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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.

7.EE.B

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

HSA.SSE.A.1.A

Interpret parts of an expression, such as terms, factors, and coefficients.

HSA.SSE.A.1.B

Interpret complicated expressions by viewing one or more of their parts as a single entity.

CSTA Standards
1B-AP-12

Modify, remix, or incorporate portions of an existing program into one’s own work, to develop something new or add more advanced features.

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-14

Construct solutions to problems using student-created components, such as procedures, modules and/or objects.

3B-AP-21

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

Oklahoma Standards
OK.A1.F.1.4

Given a graph modeling a real-world situation, read and interpret the linear piecewise function (excluding step functions).

OK.A2.F.1.8

Graph piecewise functions with no more than three branches (including linear, quadratic, or exponential branches) and analyze the function by identifying the domain, range, intercepts, and intervals for which it is increasing, decreasing, and constant.

OK.PA.A.1.1

Recognize that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable.
Textbook Alignment
  • IM.Alg1.4.12
  • IM.8.5.10
  • Function Junction: Families of Functions
  • MP.8

(Also available for WeScheme)

Students learn how to define a function so that it behaves differently depending on the input.

Lesson Goals

Students will be able to:

  • Explain what a piecewise function is.

  • Give examples of inputs and outputs of a given piecewise function.

Student-Facing Lesson Goals

  • I can describe how piecewise functions work.

Materials

Preparation

  • Make sure all materials have been gathered

  • Decide how students will be grouped in pairs

Key Points for the Facilitator

  • The Design Recipe looks a bit different for piecewise, or conditional, functions. Check that students are taking time to write examples and circle what is changing.

Language Table

Types

Functions

Values

Number

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

4, -1.2, 2/3, pi

String

string-length, string-repeat, string-contains

"hello", "91"

Boolean

<, >, <>, <=, >=, string-equal, <, >, ==, <>, >=, and, or

true, false

Image

star, triangle, circle, square, rectangle, rhombus, ellipse, regular-polygon, radial-star, text, overlay, above, beside, rotate, scale, flip-horizontal, flip-vertical

🔵🔺🔶

Click here to see the prior unit-based version

Glossary
conditional

a code expression made of questions and answers

example

shows the use of a function on specific inputs and the computation the function should perform on those inputs

function

a mathematical object that consumes inputs and produces an output

piecewise function

a function that computes different expressions based on its input

string

a data type for any sequence of characters between quotation marks (examples: "hello", "42", "this is a string!")

🔗Not Every Function is Smooth 15 minutes

Overview

Students are challenged via counterexamples to see just how far the Vertical Line Test will go: into behaviors that feel like functions but don’t act like a straight line or smooth curve!

Launch

Students should have their computer, workbook, contracts page, and pencil and be logged in to code.pyret.org and have their workbooks with a pen or pencil.

Have students stand up and put some space between themselves, as if on a number line (each student essentially represents an "x-coordinate"). Give directions to distinct groups of students. For example:

  • If you have brown eyes, wave your arms in the air.

  • If you have blue eyes, walk in place.

  • If you have green or hazel eyes, flap your arms like a chicken.

  • If you like sushi, go back to your seat.

Every student should have an activity to perform. Ask a student walking in place why they aren’t waving their arms in the air, or how they knew what to do. Their behavior is essentially the y-coordinate, though for a more direct connection you can specify that different groups sit, kneel, or stand so that their literal height represents the y-axis.

The Vertical Line Test says that to be a function, every input has to be matched with exactly one output.

Ask students: Is this activity representing a function? What is the input? What is the output? Since each student ("input") has only one action ("output"), it *is still a function*.

Up until now, almost all the functions students have seen are continuous and smooth. Make a big deal about this, so they recognize how big of a shift this is!

Explain that students have just acted out what is called a piecewise function. Even though their behavior didn’t follow a smooth pattern (or even a continuous one!), it clearly followed a set of rules and each input had exactly one output. Math has functions like this as well!

Example: Suppose I sell boxes of candy for $2 each. We could imagine that a graph of sales-v-revenue looks like a straight line with a slope of 2: a linear function! But then I want to offer a "bulk discount", where the price drops to $1 for the 21st box of candy and every box after that. Suddenly our line has a kink in it at 20 boxes, where the slope suddenly changes from 2 to 1.

Can students come up with their own examples?

Investigate

Students open the Alice’s Restaurant starter file (Pyret) and turn to Welcome to Alice’s Restaurant! (Page 75).

Students investigate the file using their workbook page as a guide.

Notice and Wonder

Have students take time to think and discuss what they Notice and Wonder about this file, which contains some new elements they haven’t seen before!

Synthesize

  • What are some familiar things you notice in this file?

    Answers vary: fun, end, a contract and purpose statement, etc.

  • What new things did you notice in this file?

    Answers vary: the ask keyword, the pipe symbols, otherwise, the general look of the order function, etc.

  • What function is being defined here? What is its contract?

    order takes in a String and produces a Number.

  • How do you think this function works?

    Answers vary - let students drive discussion!

The order function is also piecewise function! Each input has a single output, but the behavior isn’t smooth (there’s no relationship between one item’s price and another!) or continuous (there are plenty of items not on the menu!).

Partial Functions

For Algebra 2 or pre-calculus teachers, this is a useful time to address partial functions. The students who liked sushi had no rule at all, meaning that the function was undefined at those points. The candy-sales analogy can be extended to say that no one can order more than 100 boxes at a time, making the function undefined for values of x greater than 99.

🔗Defining Piecewise Functions 30 minutes

Overview

Having acted out a piecewise function and examined the code for one on their computers, students take the first step towards writing one, by modifying a function that’s already been written for them.

Launch

Students turn to Alice’s Restaurant - Explore (Page 76) and complete the exercises with their partner. Students should have added as least one extra option to the menu before moving on.

  • Why do you get an error when you try to use the sales-tax function for an item not on the menu?

    Let students discuss - move towards the realization that the contract for order is order : String -> Number, and the "catch-all" branch at the bottom returns a String instead of a Number.

  • What should we do about this?

    Since we want the program to stop if we give it an invalid input, we should just delete the last branch altogether. Think about other functions that don’t work when we give them an invalid input, like dividing by zero!

Investigate

So how do we actually write a piecewise function? And more importantly, how does the Design Recipe help us get there?

The Contract and Purpose Statements don’t change: we still write down the name, Domain and Range of our function, and we still write down all the information we need in our Purpose Statement (of course, now we might need to write a lot more, since there’s more information!).

The examples are also pretty similar: we write the name of the function, followed by some example inputs, and then we write what the function produces with those inputs.

How many examples are needed to fully test this function?

More than two! In fact, we need an example for at least every possible item on the menu!

examples:
  order("hamburger")   is 6.00
  order("onion rings") is 3.50
  order("fried tofu")  is 5.25
  order("pie")         is 2.25
end

Now we circle and label everything that is change-able, just as we always have. So what changes?

  • The input changes (the String, representing the food being ordered)

  • The price changes (the Number, representing the price of the food)

Pedagogy Note

Up until now, there’s been a pattern that students may not have noticed: the number of things in the Domain of a function is always equal to the number of labels in the example step, which is always equal to the number of variables in the definition. Make sure you explicitly draw students' attention to this here, and point out that this pattern no longer holds when it comes to piecewise functions.

If there are more unique labels in the examples than there are things in the Domain, we’re probably looking at a piecewise function.

We have two things changing (the item and the price), but only one thing is in our Domain. That’s how we know this function is piecewise function!

We start writing the definition as we normally would, using the function name and the input label from the examples step (fun order(item): …​ end. But since we know it’s a piecewise function, now we add ask: …​ end to the body of the function.

Then, for each different behavior we wrote in our examples, we add a condition to the body of our ask block. Each condition has a test, a then:, and a result, and we copy the results from our examples just as we always do.

fun​ order(item):
  ask:
    |           ...                  then:​ 6.00
    |           ...                  then:​ 3.50
    |           ...                  then:​ 5.25
    |           ...                  then: 2.25
  end
end

Finally, we fill in the tests with an expression that tells us when the function should behave that way. When should order return 6.00? when the menu item is "hamburger"!:

fun​ order(item):
  ask:
    | string-equal(item, "hamburger") then:​ 6.00
    |           ...                   then:​ 3.50
    |           ...                   then:​ 5.25
    |           ...                   then: 2.25
  end
end

Extension Activities

Option 1: Students create another function in the code that displays an image of the food instead of the price. This integrates earlier-learned skills in creating images and defining values.

Option 2: Students create a visual representation of how the computer moves through a conditional function.

Synthesize

  • Can you think of any situations in real life that can be modeled using a piecewise function?

  • Is "square root" a piecewise function? Why or Why not?

  • Is "absolute value" a piecewise function? Why or Why not?

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.