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(Also available for Pyret)

Students discover the Boolean data type. They then learn to identify solutions and non-solutions of inequalities.

Prerequisites

Defining Functions

Relevant Standards

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

Common Core Math Standards
6.EE.B.5

Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.B.8

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

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

Student-Facing Lesson Goals

  • I can find solutions and non-solutions to an inequality

Materials

Preparation

  • Make sure all materials have been gathered

  • Decide how students will be grouped in pairs

Supplemental Resources

Key Points for the Facilitator

  • A Boolean is just another data type, like Numbers or Images. But unlike those types, there are only two values: true and false. While simple to explain, this different behavior can be confusing for some students.

  • Boolean-producing functions are essentially yes-or-no questions, so the names of the functions in this lesson read like questions.

  • For example, odd? and even? (pronounced "odd-huh?" and "even-huh?" in Scheme) are both functions that ask if a given number is odd or even.

  • Role-playing can help students understand the jobs of odd? and even?.

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

🔵🔺🔶
Glossary
Boolean

a type of data with two values: true and false

data types

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

🔗Introducing Booleans 20 minutes

Launch

Students should be logged into WeScheme

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 mean, 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 outputs mean?

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 data type, called a Boolean.

Investigate

Have students open the Boolean Starter File (Wescheme)

  • Explore the five functions in this starter file: odd?, even?, less-than-one?, continent?, primary-color?

  • All five functions produce Booleans. Through your exploration, see if you can come up with an explanation of what a Boolean is.

A Boolean is just another data type, like Numbers or Images. But unlike the others there are only two values: true and false.

  • Turn to Boolean Functions (Page 68) and use the starter file to complete the questions, identifying inputs that will make each function produce true, and inputs that will make each function false.

Synthesize

  • Students will see functions on this page that they’ve never encountered before! But instead of answering their questions, encourage them to make a guess about what 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.

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. == Introducing Inequalities 20 minutes

Overview

Students discover (or expand their understanding of) inequalities by identifying solutions and non-solutions and connecting expressions to graphic representations.

Launch

Have students open the Simple Inequalities Starter File. (Wescheme)

Unlike equations, which have finite solution sets, inequalities can have infinite solutions. Inequality expressions divide all of the numbers in the universe into two categories: solutions and non-solutions. It is important that students are able to recognize that there are many possible solutions and non-solutions to an inequality and that they can identify whether or not a given number is or isn’t part of the solution set.

This starter file includes a special inequality function that takes in a function, which tests numbers in an inequality, a list of 8 numbers (to test in the function), and plots the numbers and a graph of the inequality on a number line.

When students successfully include 4 solutions and 4 non-solutions, the image returned will show 4 green dots and 4 red dots.

🖼Show image

If their list of 8 values doesn’t include an equal number of solutions and non-solutions there will be an unequal distribution of red and green dots and they will get an error message encouraging them to adjust their list.

🖼Show image

Encourage students to use negatives, positives, fractions and decimals as they generate their lists.

The starter file includes an example. Read the example code in the file carefully and click run to see the image it returns. Discuss the code with your partner.

  • What do you Notice?

  • What do you Wonder?

Hiding Example Code In order to stop seeing the examples written into the starter file code, students can comment out the example code by adding a # in front of each of the lines they want to hide.

Investigate

Have students open to the Simple Inequalities (Page 69) and complete it with a partner, identifying solutions and non-solutions to each inequality and testing them in the Simple Inequalities Starter File. (Wescheme)

Synthesize

  • What patterns did you observe in how the inequalities worked?

🔗Additional Exercises:

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.