### Distributive property

A little bit of math for you today.

Usually kids encounter these words in algebra or maybe pre-algebra. But, they use the principle long before that, say in 2nd or 3rd grade. Did you know where?

The distributive property says (as it's stated in algebra class) that

a (b + c) = ab + ac.

It's true for all real numbers - including negative ones, so it follows it's true for subtraction too:

a (b - c) = ab - ac.

We use this principle often when multiplying numbers. For example, do in your head 7 × 21. Most people take 7 × 20 and 7 × 1 and add those. In other words, you 'broke down' 21 to two parts and multiplied the parts by 7 separately:

7 × (20 + 1) = 7 × 20 + 7 × 1.

How about 8 × 98? Aren't you tempted to go 8 × 100 and subtract 16?

8 × (100 - 2) = 8 × 100 - 8 × 2.

37
×   4

Here, you should explain that this algorithm (or procedure) is based on the idea that you multiply 4 × 7 and then 4 × 30. In other words,

4 × 37 = 4 × (30 + 7) = 4 × 30 + 4 × 7.

You can even write it out this (unconventional) way, which conveys the basic idea better:

37
×  4

28
+   120

148

The idea is, of course, exactly the same if you have a 3-digit number. For example:

5 × 487 = 5 × (400 + 80 + 7) = 5 × 400 + 5 × 80 + 5 × 7.

You can again write it out this (unconventional) way, which conveys the basic idea better:

487
×  5

35
400
+   2000

2435

Or, the conventional way looks like this:

4 3
487
×  5

2435

The idea is STILL the same if you're multiplying 2-digit numbers by 2-digit numbers; you just have more 'parts' in your numbers. For example, 38 × 47. You need to do 30 × 40, 30 × 7, 8 × 40 and 8 × 7, and add all those.

How about writing it out like this for kids:

38
×  47

56
210
320
+  1200

When you explain WHY it works, and show HOW it works, you are teaching both conceptual knowledge and procedural knowledge.

There is a lot of debate as to which should be taught first. I don't really know; I don't even know if it perhaps varies from child to child. I think this is the way I would do it:

1) Teach why it works - but not expect the student to grasp it totally.
2) Teach how it works and have him practice.
3) Teach again why it works.
4) Again practice how it works.

After going back and forth a few times between the concept and the procedure, hopefully both sink in. Also I would require knowing the 'why' in a test: I would ask the student to explain, in his own words, why it works.

Remember, he's going to need to know it well in algebra.

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