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Lessons

Standards in this Lesson

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.

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

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

HSA.CED.A.1

Create equations and inequalities in one variable and use them to solve problems.

HSA.CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

HSF.BF.A.1.C

Compose functions.

HSF.BF.B

Build new functions from existing functions.

CSTA Standards
3B-AP-10

Use and adapt classic algorithms to solve computational problems.

Oklahoma Standards
OK.6.AP.C.01

Develop programs that utilize combinations of repetition, conditionals, and the manipulation of variables representing different data types.

OK.6.AP.M.01

Decompose problems into parts to facilitate the design, implementation, and review of programs.

OK.7.A.3.3

Represent real-world or mathematical situations using equations and inequalities involving variables and rational numbers.

OK.7.AP.C.01

Develop programs that utilize combinations of repetition, compound conditionals, and the manipulation of variables representing different data types.

OK.7.AP.M.01

Decompose problems into parts to facilitate the design, implementation, and review of increasingly complex programs.

OK.8.AP.M.01

Decompose problems and subproblems into parts to facilitate the design, implementation, and review of complex programs.

OK.A1.A.2

Represent and solve real-world and mathematical problems using linear inequalities, compound inequalities and systems of linear inequalities; interpret solutions in the original context.

OK.A1.A.2.2

Represent relationships in various contexts with compound and absolute value inequalities and solve the resulting inequalities by graphing and interpreting the solutions on a number line.

OK.L1.AP.A.01

Create a prototype that uses algorithms (e.g., searching, sorting, finding shortest distance) to provide a possible solution for a real-world problem.

OK.L2.AP.M.03

Create programming solutions by reusing existing code (e.g., libraries, Application Programming Interface (APIs), code repositories).

OK.PA.A.4.3

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

Textbook Alignment

IM Algebra 1
IM.Alg1.2.20

Writing and Solving Inequalities in One Variable

IM 7 Math™
IM.7.6.17

Modeling with Inequalities

IM 6 Math™
IM.6.7.10

Interpreting Inequalities
Practices in this Lesson

MP.4

Model with mathematics

(Also available in Pyret)

Students discover that inequalities have an important application in video games: keeping game characters on the screen! Students apply their understanding to edit code so that it will keep Sam the Butterfly safely in view.

Lesson Goals

Students will be able to:

  • apply their understanding of inequalities to keep a game character on the screen

Student-Facing Lesson Goals

  • I can use what I know about inequalities to define the boundaries that will keep a character on the screen.

Materials

Preparation

  • Make sure all materials have been gathered

  • Decide how students will be grouped in pairs
Glossary
coordinate

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

expression

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

🔗Introducing Sam 15 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 (Wescheme) and click "Save."

Have students turn to the Introducing Sam, click Run and use the arrow keys to investigate the program with their partner.

  • What is something you noticed about this program? As Sam moves, 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? Appropriate responses are x > -50, or (for students who notice that Sam only moves in increments of 10) x ≥ -40.

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

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!

To further support students, consider asking what three functions are defined in their starter files. Then, ask students what each function should do, when working properly.

  • 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 690

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

Investigate

Direct students to complete Left and Right with their partners. Once finished, they can fix the corresponding functions in their Sam the Butterfly file, and test them out.

Students will notice that fixing safe-left? keeps Sam from disappearing off the left side, 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.

Emphasize to students that they cannot trust the behavior of a complex system! After looking closely at examples and observing that they all pass, students should suspect that the bug is elsewhere.

Synthesize

  • Does safe-left? work correctly? How do you know?

  • Does safe-right? work correctly? How do you know?

🔗Protecting Sam on Both Sides 30 minutes

Overview

Students solve a word problem involving compound inequalities, using and to compose the simpler Boundary-checking functions from the previous lesson.

Launch

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

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.

Ask the volunteers what their name, Domain and Range are. Explain that you, the facilitator, will be providing a coordinate input. The functions safe-left? and safe-right? will respond with either "true" or "false".

The function onscreen? , however, will call the safe-left function! So the student roleplaying onscreen?should turn to safe-left and give the input to them.

For example:

  • Facilitator: "onscreen-huh 70"

  • onscreen? (turns to safe-left?): "safe-left-huh 70"

  • safe-left?: "true"

  • onscreen? (turns back to facilitator): "true"

  • Facilitator: "onscreen-huh -100"

  • onscreen? (turns to safe-left?): "safe-left-huh -100"

  • safe-left?: "false"

  • onscreen? (turns back to facilitator): "false"

  • Facilitator: "onscreen-huh 900"

  • onscreen? (turns to safe-left?): "safe-left-huh 900"

  • safe-left?: "true"

  • onscreen? (turns back to facilitator): "true"

Hopefully your students will notice that safe-right? did not participate in this roleplay scenario at all!

  • What is the problem with onscreen?? It’s only talking to safe-left?, it’s not checking with safe-right?

  • How can onscreen? check with both? It needs to talk to safe-left? AND safe-right?

Investigate

Have students complete Word Problem: onscreen?. When this function is entered into the editor, students should now see that Sam is protected on both sides of the screen.

Extension Option What if we wanted to keep Sam safe on the top and bottom edges of the screen as well? What additional functions would we need? What functions would need to change? We recommend that students tackling this challenge define a new function is-onscreen-2.

Synthesize

Bring back the three new student volunteers to roleplay those functions, with the onscreen function now working properly. Make sure students provide correct answers, testing both true and false conditions using coordinates where Sam is onscreen and offscreen.

  • How did it feel when you saw Sam hit both walls?

  • Are there multiple solutions for onscreen??

  • Is this Top-Down or Bottom-Up design?

🔗Boundary Detection in the Game 10 minutes

Overview

Students identify common patterns between two-dimensional Boundary detection and detecting whether a player is onscreen. They apply the same problem-solving and narrow mathematical concept from the previous lesson to a more general problem.

Launch

Have students open their in-progress game file and press Run. Invite them to analyze the movement of the danger and the target

  • How are the TARGET and DANGER behaving right now? They move across the screen.

  • What do we want to change? We want them to come back after they leave one side of the screen.

  • What happens to an image’s x-coordinate when it moves off the screen? An image is entirely off-screen if its x-coordinate is less than -50 and greater than 690.

  • How can we make the computer understand when an image has moved off the screen? We can teach the computer to compare the image’s coordinates to a boundary on the number line, just like we did with Sam the Butterfly!

Investigate

Have students apply what they learned from Sam the Butterly to fix the safe-left?, safe-right?, and onscreen? functions in their own code.

Since the screen dimensions for their game are 640x480, just like Sam, they can use their code from Sam as a starting point.

Common Misconceptions

  • Students will need to test their code with their images to see if the boundaries are correct for them. Students with large images may need to use slightly wider boundaries, or vice versa for small images. In some cases, students may have to go back and rescale their images if they are too large or too small for the game.

  • Students may be surprised that the same code that "traps Sam" also "resets the DANGER and TARGET ". It’s critical to explain that these functions do neither of those things! All they do is test if a coordinate is within a certain range on the x-axis. There is other code (hidden in the teachpack) that determines what to do if the coordinate is offscreen. The ability to re-use function is one of the most powerful features of mathematics - and programming!

Synthesize

  • The same code that "trapped" Sam also "resets" the DANGER and the TARGET. What is actually going on?

🔗Additional Exercises

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). 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.