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Elevator, Soda Function Machine, & Noise Canceling Headphones
Participant Handout
Elevator (Rate of change)
Rate of change is a difficult concept for students as they attempt to make sense of the variation of one variable in relation to another. Capturing rate of change in a static environment is even more challenging. Thankfully dynamic tools that allow for exploration of movement can facilitate students in understanding rate of change.
We will use simulated movement to represent real-time movement. Many students have experiences riding in elevators. These experiences can provide a basis for considering the relationship between the distance an elevator has traveled from the ground floor and the time that has elapsed.
Questions to consider:
1. Imagine that you get on an elevator on the ground floor of a 10-floor building. You ride the elevator to the 10th floor, the doors open, and you briefly peek out to the 10th floor, realize you pushed the wrong button and get back on. You then push the button for the 4th floor. The elevator travels back down to the 4th floor, stops, and you get out. Describe how your distance from the ground floor changes throughout this ride.
2. With time (in seconds) on the x-axis, and distance from the ground floor on the y-axis, sketch a graph that represents the elevator ride described in question 1.
3. Open the Elevator.ggb file. In the left Graphics window the 10 floors are indicated. The right Graphics window has the coordinate plane view. Press the “Start Elevator ” button. This should simulate the elevator trip described in question 1. Press the “Show Graph” button and compare the graph created to your sketch in question 2.
4. Imagine you got on an elevator on the 8th floor of a 10-floor building. You ride to the 3rd floor where you meet a friend who is trying to load some furniture onto the elevator, so you hold the doors open for a while to allow her to do so. Next you both ride to the ground floor to unload the furniture. With time on the x-axis and distance from the first floor on the y-axis, sketch a graph that represents this elevator ride.
5. How would your sketch from question 4 change if you were asked to include time on the x-axis and total distance traveled on the y-axis? Why?
6. Consider the sketch shown here. What might the elevator ride that resulted in this graph possibly have been? Explain. Be sure to discuss the units on the axes in your explanation.
7. Replay the elevator simulation. What are some of the assumptions that were made about the situation for this particular simulation?
8. What assumptions do you think students’ might make about the elevator ride prior to sketching their graphs? Why?
9. Imagine a student that watched the elevator simulation in the Elevator.ggb file. Prior to seeing the graph, they sketched the graph below. What might explain this student’s thinking?
Soda Function Machine
The function machine is a common metaphor used to introduce students to the concept of function. It is a useful metaphor since students often have experience with machines and it draws attention to the relationship between the input (i.e. what goes into the machine) and output (i.e. what comes out). Rather than jumping right into thinking about function with number, examining functions and non-functions in real, non-numerical contexts, can help students pay attention to the definition of function without being distracted by specific individual values.
Questions to consider:
1. Open the file Function_Machine.ggb and explore the five machines. (Press the reset button to remove the can from the machine between each input selection.) What is the input? What is the output? Which machines are functions and which are not? How do you know?
2. When discussing possible definitions of function a student suggests the following: each input is paired with a unique output. This was followed by a discussion about the use of the term “unique.” Which machines that you explored in question 1 would be helpful to a class discussion about the appropriateness of the term “unique” in the student’s description of function? How would the machines you choose be helpful in the discussion?
3. Are there any machines that you feel may not be appropriate to use with beginning algebra students? If so, which one(s) and why?
4. Common representations of function include graphs, sets of ordered pairs, mappings, and algebraic symbols. Which representations are most helpful to use when discussing the function machines? Why?
Noise Canceling Headphones
Sound waves are sinusoidal, and thus make a great context for thinking about trigonometric functions. For example, in recent years, noise-cancelling headphones have become very popular. These headphones work by calculating the ambient noise in a given area and producing a tone that negates the noise. Given a particular sound created by a sound wave (sine function), you will need to figure out how to “cancel it”.
Open GeoGebra and the file Noise_Canceling.ggb. The function rule shown, , is the musical note A. Press the “hear f (x)” button to listen to the note.
Questions to consider:
1. What do you think needs to be done to “cancel” the sound produced by the function f ?
2. Select the “show cancelling function” button. You should see another function rule, gx=sin(880πx-cπ), where the value of c is determined by a slider. Before using the slider, what do you think changing the value of c will do to the graph? To the sound produced by the function? Why?
3. Adjust the slider for c until you think you have created a function that would result in the “cancelling” of f if both f and g were played together. Describe your solution and why you think it works. (Note: GeoGebra can only play one function at a time.)
To test whether or not your “cancelling” function works we will create another “play sound button”. To create a “play sound” button:
1. Select the “button” tool (under slider drop down menu) and click on your graphics window where you want this new button to be placed.
2. In the pop up first name your button “hear f(x) + g(x)” in the caption bar.
3. In the GeoGebra script window enter the following: playsound[f(x)+g(x),1,5]. This script is indicating that the sum of the functions should be played for 5 seconds.
Questions to consider:
1. Test your “cancelling” function. If it did not work, continue to experiment until it does. Once you have a solution, explain what it is and why it works.
2. What important ideas are addressed in the Noise Cancelling Headphones task? If you were to use this task with high school students, what would your learning objectives be? Be specific.
3. The note A has a frequency of 440 hertz. How could you rewrite the function in the Noise_Cancelling.ggb file to support students in making this connection?
Adapted from: Adapted from: McCulloch, A., Lee, H. S., & Hollebrands, K. F. (In progress). Preparing to teach mathematics with technology: An integrated approach to algebra.