How to teach operations with integers
including printable summary fact sheets for download

Addition

Addition and subtraction on number lines can be presented in a few various ways and it CAN get confusing.

In addition to number line, try this:

For ADDITION only, use positive/negative markers, such as little balls with + or - sign drawn inside them.

For example,

+ + + + +
- - -

represents 5 + (-3).

The pluses and minuses cancel each other and the answer is positive 2.

Subtraction

Try split it to two cases:

1. Subtracting a positive integer.

   For example, consider 2 - 5. 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 =

   Also here you can use the number line. For example, 5 - 8. Place your finger at 5, and show or draw an arrow that is 8 units long towards the left. You will 'end up' at (-3).

   Then do the same when your starting point is a negative number, such as (-4) - 5. Start at (-4) and move 5 units to the left.

   Even when subtracting from a negative integer you can use a pattern, and ask the student to observe the answers, and then continue the pattern:

   (-4) + 2 =
   (-4) + 1 =
   (-4) + 0 =
   (-4) - 1 =
   (-4) - 2 =
   etc.

   Also use temperature dropping examples:
   5 - 9 means temperature is 5 degrees and drops 9 degrees.
   (-4) - 8 means temperature is -4 now and drops 8 degrees.

2. Subtracting a negative number.

   This is the case of 7 - (-2) or (-4) - (-3).

   Observe the pattern and see what happens:

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

   I've seen illustration on number line that if you're doing 3 - (-4), you're going to subtract, so you're initially heading 4 units left, but this extra minus sign makes you turn 180 degrees so to speak so that you head 4 units to the right instead. But this is where that can get confusing.

   I personally just remember the rule that two negatives turns into a positive.

Multiplication

Multiplying with negative numbers is EVENTUALLY quickest to do by just memorizing the little rules:

negative x negative is positive
positive x positive is positive

negative x positive is negative
positive x 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 if your student or you would like to know a little bit as to WHY it all works that way, use this:

1. 3 × (-8) or when you have positive × negative:

   This can be written as repeated addition:
   (-8) + (-8) + (-8) = -24

2. (-5) × 4 or negative times positive.

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

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

3. Negative times negative. Make a pattern:
   

   (-3) x 3 =
   (-3) x 2 =
   (-3) x 1 =
   (-3) x 0 =
   (-3) x (-1) =
   (-3) x (-2) =
   (-3) x (-3) =
   (-3) x (-4) =

   and observe how the products continually increase by 3 in each step.

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

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

   So, if a = (-1), b = 3, and c = (-3), it should still hold:

   (-1)(3 + (-3)) = (-1)(3) + (-1)(-3)

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

   So (-1)(3) + (-1)(-3) must be zero as well.

   (-1)(3) is -3. So it follows that (-1)(-3) has to be opposite of -3, or 3.

   This last part might be too difficult for 6-7th 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 now.

   The negative x negative makes positive rule has to do with the fact that IF we made it to he positive, then all these neat rules/properties of arithmetic wouldn't hold for negative numbers... but since we want them to hold, since we DO want mathematics to be a very consistent system, then we make negative x negative to be positive.

Division of integers

Division follows because it's the opposite operation of multiplication:

What is (-21) ÷ (-7) ? I call the answer A.

(-21) ÷ (-7) = A.

It follows that A × (-7) = (-21)

Knowing the multiplication rules, the only number that fits A is 3.

And so on. Just make a similar case for (-21) ÷ 7 and 21 ÷ (-7).

(In reality, mathematicians would not use specific numbers like 21 and 7 but just variables; I wrote this with specific numbers to make it easier to grasp the argument, plus this is the way you'd probably explain it to a 6th or 7th grader.)

Of course with division too the student will just memorize the little rules and use those in practical computations.

But studying the logic behind all this is very enlightening.

Integers fact sheets

You're welcome to download and print these integer operations fact sheets for your students.

All I require is that you not modify them.

Add_Subtract_Integers_Fact_Sheet.pdf

Multiply_Divide_Integers_Fact_Sheet.pdf

These fact sheets are pulled out from Math Mammoth Grade 7 Worksheets Collection.

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