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* *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 1a, Part 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

Lesson 7 of 9

## Objective: SWBAT use units to understand problems, and choose appropriate scale on a horizontal axis.

*43 minutes*

#### Patterns Quiz

*15 min*

For the first 15 minutes of today's class, students will take their second Patterns Quiz of the year.

As students arrive, I've posted the rubric from our previous Patterns Quiz, which looks like this: Patterns Quiz Rubric. I post this photograph on the board, and tell students that they'll be graded in exactly the same manner as last time, with an emphasis on trying every problem.

This is a second chance for students to take this quiz, and another opportunity for me to learn about my students. One thing I'll learn is how well they can retain information or reference prior notes. It's been two weeks since they took the previous quiz, and we haven't specifically studied these sorts of problems in the last few classes. All of my quizzes are open-notebook, so as kids take this one, I'm looking to see who can use those "older" notes, and who's completely at a loss in that regard.

Also, even though it's been a few days since we've gone over the algorithms for thinking about these problems, our work has involved patterns in other ways: on the addition and multiplication tables, we've seen all sorts of patterns, and then on Part 2c of the Number Line Project, kids have been describing those patterns in words. I'm curious to see if any of this work helps them solve pattern problems like those on this quiz, and I'm looking for kids who reference that project work while taking the quiz.

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Part 3 of the Number Line Project follows an arc similar to that of Part 1. Part 3a, which I introduce here, is fairly algorithmic and guides students with various scaffolds. Part 3b requires students to make a similar set of number lines from scratch, on loose-leaf, with no scaffolding. Part 3c is like Part 3b, with a little greater complexity. What's interesting about 3b and 3c is that they're actually pretty simple once a student makes it that far: if they've been successful on all the other parts of this assignment, these last pieces (which I describe in tomorrow's lesson) should not be too difficult.

Please check out my narrative video for an overview of Part 3a.

*expand content*

With about five minutes left in the class, I remind everyone that the whole project is due on Friday (two days from now). I tell students that when they show up for class on Friday, I'll expect them to bring everything they've done. They will fill out a reflection sheet, paper clip everything together, then submit their work. "There won't be much time for you to complete unfinished work on Friday," I say. "So make sure that you get done whatever you can tonight."

Most students started Part 3a in the 25 minutes they had to work today, so that's likely to be the homework for most kids. I tell everyone to make sure that they've recorded whatever they're assigning themselves on this week's homework sheet. I advise everyone to decide what they're going to accomplish tonight and get organized in these last few minutes of class, then I circulate to help anyone who needs it.

*expand content*

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- UNIT 1: Number Tricks, Patterns, and Abstractions
- UNIT 2: The Number Line Project
- UNIT 3: Solving Linear Equations
- UNIT 4: Creating Linear Equations
- UNIT 5: Statistics
- UNIT 6: Mini Unit: Patterns, Programs, and Math Without Words
- UNIT 7: Lines
- UNIT 8: Linear and Exponential Functions
- UNIT 9: Systems of Equations
- UNIT 10: Quadratic Functions
- UNIT 11: Functions and Modeling

- LESSON 1: Introducing the Number Line Project
- LESSON 2: Fractions and Decimals on the Number Line
- LESSON 3: Irrational (and Other!) Numbers on the Number Line
- LESSON 4: The Number Line Project, Part 2: Two Dimensional Number Lines
- LESSON 5: Patterns on Two Dimensional Number Lines
- LESSON 6: Workshop Period: The Number Line Project
- LESSON 7: The Number Line, Patterns, and Units
- LESSON 8: The Number Line Project: More Unit Lines and Finishing Up
- LESSON 9: Project Collection Day, and an Introduction