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How to teach operations with integers
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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.
Counters. These are represented as little balls with + or - sign drawn inside them, or something similar. For example:
+ + + + + − − −
This represents 5 + (-3).
Each plus-minus pair cancels,
− − − − − − − − + + +
This represents (-8) + 3.
Each plus-minus pair cancels,
You have several options how to present subtraction of integers. Personally, I think of situations where we subtract a positive integer in terms of the number line, and the situations where we subtract a negative integer ("the double negative"), I change those to additions.
One is the familiar number line. Try split it to two cases:
Number line. Here, 2 − 5 would mean that you start at 2, and you 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 you 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.
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.
Pattern to justify the rule for subtracting a negative number. This is the case with a problem such as 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) =
Students are led to discover the shortcut that two negatives turns into a positive!.
In other situations, you may not initially have the counters that you are supposed to take away. For example, in 5 − (-3), you start out with 5 positive counters, but you are supposed to take away 3 negative counters. How to do that? The trick is to first add enough pairs of negative-positive counters to the situation - this amounts to adding zero, so it is alright. Then you can take away what you need.
+ + + + +
5 − (-3).
We cannot take away three
+ + + + + + + + − − −
Now, taking away three
Using this model, (-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 order of having the greater number first. Similarly, 4 − (-2) would be 6 since that is the distance between 4 and -2. -6 − (-3) would have the numbers in the "wrong" order, so we'd take their distance as a negative number and the answer would be -3.
The video below shows how to use THREE of the different models for subtraction of integers: 1) the number line model, 2) Concept of difference, and 3) counters.
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:
This can be written as repeated addition:
(-8) + (-8) + (-8) = -24
See also this clever animation about a pattenr in multiplying 2 x (a number), and when it goes to the negative numbers.
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) 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. 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.
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 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.
You're welcome to download and print these integer operations fact sheets for your students.
All I require is that you not modify them.
These fact sheets are pulled out from Math Mammoth Grade 7 Worksheets Collection.