# Functions and Finance

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## Objective

SWBAT write a function that describes a relationship between two quantities.

#### Big Idea

Look at functions as a way of modeling real situations like living expenses after high school.

## Set the Stage

15 minutes

I have my students use chromebooks to look up information for this lesson, but you can also print out the latest data from these reference websites.  I begin this lesson by asking my students to brainstorm what expenses they think they'll have in college, assuming they live off-campus.  This basic question also works well for students who are planning on working immediately after graduation, with only a few minor changes like rent and utility deposits instead of tuition and books.  After a moment or two I have them "popcorn" share their ideas while I act as scribe to write them on the front board.  When everyone has had a chance to contribute, we summarize and consolidate the list into occasional and monthly bills, and then into categories for each group.  Tuition, textbooks, and fees would all be on the occasional list, while food, rent, utilities, clothing, incidentals, and transportation would be on the monthly list.  I tell my students that today's challenge will be to figure out a monthly budget equation or equations that they can actually use and that first we're going to be working through an example for food costs to help them understand the process.

This USDA website http://www.cnpp.usda.gov/USDAFoodCost-Home.htm has food cost estimates that are updated monthly and separated into age, gender, and level of living lists.  I have my students use chromebooks to look up their personal estimated food expense.  We use the weekly cost option so that we can calculate daily and then caculate each month specifically depending on the number of days in the month.  I challenge them to figure out how much they would need to spend on food for the current month, with the condition that they need to be able to show and explain their arithmetic. (MP1) When everyone is done, I randomly select students to post their work on the board and explain it, asking three or four to go up at the same time.  I ask the class to look for similarities in the way these students found their answers, then ask for volunteers to discuss what they see. (MP3) If nobody notices that everyone had to first divide by seven and then multiply that result by the number of days in the month I ask leading questions like "What did all (or most) of the students do first?" or "How did they switch from a weekly to a monthly estimate?"  The final part of this exercise is to have my students remove the person-specific information from the calculation and see what kind of equation they can come up with that anybody could use. (MP2) The final result should look something like this: C/7 x D = E where C is the weekly cost, D is the number of days, and E is the monthly expense.   As my students are working, I give assistance as needed to ensure that everyone has a workable equation.

## Put It Into Action

30 minutes

To start this part of the lesson I post this equation on the board: C/7 x D = f(C) as a summary of the work we just completed.  Some of my students will understand the use of function notation, but others will be less familiar and/or comfortable with it as I discuss in my video.  I don't directly discuss this unless someone raises a question, but tell my students that from here on we'll be using function notation.  If there are any questions or comments I address them and then say that they get to work with other data to estimate their remaining monthly expenses next.  I distribute the Living Expenses Worksheet and tell them they'll be working with their front partner to complete this part of the lesson but that each person should complete his/her own worksheet.  (MP2)

When everyone is done I have all my students post their final equations on the board.  Since I have a small class, we can view them all at once and it only takes a few minutes to post, but if your class is larger you might randomly select several students to post or select students based on your observations while they were working.  I expect a variety of equations because this lesson isn't about finding the one "right" way to work this out, it's about students becoming confident and comfortable with their own ability to generate appropriate equations.  To that end I have the class critique the posted equations for completeness and feasibility rather than doing the critiquing myself. (MP3)  Because the equations are all on the board simultaneously it provides a certain level of anonymity to my students and makes the critique process less stressful.  To ensure nobody coasts through this part of the activity I tell my students that they each need to find at least one part of any of the equations that either needs to be explained or needs to be changed.

## Wrap It Up

10 minutes

To close this lesson I post the following on the board: the average work-study award varies from about \$1200 to \$1800 per semester and that the maximum Pell Grant per semester is \$5550.  I tell my students their final challenge is to figure out if they'll have any money left over after paying all their expenses if they calculate a semester as five months.  I remind them to define variables, show their work and explain their solution. (MP1, MP2)