Integrating Technology into

Mathematics (6-12)

MELT 2015

Appalachian State University

Kayla Chandler

DAY 2

Agenda

Time Activity

8:30 – 10:00

Additional Algebra Tasks• Elevator• Soda Function Machine• Noise Canceling Headphones

10:00 – 10:15 Break

10:15 – 11:45 • Leadership Development Session

11:45 – 1:00 Lunch

1:00 – 2:30 • Drawing vs. Constructing

2:30 – 2:45 Break

2:45 – 4:15 • Partitioning Land

4:15 – 4:30 Wrap up

SESSION 1

Rate of Change - Elevator

Difficult for students to understand Hard to reason about statically Draw upon students’ experiences on

elevators Simulation of an elevator in a 10-floor

building

Questions to Consider

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.

Questions to Consider

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 Q1.

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 Q1. Press the “Show Graph” button and compare the graph created to your sketch in Q2.

Questions to Consider

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.

How would your sketch from Q4 change if you were asked to include time on the x-axis and total distance traveled on the y-axis? Why?

Questions to Consider

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.

Questions to Consider

Replay the elevator simulation. What are some of the assumptions that were made about the situation for this particular simulation?

What assumptions do you think students’ might make about the elevator ride prior to sketching their graphs? Why?

Questions to Consider

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

Function machine is a common metaphor used to introduce students to the concept of function.

Draws attention to the relationship between the input (i.e. what goes into the machine) and output (i.e. what comes out).

Helpful for students to begin by examining functions and non-functions in real, non-numerical contexts

Questions to Consider

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?

Questions to Consider

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 Q1 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?

Questions to Consider

Are there any machines that you feel may not be appropriate to use with beginning algebra students? If so, which one(s) and why?

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”.

Noise Canceling Headphones

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

What do you think needs to be done to “cancel” the sound produced by the function f ?

Select the “show cancelling function” button. You should see another function rule,

, 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?

Questions to Consider

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.)

Testing your canceling function

Select the “button” tool and click on your graphics window where you want this new button to be placed.

In the pop up first name your button “hear f(x) + g(x)” in the caption bar.

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

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.

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.

Questions to Consider

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?

SESSION 2

Leadership Development Session

This session is devoted to a talk led by Dr. Anita Kitchens.

SESSION 3

Introduction to GSP

See what you can do in GSP. Try creating points, lines, segments, and

circles. Determine how to measure, drag, or

transform objects.

Questions to Consider

What types of things were you able to do in GSP?

Discuss the dependent and independent relationships of objects in GSP. How can you determine if an object is dependent or independent?

Questions to Consider

Use GSP to determine the angle sum of a triangle. (This is question 3.)

How would you classify the cognitive demand of the task in question 3?

What other geometric properties of triangles could you have students explore using your sketch from question 3 or with a slight modification of the sketch?

Drawing vs. Constructing

Create a right triangle. Measure the angles and then drag a vertex to

determine if the triangle always remains right.

Questions to Consider

How can we create figures so that they pass a drag test (i.e. keep the properties we want it to have) and don’t break?

Can you create a triangle that is a right triangle and will always be a right triangle when dragged?

Drawing vs. Constructing

What’s the difference? A construction keeps the properties of the

figure that we want it to have, thus we create our figure based on these properties.

A drawing may be able to represent the particular properties we desire but will not maintain these properties under a drag test.

Questions to Consider

Construct the following objects: Isosceles triangle Isosceles right triangle Equilateral triangle Rectangle Square Rhombus Kite Trapezoid Isosceles trapezoid Cyclic quadrilateral

Questions to Consider

In what ways is constructing in GSP similar to and different from using a compass and straightedge?

Why is it important for students to understand the difference between drawing and constructing?

How would you characterize the cognitive demand of the construction task? Why?

How can you use GSP to have students explore properties of particular objects? Provide an example.

SESSION 4

Partitioning Land

Two brothers are part of an expedition and they have discovered a new island. From the sky, they notice that the island is shaped like an irregular convex quadrilateral. They are not sure of the actual dimensions of the island, but they want to determine a way that they can divide the island so they both receive equal amounts of land. How should the brothers divide the island?

Questions to Consider

Use GSP to determine a method for dividing the land fairly among the brothers. Be prepared to explain why your method works.

Suppose the brothers want to make sure they each have land that is on the water. Devise a method that will satisfy each brother. Be prepared to explain your solution.

What prerequisite knowledge would students need to know in order to solve this problem and be able to explain why their approach works?

Midpoint Quadrilaterals

The interior quadrilateral formed by joining the consecutive midpoints of any convex quadrilateral.

Questions to Consider

Using GSP, create a generic, irregular convex quadrilateral and label it ABCD. Construct the midpoints of each of the sides and label them E, F, G, and H. Now construct segments joining consecutive pairs of midpoints to create a new quadrilateral, EFGH, also known as the midpoint quadrilateral. What type of quadrilateral is the midpoint quadrilateral? Prove your conjecture.

Questions to Consider

Compare the areas of the two quadrilaterals. Prove your conjecture.

Form and prove at least one additional conjecture regarding the midpoint quadrilateral.