Math For Elementary Teachers

Part 1viii: Introduction to Multiplication

Further Reading:

Using Properties of Arithmetic to Learn Multiplication Facts

At the end of the previous clip I discussed how one can use arithmetic properties of multiplication such as the commutative, identity, and distributive properties to teach multiplication facts. I would like to expand on that discussion here by summarizing the discussion given in section 1.5 of Elementary Mathematics for Teachers by Thomas Parker and Scott Baldridge.

As mentioned in the clip, there are 121 multiplication facts from 0 \times 0 to 10 \times 10. Having to memorize 121 facts can be daunting for anyone, but if we use the properties of multiplication discussed in the previous clip, we can bring this down to a very manageable number.

For starters, if we understand that the commutative property is essentially a “2 for 1” property, in that facts like 4 \times 7 and 7 \times 4 are essentially the same (if you know one, you automatically know the other), the original number of 121 facts reduces to 66 (the original number doesn’t quite reduce in half, as the commutative property doesn’t help us with facts like 3 \times 3 and 8 \times 8).

Any multiplication fact involving a factor of 1 should be automatic from the identity property discussed in the clip. And even though we haven’t officially discussed multiplication by 0 or 10, any fact involving these numbers as factors should be automatic as well. It is easy to justify why multiplication by 0 always returns 0 (for example, 4 \times 0 = 0 because a) 0 added to itself 4 times is 0, b) 4 groups of 0 objects is a total of 0 objects or, c) if you take 4 jumps of 0 on a number line you end up at 0). And from our discussion of place value, it is easy to see that multiplication by 10 simply involves a place value shift (3 \times 10 = 3\ tens = 30). Also, any fact involving a factor of 2 is already known by a student who knows her addition facts (for example, 6 \times 2 = 6+6). So if we combine these strategies for multiplying by 0,1,2 and 10 with the commutative property discussed above, the original list of 121 facts is reduced to 28!

In Part 2 we will discuss mental math in detail, and while the problems we will solve involve larger numbers, several of the strategies can be applied to multiplication facts. For facts involving a factor of 5, since 5=10 \div 2, to multiply by 5 one can first multiply by 10 and then divide by 2 (for example, 7 \times 5 = (7 \times 10) \div 2 = 35. Note that these operations can be interchanged; one can first divide by 2 and then multiply by 10 (for example, 8 \times 5 = (8 \div 2) \times 10 = 40). And finally, we will see that the distributive property can prove quite useful when performing mental math. In the context of multiplication facts, we can use the distributive property on any fact involving a factor of 9. Since 9 = 10-1, we can, for example, think of 7 \times 9 = 7 \times (10-1) = 70-7=63 (see clip Part 2 vii). If we add these mental math strategies to the techniques discussed earlier, the original list of 121 facts is reduced to 15!

Admittedly your students’ mental math abilities will vary and perhaps not all of them will completely grasp the strategies outlined in the previous paragraph. Regardless, approaching facts from the perspective of mathematical understanding as opposed to pure memorization will not only help your students with their facts, but it will also strengthen their mental math skills and general number sense.

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