## Reflection: Developing a Conceptual Understanding The Number Line, Patterns, and Units - Section 2: The Number Line Project, Part 3

Throughout the Number Line Project, students have to make sense of fractions.  On different parts of the project, kids have opportunities to develop their own conceptual understanding of the order of fractions and what these numbers really mean.  (To see what preceded today's activity, check out Part 1aPart 1b and Part 1c.)

Scale on a number line - and eventually on vertical and horizontal axes - will be so important when we move on to investigating slope, data representations, and the shapes of other functions.  A deep understanding that fractions are real, useful tools can be cultivated by looking at number lines, and anything students can learn about fractions now will make it easier for them to understand those upcoming topics.

Today, I got to watch a group of students make some exciting breakthroughs as they worked on Part 3a of the project.  They were working on the "Parts of an Hour" number lines.  They quickly acknowledged that 1/2 hour is equivalent to 30 minutes, and they were excited to be able to recognize that 1/4 of an hour is equivalent to 15 minutes, so we got to talking about other fractions of an hour.  Soon, we noted that 20 minutes is equivalent to 1/3 of an hour, and the group filled in their number lines accordingly.  Then lightning struck!  A student looked at here paper, which I've snapped a picture of here, and said, "Wait!  Why is 1/3 closer to 1/4 than 1/2?"

Think about that question.  What does she understand so far?  What is she just noticing for the first time?  I note that she understands scale and how numbers are spaced on a number line, but that she's still coming at these numbers from the perspective of whole numbers.  The integers 2, 3, and 4 must be evenly spaced on a number line, but the same cannot be said of their inverses.  The minutes number line helps to make that clear, and in this moment, I took the approach of talking about the minutes line.  "We can see that 15, 20, and 30 minutes are spaced correctly on the minutes line, so fractions of an hour should be spaced the same way," I told the group.  There is a lot more to say, but for now I left it at that, to see what the group would do with it.

As kids often do in my class, the group moved to the board to have their own conversation about other values on the "parts of an hour" line.  Here is the result of their work.  What they realized is that "2/12 of an hour" is equivalent to 10 minutes.  Thinking is sets of 10 comes naturally to us humans.  When we want to think of the quantity 40, we're pretty comfortable thinking of "four 10's".  This group used that understanding to see that sets of fractions can be combined in the same way.  When kids develop the concept that fractions can be component parts of numbers, they're able to see fractions in a new light: not as a needlessly confusing construction, but as an indispensable tool.

It is also worth noting that the same breakthrough can - and often does - happen on Part 1b of the project, but for this particular group of students it took until now for that to happen.  That's a key component of Standards-Based Learning: kids need multiple opportunities to develop important ideas on their own terms.

Developing a Conceptual Understanding: Why is 1/3 closer to 1/4 than 1/2?

# The Number Line, Patterns, and Units

Unit 2: The Number Line Project
Lesson 7 of 9

## Big Idea: Units of measurement add context to the number lines that we've been creating.

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Subject(s):
Math, Measurement, proportions, Units and Systems of Measurement, Number lines , Number Sense and Operations, unit conversions (within a system), Project Based Learning
43 minutes

### James Dunseith

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