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Students take a closer look at how functions can work together by investigating the relationship between revenue, cost, and profit.

Prerequisites

Restating the Problem

Relevant Standards

Select one or more standards from the menu on the left (⌘-click on Mac, Ctrl-click elsewhere).

Common Core State Statements
HSF.BF.A.1

Write a function that describes a relationship between two quantities.

MP.1

Make sense of problems and persevere in solving them

MP.6

Attend to precision

Older Statements
BS-DR.1

The student is able to translate a word problem into a Contract and Purpose Statement

BS-DR.2

The student can derive test cases for a given contract and purpose statement

Lesson Goals

Students will be able to:

  • Write a function that explicitly uses another function.

  • Explain the benefits and drawbacks of functions that depend on each other.

  • Explain the difference between bottom-up and top-down strategies.

Student-Facing Lesson Goals

  • I can explain the benefits and drawbacks of functions that use other functions.

  • I can write a function that uses another function.

Materials

Bootstrap Formative Assessments

  • Bootstrap: Algebra - Coordinates, Circles of Evaluation, & Code (Quizizz)

  • Bootstrap:Algebra - Data Types & Circles of Evaluation (Desmos Activity)

  • Bootstrap:Algebra - Circles of Evaluation Review(Blank Template) (Desmos Activity)

  • Bootstrap:Algebra - Contracts, Domain/Range, Data Types, & Functions (Quizizz)

  • Bootstrap:Algebra - Data Types, Circles of Evaluation, and Contracts (Desmos Activity)

Preparation

  • Make sure all materials have been gathered

  • Decide how students will be grouped in pairs

Supplemental Resources

Key Points for the Facilitator

  • There are several ways to write the profit function - use this opportunity for discussion and to promote higher-order critical thinking.

  • If students are struggling with understanding the basics of the problem, start by coming up with examples of cost and revenue. If Sally sells one glass, what is her total revenue? How much does it cost her to produce that one glass?

  • Ensure students understand the difference between "revenue" and "profit", and that "cost" refers to the cost of making the lemonade, not the amount Sally is charging.
Glossary
function

a mathematical object that consumes inputs and produces an output

Warmup

Students should have their workbook, pencil, and be logged into WeScheme and have their workbooks with a pen or pencil.

Problem Decomposition 30 minutes

Overview

Students are introduced to word problems that can be broken down into multiple problems, the solutions to which can be composed to solve other problems. They adapt the Design Recipe to handle this situation.

Launch

Display the following image:

Lemonade Stand Ideas Lemonade Stand Ideas🖼Show image

Notice and Wonder

What do you notice? What do you wonder?

One example of a relatioship we can find in this situation is that Sally takes in $1.75 for every glass she sells:

What other relationships can you find here?

(Give students a chance to discuss and brainstorm)

  • Every glass sold brings in $1.75 in revenue

  • Every glass sold costs $0.30 in costs, such as lemonds, sugar and water

  • Every glass sold brings in some amount of profit: it costs a certain amount to make, but it brings in another amount in revenue

Investigate

Students form groups and brainstorm their ideas for functions. Students can use any strategies they’ve learned so far.

Strategies for English Language Learners

MLR 7 - Compare and Connect There are several correct ways to write the functions needed for Sally’s Lemonade. Have students compare methods and develop understanding and language related to mathematical representation and methods. What are the advantages of the different solutions? What are some drawbacks?

  • What is the difference between revenue and profit? Revenue is the total amount of money that comes in, profit is the remaining money after cost has been subtracted.

  • How could Sally increase her profits? By decreasing her costs, raising her prices (which increases revenue), by selling more lemonade.

  • What is the relationship between profit, cost, and revenue? Profit = Revenue - Cost

Students work with their partners to develop their function models for revenue (Page 44), cost (Page 45), and profit (Page 46), using the Design Recipe.

While students are working, walk the room and gauge student understanding. There is more than one correct way to write the profit function! Encourage discussion between students and push students to develop their thinking on the advantages and disadvantages of each correct solution.

Synthesis

This activity started with a situation, and students modeled that situation with functions. One part of the model was profit, which can be written several ways, for example:

(define (profit g) (- (* 1.75 g) (* 0.30 g)))
(define (profit g) (* (- 1.75 0.30) g))
(define (profit g) (* 1.45 g))
(define (profit g) (- (revenue g) (cost g)))

  • Which way is "best", and why?

  • If lemons gets more expensive, which way requires the least amount of change?

  • If sugar gets less expensive, which way requires the least amount of change?

Big Ideas

  1. profit can be decomposed into a simple function that uses the cost and revenue functions.

  2. Decomposing a problem allows us to solve it in smaller pieces, which are also easier to test!

  3. These pieces can also be re-used, resulting in writing less code, and less duplicate code.

  4. Duplicate code means more places to make mistakes, especially when that code needs to be changed.

Top-Down vs. Bottom-Up 20 minutes

Overview

Students explore problem decomposition as an explicit strategy, and learn about two ways of decomposing.

Launch

Top-Down and Bottom-Up design are two different strategies for problem decomposition.

When thinking Bottom-Up, we start with the small, easier relationships first and then build our way to the larger relationships. In the Lemond Stand example, we had you write the lower-level functions &emdash; cost and revenue &emdash; first, and then gave you the chance to use them in the higher-level profit function. This is called Bottom-Up design.

When thinking Top-Down, we start with the "big picture" and then worry about the details later. For example, we could have started with profit, and kept track of all the lower-level functions we would need to write. This is called Top-Down design.

Investigate

Consider the following situation:

Jamal’s trip to Thailand requires him to drive 20 miles to the airport, take a plane 9,000 miles to Thailand, and then a bus 6 miles to his hotel. The average speed when driving to the airport is 40mph, the average speed of an airplane is 575mph, and the average speed of his shuttle bus is 15mph

Setting aside time spent waiting at the airport or for the bus, how long is Jamal in transit?

This problem can be decomposed in Top-Down or Bottom-Up fashion. Describe what your steps would be in each solution (for extra credit, you can actually compute the answer!).

Synthesize

Make sure that students see both strategies, and have them discuss which they prefer and why.

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). CCbadge BS:Games by Jen Poole, Emmanuel Schanzer, Ed Campos Jr, and Dorai Sitaram is licensed under a Creative Commons 4.0 Unported License. Based on a work at www.BootstrapWorld.org. Permissions beyond the scope of this license may be available by contacting schanzer@BootstrapWorld.org.