Unit 7:   Conditional Branching

imageUnit 7Conditional Branching
Unit Overview

Students use piecewise functions to move their players in response to key-presses.

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Product Outcomes:
  • Students will write update-player, which moves their player in response to key-presses

Standards and Evidence Statements:

Standards with prefix BS are specific to Bootstrap; others are from the Common Core. Mouse over each standard to see its corresponding evidence statements. Our Standards Document shows which units cover each standard.

  • A-SSE.1-2: The student interprets the structure of expressions to solve problems in context

    • interpretation of expressions that represent a quantity in context

    • interpretation of terms, factors, and coefficients of an expression

  • BS-DR.1: The student is able to translate a word problem into a Contract and Purpose Statement

    • given a word problem, identify the domain and range of a function

  • BS-DR.2: The student can derive test cases for a given contract and purpose statement

    • given a Contract and a Purpose Statement, write multiple examples or test cases

    • given multiple examples, identify patterns in order to label and name the variables

  • BS-DR.3: Given multiple test cases, the student can define a function

    • given examples and labeled variable(s), define the function

  • BS-PL.4: The student is familiar with the syntax for conditionals

    • defining and using functions than involve conditionals

Length: 50 Minutes
Glossary:
  • clause: a question and its corresponding answer in a conditional expression

  • conditional: a code expression made of questions and answers

  • piecewise function: a function that computes different expressions based on its input

Materials:
  • Computer for each student (or pair), running WeScheme or DrRacket with the bootstrap-teachpack installed

  • Student workbooks, and something to write with

  • Class poster (List of rules, language table, course calendar)

  • Overhead projector

Preparation:

Types

Functions

Values

Number

+ - * / sqr sqrt expt

1 ,4 ,44.6

String

string-append string-length

"hello"

Image

rectangle circle triangle ellipse star scale rotate put-image

(circle 25 "solid" "red")

Boolean

= > < string=? and or

true false



Luigi’s Pizza

Overview

Students analyze the behavior of a piecewise function

Learning Objectives

  • Students learn the concept of piecewise functions

  • Students learn about conditionals (how to write piecewise functions in code)

Evidence Statementes

  • Students will understand that functions can perform different computations based on characteristics of their inputs

  • Students will begin to see how Examples indicate the need for piecewise functions

  • Students will understand that cond statements capture pairs of questions and answers when coding a piecewise function

Product Outcomes

    Materials

    • Computer for each student (or pair), running WeScheme or DrRacket with the bootstrap-teachpack installed

    • Student workbooks, and something to write with

    • Class poster (List of rules, language table, course calendar)

    • Overhead projector

    Preparation

    Luigi’s Pizza (Time 20 min)

    • Luigi’s Pizza

      To get started with this lesson, complete Luigi’s Pizza Worksheet.

      Review students’ answers to the exercise. You can see a video demonstration of this intro at these links: 1, 2

    • The code for the cost function is written below:  

    • Up to now, all of the functions you’ve seen have done the same thing to their inputs:

      • green-triangle always made green triangles, no matter what the size was.

      • safe-left? always compared the input coordinate to -50, no matter what that input was.

      • update-danger always added or subtracted the same amount

      • and so on...

      This was evident when going from EXAMPLEs to the function definition: circling what changes essentially gives away the definition, and the number of variables would always match the number of things in the Domain.

      Turn to Page 67, fill in the Contract and EXAMPLEs for this function, then circle and label what changes.

      It may be worthwhile to have some EXAMPLEs and definitions written on the board, so students can follow along.

    • The cost function is special, because different toppings can result in totally different expressions being evaluated. If you were to circle everything that changes in the example, you would have the toppings circles and the price. That’s two changeable things, but the Domain of the function only has one thing in it - therefore, we can’t have two variables.

      Have students refer back to prior Design Recipe pages - it is essential that they realize that this is a fundamentally new situation, which will require a new technique in the Design Recipe!

    • Of course, price isn’t really an independent variable, since the price depends entirely on the topping. For example: if the topping is "cheese" the function will simply produce 9.00, if the topping is "pepperoni" the function will simply produce 10.50, and so on. The price is still defined in terms of the topping, and there are four possible toppings - four possible conditions - that the function needs to care about. The cost function makes use of a special language feature called conditionals, which is abbreviated in the code as cond.

    • Each conditional has at least one clause. Each clause has a Boolean "question" and a "result". In Luigi’s function, there is a clause for cheese, another for pepperoni, and so on. If the "question" evaluates to true, the "result" expression gets evaluated and returned. If the "question" is false, the computer will skip to the next clause.

      Look at the cost function:

      • How many clauses are there for the cost function?

      • Identify the question in the first clause.

      • Identify the question in the second clause.

      Square brackets enclose the question and answer for each clause. When students identify the questions, they should find the first expression within the square brackets. There can only be one expression in each answer.

    • The last clause in a conditional can be an else clause, which gets evaluated if all the questions are false.

      In the cost function, what gets returned if all the questions are false? What would happen if there was no else clause? Try removing that clause from the code and evaluating an unknown topping, and see what happens.

      else clauses are best used as a catch-all for cases that you can’t otherwise enumerate. If you can state a precise question for a clause, write the precise question instead of else. For example, if you have a function that does different things depending on whether some variable x is larger than 5, it is better for beginners to write the two questions (> x 5) and (<= x 5) rather than have the second question be else. Explicit questions make it easier to read and maintain programs.

    • Functions that use conditions are called piecewise functions, because each condition defines a separate piece of the function. Why are piecewise functions useful? Think about the player in your game: you’d like the player to move one way if you hit the "up" key, and another way if you hit the "down" key. Moving up and moving down need two different expressions! Without cond, you could only write a function that always moves the player up, or always moves it down, but not both.

    • Right now the else clause produces a String, even though the Range of the function is Number. Do you think this is a problem? Why or why not? As human beings, having output that breaks that contract might not be a problem: we know that the functions will produce the cost of a pizza or an error message. But what if the output of this code didn’t go to humans at all? What if we want to use this function from within some other code? Is it possible that that code might get confused? To find out, uncomment the last line of the program (start cost by removing the semicolon. When you click "Run", the simulator will use cost function to run the cash register. See what happens if you order off the menu!

      To fix this, let’s change the else clause to return a really high price. After all, if the customer is willing to pay a million bucks, Luigi will make whatever pizza they want!

    Player Movement

    Overview

    Students write a piecewise function that allows their player to move up and down using the arrow keys.

    Learning Objectives

    • Students learn that handling user input needs piecewise functions

    • Students learn which questions to ask to detect key presses

    • Students learn how to write their own conditional expressions

    • Students reason about relative positioning of objects using mathematics

    Evidence Statementes

    • Students will understand how to write different expressions that compute new coordinates in different directions

    • Students will be able to write expressions that check which key was pressed

    • Students will be able to write a conditional connecting key presses to different directions of movement

    • Students will learn to write examples that illustrate each condition in a piecewise function

    • Students will be able to experiment with using functions to change the speed or nature of character movement in a game

    Product Outcomes

    • Students will write update-player, which moves their player in response to key-presses

    Materials

      Preparation

        Player Movement (Time 25 min)

        • Player MovementNow that we know cond, we can write update-player.

          image Look at the following picture, which describes what happens when the "up" arrow key is pressed.

          • What is the player’s starting x-coordinate?

          • What is the player’s starting y-coordinate?

          • What about after the player moves?
            • What are the new x and y coordinates?

            • What has changed and by how much?

            • What happens when we press the "down" arrow key?

            • What should the new coordinates be then?

            • What should happen if a user presses a key other than the up or down arrows?

          Draw a screen on the board, and label the coordinates for a player, target and danger. Circle all the data associated with the Player.

        • The following table summarizes what should happen to the player for each key:

          When...

          Do...

          key is "up"

          add 20 to player-y

          key is "down"

          subtract 20 from player-y

          key is anything else

          return y unchanged

          On Page 68 in your workbook, you’ll find the word problem for update-player.

          (If you don’t like using the arrow keys to make the player move up and down, you can just as easily change them to work with "w" and "x"!)

          Be sure to check students’ Contracts and EXAMPLEs during this exercise, especially when it’s time for them to circle and label what changes between examples. This is the crucial step in the Design Recipe where they should discover the need for cond.

        • You can also add more advanced movement, by using what you learned about boolean functions. Here are some ideas:

          • Warping: instead of having the player’s y-coordinate change by adding or subtracting, replace it with a Number to have the player suddenly appear at that location. (For example, hitting the "c" key causes the player to warp back to the center of the screen, at y=240.)

          • Boundary-detection Change the condition for moving up so that the player only moves up if key="up" AND player-y is less than 480. Likewise, change the condition for "down" to also check that player-y is greater than 0.

          • Wrapping: Add a condition (before any of the keys) that checks to see if the player’s y-coordinate is above the screen (y > 480). If it is, have the player warp to the bottom (y=0). Add another condition so that the player warps back up to the top of the screen if it moves below the bottom.

          • Challenge: Have the player hide when the "h" key is pressed, only to re-appear when it is pressed again!

          Hint on the challenge: multiply by -1.

        Closing

        Overview

        Learning Objectives

          Product Outcomes

            Materials

              Preparation

                Closing (Time 5 min)

                • ClosingCongratulations - you’ve got the beginnings of a working game! What’s still missing? Nothing happens when the player collides with the object or target! We’re going to fix these over the next few lessons, and also work on the artwork and story for our games, so stay tuned!

                  • Have students volunteer what they learned in this lesson

                  • Reward behaviors that you value: teamwork, note-taking, engagement, etc

                  • Pass out exit slips, dismiss, clean up.