” See bullet note for possible student responses—be sure to include
ideas that students may have missed.
• [weight gain/loss, football yards gained/lost, stock market
gains/losses, bank deposits and withdrawals, temperatures, elevation
above and below sea level, elevators, escalators, parking garages…]
Show a number line and ask:
“Where are the bigger numbers on the number line?” [right, right of zero]
“The smaller numbers?” [left, left of zero]
“Where are the positive numbers on the number line?” [to the right of
zero]
“The negative numbers?” [to the left of zero]
“Do positive numbers have a greater value than negative numbers?” [yes]
“How do you know?” [they are to the right of negative numbers, any number
to the right of another number is greater, when you move to the right on a
number line the numbers increase in value]
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“Which number is greater: 3 or -2?” [3]
“How do you know?” [3 is to the right of -2]
“Which number is greater: -7 or -2?” [-2]
“How do you know?” [-2 is to the right of -7]
Students should be able to articulate and justify their responses: [the
number that is the furthest to the right on a number line has a greater
value, if both numbers are negative, the number closest to zero has the
greatest value]
Check for Understanding: List several pairs of numbers and have students
state either the bigger or the smaller of the two. List three numbers and
have students order the numbers from greatest to least, least to greatest.
Number Line Concept Development: “If I were to add two integers, 3 and
5, using a number line, what would be a good strategy for me to do this?”
[start at 3 and move 5 to the right, end up at 8; 3 + 5 = 8]
“So, if I am adding positive integers, I move to the right. What if I am
adding negative integers? Which direction would I move?” [to the left]
“So, let’s add a positive 3 with a negative 5: 3 + (-5). What number do I
start with?” [3]
+ (-­‐5) “What am I adding to that?” [-5]
“Is this a positive or negative?” [negative]
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“If I were adding a positive 5, I would move to the right, but I am not. I’m
adding a negative 5; which direction do I move?” [left]
“Where do I end up on the number line?” [negative 2]
“Let’s try again: 7 + (-3). Where do I start on the number line?” [7]
+ (-­‐3) “What am I adding to that?” [-3]
“Is that a positive or negative?” [negative]
“If I were adding a positive 3, I would move to the right, but I am not. I’m
adding a negative 3; which direction do I move on the number line?” [left]
“Where do I end up on the number line?” [4]
“So, 7 + (-3) = 4.”
“Again: -4 + (-5). Where do I start on the number line?” [-4]
+ (-­‐5) “What am I adding to that?” [-5]
“Is that a positive or a negative?” [negative]
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“If I were adding a positive 5, I would move to the right, but I am not. I am
adding the opposite of 5, or a negative 5, so which direction do I move on
the number line?” [left]
“Where do I end up?” [-9]
“So, what is our equation?” [-4 + (-5) = -9]
“Let’s try another: -6 + 5. Where do I start on the number line?” [-6]
+5 “What am I adding to that?” [5]
“Is that a positive or a negative number?” [positive]
“Which direction do I move on the number line?” [right]
“Where do I end up on the number line?” [-1]
“What is our equation?” [-6 + 5 = -1]
“Let’s try a word problem together: ‘One morning, the temperature was 5°
below zero. By noon, the temperature rose 20° Fahrenheit (F) and then
dropped 8°F by evening. What was the evening temperature?’
“What integer represents five degrees below zero?” [-5]
“What does it mean when a temperature rose twenty degrees? What math
operation would that be?” [adding 20, positive 20]
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“Finally, what math operation represents dropping eight degrees?” [negative
8, subtracting 8]
“So, what is the equation for this word problem?” [-5 + 20 + (-8) =]
20 15 10 15 + (-­‐8) = 7 -­‐5 + 20 = 15 5 0 -­‐5 “Where do I start on the number line?” [-5]
“What are we adding to that?” [20]
“Is that a positive or negative number?” [positive]
“Since this number line is vertical, which direction do we move on it?”[up]
“Where are we on the number line?” [15]
“What are we adding to that?” [-8]
“Which direction do we move?” [down]
“Where are we on the number line?” [7]
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“What is our equation?” [-5 + 20 + (-8) = 7]
You Tries:
a) 7 + (-9) = -2
+ (-­‐9) b) -15 – 22 = -37 (Holt 7, L 1-5)
+ (-­‐22) *Note: At this point, some students may see that they will keep the sign of
the number that has the greatest absolute value.
Take a moment to frontload students with the concept of ‘zero pairs’.
Zero Pair Concept Development:
“Imagine I am standing at zero on a number line. If I take one step to the
right, where would I be on the number line?” [positive 1]
“Now I will take one step to the left. Where am I on the number line?”
[zero]
“Now imagine you are on the ground floor in an elevator and you take it one
floor up (+1). You stay on the elevator and ride it one floor down (-1). Where
do you end up? [ground floor, back where you started, zero]
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“What if you are at the top of the staircase, and you take 7 steps
down (-7); you forgot something upstairs and took 7 steps up (+7). Where
did you end up?” [back where you started, zero]
“These examples illustrate ‘zero pairs’. A working definition for the concept
of zero pairs is: any number or variable and its opposite equals zero.
The value of a Positive 1 and Negative 1 create a zero; therefore we have a
zero pair. The value of a Positive 27 and Negative 27 create a zero; zero
pair. The value of a Positive x and a Negative x create a zero: zero pair.”
Ask for students to give examples of ‘zero pairs’.
Tile Spacers Concept Development:
“Let’s work with integers in a different way. If I am adding
3 + 5, how many positives do I need to represent 3?” [3] Draw three
positives.
+
+
+
“How many positives do I need to represent 5?” [5] Draw five positives
next to the three positives.
+
+
+
+
+
+
+
+
“How many positives do I have altogether?” [8]
“What is our equation?” [3 + 5 = 8]
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“If I am adding -3 + 5, how many negatives do I need to represent -3?” [3]
Draw three negative symbols.
“How many positives do I need to represent 5?” [5] Draw five positive
symbols next to the three negative symbols.
-
+
+
+
+
+
“I now have 3 negatives and 5 positives. Remember our examples with going
up one floor and then down one floor…what number do we have when there
is a number and its opposite, like a positive 7 and a negative 7?” [zero]
“If I matched one negative with one positive, what concept would that
represent?” [zero, zero pair]
“How many zero pairs do we have?” [3]
“Take them away. What is left?” [2 positives]
-
+
+
+
+
+
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“We keep the sign of what we have the most of; since we have more
positives than negatives, our answer is positive 2.”
“What is our equation?” [-3 + 5 = 2]
“Let’s try another: 3 + (-5). How many positives do I have?” [3] Draw three
positives.
+
+
+
“How many negatives?” [5] Draw five negatives next to the three positives.
+
+
+
-
“Are there any zero pairs?” [3]
“Take them away. What is left?” [2 negatives]
+
+
+
“Since we keep the sign of what we have the most of, what is our equation?”
[3 + (-5) = -2]
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You Tries:
a) -4 + (-3)
-
-
-4 + (-3) = -7
-
-
-
-
We’ve added (combined) the numbers and kept
the sign of what we have the most of.
b) 18 + (-7)
18 + (-7) = 11 (Holt 7, L 1-4)
+
+
-
+
+
-
+
+
-
+
+
-
+
+
-
+
+
-
+
+
-
+
+
We found zero pairs (combined the
numbers) and kept the sign of what we
have the most of.
+
+
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Concept Closure: Give students the following two stems and have them
work with an elbow partner to see if they can articulate the ‘rules’ for
adding integers:
• “If the signs are the same…”
• “If the signs are different…”
Give students time to process, and then debrief. Ask groups to share their
ideas.
• [If the signs are the same, “add the numbers and keep the sign.”]
• [If the signs are different, “subtract the numbers and keep the sign
of what you have the most of.”] Using this language is key.
Subtracting Integers
Have students compare the following two problems:
3 + (-5) and 3 – 5
Use the number line and tile spacers to model.
“Traditional: Signs are
different; subtract
and keep the sign of
what you have the
most of.”
+ (-­‐5) 3 + (-5) = -2
+
-
+
-
+
-
= -2
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3 – 5 = -2
+ (-­‐5) “The rule for subtracting integers is to re-write the subtraction problem
into an addition problem, and then add the opposite of the integer.
So,
=
3–5
becomes three plus the opposite of five:
3 + (-5)
which are the same two problems we’ve compared.
We will need to re-write all of our subtraction problems so that they are
addition ones.”
You Try Rewriting:
“What is the opposite of -8?” [8]
-5 – (-8)
= -5 + 8
“What is the opposite of 35?” [-35]
32 – 35
= 32 + (-35)
“What is the opposite of -25?” [25]
75 – (-25)
= 75 + 25
“But, why is subtracting a negative the same as adding a positive?” Check
student responses.
“If I have gone into debt in my checking account because I am $25
overdrawn, do I have a positive balance or a negative balance?” [negative,
-25 dollars]
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“If someone was going to take my debt away as a gift to me, what would
they have to do?” [they would take away negative $25]
“What would the balance be in my account?” [zero dollars]
“Even though I don’t have any money, is that better than having -$25 in my
account?” [yep!]
“What would that look like as a math problem?” Write the following:
-25 – (-25) = 0
“If someone was willing to help me out like this, what would they have to do
in order to bring my checking account balance to zero if I have - $25 ?
They would have to give me $25! Positive 25.”
“So, knowing the integer rule for subtraction is to re-write a subtraction
problem into an addition one and then adding the opposite integer, we would
need to think about my banking situation like this”: Write the following:
-25 – (-25) = 0
-25 + (+25) = 0
“Does anyone see an example of a zero pair?” [yes!]
“Where is it?” [-25 and +25, negative 25 and positive 25]
“If I were using a number line for this situation,” Write the following:
-25 – (-25) = 0
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“Where would I start on the number line?” [-25]
“If I was going to add negative 25, which direction would I go?” [left]
“But I’m not, I’m doing the opposite of adding negative 25, I’m going to
subtract negative 25, so which direction do I go?” [right]
“Where do I end up on the number line?” [0]
When adding a negative number, move to the left. -­‐ (-­‐25) Since we are subtracting a negative number, we move to the right. “Let’s try one more guided practice,” Write the following:
14 – 8 =
“We need to change our subtraction to addition and add the opposite
integer.” Write the following, and ask for students’ response on rewriting:
14 – 8
= 14 + (-8)
[14 plus -8]
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“Now, where do we start on the number line? [14]
“We are adding the opposite of 8; is this a positive or negative 8?
[negative]
“Which direction do we move on the number line?” [left]
“Where do we end up?” [6]
+ (-­‐8) You Tries
-17 – (-10)
= -17 + 10
+10 = -7
“Traditional: Signs
are different;
subtract and keep
the sign of what you
have the most of.”
-
-
-
-
-
-
-
-
-
-
+
+
+
+ +
+
+
+
+
+
Page 19 of 26
-
-
-
-
-
-
-
= -7
MDC@ACOE 10/17/10
You Tries (continued…)
-14 – (-14)
+ (-­‐14) = -14 + 14
=0
“Traditional: Signs
are different;
subtract and keep
the sign of what you
have the most of.”
+ + + + + + + + + + + + + +
- - - - -
- - - -
- - - -
-
= 0
Decomposing Concept Development:
“Let’s try one more strategy with some problems we’ve already worked
through: decomposing.”
3–5
3–5
Given
= 3 + (-5)
= 3 + (-5)
Rewrite as addition problem
= -2
= 3 + (-3) + (-2)
Decompose -5 to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
= 0 + (-2)
Identity Property of Addition
= -2
Answer
*Note: The use of the third column frontloads students for success in
geometry, where they will be required to justify their work with twocolumn proofs. In this context, having students write the justifications is
an option, but explicit discussion of the math reasoning and properties is
highly recommended.
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-17 – (-10)
-17 – (-10)
Given
= -17 + 10
= -17 + 10
Rewrite as addition
= -7
= -7 + (-10) + 10
Decompose to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
= 0 + 10
Identity Property of Addition
= -7
Answer
14 – 8
14 – 8
Given
= 14 + (-8)
= 14 + (-8)
Rewrite as addition
=6
= 6 + 8 + (-8)
Decompose to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
=6+0
Identity Property of Addition
=6
Answer
Same problem…
14 – 8
14 – 8
Given
= 14 + (-8)
= 14 + (-8)
Rewrite as addition
=6
= 8 + 6 + (-8)
Decompose to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
= 6 + 8 + (-8)
Commutative Property of Addition
=6+0
Identity Property of Addition
=6
Answer
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You Tries
(Holt 7, L 1-5)
-18 – (-25)
-18 – (-25)
Given
= -18 + 25
= -18 + 25
Rewrite as addition
=7
= -18 + 18 + 7
Decompose to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
=0+7
Identity Property of Addition
=7
Answer
-88 – (-10)
-88 – (-10)
Given
= -88 + 10
= -88 + 10
Rewrite as addition
= -78
= -78 + (-10) + 10 Decompose to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
= -78 + 0
Identity Property of Addition
= -78
Answer
-15 – x; x = -10
-15 – (-10)
-15 – (-10)
Given
= -15 + 10
= -15 + 10
Rewrite as addition
= -5
= -5 + (-10) + 10
Decompose to create a zero pair
“Traditional: Signs are
different; subtract and
keep the sign of what
you have the most of.”
= -5 + 0
Identity Property of Addition
= -5
Answer
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“Now let’s practice it all!”
Traditional
Number Line
Tile Spacers
-15 – (-10)
-15 – (-10)
-15 – (-10)
= -15 + 10
(Holt 7, L 1-5)
= -15 + 10
= -5
Decompose
-15 – (-10)
= -15 + 10
+10 “Traditional: Signs
are different;
subtract and keep
the sign of what
you have the most
of.”
= -15 + 10
-
-
-
-
-
= -5 + (-10) + 10
-
-
-
-
-
= -5 + 0
-
-
-
-
-
= -5
+
+
+
+
+
+
+
+
+
+
= -5
-5 – (-15)
-5 – (-15)
= -5 + 15
= -5 + 15
= 10
-5 – (-15)
= -5 + 15
+15 “Traditional: Signs
are different;
subtract and keep
the sign of what
you have the most
of.”
-5 – (-15)
= -5 + 15
- - - - -
= -5 + 5 + 10
+ + + + +
= 0 + 10
+ + + + +
= 10
+ + + + +
= 10
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-15 – (-12)
-15 – (-12)
= -15 + 12
= -15 + 12
= -3
“Traditional: Signs
are different;
subtract and keep
the sign of what
you have the most
of.”
-15 – (-12)
= -15 + 12
-15 – (-12)
= -15 + 12
= -3 + (-12) + 12
+12 -­‐3 - - + + -
= -3 + 0
- - + + -
= -3
- - + + - - + +
- - + +
- - + +
= -3
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-18 – (-25)
-18 – (-25)
= -18 + 25
= -18 + 25
=7
“Traditional: Signs
are different;
subtract and keep
the sign of what
you have the most
of.”
-18 – (-25)
-18 – (-25)
= -18 + 25
= -18 + 25
= -18 + 18 + 7
+25 - - - + + + =0+7
7 - - - + + + =7
- - - + + +
- - - + + +
- - - + + +
- - - + + +
+ + +
+ + +
+
=7
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15 – (-20)
15 – (-20)
= 15 + 20
= 15 + 20
= 35
“Traditional: Signs
are the same; add
and keep the sign.”
15 – (-20)
= 15 + 20
+20 35 15 – (-20)
= 15 + 20
+ + + + +
= 10 + 5 + 20
+ + + + +
= 10 + 20 + 5
+ + + + +
= 30 + 5
+ + + + +
= 35
+ + + + +
+ + + + +
+ + + + +
= 35
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