# Lesson 7: Composite Functions

## Overview

In the past lessons students have defined Variables and written Fast Functions. In this stage, they will continue to explore function writing with ever increasing complexity.

## Agenda

### Getting Started

### Composite Functions

## Anchor Standard

### Common Core Math Standards

**8.F.1**- Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

## Objectives

### Students will be able to:

- Analyze and use existing functions.
- Modify existing functions.
- Create new functions.
- Create similar shapes by changing size parameters on functions.

## Vocabulary

**Parameter**- A value or expression belonging to the domain of a function.

## Support

### Report a Bug

# Teaching Guide

## Getting Started

### Introduction

In the previous lessons students began composing functions together to make their lives easier. This Stage should be used as a way to deepen the understanding of students. There are assessments built into the later levels, so be sure to use probing questions to have students think deeply about their work.

Level 1 through 5 are composite image functions and beware of `aqua-star`

as this is where most students struggle transferring their learning. Pause at Level 4 to make sure all students have a firm understanding.

Levels 6 though 9 bring the content back to math. It is crucial that students continue to use their Fast Function process for these problems because that will help make this Design Recipe more applicable to Algebra problems in the future.

Assessment Levels 10 through 15 are a great way to have students check their understanding and are by no means required.

This is a Stage that, as a teacher, you have full flexibility as to which levels you want students to complete.

## Composite Functions

### Online Puzzles

In this stage you'll define simple functions. Head to Course A stage 7 in Code Studio to get started programming.

Be sure that students have plenty of space on their Fast Functions form so they can build these new functions from the Contract to the Examples to the Definition. Great code starts on paper first!

### Student Instructions

Here's a new function called `green-triangle`

that takes a single Number for size and produces a green triangle. Use the new function to create a 125 pixel green triangle

### Student Instructions

Let's look inside that `green-triangle`

function to see how it works. Can you modify it so that the `green-triangle`

function always draws outlined green triangles? **Don't forget to change the examples too!**

### Student Instructions

Here's the start of another new function called `purple-circle`

with a domain of one Number (the radius) and a range of Image. The body of the function is broken though, it always draws a circle of 50 pixel radius instead of using the radius parameter. Replace the current Number with the `radius`

block from the domain.

When you've fixed the function, test your new function with a 150 pixel radius.

### Student Instructions

Create a new function called `aqua-star`

that has a Domain of a single Number for radius and returns an aqua star of the given radius. Run your new function with a radius of 75.

### Student Instructions

Let's use that `aqua-star`

function to make a row of different sized stars. From left to right, the stars should have radii of 25, 50, and 75.

### Student Instructions

Let's look at some more traditional algebraic functions now. Here's a function translated from the simple algebraic function `f(x) = x + 1`

. What would you expect this function to output if provided an input of 15?

### Student Instructions

Write a function `f`

that takes a parameter `x`

and returns `x - 10`

. Test your function using `f(15)`

.

### Student Instructions

We don't have to call our function `f`

every time; create a function called `times-ten`

that should take a parameter `x`

and return `x * 10`

. Once you've created the function, run `times-ten(50)`

.

### Student Instructions

Write a function called `half`

that takes a number `x`

and returns half that number. Once you've written the function, use it to calculate `half(21)`

### Student Instructions

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## Standards Alignment

#### View full course alignment

#### Common Core Math Standards

**CED** - Creating Equations

**A.CED.1**- Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.**A.CED.2**- Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

**F** - Functions

**8.F.1**- Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1**8.F.2**- Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented

**G** - Geometry

**7.G.1**- Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

**IF** - Interpreting Functions

**F.IF.1**- Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f c**F.IF.2**- Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.**F.IF.3**- Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.**F.IF.4**- For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercep**F.IF.5**- Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers w**F.IF.6**- Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★

**LE** - Linear, Quadratic, And Exponential Models★

**F.LE.1**- Distinguish between situations that can be modeled with linear functions and with exponential functions.

**MP** - Math Practices

**MP.1**- Make sense of problems and persevere in solving them**MP.2**- Reason abstractly and quantitatively**MP.3**- Construct viable arguments and critique the reasoning of others**MP.4**- Model with mathematics**MP.5**- Use appropriate tools strategically**MP.6**- Attend to precision**MP.7**- Look for and make use of structure**MP.8**- Look for and express regularity in repeated reasoning

**OA** - Operations And Algebraic Thinking

**5.OA.1**- Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.**5.OA.2**- Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three t

**Q** - Quantities

**N.Q.1**- Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.**N.Q.2**- Define appropriate quantities for the purpose of descriptive modeling.

**SSE** - Seeing Structure In Expressions

**A.SSE.1**- Interpret expressions that represent a quantity in terms of its context.**A.SSE.1.a**- Interpret parts of an expression, such as terms, factors, and coefficients.