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* *Reflection: Developing a Conceptual Understanding
The Spending & Saving Project, Day 3 - Section 2: It's a Work Day, and Here's Part 2

One major goal for this project is to help students get comfortable transitioning back and forth between mathematical notation and the sort of everyday language that we'd use to talk about these situations. So on Part 1b of the project, for example, I want students to be able to read f(x) > 20 as "How many years will I have to work before I'm making twenty bucks an hour?" Many people might live their entire adult lives without asking themselves, "Hmm is f(x) greater than 20?" On the other hand, how many people will make it their adult lives *without* wondering when their pay will cross a certain threshold?

So today, a student was working on Part 1b when she called me over to say, "In 8 years, the car will be worth $1001. So *after* eight years, it will be worth less than a thousand dollars." That's what we're going for on this project! If students can translate all this math notation into something conversational, then the project is a success. We have so many conversations about context throughout the project. I'll often rephrase the math for students, to scaffold their understanding of appropriate answers to the question *So what do these numbers mean?* Then I'll ask, "Does that sound reasonable?"

*Couldn't have said it better!*

*Developing a Conceptual Understanding: Couldn't have said it better!*

# The Spending & Saving Project, Day 3

Lesson 12 of 19

## Objective: SWBAT graph functions represented symbolically, and interpret key features of those graphs in terms of a context.

One purpose of this week's sequence of lessons is to encourage students to move fluently between the representations of functions. So while yesterday's opener asked students to evaluate a few rules for two different inputs, today they get two tables of values, and must write the rules. By moving between the different representations each day, I want to give students their own space to see which tasks are easiest for them, and which are more challenging; I want them to be aware of which representation works best for them. I pay attention to this. For example, when a student is quiet during one opener, but eager to contribute to our work on another, that's important. I try to show students how to pay attention to their own thinking as well

Today's opener is pretty straightforward, and as a group, my students should be able to handle this task pretty quickly, so I don't expect to spend too much time on it. There are couple of blanks in each table that need to be filled in, and then the task is to write algebraic rules for each. The first table is generated by a linear function, and the second one is exponential. I want to make sure that students notice that the input column is counting by 2's in the first table, and I check to make sure that they can accurately determine the slope of this line.

Kids may slip back into the language of arithmetic and geometric sequences. When this happens, I make sure to provide the equivalent terminology of linear and exponential functions.

#### Resources

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**It's a Work Period**

Today is the third day that students will work on the Spending & Saving Project, and first and foremost, it's a work day. So if kids need time to work on Part 1 of the project, they'll get it (you can read about Part 1 of the project here and here). When a student is ready to move on, they let me know, and I introduce them to Part 2. Wherever they stand, I try to check in with each student to answer questions and to continue to encourage them to interpret the mathematics in the context of each problem situation.

I use today's agenda to guide students through what they've done so far and should do next. Students must complete each part of the project before receiving the next. I've posted each part of the project at the bottom of the agenda, which gives kids a picture of how the project works.

**Part 2 of the Project**

When they're ready, students start on Part 2 of the Spending & Saving Project. The task is to create a table and a graph for each of the function situations that was introduced in Part 1. Here is my video description of this part of the project. When students get to work, there's this beautiful array of cross-referenced documents, that lead to all sorts of productive conversations and idea sharing. Often, I'm able to stand back and let it happen. As in previous days, I just circulate and make sure that kids are able to provide a contextualized interpretation of what they're doing.

**Scale on Part 2**

Determining how to appropriately scale the axes is an important part of this project. Students have been working with number lines and scale since the first month of school (please see September's Number Line Project), and they continue to apply these skills now.

I have two go-to pieces of advice for kids. First of all, I want to be able to see the y-intercept on each graph. For the linear functions, I want to see the x-intercept as well, so both axes should be scaled accordingly. For the exponential functions, the y-intercept will do (we're dancing around the idea that the x-axis is an asymptote on the exponential decay problem), and it's appropriate to count by 1's on the x-axis.

If they need the second piece of advice, I tell students to count the length (in number of grid squares) on each axis (they're both 30). Students should divide the number they need to get up to (it depends on the problem) by 30, and then round up to a nice counting number. It's interesting to see that some students will, for example, divide 8000 by 30, and then take the time to carefully count by 267's on their vertical axis. I want kids to see that this is not wrong, but it's also not too efficient.

I am including here two photographs of notes that I've given on the board, a day or two after today's lesson: Part 2 Scaling i and Part 2 Scaling ii. When all students have had a little time to get started on this part of the project, I'll take a little bit of class time to work through this example. I want students to be able to provide context for each of the intercepts, and I also want them to see that it's unnecessary to plot every single point on a linear function.

**Recursion**

One additional note about filling in the tables: my kids don't have access to a graphing calculator like the TI-83. If they did, the task of filling in the table of values would be different. Most kids come equipped with simple arithmetic calculators, scientific calculators like the TI-30, or the built in calculator programs on their cell phones. I show them how, on most calculators, you can use a recursive definition of a function by entering a starting value, then an operation, and repeatedly pressing the "=" button. This way, as long as you don't lose track of how many times you've hit the equal sign, you can pretty quickly get a set of consecutive outputs for a function.

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With about five minutes left in the class, I ask for everyone's attention. I applaud students on the work they've done today, and I might give a few shout outs to individual students who have worked particularly hard, helped others, or gained new insights.

Then, I tell students to take a moment to gather their work, assess what they've done so far, and decide what they need to work on tonight. "You all have homework tonight!" I say. "It's different for each of you, because you're each at a different point in the project, but I expect that everyone can finish whatever you're currently working on tonight." Sometimes, I give a specific time suggestion, like, "Everyone should do 30 minutes of math homework tonight." I find that suggesting a *maximum* amount of work can make homework seem less daunting to kids. "Sure, it's hard to do piles and piles of homework, for hours upon hours, but what if you commit to working for 30 minutes tonight, and see how much you can accomplish?"

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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: How Will Your Salary Grow?
- LESSON 2: Sequences, Spreadsheets, and Graphs
- LESSON 3: Sequences Gallery Walk (Day 1 of 2)
- LESSON 4: Sequences Gallery Walk (Day 2 of 2)
- LESSON 5: Graphing Stories
- LESSON 6: More Graphing Stories
- LESSON 7: Parameters on Desmos
- LESSON 8: Functions: When One Representation Sheds Light on Another
- LESSON 9: Using Points to Determine the Shape of a Graph
- LESSON 10: Introduction to the Spending & Saving Project
- LESSON 11: Spending & Saving, and Function Notation
- LESSON 12: The Spending & Saving Project, Day 3
- LESSON 13: The Spending & Saving Project, Day 4
- LESSON 14: Earning Interest, Paying Interest
- LESSON 15: The Cost of Debt (Part 3 of the Spending & Saving Project)
- LESSON 16: Finishing Up the Spending & Saving Project, With Extensions
- LESSON 17: Tying a Bow on Unit 4 (Review Day 1 of 2)
- LESSON 18: Tying a Bow on Unit 4 (Review Day 2 of 2)
- LESSON 19: Unit 4 Exam