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 I read your response to How is the square root of 3/4 is greater than 3/4? What I'm hoping for is a way for my students to use their own experience and number intuition to be able to make sense of the issue. As soon as my kids see "if y is this and x is this then..." their little eyes glaze over. Unfortunately, I can't come up with a way myself. Thank you for your help. Blaine

We have two responses for you

Blaine,

I would try money.

What's half of sixty cents. It's thirty cents. That is

0.50 × 0.60 = 0.30.

The answer, 0.30, is smaller that 0.60 since one half is a fraction less than 1.

What's half of fifty cents. It's twenty-five cents. That is

0.50 × 0.50 = 0.25.

So now let's turn this arithmetic problem around

 0.25 = ×

I want the same positive number in each of the green squares. I know this is a number less than 1 and I know from my experiments above that multiplying by a fraction less than 1 results in a reduction. Thus the number in the green squares must be larger than 0.25. Similarly for

 0.75 = ×

I hope this helps,
Penny

Hi Blaine.

You might take the angle that when we multiply or divide by 1, the number doesn't change.

In fact, When we multiply a number times itself, it always gets further from 1 (unless of course it IS 1). Students may have to try this to convince themselves.

Similarly, when we take a square root, we are doing the opposite of multiplying it by itself, so that moves it closer to 1.

Of course, you want to avoid any discussion of negative numbers in this!

Hope it helps,
Stephen La Rocque.

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