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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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Model each subtraction.

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


.

.

.

.

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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.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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