##
* *Reflection: Relevance
Introduction to the Spending & Saving Project - Section 3: Spending and Saving Part 1a: Saving Money

Luis anecdote: "This doesn't make sense," he said. He was trying to do all the work in his head. He got to a problem that he couldn't think about intuitively in his head. This was finally a moment where algebraic moves seemed to be a useful tool to him. Even after he solved the problem, he was having trouble deciding if the solution made sense. This took us back to the definitions of x and f(x), and he had to slow down again to read that. I share this story because, in general, this assignment is designed to get kids interpreting, paying attention to detail, and using what they know about math. This is indeed what happened!

*Why Learn Algebraic Manipulation?*

*Relevance: Why Learn Algebraic Manipulation?*

# Introduction to the Spending & Saving Project

Lesson 10 of 19

## Objective: SWBAT use function notation, evaluate functions for inputs on their domains, and interpret statements that use function notation in the context of several financial situations.

## Big Idea: Interpreting mathematics in context lends value to our discipline and helps students make sense of the tools we ask that they learn to use.

*43 minutes*

#### Opener: Evaluating Functions

*10 min*

Today's opener primes my students to use function notation throughout the week. My students have seen function notation before, but more as a side-note than as a focal point to a lesson. This is by design. I consider function notation a tool that makes the most sense in context. If students can use function notation in a variety of ways, then they'll be more comfortable with it than they'd be if they just practiced a series of exercises without context. With that in mind, I want to move quickly through the decontextualized exercises that comprise this opener. For everyone who understands this task, it's a great warm-up, and anyone who doesn't get it will soon see what it means to apply this tool on today's assignment.

As students get started, I circulate to see how they're doing. At each group, I try to identify an expert who's got this down, and if I can't, I stick around to show kids how to work the first example. When I can see that everyone has at least given this a try, I begin to write the solutions on the board. I take the reins here because I want to neatly and consistently model what it looks like to input each value of x into the function rule. After the first example, I skip "showing" the intermediate step, because I want students to understand that "showing work" doesn't have to mean the same thing as "writing a lot".

To reiterate the idea that * x* is an input to a function rule and

*is the output for a given value of*

**f(x)***, I also make a table of values off to the side of this work, and fill it in with the values from each of these four exercises. This move results in a satisfying "aha!" from a lot of kids, who hadn't yet really made sense of the notation.*

**x**Finally, I point to the last exercise, e, and ask for a volunteer to interpret what we're being asked to do here. There are several possibilities for what they might say, and the key is to listen for the mathematical truth and use it. For example, in one of my classes, a student was excited to point out that "it's the same as the other thing". Embedded in that mess of pronouns is something important, and I made sure to dig into this idea. After writing that student's words on the board, I pressed him to define "it" and "the other thing". Once he did, we got to the idea that ** f(x)** was the repeated notation, and that if we're told that

**and that**

*f(x)=37***, then we can use the transitive property of equality to get the equation**

*f(x)=10+18x***, which isn't such a big deal to solve.**

*37=10+18x*My other approach to the final problem is to write 37 in the output column of the table of values, and this again lends clarity to a lot of kids about what's going on here.

*expand content*

Today is the first day of a new project called the "Spending & Saving Project".

A major focus of this project, and the primary learning target on today's assignment is CCSS F-LE.5:

**I can interpret the parameters in a linear or exponential function in terms of a context.**

Additionally, students will work on evaluating functions, distinguishing between linear and exponential functions, and they'll even get some practice solving - and interpreting the solutions to - inequalities. I describe the first part of the project, and how I emphasize interpretation as kids work, in this narrative video.

There are three versions of Part 1a of the project, Easy, Medium, and Hard, in the resources folder here. What differs between the three versions is how "nicely" the numbers work out in each scenario, and a few of the "Interpretation Questions" are different from one version to the next.

In addition to the mathematical goals of this project, students will also gain some financial literacy. I like to be able to start conversations by saying things like, "If you've ever wondered how to save up $1000, here's one way to make a plan" Kids realize that that's not easy to do, but this also makes it seem a little more achievable to some of them.

**Ways to Interpret a Solution**

Over the next few lessons, I'll spend a lot of time helping students distinguish between how they'd say something in a math context, and how people actually talk in real life. Compare, for example, this student's answer to question #2a to what this student writes in the same place. They've both done the math right. The former student simply takes an answer and writes it, while the latter uses the conversation tone that someone might actually use. So for the first student, my role is to ask whether or not she's ever heard anyone answer a question by saying *88.88 weeks*, which is a fun conversation to have, and which will help her build an understanding of what it means to interpret a function, its parameters, or a solution to a problem. For the second student, I might want to touch on the idea of rounding up, but I can also see that he's got a pretty good grasp on what we're actually doing.

Today's work sets the stage for conversations of this kind for the next week or so.

##### Resources (10)

#### Resources

*expand content*

##### Similar Lessons

###### Graphing Radical Functions Day 2

*Favorites(2)*

*Resources(7)*

Environment: Suburban

###### Inequalities: The Next Generation

*Favorites(3)*

*Resources(19)*

Environment: Suburban

###### Addition and Subtraction One Step Equations

*Favorites(100)*

*Resources(19)*

Environment: Urban

- 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