Students discover functions as an abstraction over an arithmetic pattern, applying the Design Recipe to traditional word problems.
Lesson Goals 
Students will be able to:


StudentFacing Lesson Goals 


Materials 

Preparation 


Supplemental Resources 

Key Points for the Facilitator 


Language Table 

Click here to see the prior unitbased version
 contract

a statement of the name, domain, and range of a function
 data types

a way of classifying values, such as: Number, String, Image, Boolean, or any userdefined data structure
 design recipe

a sequence of steps that helps people document, test, and write functions
 function

a mathematical object that consumes inputs and produces an output
 purpose statement

a concise, detailed description of what a function does with its inputs
🔗The Design Recipe 10 minutes
Overview
In this lesson students build on what they already know about multiple representations of functions (contracts, examples and definitions) to write purpose statements and gain fluency with the Design Recipe.
Launch
Have students turn to Creating Contracts From Examples (Page 47) and write contracts for the examples provided.
Investigate
You’ve started to master most of the steps of the Design Recipe, but there’s one part you haven’t seen yet: writing a purpose statement. On its own, a contract is not enough information to generate examples and write a function definition. We need to know what the function is supposed to do with what it takes in. Programmers and Mathematicians alike find it helpful to restate a problem in their own words. After all, if you can’t explain a problem to someone else, you probably don’t understand it yourself!
Turn to Writing Examples from Purpose Statements (Page 48) and read the purpose statements. What do you notice? What do you wonder?
Debrief the notice and wonder as a class. Then have students write examples that satisfy each of the contracts and purpose statements on Writing Examples from Purpose Statements (Page 48).
OPTIONAL: For more practice, have students complete Writing Examples from Purpose Statements 2.
Synthesize
What are the important elements of purpose statements? Why are purpose statements useful?
🔗Writing Linear Functions 25 minutes
Overview
Students are given a nonworking program, which uses a linear function to determine the height of a rocket after a given length of time. The "broken" code is provided to lower cognitive load, allowing students to focus on comprehension (reading the code) and making use of structure (identifying where it’s broken).
Launch
Students should have their workbook, pencil, and be logged into WeScheme on their computer.
Ask students to open the
rocketheight Starter File (Wescheme) and click "Run". By typing (start rocketheight)
, they will see the simulation start to run on their computer.
Notice and Wonder What do you notice about this program? What do you wonder? 
Survey the class on their "Notices" and "Wonders" and record on the board before moving on to the discussion.

Is
rocketheight
working? 
Why do you think it’s not working?

What do you think the purpose of this function is? How do you know?

What is the domain of
rocketheight
? Number 
What is the range of
rocketheight
? How do you know? Number, we can tell by looking at the contract for the function. 
As the program is currently written, what happens when I give the function an input of 5? 15? One million? It always returns 0.
Investigate
Let’s use the Design Recipe to fix rocketheight
, and get comfortable with writing purpose statements.
Have students turn to Word Problem: rocketheight (Page 49), read the problem statement with their partner and write down the Contract and purpose statement. Then, given the contract and purpose statement, write two examples of how rocketheight
should work after two different lengths of time.

Circle and label what’s changing in the two examples, just as you did with the green triangle function before.

Choose a good variable name for what’s changing.

Write the function definition using the variable name.
Once the Design Recipe has been completed in the workbook, students can type the code into the rocketheight
program, replacing any incorrect code with their own code.
Synthesize

What was the problem?

What mistake(s) did the programmer make?

Where in the Design Recipe did they first go astray?
The Design Recipe allows us to trace mistakes back to the source!
🔗More Interesting Functions flexible
Overview
For teachers who cover quadratic and exponential functions, this activity deepens students' understanding of functions and extends the Design Recipe to include those. This can also be a useful activity for students who finish early, or who need more of a challenge.
Launch
Now that rocketheight
is working correctly, explore the rest of the file and try the following:

Remove the comment from before the
(start rocketheight)
and test the program. 
Put the comment back in front of
(start rocketheight)
, remove the comment from(graph rocketheight)
, and test the program. 
Try out
(space rocketheight)

Try out
(everything rocketheight)
Investigate

Can you make the rocket fly faster? Slower?

Can you make the rocket sink down instead of fly up?

Can you make the rocket accelerate over time, so that it moves faster the longer it flies?

Can you make the rocket blast off and then land again?

Can you make the rocket blast off, reach a maximum height of exactly 1000 meters, and then land?

Can you make the rocket blast off, reach a maximum height of exactly 1000 meters, and then land after exactly 100 seconds?

Can you make the rocket fly to the edge of the the universe?
Synthesize
Debrief  what did students try? Have students share their experiments with one another!
🔗Additional Exercises:
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). Bootstrap:Algebra by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.