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Model the expression:

−6 4


This figure shows a row of 10 dark pink circles, representing negative counters. The first six counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters.  These four positive counters are circled.
−10

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Model the expression:

−7 4


This figure shows a row of 11 dark pink circles, representing negative counters. The first seven counters are separated from the last four. Below the dark pink circles are four light pink circles, representing positive counters.  These four positive counters are circled.
−11

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Model: 5 ( −3 ) .

Solution

Interpret the expression. 5 ( −3 ) means 5 take away −3 .
Model the first number. Start with 5 positives. .
Take away the second number, so take away 3 negatives.
But there are no negatives to take away.
Add neutral pairs until you have 3 negatives.
.
Then take away 3 negatives. .
Count the number of counters that are left. .
The difference of 5 and −3 is 8 . 5 ( −3 ) = 8
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Model the expression:

6 ( −4 )


This figure shows a row of 10 light pink circles, representing positive counters. The first six counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters.  These four negative counters are circled.
10

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Model the expression:

7 ( −4 )


This figure shows a row of 11 light pink circles, representing positive counters. The first seven counters are separated from the last four. Below the light pink circles are four dark pink circles, representing negative counters.  These four negative counters are circled.
11

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Model each subtraction.

  1. 8 − 2
  2. −5 − 4
  3. 6 − (−6)
  4. −8 − (−3)

8 2
This means 8 take away 2 .
Start with 8 positives. .
Take away 2 positives. .
How many are left? 6
8 2 = 6
−5 4
This means −5 take away 4 .
Start with 5 negatives. .
You need to take away 4 positives. .
Add 4 neutral pairs to get 4 positives. .
Take away 4 positives. .
How many are left? .
−9
−5 4 = −9
6 ( −6 )
This means 6 take away −6 .
Start with 6 positives. .
Add 6 neutrals to get 6 negatives to take away. .
Remove 6 negatives. .
How many are left? .
12
6 ( −6 ) = 12
−8 ( −3 )
This means −8 take away −3 .
Start with 8 negatives. .
Take away 3 negatives. .
How many are left? .
−5
−8 ( −3 ) = −5
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Model each subtraction.

  1. 7 - (-8)
  2. -2 - (-2)
  3. 4 - 1
  4. -6 - 8


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.

.

.

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Model each subtraction.

  1. 4 - (-6)
  2. -8 - (-1)
  3. 7 - 3
  4. -4 - 2


.

.

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.

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Model each subtraction expression:

  1. 2 8
  2. −3 ( −8 )

Solution


We start with 2 positives.
.
We need to take away 8 positives, but we have only 2.
Add neutral pairs until there are 8 positives to take away. .
Then take away eight positives. .
Find the number of counters that are left.
There are 6 negatives.
.
2 8 = −6

We start with 3 negatives.
.
We need to take away 8 negatives, but we have only 3.
Add neutral pairs until there are 8 negatives to take away. .
Then take away the 8 negatives. .
Find the number of counters that are left.
There are 5 positives.
.
−3 ( −8 ) = 5
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Model each subtraction expression.

  1. 7 9
  2. −5 −9 )


This figure shows a row of 9 dark pink circles, representing negative counters. The first five are separated from the last four. The entire row is circled.  Below the last four dark pink circles is a row of four light pink  circles, representing positive counters.
4

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Model each subtraction expression.

  1. 4 7
  2. −7 ( −10 )


This figure shows a row of 10 dark pink circles, representing negative counters. The first seven are separated from the last three. The entire row is circled.  Below the last three dark pink circles is a row of three light pink circles, representing positive counters.
3

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Simplify expressions with integers

Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.

  • Subtract −23 7 .
    Think: We start with 23 negative counters.
    We have to subtract 7 positives, but there are no positives to take away.
    So we add 7 neutral pairs to get the 7 positives. Now we take away the 7 positives.
    So what’s left? We have the original 23 negatives plus 7 more negatives from the neutral pair. The result is 30 negatives.
    −23 7 = −30

    Notice, that to subtract 7, we added 7 negatives.
  • Subtract 30 ( −12 ) .
    Think: We start with 30 positives.
    We have to subtract 12 negatives, but there are no negatives to take away.
    So we add 12 neutral pairs to the 30 positives. Now we take away the 12 negatives.
    What’s left? We have the original 30 positives plus 12 more positives from the neutral pairs. The result is 42 positives.
    30 ( −12 ) = 42

    Notice that to subtract −12 , we added 12 .

While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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