Lesson Overview
Using the Design Recipe, students will work through a series of word problems about calculating the height of a rocket after a given number of seconds from launch. The functions they write will be used to animate the rocket launch.
Lesson Objectives
Students will:
Design functions to solve word problems.
Use the Design Recipe to write contracts, test cases, and function definitions.
Anchor Standard
Common Core Math Standards
- F.LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions.
Additional standards alignment can be found at the end of this lesson
Teaching Guide
Materials, Resources, and Prep
For the Student
- Rocket-Height Design Recipe (in the student workbook)
Getting Started
1) Introduction
Functions are a key part of animation in computer programs. A function that draws a static picture of a bat, for example, can place the bat at a different location based on the input. When that input changes slightly based on time or user-interaction, the bat will appear to move. This is similar to the way that flip-book animations work, in which each page draws a static image that has changed by a small amount. When the pages are displayed quickly, the images appear to change smoothly.
Putting these images together, we arrive at an animation of the bat turning around.
In the online puzzles, students will find a black block for each function they create, in addition to the colored blocks they are used to. The black function box, which has no parameter inputs, represents the function as a Type of data. This allows you to pass your function into the 'start' function, where it can be used to control the rocket animation.
Another curiosity with this program is that the rocket-height function will be executed multiple times. The periodic execution creates the flip-book effect. As each second passes, the rocket-height function is executed again, the new location is calculated, and the rocket is re-drawn in its new location. This drawing and re-drawing in different locations gives the appearance of motion.
Lesson Tip
After creating simple linear movement, students will be asked to write functions to animate simple acceleration. Students will be given an input/output table from which to write their new function. You may want to work through these problems as a whole class, so that students can see how you might analyze an input/output table in order understand the relationship between input and output values.
Activity: Rocket Height
2) Online Puzzles
In this stage you'll write functions that manipulate images to create animations. Head to CS in Algebra stage 10 in Code Studio to get started programming.
Extension Activities
3) Non-linear Animation
The final puzzle of this stage is a Free Play puzzle that will allow you amd your students to experiment with other variations on the rocket-height formula. One activity that students find particularly interesting (and often challenging) is to write functions that produce non-linear acceleration. If your students are familiar with quadratics then you can call this out as such, but even younger students who haven't yet seen quadratics can enjoy this extension challenge.
Place the following input/output tables on the board and see if students can come up with functions that will produce the appropriate animation.
Challenge 1
Input | Output |
---|---|
1 | 10 |
2 | 40 |
3 | 90 |
4 | 160 |
Challenge 2
Input | Output |
---|---|
1 | 15 |
2 | 45 |
3 | 95 |
4 | 165 |
Once students have figured out the provided Input Output tables, encourage them to come up with non-linear animation functions of their own.
Standards Alignment
Common Core Math Standards
- 5.OA.1 - Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
- 5.OA.2 - Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
- 6.NS.8 - Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
- 6.EE.9 - Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
- 7.EE.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.
- 8.F.1 - Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1
- 8.F.2 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
- A.CED.1 - Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- A.CED.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- A.CED.3 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
- A.CED.4 - Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
- F.IF.1 - Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
- F.IF.2 - Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
- F.IF.3 - Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
- F.IF.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★
- F.IF.5 - Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★
- F.IF.6 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★
- F.IF.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
- F.IF.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
- F.BF.1 - Write a function that describes a relationship between two quantities.
- F.BF.2 - Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★
- F.LE.1 - Distinguish between situations that can be modeled with linear functions and with exponential functions.
- F.LE.2 - Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Common Core Math Practices
- MP.1 - Make sense of problems and persevere in solving them.
- MP.2 - Reason abstractly and quantitatively.
- MP.3 - Construct viable arguments and critique the reasoning of others.
- MP.4 - Model with mathematics.
- MP.5 - Use appropriate tools strategically.
- MP.6 - Attend to precision.
- MP.7 - Look for and make use of structure.
- MP.8 - Look for and express regularity in repeated reasoning.