Modeling Addition of Integers

We will first 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 [latex]2 - 1[/latex]. 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.

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.

A blue counter represents [latex]+1[/latex]. A red counter represents [latex]-1[/latex]. Together they add to zero.

Doing the Manipulative Mathematics activity “Addition of signed Numbers” will help you develop a better understanding of adding integers.
We will model four addition facts using the numbers [latex]5,-5\text{ and }3,-3[/latex].

[latex]5+3 - 5+\left(-3\right)-5+35+\left(-3\right)[/latex]

 

 

 

 

 

The first and second examples are very similar. The first example adds [latex]5[/latex] positives and [latex]3[/latex] positives—both positives. The second example adds [latex]5[/latex] negatives and [latex]3[/latex] negatives—both negatives. In each case, we got a result of [latex]\text{8-either}8[/latex] positives or [latex]8[/latex] 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.

 

 

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

 

 

 

 

In the following video we present more examples of using color counters to model addition of integers.

Modeling Subtraction of Integers

You learn as a child how to subtract numbers through everyday experiences. For example, if you have 10 animal cookies and eat 6 of them, you will have 4 animal cookies left.

Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.
We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

Perhaps when you were younger, you read [latex]5 - 3[/latex] as five take away three. When we use counters, we can think of subtraction the same way.
We will model four subtraction scenarios using the numbers [latex]5[/latex] and [latex]3[/latex].

• [latex]5 - 3[/latex]
• [latex]- 5-\left(-3\right)[/latex]
• [latex]-5 - 3[/latex]
• [latex]5-\left(-3\right)[/latex]

Now you can try a similar problem.

In teh previous example we subtracted [latex]3[/latex] positives from positive [latex]5[/latex]. Now we will subtract [latex]3[/latex] negatives from negative [latex]5[/latex]. Compare the results of this example to the previous example after you read through it.

You can try a similar problem.

Notice that the previous two examples are very much alike.

• First, we subtracted [latex]3[/latex] positives from [latex]5[/latex] positives to get [latex]2[/latex] positives.
• Then we subtracted [latex]3[/latex] negatives from [latex]5[/latex] negatives to get [latex]2[/latex] negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.

Now you can try a similar problem.

Now we will subtract a negative number from a positive number. Think of this as taking away the negative.

Now you can try a similar problem.

 

Now we will do an example that summarizes the situations above, with different numbers. Recall the different scenarios:

• subtracting a positive number from a positive number
• subtracting a positive number from a negative number
• subtracting a negative number from a positive number
• subtracting a negative number from a negative number

Now you can try a similar problem.

Each of the examples so far have been carefully constructed so that the sign of the answer matched the sign of the first number in the expression.  For example, in  [latex]−5 − 4[/latex], the result is [latex]-9[/latex], which is the same sign as [latex]-5[/latex]. Now we will see subtraction where the sign of the result is different from the starting number.

Now you can try a similar problem.

When you subtract two integers, there are two possibilities, either the result will have a different sign from the starting number, or it will have the same sign.

Watch the video below to see more examples of modeling integer subtraction with color counters.

Modeling Multiplication of Integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.
We remember that [latex]a\cdot b[/latex] means add [latex]a,b[/latex] times. Here, we are using the model shown in the graphic below just to help us discover the pattern.

Now consider what it means to multiply [latex]5[/latex] by [latex]-3[/latex]. It means subtract [latex]5,3[/latex] times. Looking at subtraction as taking away, it means to take away [latex]5,3[/latex] times. But there is nothing to take away, so we start by adding neutral pairs as shown in the graphic below.

In both cases, we started with [latex]\mathbf{\text{15}}[/latex] neutral pairs. In the case on the left, we took away [latex]\mathbf{\text{5}},\mathbf{\text{3}}[/latex] times and the result was [latex]-\mathbf{\text{15}}[/latex]. To multiply [latex]\left(-5\right)\left(-3\right)[/latex], we took away [latex]-\mathbf{\text{5}},\mathbf{\text{3}}[/latex] times and the result was [latex]\mathbf{\text{15}}[/latex]. So we found that

[latex]\begin{array}{ccc}5\cdot 3=15\hfill & & -5\left(3\right)=-15\hfill \\ 5\left(-3\right)=-15\hfill & & \left(-5\right)\left(-3\right)=15\hfill \end{array}[/latex]

Notice that for multiplication of two signed numbers, when the signs are the same, the product is positive, and when the signs are different, the product is negative.

 

 

Watch the following video for more examples of how to multiply integers with different signs, and the same sign.

When we multiply a number by [latex]1[/latex], the result is the same number. What happens when we multiply a number by [latex]-1?[/latex] Let’s multiply a positive number and then a negative number by [latex]-1[/latex] to see what we get.

[latex]\begin{array}{ccc}\hfill -1\cdot 4\hfill & & \hfill -1\left(-3\right)\hfill \\ \hfill -4\hfill & & \hfill 3\hfill \\ \hfill -4\text{ is the opposite of }\mathbf{\text{4}}\hfill & & \hfill \mathbf{\text{3}}\text{ is the opposite of }-3\hfill \end{array}[/latex]
Each time we multiply a number by [latex]-1[/latex], we get its opposite.

 

 

Division with Integers

Division is the inverse operation of multiplication. So, [latex]15\div 3=5[/latex] because [latex]5\cdot 3=15[/latex] In words, this expression says that [latex]\mathbf{\text{15}}[/latex] can be divided into [latex]\mathbf{\text{3}}[/latex] groups of [latex]\mathbf{\text{5}}[/latex] each because adding five three times gives [latex]\mathbf{\text{15}}[/latex]. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

[latex]\begin{array}{ccccc}5\cdot 3=15\text{ so }15\div 3=5\hfill & & & & -5\left(3\right)=-15\text{ so }-15\div 3=-5\hfill \\ \left(-5\right)\left(-3\right)=15\text{ so }15\div \left(-3\right)=-5\hfill & & & & 5\left(-3\right)=-15\text{ so }-15\div -3=5\hfill \end{array}[/latex]
Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Remember, you can always check the answer to a division problem by multiplying.

 

Just as we saw with multiplication, when we divide a number by [latex]1[/latex], the result is the same number. What happens when we divide a number by [latex]-1?[/latex] Let’s divide a positive number and then a negative number by [latex]-1[/latex] to see what we get.

[latex]\begin{array}{cccc}8\div \left(-1\right)\hfill & & & -9\div \left(-1\right)\hfill \\ -8\hfill & & & 9\hfill \\ \hfill \text{-8 is the opposite of 8}\hfill & & & \hfill \text{9 is the opposite of -9}\hfill \end{array}[/latex]
When we divide a number by [latex]-1[/latex] we get its opposite.

 

Watch the following video for more examples of how to divide integers with the same and different signs.

Absolute Value

We saw that numbers such as [latex]5[/latex] and [latex]-5[/latex] are opposites because they are the same distance from [latex]0[/latex] on the number line. They are both five units from [latex]0[/latex]. The distance between [latex]0[/latex] and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.
The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of [latex]5[/latex] is written as [latex]|5|[/latex], and the absolute value of [latex]-5[/latex] is written as [latex]|-5|[/latex] as shown below.

 

 

 

In the video below we show more example of how to find the absolute value of an integer.

We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.

 

Notice that the result is negative only when there is a negative sign outside the absolute value symbol.

 

 

In the video below we show more examples of how to compare expressions that include absolute value and integers.

Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.

 

 

 

 

 

Watch the following video to see more examples of how to simplify expressions that contain absolute value.

Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

[latex]4+3\cdot 7[/latex]

[latex]\begin{array}{cccc}\hfill \text{Some students say it simplifies to 49.}\hfill & & & \hfill \text{Some students say it simplifies to 25.}\hfill \\ \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }4+3\text{ gives 7.}\hfill & & \hfill 7\cdot 7\hfill \\ \text{And }7\cdot 7\text{ is 49.}\hfill & & \hfill 49\hfill \end{array}& & & \begin{array}{ccc}& & \hfill 4+3\cdot 7\hfill \\ \text{Since }3\cdot 7\text{ is 21.}\hfill & & \hfill 4+21\hfill \\ \text{And }21+4\text{ makes 25.}\hfill & & \hfill 25\hfill \end{array}\hfill \end{array}[/latex]

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

 

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally.

Order of Operations
Please	Parentheses
Excuse	Exponents
My Dear	Multiplication and Division
Aunt Sally	Addition and Subtraction

It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

 

 

 

 

 

 

In the video below we show another example of how to use the order of operations to simplify a mathematical expression.

 

When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.

 

In the video below we show another example of how to use the order of operations to simplify an expression that contains exponents and grouping symbols.

 

 

 

Watch the following video to see another example of how to use the order of operations to simplify an expression that contains integers.

In our next example we will simplify expressions with integers that also contain exponents.

Now you try it.

In the following example we provide an explanation for the difference between [latex](-3)^2[/latex] and [latex]-3^2[/latex].

 

 

 

 

 

In the following video we show more examples of how to evaluate expressions with integers using the order of operations.