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Lessons

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    (Also available in Pyret)

    Students identify solutions and non-solutions of inequalities using an interactive starter file. This lesson also reviews the Boolean data type.

    Lesson Goals

    Students will be able to:

    • Describe the solution set of a simple inequality

    Student-Facing Lesson Goals

    • Let’s find solutions and non-solutions to inequalities.

    Materials

    Supplemental Materials

    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.

    🔗Booleans 20 minutes

    Overview

    While students have been cursorily introduced to the Boolean data type in our Simple Data Types lesson, Booleans are so different from other data types and so foundational to inequalities that this section reintroduces students to Booleans (by way of Circles of Evaluation and a starter file) in preparation for working with inequalities in the second section of the lesson.

    Launch

    To get ourselves warmed up, evaluate these Circles of Evaluation, convert them into code, and then log into WeScheme and test them out.

    (+ 1 2)

    (​- 4 5)

    (​* 8 0)

    Take a minute to think about the Circles of Evaluation below: What do you think the expressions mean? What do you expect them to evaluate to?

    Then convert these Circles of Evaluation to code and test them out.

    (> 1 2)

    (< 4 5)

    (= 8 0)

    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.

    Boolean is just another data type, like Number or Image. There are only two Boolean values: true and false.

    Investigate

    Turn to Boolean Functions and use the Boolean Starter File to complete the questions.

    • Students will see functions in this activity that they’ve never encountered before ( odd?, even?, less-than-one?, continent?, and primary-color?)! All five functions produce Booleans.

    • 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 by explaining that our Boolean-producing functions should sometimes return false.

    Synthesize

    How would you describe what a Boolean is?

    🔗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

    How are equations different from inequalities?

    Equations typically have finite solution sets: there’s only one answer for an unknown, or perhaps several answers.

    Inequalities, on the other hand, 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 learn to recognize that there are many possible solutions and non-solutions to an inequality and are able to identify whether or not a given number is or isn’t part of the solution set.

    • We are going to practice identifying whether or not a given number is part of the solution set.

    • Open the Simple Inequalities Starter File and click "Run".

    • This starter file includes a special inequality function that takes in a function (which tests numbers in an inequality) and a list of 8 numbers (to test in the function).

    • Look at the graph that appears (image below), as well as the provided code. A picture of an inequality displayed on a number line, with four red dots and four green dots

    • What do you Notice? What do you Wonder?

    Be sure to surface the following:

    • When we click "Run", we see a graph of the inequality on a number line.

    • The solution set is shaded in blue.

    • The 8 numbers provided in the list are shown as dots on the number-line. They will appear:

      • green when they’re part of the solution set

      • red when they are non-solutions

    • Find line 17 in the Simple Inequalities Starter File. Edit the list of values by changing one of the negative values to positive.

    • Hit "Run". Examine the graph that appears (sample image below). A picture of an inequality displayed on a number line, with three green dots and five red dots

    • How is this graph different from the one you first produced?

    A successful input in this starter file will include 4 solutions and 4 non-solutions; in other words, the image returned will show 4 green dots and 4 red dots.

    When students modify the list of numbers, they will see there are now 3 green dots and 5 red dots - along with a message that says, "Challenge yourself: Find 4 true examples and 4 false".

    Investigate

    • Complete Simple Inequalities with a partner, identifying solutions and non-solutions to each inequality and testing them in the Simple Inequalities Starter File.

    • For each inequality, you must find four solutions and four non-solutions.

    • Try using negatives, positives, fractions and decimals as you generate your lists.

    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, 1738598, 2031479, and 1501927). CCbadge Bootstrap 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.