Modeling Addition of Integers

Modeling Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of \(2-1.\) Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

Modeling Addition of Integers

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in the figure below.

Modeling Addition of Integers

A blue counter represents \(+1.\) A red counter represents \(-1.\) Together they add to zero.

We will model four addition facts using the numbers \(5,-5\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}},-3.\)

\(5+3\phantom{\rule{1em}{0ex}}-5+\left(-3\right)\phantom{\rule{1em}{0ex}}-5+3\phantom{\rule{1em}{0ex}}5+\left(-3\right)\)

Example

Model: \(5+3.\)

Solution

Interpret the expression.\(5+3\) means the sum of \(5\) and \(3\).
Model the first number. Start with 5 positives.Modeling Addition of Integers
Model the second number. Add 3 positives.Modeling Addition of Integers
Count the total number of counters.Modeling Addition of Integers
The sum of 5 and 3 is 8.\(5+3=8\)

Optional Video: Adding Integers with Same Sign Using Color Counters by Mathispower4u

Example

Model: \(-5+\left(-3\right).\)

Solution

Interpret the expression.\(-5+\left(-3\right)\) means the sum of \(-5\) and \(-3\).
Model the first number. Start with 5 negatives.Modeling Addition of Integers
Model the second number. Add 3 negatives.Modeling Addition of Integers
Count the total number of counters.Modeling Addition of Integers
The sum of −5 and −3 is −8.\(-5+-3=-8\)

Both examples above are very similar. The first example adds \(5\) positives and \(3\) positives—both positives. The second example adds \(5\) negatives and \(3\) negatives—both negatives. In each case, we got a result of \(\text{8—either}\phantom{\rule{0.2em}{0ex}}8\) positives or \(8\) negatives. When the signs are the same, the counters are all the same color.

Now let’s see what happens when the signs are different.

Example

Model: \(-5+3.\)

Solution

Interpret the expression.\(-5+3\) means the sum of \(-5\) and \(3\).
Model the first number. Start with 5 negatives.Modeling Addition of Integers
Model the second number. Add 3 positives.Modeling Addition of Integers
Remove any neutral pairs.Modeling Addition of Integers
Count the result.Modeling Addition of Integers
The sum of −5 and 3 is −2.\(-5+3=-2\)

Notice that there were more negatives than positives, so the result is negative.

Optional Video: Adding Integers with Different Signs Using Counters by Mathispower4u

Example

Model: \(5+\left(-3\right).\)

Solution 

Interpret the expression.\(5+\left(-3\right)\) means the sum of \(5\) and \(-3\).
Model the first number. Start with 5 positives.Modeling Addition of Integers
Model the second number. Add 3 negatives.Modeling Addition of Integers
Remove any neutral pairs.Modeling Addition of Integers
Count the result.Modeling Addition of Integers
The sum of 5 and −3 is 2.\(5+\left(-3\right)=2\)

Modeling Addition of Positive and Negative Integers

Model each addition.

  1. 4 + 2
  2. −3 + 6
  3. 4 + (−5)
  4. -2 + (−3)

Solution

  
 \(4+2\)
Start with 4 positives.Modeling Addition of Integers
Add two positives.Modeling Addition of Integers
How many do you have?\(6\). \(4+2=6\)
  
 \(-3+6\)
Start with 3 negatives.Modeling Addition of Integers
Add 6 positives.Modeling Addition of Integers
Remove neutral pairs.Modeling Addition of Integers
How many are left?Modeling Addition of Integers
 \(3\). \(-3+6=3\)
  
 \(4+\left(-5\right)\)
Start with 4 positives.Modeling Addition of Integers
Add 5 negatives.Modeling Addition of Integers
Remove neutral pairs.Modeling Addition of Integers
How many are left?Modeling Addition of Integers
 \(-1\). \(4+\left(-5\right)=-1\)
  
 \(-2+\left(-3\right)\)
Start with 2 negatives.Modeling Addition of Integers
Add 3 negatives.Modeling Addition of Integers
How many do you have?\(-5\). \(-2+\left(-3\right)=-5\)

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This is a lesson from the tutorial, Introducing Integers and you are encouraged to log in or register, so that you can track your progress.

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