What number is this?

123? 12.3? 1.23? One has to ask oneself one question: Which one is one?

Earlier this year, I was invited into a classroom to introduce decimals. We had been representing and describing tenths concretely, pictorially, and symbolically. We finished five minutes short, so I gave the students a blank hundred-frame and asked them to show me one half and express this in as many ways as they could.

As expected, some expressed this as 5/10 and 0.5. They used five of the ten full ten-frames it takes to cover an entire hundred-frame. Others expressed this as 50/100 and 0.50. They covered the blank hundred-frame with fifty dots. I was listening *for* these answers.

One student expressed this as 2/4. I assumed he just multiplied both the numerator and denominator of 1/2 by 2. And then he showed me this:

One student expressed this as 500/1000 and 0.500. I assumed he was just extending the pattern(s). “Yeahbut where do you *see* the 500 and 1000?” I ~~asked~~ challenged. “I imagine that inside every one of these *points to a dot* there is one of these *holds up a full ten-frame*,” he explained. As his teacher and I listened *to* his ideas, our jaws hit the floor.

In my previous post, I discussed fractions, decimals, place value, and language. To come full circle, what if we took a closer look at 0.5, 0.50, and 0.500? These are *equivalent* decimals. That is, they represent equivalent fractions: “five tenths,” “fifty hundreds,” “five hundred thousandths,” respectively. From a place-value-on-the-left-of-the-decimal-point point of view, 0.5 is five tenths; 0.50 is five tenths and zero hundredths; 0.500 is five tenths, zero hundredths, zero thousandths. *Equal*, right?

**Hat Tip:** Max Ray‘s inductive proof of Why 2 > 4

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Thanks for the Hat Tip. I love seeing blog posts with student thinking (especially with visuals). Andrew Stadel’s QOTW (http://mr-stadel.blogspot.com/2013/06/qotw-2nd-semester-2013.html) is another great example of student voices shining through on blogs. Even Math Mistakes gives us a chance to listen to students online (even though it’s hard to train myself not to listen for the right answer there…)

Love it, thanks for sharing, Chris. I’m sending this off to my colleagues who introduce decimals to give them some ideas.