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Lesson Pathway, Standards and Practices
Lessons
Prerequisites only (★)

Standards (click one)

Common Core Math Standards
6.EE.B.6

Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

HSF.BF.A

Build a function that models a relationship between two quantities.

HSF.BF.A.1.C

Compose functions.

CSTA Standards
2-AP-11

Create clearly named variables that represent different data types and perform operations on their values.

2-AP-17

Systematically test and refine programs using a range of test cases

2-AP-19

Document programs in order to make them easier to follow, test, and debug.

3B-AP-21

Develop and use a series of test cases to verify that a program performs according to its design specifications.

Iowa Standards
IA.HSF.BF.A.1.B

Combine standard function types using arithmetic operations.

IA.HSF.BF.A.1.C

Compose functions.

Oklahoma Standards
OK.A1.F.1.3

Write linear functions, using function notation, to model real-world and mathematical situations.

OK.A1.F.3.2

Use function notation; evaluate a function, including nonlinear, at a given point in its domain algebraically and graphically. Interpret the results in terms of real-world and mathematical problems.

OK.A1.F.3.3

Add, subtract, and multiply functions using function notation.

OK.PA.A.3.1

Use substitution to simplify and evaluate algebraic expressions.
Textbook Alignment: None
  • MLR.7
  • MLR.3
  • MLR.1
  • SEP.5
  • MP.5
  • MP.1

(Also available for Pyret)

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

Preparation

  • Make sure all materials have been gathered

  • Decide how students will be grouped in pairs

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.

Language Table

Types

Functions

Values

Number

+, -, *, /, expt, sqr, sqrt

4, -1.2, 2/3, pi

String

string-length, string-repeat, string-contains?

"hello", "91"

Boolean

<, <>, <=, >=, string<?, string>?, string=?, string<>?, string>=?

true, false

Image

star, triangle, circle, square, rhombus, ellipse, regular-polygon, radial-star

🔵🔺🔶

Click here to see the prior unit-based version.

Glossary
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 user-defined data structure

definitions area

the left-most text box in the Editor where definitions for values and functions are written

image

a type of data for pictures

interactions area

the right-most text box in the Editor, where expressions are entered to evaluate

🔗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. 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 difficulty). 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 ok!

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

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
+
-
x1
6
(* 3 (+ (- x 1) 6))

The circles show us that in order to evaluate f(g(h4)))

  • First we would have to evaluate h4, subtracting 1 from 4 to get 3

  • Then we would evaluate g3, adding 6 to 3 to get 9

  • Then we would evaluate f27, tripling 9 to get 27

Investigate

Have students turn to Diagramming Function Composition (Page 25) to practice writing, translating and evaluating Circles of Evaluation of composed functions.

Synthesize

Do f(g(hx)) and g(h(fx)) evaluate to the same thing? No!

Why 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

The contracts page in your workbook is just like the Function Cards from this activity. Your job as a programmer is to figure out how to compose those functions to get where you want to go, in the most clever or elegant way possible.

Investigate

Have students log into WeScheme open a new program and save it as Function Composition.

Have students open to Function Composition — Green Star (Page 26), in which they will be drawing circles of evaluation to help them write expressions to compose a series of images.

  • Make sure students are using the Definitions area (left side) for code they want to keep and are using 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 your choosing.

Then have students open to Function Composition — Your Name (Page 27) in which they will create a text image of their 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 (Page 28) 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 I 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 Flags lesson 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 (star 50 "solid" "red")).

What’s another line of code I could write that would produce the exact same image? (star 150 "solid" "red")

Investigate

Students complete More than one way to Compose an Image! (Page 29).

Synthesize

There is a special function in WeScheme that let’s us test whether or not two images are equal.

image=?:: Image, Image -> Boolean

Use it to test whether all of the expressions you wrote successfully build the same images.

  • 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). CCbadge 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.