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# Patterns on Two Dimensional Number Lines

Lesson 5 of 9

## Objective: SWBAT describe the patterns they've seen in words.

## Big Idea: It's one thing to have a great idea, but real power comes in the ability to take that idea and share with somebody else!

*42 minutes*

Today's opener is my response to the exit I used in the previous class. On that exit slip, I asked students to describe some patterns they noticed while working on Part 2 of the Number Trick Project. This turned out to be very difficult for many of them; although many of my students are coming in with strong computational skills, their ability to describe mathematical ideas in words will need improvement this year.

So with today's opener, I show students a simple, formulaic way to describe a pattern, in hopes that this will give them a starting point on the work that's coming up in today's class. First of all, the opener is on the first of these two pages: Opener and Framing Notes (the second page is for my framing of the lesson, which follows).

For each of these patterns, I tell students that I'd like for them to be able to describe *what the pattern is doing, *and *where each pattern starts. *In addition for giving kids a starting place to talk about patterns, this language will give us the ability to transition from talking about these as number patterns to seeing them as linear functions. Look at the wording we use (U1L18 Opener Complete.jpg), and you can see how we'll transition to "slope" and "y-intercept".

The last one is different, of course, and it's always interesting to see how few of my students notice that this is the list of perfect squares. So it goes, but at least it's fun to get kids thinking about how the "increase is increasing". It's fun to let them know that the idea of "changing change" is central to what they'll see in a calculus course. This pattern also forms a diagonal on the multiplication tables that students will see again today, so that will make for an opportunity to help kids notice what these numbers are and how they're related to each other.

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After the opener, we make a quick check of today's agenda. I remind students that the goal for the Number Line Project is for everyone to improve on their results from the previous project. Today we will continue that work, with the goal of finishing all parts of the project by Friday, when it's due.

I distribute this week's homework sheet, which is just a half-page for this week, simply providing room for students to record what they'll do for homework each night, as they complete their projects. Last week's homework sheet, and the checklist of all parts of the project is still an important place for students to record and review their progress.

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Here's a video overview of this part of the project: About Part 2c.mov

As in previous days, different students may have different work to do today. For the most part, however, almost all students have completed Parts 2a and 2b of the Number Line Project as prerequisites to receiving Part 2c: Patterns Questions (NLP Part 2c.pdf). This new handout is a series of questions about the addition and multiplication tables that students produced on Parts 2a and 2b.

I love teaching this part of the project, because it gives students a chance to really slow down and think, and it provides for them an example of the kind of think I aspire for them to be able to do. It would be impossible to try to recount all of the great conversations I've had with kids while they've worked on this part of the project, so here's what I'll say: if you'd like to try a student-centered activity that challenges students, lets them feel the thrill of "getting it," and promotes all sorts of interesting conversations in your classroom, then try this activity.

**What Groundwork is Laid Here?**

Moving forward, these arithmetic tables and the questions on this handout will give us a way to talk about inequalities, arithmetic properties, linear functions, arithmetic sequences, and number theory, to name a few topics. A week from now, when I return this project to students after grading, I tell them all to place these tables prominently in their binders, because we'll reference them frequently throughout the term.

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With a few minutes left in class, I call everyone to attention and ask, "What's your homework tonight?" As was the case last week, students are responsible for figuring out what they need to accomplish each night this week. Today, that will mean completing Part 2c for most students, but there may be other incomplete work that they'd each like to finish.

This is also an opportunity to remind students to stay organized. "There's nothing worse than doing all your work but then losing it," I say. "Remember that I'm not collecting and grading everything until this Friday, when you'll clip it all together and hand it in for a grade." As important as all of these number line concepts are, the ability to set and accomplish goals and to stay organized is something I'd like to cultivate in my students as they begin their high school careers.

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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