## Integers:

There are a few ways you can use to teach students about adding integers.

1. Number line. Addition of integers is presented as a movement of so many units either to the right or to the left. The first number in the expression is your "starting point". If you add a positive integer, you move that many units right. If you add a negative integer, you move that many units left.

For example, 5 + (−6) means you start at 5, and you move 6 units to the left. −9 + 5 means you start at −9, and move 5 units to the right.

This idea is usually relatively simple for students to grasp.

2. Counters. These are represented as little circles with + or − signs drawn inside them, or something similar. For example:

+ + + + +
− − −

This represents 5 + (−3).
Each plus-minus pair cancels, so the answer is positive 2.

− − − − − − − −
+ + +

This represents (−8) + 3.
Each plus-minus pair cancels, so the answer is −5.

### Subtraction

You have several options how to present subtraction of integers. Personally, when subtracting a positive integer, I think in terms of jumps on the number line, and when subtracting a negative integer ("the double negative"), I change those to additions.

1. Number line. Here, 2 − 5 would mean that you start at 2 and move 5 units to the left, ending at −3. This is identical to interpreting the addition 2 + (−5) on the number line.

Similarly, −4 − 3 would mean that you start at −4 and move 3 units to the left, ending at −7. This is identical to interpreting the addition −4 + (−3) on the number line.

Subtracting a negative integer using number line movements is a bit trickier. Problem such as −4 − (−8) would mean that you start at −4, you get ready to move 8 units to the left (the "minus sign"), but the second minus sign reverses your direction, and you go 8 units to the right instead, ending at 4.

Please also see these animations that illustrate adding and subtracting integers on a number line.

2. Patterns can be used to justify the common rules for subtracting integers. First, consider subtracting a positive integer. Do a little pattern for the student to solve, and observe what happens with the answers:

3 − 1 =
3 − 2 =
3 − 3 =
3 − 4 =
3 − 5 =
3 − 6 =

Here's another, similar pattern. Ask the students to observe the answers, and then continue the pattern:

(−4) + 2 =
(−4) + 1 =
(−4) + 0 =
(−4) − 1 =
(−4) − 2 =
(−4) − 3 =
etc.

Another great idea is to use a change in temperature: 5 − 9 means the temperature is 5° and drops 9 degrees.

(−4) − 8 means the temperature is −4° now and drops 8 degrees. This is of course conceptually the same as number line jumps.

The last pattern I show here actually justifies the rule for subtracting a negative integer, such as 7 − (−2). Observe the pattern and see what happens:

3 − 3 =
3 − 2 =
3 − 1 =
3 − 0 =
3 − (−1) =
3 − (−2) =
3 − (−3) =
3 − (−4) =

Students are led to discover the shortcut that two negatives turns into a positive!

3. Counters are trickier to use with subtraction, but we can do it. The basic idea is to interpret subtraction as "taking away". For example, with (−4) − (−2), you start out with 4 negative counters and you take away two negative counters. You are left with 2 negative counters.

4. Difference. Remind the students that 5 − 2 denotes the difference of 5 and 2, which is 3. You can think of the difference as the distance between the two numbers on the number line. However, you need to write the greater number first! If we wrote 2 − 5 instead, it wouldn't work, because distance can't be negative.

Using this idea, (−2) − (−9) would mean the distance between −2 and −9, which is 7. However, (−9) − (−2) would be −7, because the numbers wouldn't be in the correct order where the greater number would be first. Similarly, 4 − (−2) would be 6 since that IS the distance between 4 and −2. In −6 − (−3), the numbers are in the wrong order for calculating distance, so we take their distance as negative, and the answer is −3.

The video below shows how to use THREE of these different models for subtraction of integers: 1) the number line model, 2) the concept of difference, and 3) counters.

### Multiplication

The quickest way to multiply negative numbers is ny memorizing these little rules:

negative × negative is positive

positive × positive is positive

negative × positive is negative

positive × negative is negative.

In other words, if the two integers have a different sign, then the product is negative, and otherwise it's positive.

But let's also explain WHY it works that way.

1. A positive × a negative integer, for example 3 × (−8).

This can be written as repeated addition:

(−8) + (−8) + (−8) = −24

2. A negative times a positive, such as (−5) × 4.

By the fact that multiplication is commutative, you can turn this around and then by (1) above, the answer is negative:

(−5) × 4 = 4 × (−5)
= (−5) + (−5) + (−5) + (−5) = −20.

3. A negative times a negative. Make a pattern:

(−3) × 3 =
(−3) × 2 =
(−3) × 1 =
(−3) × 0 =
(−3) × (−1) =
(−3) × (−2) =
(−3) × (−3) =
(−3) × (−4) =

Observe how the products continually increase by 3 in each step. Therefore, we get (for example) that (−3) × (−4) = 12. So, a negative times a negative is a positive!

You can also see it in this animation.

Another justification for this rule can be seen using distributive property.

Distributive property of arithmetic states that a(b + c) = ab + ac.

If we choose a = (−1), b = 3, and c = (−3), the distributive property gives us:

(−1)(3 + (−3)) = (−1)(3) + (−1)(−3)

Now, since 3 + (−3) is zero, the whole left side is zero.

So the right side, or (−1)(3) + (−1)(−3), must be zero as well.

(−1)(3) is −3. It follows that (−1)(−3) has to be the opposite of −3, or 3.

This last part might be too difficult for 6th graders to grasp. But they don't have to grasp it all; you can say that sometimes we have to just follow the rules and understand the "why" fully later. They can probably understand it partially

### Division of integers

The rules for division with negative numbers follow because division is the opposite operation of multiplication.

For example, what is (−21) ÷ (−7) ? Let's call the answer to that A.

So (−21) ÷ (−7) = A. It follows that A × (−7) = (−21)

Knowing the multiplication rules, the only number that fits is 3. So (−21) ÷ (−7) = 3.

You can make similar cases for (−21) ÷ 7 and 21 ÷ (−7).

In reality, mathematicians would not use specific numbers like 21 and 7, but variables. I used specific numbers to make the argument easier to grasp, plus this is the way you'd probably explain it in 6th or 7th grade.

Of course the students will memorize the little rules for division of integers and use those in computations, but studying where those rules come from is very enlightening, and also needful, I feel.

1. https://www.math-drills.com/integers.php
5. Integer Arithmetic http://faculty.kutztown.edu/fowler/Handouts/Math/MAT103/IntegerArithmeticPre-TestWorksheet.pdf

1. Integers, Decimals, Fractions https://youtu.be/BX0ztZGKQiU