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# The Number Line Project, Part 2: Two Dimensional Number Lines

Lesson 4 of 9

## Objective: SWBAT use horizontal and vertical number lines to create addition and multiplication tables that may be used to answer such questions as "why is a negative times a negative a positive?"

## Big Idea: The mathematical structures we build today will be used to justify many algebraic ideas in the coming months.

*43 minutes*

As students enter the classroom today, I welcome them and hand each an index card. On the screen at the front of the room is this image, Index Card Part 1.pdf, which shows students what to write on the card as today's opener. In a week that has been filled with a lot of independent work time, this is another check in. On the front of the card, students will write what they've accomplished so far on each part of the project, then, at the end of class, they will respond to another prompt on the back of the card and turn them in.

I give students 2-3 minutes to write what they have done on their index cards, and I circulate to make sure that they're paying attention to the details of how I'd like the information arranged on the card. After that time is up, I say, "Make sure not to lose your index card! I'm going to collect these at the end of class."

#### Resources

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I spend a few moments running through today's agenda, which includes some practical information about the Number Line Project. Today, everyone will begin Part 2 of the project, no matter what they've accomplished so far on Part 1. Anyone who needs to go back to Part 1 will have that opportunity after completing Part 2. In total, the project has three parts (each of which has three sub-sections, a, b, and c), and all of this will be due in a week, next Friday.

I remind students of their weekly homework sheet, which includes a checklist of all parts of the project and a place to record nightly homework. The homework that students do each night is up to them; it depends on what they have accomplished so far and what they still need to do. I make the point that everyone should be doing *something* for homework every night, and that everyone should be working to stay organized. Hard work and organization are central to success in this class, and as much as the Number Line Project helps students develop the number sense that will help them as this class continues, habits of work, organization, and attention to detail are just as important to cultivate here.

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Part 2 of the Number Line Project is called "Two-Dimensional Number Lines". I love this activity, and it lays a fantastic foundation for many of the algebra topics that we'll study in the coming months. We will refer back to the documents that students produce today throughout the year. This activity is all about **looking for, finding, and using structure (MP7)**, and being able to **describe the repeated reasoning (MP8)** that all students employ, whether consciously or unconsciously, as they complete these tasks. A big part of my teaching role today is to help students notice when they're using repeated reasoning, and then to help them describe what they're doing.

To begin, I distribute this double-sided handout. I tell students to fill in the blanks at the top right. Where the document says "Part 2__", I tell students to write an "a", then I ask for the name of Part 2a. By referencing their project checklists, students can see that this is called "Addition Tables".

**One, Two and (maybe even) Three Dimensions**

For students, the set-up looks like this. At the top of the page, this student noted that "a line is one-dimensional" and "a rectangle is two-dimensional". That's part of what I explain to students as we transition from Part 1 to Part 2. What we have here are two number lines: one horizontal and one vertical. I say that I expect students have seen this before. We talk about how lines are one-dimensional shapes, and that this includes the number lines that we saw in Part 1. When we use two perpendicular number lines, now we're talking about two dimensions. We can refer to these dimensions as "left and right" and "up and down", or we can call them the horizontal and vertical dimensions. "You might also know these two lines as the x-axis and the y-axis," I say, and I label the axes as such on the board. It's also worth noting that we live in a three-dimensional world. "What would that third dimension be?" I ask. "If I can move from left to right, or if I can jump up and down, how else can I move in this room?" We note that I can also move forward and backward, and I point to the flat image on the board. "Where would the third axis go, and what would we call it?" Pointing to the origin, I say that the third axis would come straight out from the wall and travel through the room. I can usually count on a clever student calling this the "z-axis" without really believing what they're saying, and it's always fun to say that they're exactly right: in three dimensions, we have x-axis, y-axis and z-axis. But that's all getting a little ahead of what we're doing today.

**Labeling the Axes**

With this background established, it's time to label the axes with numbers. The bold arrows on this handout represent the x-axis and y-axis, and these we will label "0", on the outside of the diagram. Then, we just count up on both number lines. I'm careful to make sure that students are labeling the outsides of their axes - not yet writing anything in the circles.

When that's done, it's time to fill in the circle, each of which lies at the intersection of two lines. "In each circle," I say, "write the sum of the numbers on the outside of your table." So in the circle at the bottom left, I ask for the sum of zero and zero. We fill that one in. Next to that is the sum of 0 and 1. In the top right corner is the sum of 13 and 14, and I tell everyone that they should fill that in too. I ask if anyone has clarifying questions, then I say, "Alright, go for it - see how quickly you can fill in the rest of your addition table."

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In one sense, this is just about the busiest busy work I'm going to give students all year. In another sense, to see this as busy work is to miss the point. "Did I give you this assignment because I think you don't know how to add?" I ask, and allow students to think about it. "I think that you know how to add - so then, what's the point of doing this?" I tell students to see how quickly they can do this by looking for patterns and repeated numbers - there are many, and I think the best way to find them is to look for yourself. In individual conversations with kids, I like to see what they can tell me. I might prompt them with the question, "Did you really have to think, 'ok, what's 5+1, 5+2, 5+3, 5+4...,' or did you find another way to think about it?" Students know that they're using patterns here, so then I ask them to try to describe what patterns they're using.

**Another Addition Table, then Repeat for Multiplication**

As each student finishes up on the front of Part 2a, I ask them to describe how the other side of this document is different. On the other side is another table to fill in, but the axes have been moved. Again, zero goes on each axis, but then we also have room to fill in some numbers to the left and bottom of that number. In other words, we have room for some negative integers. Students should again label this "Part 2a: Addition Tables", label their axes, and fill in the circles. For some this comes naturally, and for others it's now a little more confusing. Either way, I say that students should continue to use the patterns they noticed on the front. If they're stuck, I have them fill in the axes first, by finding sums of zero and each integer on the number line. Then, by continuing up and down from each number they have, they can fill in the rest.

When each student is done with Part 2a, front and back, I provide another copy of the same document, saying, "This is for Part 2b. What is this part called?" On the project overview, kids see that 2b is called "Multiplication Tables". I ask students what they think they're going to do here. The task is to set everything up the same way, on the front and back, and then to fill in the circles with products instead of sums. "Again," I say, "this is about patterns. See what patterns you can find, and see how quickly those patterns can help you to fill these in."

Here is what the work looks like when it's done:

**Building Urgency**

This is relatively simple work, but there's a lot of it. This offers a nice opportunity to build urgency. I expect that all students will be able to finish all four tables today. There will usually be a few students who manage to distract themselves into not finishing, but for the most part, the majority of every class will get it done. I circulate constantly during the class, making sure that everyone has a pencil to paper and is progressing toward that goal. For students, this is a first-hand example of what it looks like to work hard, and we're not encumbered by overly difficult mathematics.

If anyone finishes with a lot of time left, they can go back and make sure that all their work for Part 1 is complete. If that's done too, I have copies of Part 2c available, which I fully introduce in the next lesson.

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With about five minutes left in class, I tell everyone to find their index cards from the beginning of class. "On the back," I say, "Describe three patterns you saw today." All students can say that they saw patterns, but describing these patterns in words is challenging for many. I say, "That's exactly the point: I want you to practice being able to talk about what you see in this class."

As class ends, I collect the index cards. Even though we'll work on this project for another week, I'm getting a lot of info on these cards! I get a quick snapshot of where all of my students stand on the project and how well they're able to use words to describe what they see when they look for patterns. All of this information helps to inform the scaffolds I'll need to provide next week.

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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