One way I get around that is to try to hold myself still with a problem for a bit and look for the hidden connections. With kids, I might tell them we're looking for the

**Hidden Wows**.

Suppose I am preparing a lesson for sixth graders to introduce problems of this form:

My planning might start out sounding a bit like this:Solve:ax=b

**First of all:**These are so easy, you just divide both sides by

*a*. How will I make this interesting? And what in the world are they going to struggle with? The problems are all the same!

Well, how about (1/2)x = 5. I actually wouldn't divide by (1/2) in that one, I would you multiply by 2. Interesting. I wonder how else you could solve that.

Actually, you don't need to divide or multiply both sides by anything... you can just use your reasoning:

"Half of what number is 5? Well, 10 of course!!"

**Wow!**I'll try to remember to value that if anyone suggests that.

But the book tells us we are supposed to "divide by 1/2" on both sides.

Ok, so what if I do that? We get:

x = (5 div 1/2).Uh oh, they'll need to understand what it means to divide 5 by 1/2? How would they think about that? Well last year they learned long division of decimals, I wonder if any of them will go there: I could write 1/2 as 0.5, and then do long division:

___Oh, and that makes sense! The quotient of 5 / 0.5 should be 10 because there are ten 0.5's in 5.

0.5 | 5 --> Yuck. No "Wows" yet. We need to "move the decimal points".

____

5 | 50 --> Well, that's easier, anyway. The answer is 10.

**Wow!**That's a neat connection.

I wonder why we can move the decimal points like that. Hmm...

Let's see... 5 / 0.5 can be renamed 50 / 5 (multiply numerator and denominator by 10). Hey, that's neat! Moving the decimal point in a long division problem is just another way of renaming fractions. That's cool!

**Wow!**I wonder if there's an easier way for them to divide 5 by 0.5...

So moving the decimal point in long division is just like renaming the fraction. I wonder... what if instead of renaming using 10s (which moves the decimal), what if I renamed using 2s (double both the divisor and dividend). Then I could "rename the quotient" this way:

___

0.5 | 5

___

1 | 10

**Wow!**It's just like rewriting 5 / 0.5 as 10 / 1. My teachers never showed me that!

I wonder what else I can notice.

Well, the numerator and denominator of 5 / 0.5 are the numbers from the original problem, (1/2)x = 5. And if we multiply both sides of that equation by 2 (just like we did when renaming the fraction) the whole equation becomes much simpler: in fact, it becomes 1x = 10. Well that's easy!

**Wow!**I wonder if I could multiply both sides by 10, too, like we did to rename 5/0.5 as 50/5? Let's see:

(1/2)x = 5 ----> (10)(1/2)x = (10)(5) -----> 5x = 50.Well, that's not much easier, but at least the fraction is gone now. I could think of solving 5x=50 as "5 times what equals 50," which is a little easier than "(1/2) times what equals 5".

Hey, it's another

*a*x =

*b*problem, except without the fraction! The book tells us we need to solve

*a*x=

*b*by dividing both sides by

*a.*In this case, 5x = 50, that's probably what I would do: divide both sides by 5. But.... I could also multiply both sides by the reciprocal of 5: (1/5)5x = (1/5)50.

So solving (1/2)x = 5 can be done by:

- multiplying both sides by 10 and then dividing both sides by 5, or
- multiplying both sides by 2, or
- dividing both sides by (1/2).

**Wow**, you know what? This 6th grade algebra is kind of fun...

**Here's one for you to try:**

Let us know what you find! Happy hunting!Find the "Hidden Wows" in the problem

x - 5 = -3

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