Recognize equivalent representations

The title of today's blog is a phrase that describes one set of skills that students should be able to exercise by grades 3-5. The full phrase goes like this

... all students should recognize equivalent representations for the same number and generate them by decomposing and composing numbers ...

I know this is where adults say "They didn't teach math this way when I was a kid!" and I sympathize.

We probably learned what these words mean but we'd never explain it in this jargon. Let's see if we can do something with the concept to prove "we got it."

First, we need a number to work with. I nominate this complicated number, which I chose out of thin air.

Here's the numerical form:

243,615.78

In word form, that's

243,615.78 = two hundred forty-three thousand, six hundred fifteen point seven eight

Everyone with me so far? Good. Here comes expanded notation:

243,615.78 = (2 x 100,000) + (4 x 10,000) + (3 x 1,000) + (6 x 100) + (1 x 10) + (5 x 1) + (7 x .1) + (8 x .01)

Let's try a number sentence.

243,615.78 = 250,000 - 6384.22


A more complex number sentence?

243,615.78 = (730,847.34 ÷ 3 ) + 0
 
How about as a fraction rather than a decimal?

243,615 and 78/100

Can you think of another representation? How about an inequality?

243,615.78 ≠ 250,000


How about a graphical representation? This isn't decomposing, it's lateral thinking.

Here's a chart from Microsoft's Excel ...

How about showing this value on a number line?







Here it is in Hexadecimal.

3B79F.C7AE147AE

All of these are the same number. They have the same value. They are equivalent.
For more on equivalent representations of fractions plus a FREE math worksheet, visit our post from April 2012.