Reflection: Developing a Conceptual Understanding Fractions and Decimals on the Number Line - Section 2: The Number Line Project, Part 1b: From -1 to 1


Students finish Part 1a at their own pace, and move on to Part 1b at different times.  Some breeze through Part 1b, and others struggle to make sense of where each fraction goes.

For some students, just understanding that 1/3 is less than 1/2, and should therefore be placed in-between 0 and 1/2, rather than to the right of 1/2 is a new breakthrough.  That's exactly why it's so important for students to try this activity early in the year.  Imagine trying to compare the slopes of lines without that prior knowledge?

I rarely teach a whole-group lesson about Part 1b of the project during today's lesson, but during an upcoming work-period, I'll usually pull a small group of struggling students aside and say that I want to help them make this number line perfect.

First, I guide students to recognize that 1/2 should be placed exactly halfway in-between 0 and 1.  If there are 20 graph paper boxes separating 0 and 1, then 1/2 should be 10 boxes from each number.  With that in mind, figuring out where 1/4 and 1/5 should go can follow the same logic, and students gain confidence by doing that with minimal guidance.  One-tenth is similarly just a whole number of boxes (2) away from 0.

Then we get to 1/3.  I'll recap for students: "We know that half of 20 is 10, so 1/2 is 10 boxes away from 0.  We know that 1/4 of 20 is 5, so 1/4 is 5 boxes away from 0.  So what about 1/3?"  Kids are quick to recognize that it's not a whole number, and sometimes we end up using guess and check to find the best possible answer to this question.

In the bottom left corner of this photo, you'll see a record of what kids tried today, and you can see some of the notes I give to recap important ideas.  

An important moment comes after we figure out how many times 3 goes into 20.  We end up with 6.7 or 6.67 or "6-point-6 repeating" and then there's a pause in which kids try to remember why we were doing this in the first place.  It's great moment, and I let kids think for a moment before pointing to our result and saying, "What does this number tell us, anyway?  6.7 whats?"  It takes a little while for everyone to re-situate themselves within the context that led us down this wormhole in the first place.  If kids are really struggling, I'll recap once again how we decided to place 1/2, 1/4 and 1/5 where we did.

Finally, we see that since 20/3 is ~6.7, 1/3 should be 6.7 boxes away from 0.  More often we'll say this as, "between 6 and 7 boxes from 0, and a little closer to 7."  From there, we can talk about the scale of the number line, and that each vertical grid line represents 1/20 or 0.05.  Counting by 0.05's, students see that the decimal equivalent to each fraction makes sense.

  Developing a Conceptual Understanding: 6.7 Whats?
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Fractions and Decimals on the Number Line

Unit 2: The Number Line Project
Lesson 2 of 9

Objective: SWBAT locate rational numbers on the number line, and get comfortable expressing these numbers as both fractions and decimals.

Big Idea: Counting is easy, right? We hope to find that counting by fractions is as well!

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