Fall 2016 - Professor Kate Poirier

# Category: Uncategorized(Page 2 of 3)

One of the most familiar technological tools in the classroom is the last we’ll discuss in this class: the graphing calculator. You are probably already aware of at least the basic functions of whatever calculator you have used in your own classes, but you might not have thought about the calculator as a pedagogical tool.

I’m of two minds about graphing calculators. On one hand, they are surprisingly powerful machines and, when used the right way, they can help a student understand a concept or an example without being distracted by rote computation. On the other, they’re clunky and old fashioned; we have much more powerful and user-friendly tools available now (for example, the software we’ve discussed in this class).

I found this 20-year-old report from Texas Instruments about the role of the calculator in math education, and figured I would hate-read it while I was procrastinating. After all, the report was put out by the same company that has had a near monopoly on calculators in classrooms for years…so it’s not exactly unbiased. However, the report discusses the exact same themes we’ve been discussing all semester! Take a look at the five myths mentioned near the beginning of the report; do they sound familiar? Familiarize yourself with the content of the rest of the report; you could replace the words “graphing calculator” with any other kind of technology essentially throughout the whole piece.

In addition to the benefits of using the calculator as a pedagogical tool, you should become familiar with the pitfalls as well. There is a nice chapter on Lies My Calculator and Computer Told Me from Stewart’s Calculus book. The examples listed in it aren’t the most relevant for us (many of them deal with rounding errors) but the chapter contains a nice quote:

Computers and calculators are not replacements for mathematical thought. They are just replacements for some kinds of mathematical labor, either numerical or symbolic. There are, and always will be, mathematical problems that can’t be solved by a calculator or computer, regardless of its size and speed. A calculator or computer does stretch the human capacity for handling numbers and symbols, but there is still considerable scope and necessity for “thinking before doing.”

Complete the following exercises:

1. Imagine you are trying to help your students understand $\lim_{n \to \infty} (1+ \frac{1}{n})^n$. Try substituting larger and larger numbers for $n$ in your calculator. What do you expect to see? What do you notice?
2. Graph the function $f(x)=\sqrt{4-\ln(x)}$ on your calculator. What behavior do you expect near the $y$-axis? Do you see it on the calculator’s graph? Compare the graph your calculator gives you with the graph Desmos gives you.
3. Graph the functions $f(x) = \sin(10x), g(x)=\sin(100x), h(x)=\sin(1000x)$ on your calculator. Do you see what you expect to see? Do you notice anything weird? What happens if you graph the same functions on Desmos?
4. Graph the function $f(x) = \sin(\ln(x))$ on your calculator in the window $[0,1]$ for $x$ and $[-1,1]$ for $y$. How many roots does it look like there are in $[0,1]$? Change the window to $[0,0.1]$ for $x$ and then to $[0,0.01]$ for $x$ with the same $y$-values. What has happened to the roots? Try graphing the same function in Desmos.
5. In the standard window on your calculator, graph the piecewise defined function $f(x)= 3x - 2$ if $x < 1.5$ and $x^2$ if $x \geq 1.5$. Ask the calculator to tell you the derivative at $x=1.5$. Is this what you were expecting? Try graphing the function on Desmos.
6. Use the equation solver on your calculator to solve $\frac{\sin(x)}{x} = \frac{1}{x}$. How many solutions do you expect?

My audience is trigonometry students in high school or college. I am assuming that they know and understand how to graph trig functions and the basic form for a sinusoidal function and its components:

|A| = amplitude
B = cycles from 0 to
period =
D = vertical shift
C = horizontal shift
(“phase shift” when B = 1)
The content is delivered, supposedly following a lesson in trigonometric functions, using maple as a pedagogical tool for expressing the translations of the sine and cosine functions with a change in C representing a horizontal shift, demonstrating how to transform each of the trigonometric functions to the other.

Maple is an appropriate tool for demonstrating this concept of trigonometry through an in-class lesson, because it enables you to do calculations, graphing, labels, and input text without any problems and easily correct errors, or change in equations, functions, or graphs.

Save

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In this presentation, my audience will be my classmates who took the mid-term exam. I am sure that all of us took it and my topic is how to use Maple to graph trigonometric functions with various amplitude, period, phase shift, and vertical shift. Also, how can we graph them in the same x-y coordinate.

I assume that my audience still remembered how they used sliders in Desmo activity to change the features of a trigonometric function. My content will be using a different software to solve a same problem. So, after this class, I believe my classmates will have some insights for further tasks associated to Maple. Maple is an appropriate tool for this activity because it might be new to some of us and using it show something we learned is a good practice.

If anyone is interested in using the Desmos Activity Builder for their final project, log into teacher.desmos.com with your Desmos account. It’s pretty straightforward to start building activities. Let me know if you have any questions.

On the midterm, you experienced the Desmos Activities from the student side; I created the activity using the Activity Builder on the teacher side.

There are different Maple tutorials available around the web, both for beginners and for advanced users. Of the tutorials I found, I thought Maple’s own was the clearest and easiest to use.

### Midterm corrections

You may re-submit one of the four components of the midterm by Tuesday. Remember that you are to work on your own, without assistance from other humans or the internet.

Also remember, in order for your old grade for that component to be replaced by the grade for your re-submission, your re-submission must be perfect. Remember that we use the rubric here (and adapt it to individual tasks) to assign scores. “Perfect” means scores of 4 for each task, except GeoGebra #8, which is out of 2 only. In general, the difference between a 3 (proficient) and 4 (exemplary) has to do not necessarily with the correctness of the work itself, but its presentation. Make sure your work is clear, logical, and precise; be sure to proof-read it for clarity as well as for spelling and grammar.

### Project #3

In class on Thursday, we discussed your next project on Maple. Presently, there are few rules for this assignment–you have a lot of freedom–but we will add details and rules if need be. The project is due the Tuesday after Thanksgiving. It must satisfy two conditions:

1. It must demonstrate knowledge and/or skills in the software itself. (For example, it must use both the computation and typesetting features, not just one or the other.)
2. It must use Maple in a “pedagogical” way. (This condition is somewhat vague, but you should ask yourself, “How am I using this as an instructional tool?)

Some ideas for your Maple project:

• lesson (like Project #1)
• In-class activity (like Project #2)
• Homework assignment
• In-class assessment

If you are not familiar with the software, you should spend some time playing around in it to see what it is capable of. Because the rules for the project are somewhat vague, be sure to discuss your idea with me before the due date.

The test has four components. CHOOSE THREE OF THE FOUR TO COMPLETE. Read the instructions for each component carefully. While some components require the use of a computer with access to the internet, you may not access webpages other than the ones linked below and you may not communicate with your peers or anyone else during the test.

### 1. Geometry component

Answer the questions in the space provided on the paper. (Here’s a link link to the PDF if you’d like to view the paper on the computer.)

### 3. Desmos component

1. Click on this link to access the Desmos activity.
3. Complete the activity by answering the questions on each page. Your work is automatically saved. You will be able to go back and edit your previous answers during the test as long as you keep the browser tab open.

### 4. GeoGebra component

Complete using the GeoGebra desktop app. Save your response as a GeoGebra (.ggb) file with your name as its filename. Email your file as an attachment to kate.poirier@utoronto.ca. (Save a copy of your file for your records.)

1. Place 6 points A, B, C, D, E, F in the plane so that A, C, and E lie on one line and B, D, and F lie on another line.
2. Color the two lines black and color the 6 points A, B, C, D, E, F gray.
3. Create the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{DE}$ and color them red. Let L be their intersection point. Color the point L red.
4. Create the lines $\overleftrightarrow{BC}$ and $\overleftrightarrow{EF}$ and color them green. Let M be their intersection point. Color the point M green.
5. Create the lines $\overleftrightarrow{CD}$ and $\overleftrightarrow{AF}$ and color them blue. Let N be their intersection point. Color the point N blue.
6. Use the drag test to see how the configuration of lines and points changes as you move the free points around. Pay special attention to the points L, M, and N.
7. Make a conjecture about the relationship between the points L, M, and N. (Hint, it may be helpful to hide all the lines and perform the drag test again.) Create one text box containing the statement of your conjecture. Be explicit and precise. Use full sentences.
8. Does the drag test consist of a proof of your conjecture? Why or why not? Create a second text box containing your answer.
1. If you would like to update your GeoGebra dynamic worksheet based on your peers’ feedback, you have until Friday night at 11:59pm to do so.
2. Here is the 3D GeoGebra worksheet that I was showing in class today. You can download it to see what I entered where and to change the perspective to 3D glasses.
3. Josiel and Mei have kindly allowed me to share their HW #4 and #6 solutions. (Mei’s got cut off when I was scanning, but you can still see most of her steps.) HW #4  HW #6

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