Tuesday, August 1, 2017

Discovering Math Tricks, Secrets, and Facts To Really Learn Algebra Mathematical Principles

"Dy, quick, what is 5 times 484,682?"
[about 1 second later]
"2,4,2,3,4,1,0 ....  that's an easy one dad ... give me toughie"

I love these kinds of interactions with my 9 year old daughter.   It allows her to show off her abilities, add to her confidence, flex her mental muscle, and work on her mental math skills.  But how did she know the answer so quick?  By applying a simple math fact*.

Cool as this is, what is more powerful and important is that an awesome mathematical fact (like the one she applied here) provides a vector for learning bigger, more important Mathematical principles.

To explain this more plainly, let's look at the math fact that Dy used.  You may have already figured it out, based on the way I wrote her response.

Here is the math fact she used:
5 times any rational whole number is simply that number divided by 2, times 10 (which is simply tacking on a zero if the starting number is even)

So, for her I made the problem easy.  First, I led off with "5 times".  At that point, Dy triggered into her 5 times facts.  Then I gave her a number that she could divide by 2 in her mind easily, simply going down the line of the even digits that made up the number 484,682.  So, she retained that number in her mind (a good mental exercise in its own right) and spoke out the result of dividing each digit by 2.  So, 4 ÷ 2 = 2, 8 ÷ 2 = 4, 4 ÷ 2 = 2, 6 ÷ 2 = 3, 8 ÷ 2 = 4, 2 ÷ 2 = 1, and times 10 so 0. So, 2423410.

She knows this fact now, which is cool.... and maybe during some test for college, a problem will come up with a 5 times in it and her speed will be helped if she recalls this fact .... but the real cool part is that in "discovering" this fact and how it works, she has nicely teed up algebra.

Here is the flow of her discoveries.

First, I challenged her to demonstrate her ability to multiply anything by 10 as I gave her a series of numbers (2, 4, 42, 544, 18, 20, 8, 532934, ....), which she did like a champ. Then I asked her to describe for me all the ways she could think of to write the number 10. I prompted her with "Like, 20 ÷ 2, or 10 x 1"  She paused, thought about it, and blurted "100 ÷ 10".  Awesomesauce! I kept poking her to produce more, and then she shared "2 x 5." Ahaha!  The one I wanted.

Second, I asked her to write down "4 x 10", which she did.  Then I asked her to replace the 10 with her example of the 10 equivalent, 2 x 5.  This resulted in "4 x 2 x 5."  We then explored multiplying in different orders, and to her amazement it always worked.  4 x 2 is 8 and 8 x 5 is 40, etc.

Third, I asked her if she could re-write 10 but with addition.  She wrote down 3 + 7.  I asked her to replace her 10, in her 4 x 10, with her 3 + 7,  which resulted in her writing 4 x 3 + 7.  I then asked her to solve, which yielded 19! (4x3=12, 12 + 7 = 19).  "Dad, that doesn't work." "No baby, it doesn't. The order you do this, the operation, matters."  We then launched into a nice discussion of order of operations, parenthesis, etc. for 30 minutes. :)

Fourth, I then asked her to go back to her original equation, 4 x 10 and write it out.
"4 x 10 = 40".  Next, I had her replace the 10 with her 2 x 5 version.  She wrote, "4 x 2 x 5 = 40".  Good.  Now I asked her "well, let's do the 4 x 5 first of the 4 x 2 x 5, since the order doesn't matter."  "That would be 20 then dad, so we have 20 x 2"  Me: "Well, that's interesting because 20 is half of 40, so that that makes total sense."

"Using this stuff, Dy, here is something that pops out that I'd like you to think about.   For any number that you multiply by 5, you could simply multiply it by 10 and cut it in half .... or cut it in half first and then multiply it by 10."  It wasn't instant that she understood it, but after 3 examples, she knew the trick.  After 10 examples, she was convinced. Trick learned.

Fifth, "Dy, let's replace the 40 with 20 x 2"  4 x 2 x 5 = 2 x 20  "Dy, do you see that both sides has the 2?  Does the equation work if we simply pull the 2 out?" It sure does.  

Sixth, "Let's re-write our equation but instead of writing the 2, let's replace it with a star"  Dy then writes out 4 x ☆ x 5 = ☆ x 20.  "I wonder if we can replace that star with anything?" Dy then starts trying numbers out.  2 works of course.  3 works.  How about 10.  Wow, she finds that everything works.  Neato!

Seventh, "We could always simply pull out that ☆ and it would be true right?" Dy then erases the star and sees 4 x x 5 = 20.  Big smile on her face.

A couple of wonderful hours with my little girl, and she now has an amazing foundation start with Algebra.

[Sidebar *: Early in this post, I used the phrase math fact.  I am using the word fact here because my wife hates when I call it a trick.  She likes to point out that it isn't a trick, just an application of some facts.  She is, of course, right (aren't they always! ;)  but many kids like to have secrets or tricks, and calling it a math secret could make your kid even more interested)]

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