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(Also available for Pyret)

Students discover the Boolean data type, and apply knowledge of inequalities to simple programming problems.

Prerequisites

Problem Decomposition

Relevant Standards

Select one or more standards from the menu on the left (⌘-click on Mac, Ctrl-click elsewhere).

Common Core Math Standards
7.EE.B.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

HSF.IF.A.2

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

MP.1

Make sense of problems and persevere in solving them

MP.4

Model with mathematics

Oklahoma Standards
OK.PA.A.4.3

Represent real-world situations using equations and inequalities involving one variable.

Lesson Goals

Students will be able to:

  • Describe the solution set of a simple inequality

  • Explain the 'Boolean' datatype

Student-Facing Lesson Goals

  • I can use two or more inequalities together and describe the area they enclose.

  • I can explain to someone else what a Boolean is.

Materials

Preparation

  • Make sure all materials have been gathered

  • Decide how students will be grouped in pairs

Key Points for the Facilitator

  • A Boolean is just another data type, like Number, or Image, but unlike the others there are only two values: true and false. While simple to explain, this different behavior can be confusing for some students.

  • Functions that produce Booleans are typically questions, so the names of the functions in this lesson read like questions.

For example, safe-left?, onscreen? are both functions that are asking if a condition, such as an image being on the screen, is true or false. * Role-playing can help students understand the jobs of safe-left? and safe-right?.

Click here to see the prior unit-based version

Glossary
Boolean

a type of data with two values: true and false

coordinate

a number or set of numbers describing an object’s location

datatypes

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

expression

a computation written in the rules of some language (such as arithmetic, code, or a Circle of Evaluation)

🔗Introducing Booleans 15 minutes

Overview

Students discover the concept of inequalities (or apply it, if they’ve seen it before) in programming, and extend their knowledge of data types, Contracts, and Circles of Evaluation.

Launch

Students should have their workbook, pencil, and be logged into WeScheme and have their workbooks with a pen or pencil.

Ask students to evaluate Circles of Evaluation for simple expressions they’ve seen before, and ask them to convert them into code.

  • +
    12
    (+ 1 2)

  • ​-
    45
    (​- 4 5)

  • ​*
    80
    (​* 8 0)

Then show them unfamiliar Circles of Evaluation, and ask them to hypothesize what they think they means, what they will evaluate to, and what the code would look like.

  • >
    12
    (> 1 2)

  • <
    45
    (< 4 5)

  • =
    80
    (= 8 0)

Have students convert these Circles to code and type them in. What did they evaluate to? What do they think the output means?

Values like true and false obviously aren’t Numbers or Images. But they also aren’t Strings, or else they would have quotes around them. We’ve found a new datatype, called a Boolean. Booleans are the answers to any yes-or-no question (for example: "Is five greater than two?", "Did a character hit a wall?", etc.)

Investigate

Have students open to the Inequalities — Launch (Page 42) worksheet and complete with a partner.

Synthesize

  • Students will see functions on the worksheet that they’ve never encountered before! But instead of answering their questions, encourage them to make a guess about they do, and then type it in to discover for themselves.

  • Explicitly point out that everything they know still works! They can use their reasoning about Circles of Evaluation and Contracts to figure things out.

🔗Introducing Sam 30 minutes

Overview

Students are introduced to Sam the Butterfly: a simple activity in which they must write 1-step inequalities to detect when Sam has gone too far in one dimension.

Launch

Have students open the Sam The Butterfly starter file and click "Save."

Have students turn to the Introducing Sam (Page 43) and investigate the program with their partner.

Let students figure out that they need to press "Run" to see what the program does, and that the arrow keys control Sam.

  • What is something you noticed about this program? Sam can be moved with the arrow keys, the coordinates are displayed at the top of the screen, the coordinates are all in the 1st quadrant, etc.

  • What do you see when Sam is at (0,0)? Why is that? You only see part of Sam’s wing. Sam’s position is based on the center of Sam’s image.

  • How far can Sam go to the left and stay on the screen? Up to, but not beyond, an x of -40.

  • How could we write this as an expression? x ≥ -40, or x > -50

Every time Sam moves, we want to check and see if Sam is safe.

  • There are three functions defined in this file. What are they?

Note: In this programming language, question marks are prounced "huh?". So safe-left? would be prounounced "safe left huh?" This can be a source of some amusement for students!

Optional: For extra scaffolding…​

  • What should our left-checking function do? Check to see if x is greater than -50

  • What should our right-checking function do? Check to see if x is less than 490

  • What should onscreen? do? Answers may vary, let students drive the discussion, and don’t give away the answer

Investigate

With their partners, students complete Left and Right (Page 44). Once finished, students can fix the corresponding functions in their Sam the Butterly file, and test them out.

Students will notice that fixing safe-left? keeps Sam from disappearing off the left, but fixing safe-right? doesn’t seem to keep Sam from disappearing off the right side! When students encounter this, encourage them to look through the code to try and figure out why. The answer will be revealed in the next lesson.

  • Recruit three new student volunteers to roleplay those same functions, which have now been corrected. Make sure students provide correct answers, testing both true and false conditions using coordinates where Sam is onscreen and offscreen.

Common Misconceptions

  • Many students - especially traditionally high-achieving ones - will be very concerned about writing examples that are "wrong." The misconception here is that an expression that produces false is somehow incorrect. You can preempt this in advance, by explaining that our Boolean-producing functions should sometimes return false, such as when Sam is offscreen.

  • Push students to think carefully about corner-cases, such as when Sam is exactly at -50 or 690.

Synthesize

  • Recruit three student volunteers to roleplay the functions safe-left?, safe-right? and onscreen?. Give them 1 minute to read the contract and code, as written in the program.

  • For each of them, ask the volunteers what their name, Domain and Range are, and then test them out by calling out their name, followed by a number. (For example, "(safe-left? 20)!", "(safe-right? -100)!") Note: Do not ask onscreen? to roleplay beyond their contract! They’ll get involved in the next lesson…​

🔗Additional Exercises

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). CCbadge Bootstrap:Algebra by Emmanuel Schanzer, Jen Poole, Ed Campos Jr, Flannery Denny, and Dorai Sitaram is licensed under a Creative Commons 4.0 Unported License. Based on a work at www.BootstrapWorld.org. Permissions beyond the scope of this license may be available by contacting schanzer@BootstrapWorld.org.