(Also available in WeScheme)

Students learn to combine image transformation functions as well as to describe the order of operations involved in algebraic function compositions such as f(g(h(x))) using Circles of Evaluation.

 Lesson Goals Students will be able to: Use functions as building-blocks, composing them to achieve a desired affect Diagram function composition using the Circles of Evaluation Compose functions when programming Student-facing Goals I can map a path from one number to another by composing functions I can use Circles of Evaluation to show how functions can be composed Materials Supplemental Resources Key Points For The Facilitator Check frequently for understanding of data types and contracts during this lesson and throughout subsequent lessons. When students encounter errors, encourage them to check their Contracts page and show their work using Circles of Evaluation. Students will use their Contracts page frequently, so it should be kept in an accessible, convenient location.
Glossary
contract

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

data type

a way of classifying values, such as: Number, String, Image, Boolean, or any user-defined data structure

image

a type of data for pictures

## 🔗Composing Functions 10 minutes

Students are given a scaffolded activity that forces them to use the output of one function as the input to another - to compose them.

### Launch

Divide students into groups of 3-4, and distribute a set of Function Cards to each group. (If you’re teaching remotely you can use Function Cards (Desmos) instead.) Write down pairs of integers on the board, representing the "starting numbers" and "ending numbers". These integers should range from -50 to +50, but you can change the difficulty of the activity by making that span wider (more difficult) or more narrow (less difficult). You can find a random integer generator here.

• Each group has a set of functions, each of which takes an input and produces an output. I can start with the number 4, for example, and give it to the function add6. What will the output be?

• 10

• I can also compose functions, meaning that the output of one is immediately passed into another. For example, I could compose add6 and double, so the 10 gets passed into the next function, and doubled to produce 20. What would happen if I composed add6 with double and with half?

• 10

• For each of the starting numbers on the board, your job is to figure out which functions to compose in order to get to the end.

• You will need to use some functions more than once, and that’s okay!

### Investigate

Give students time to experiment with this. You can make the activity more challenging by asking them to find the shortest path from start to end, using the smallest number of compositions.

### Synthesize

If two groups come up with different compositions that achieve the same end result, have them share their ideas!

## 🔗Diagramming Function Composition 15 minutes

### Overview

The Circles of Evaluation are extended to provide a visual-spatial metaphor for function composition, in addition to Order of Operations.

### Launch

Three of the function cards we just used were for the functions f, g and h:

• f multiplied its input by 3

• g added six to its input

• h subtracted one from its input

We can compose those function in any order. If we composed them as f(g(h(x))) and evaluated them for x = 4 what would happen?

We can diagram the function composition using Circles of Evaluation (see first column, below). In the second column, we’ve replaced the function names in each Circle of Evaluation with what each function does:

Function Composition Order of Operations

f
g
h
 x
(f (g (h x)))

​*
3
+
​-
 x 1
6
(​* 3 (+ (​- x 1) 6))

The circles show us that in order to evaluate f(g(h4)))$\displaystyle f(g(h(4))))$

• First we would have to evaluate h4$\displaystyle h(4)$, subtracting 1 from 4 to get 3

• Then we would evaluate g3$\displaystyle g(3)$, adding 6 to 3 to get 9

• Then we would evaluate f27$\displaystyle f(27)$, tripling 9 to get 27

### Investigate

Turn to Diagramming Function Composition to practice writing, translating and evaluating Circles of Evaluation of composed functions.

### Synthesize

• Do f(g(hx))$\displaystyle f(g(h(x)))$ and g(h(fx))$\displaystyle g(h(f(x)))$ evaluate to the same thing? Why or why not?

• No, they do not. Order matters!

## 🔗Composing Functions in Code 20 minutes

### Overview

The Circles of Evaluation are extended to functions that do more than compute values.

### Launch

Explain to students that their contracts page is just like the Function Cards from the activity they just completed. Their job as programmers is to figure out how to compose those functions to get where they want to go, in the most clever or elegant way possible.

### Investigate

• This will take them to the "Programs" page. This page is empty - they don’t have any programs yet!

• Have them open a new program by clicking "File" -> "New" and save it as "Function Composition"

• Complete Function Composition — Green Star, in which you will draw circles of evaluation to help you write expressions to compose a series of images.

• Be sure to use the Definitions Area (left side) for code you want to keep and the Interactions Area (right side) to test code or try out new ideas.

When students are finished, check their work, and ask them to change the color of all of the stars to “gold” or another color of their choosing.

Now, turn to Function Composition — Your Name in which you will create a text image of your name and experiment with other functions.

 Strategies for Facilitation While students are exploring, be available for support but encourage student discussion to solve problems. Many student questions can be addressed with these responses: Did you try drawing the Circle of Evaluation first? Did you check the contract? Have you pressed the "Run" button to save your Definitions changes? Encourage students to practice writing comments in the code to describe what is being produced, using # at the beginning of the line.

If you have time, you can also have students work with Function Composition — scale-xy and/or Function Composition Matching Activity (Desmos)

### Synthesize

• What do all of these functions have in common?

• They all produce images, they all change some element of the original image

• Does using one of these functions change the original image?

• No, it creates a whole new image

• What does the number in scale represent?

• The scale factor by which the image should grow or shrink

• What does the number in rotate represent?

• The rotation angle, measured counterclockwise

• The Domain and Range for flip-horizontal is Image -> Image. Why can we use the output of the text function as an input for flip-horizontal?

• Because the text function produces an Image, which is then used as the input for flip-horizontal.

 Strategies for English Language Learners MLR 1 - Stronger and Clearer Each Time: As an alternative, display the discussion questions during the last 5 minutes of the Explore and ask students to discuss the questions with their partner, asking each other for explanation and details and coming up with the clearest, most precise answer they can. Student pairs can then share with another pair and compare their responses before moving into a full class discussion.
 Fun with Images! Now that students have learned how to use all of these image-composing functions, you may want to give them a chance to create a design of their own, tasking them with using at least 4 functions to create an image of their choosing. Our Making Flags also dives deeper into image composition.

## 🔗Composing Multiple Ways Optional

### Overview

Students identify multiple expressions that will create the same image, and think about the merits of one expression over another.

### Launch

As is often true with solving math problems, there is more than one way to get the same composed image.

• Suppose I wrote the code: scale​(​3, circle​(​50, "solid", "red"​)​).

• What’s another line of code I could write that would produce the exact same image?

• circle​(​150, "solid", "red"​)

### Investigate

When students have completed the worksheet, explain that there is a special function that lets us test whether or not two images are equal:

images-equal:: Image, Image -> Boolean

Invite students to use the above function to test whether all of the expressions that they wrote successfully build the same images.

### Synthesize

• Could we have written more expressions to create the same images?

• Are all of the ways to write the code equally efficient?

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). Bootstrap 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.