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Lessons From the Classroom

Fawn Nguyen Fawn Nguyen

Comparing Fractions

I was taught to find a common denominator when comparing or ordering fractions. Find the LCM. Even change them into decimals.

Instead, I want to compare fractions in these three ways:

  1. Using number sense and the fraction 1/2

  2. Finding a common numerator

  3. Thinking of them as perfect pinks or not

Compare 3/4 and 5/12.

3/4 is more than 1/2, and 5/12 is less than 1/2, hence 3/4 > 5/12.

Compare 1/5 and 2/7.

1/5 is the same as 2/10, and 2/10 is smaller than 2/7 because pizza.

Compare 5/7 and 6/9.

A perfect pink color is made from 1 part red and 2 parts white, expressed as 1/2. I have a visual sense of this color, this red/white ratio of 1/2. Anything greater than 1/2 is dark pink, and anything less is light pink.

So 5/7 is darker, and to get to the color of 6/9, I’d need to do this: 5/7 + 1/2 = 6/9. If I’m adding something less dark than what I’d started with, then I’m going to dilute the color, so 6/9 is less dark than 5/7. I conclude 5/7 > 6/9.

Of course if I add the exact same color to itself, then there should be no change: 2/3 + 2/3 = 4/6.

Compare 19/60 and 21/55.

19/60 is light pink, so is 21/55. To get from 19 to 21, I’d need to add 2 reds, and to get from 60 to 55, I’d take away 5 whites. The net effect is more red, making 21/55 darker, or 19/60 < 21/55.

This is not unlike how batting averages work.

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Fawn Nguyen Fawn Nguyen

Common Denominator

I already wrote about dividing fractions here and here.

I use the explanation of "dividing by one" to explain why 5/6 divided by 2/3 is the same as 5/6 times 3/2.

But when I was asked recently about how the "common denominator" strategy worked, my muted response was, "Because it does." I didn't mean to be a jerk, rather I just hoped she'd go along with me.

I grabbed a piece of paper and wrote 5 ÷ 3 = 5/3. She was already bored with me. Then I added 1s under the numbers to show 5/1 ÷ 3/1 = 5/3. Right, right? I then changed the problem to 10/2 ÷ 6/2 = 10/6... = 5/3. Still okay, right?

Before I could give another example, she took the paper and rubbed it on my head. Rude.

***

The real common denominator is we're all in this together to #flattenthecurve. This tweet is like rainbow.

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Teach Teach

Dividing Fractions

My 6th graders have been working with dividing fractions for the last two weeks. We explore these four ways, in this order:

  1. Number line

  2. Rectangles — I wrote about this here.

  3. Dividing by one

  4. Common denominator

It's completely intentional that we work with the number line and rectangles first. I want my kids to see the answer and that it should match their intuition and understanding.

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