Fall 2017 - Professor Kate Poirier

# Category: Projects

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 awareÂ of at leastÂ 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).

In addition to the benefits of using the calculator as a pedagogical tool (see the graphing calculators homework), 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 canâ€™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 â€śthinking before doing.â€ť

For project #3, choose one of the following exercises. (Comment on this post to indicate which exercise you have selected—do not choose an exercise that has already been selected by someone else.) Prepare a short lesson introducing the topic and explaining the issue the calculator encounters in this case. Your presentation must include the use of the virtual TI calculator (available on our classrooms’ desktop). You may also include hand-drawn graphs and/or Desmos graphs if they are relevant for your topic. In addition to your presentation, include a post on your ePortfolio outlining your presentation. Include screenshots or photographs of the graphing calculator.

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. DoÂ 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.Â 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?

## Due date: Tuesday, November 28

Between presentations, you will use this form to provide feedback to your classmates.

Use this rubric to fill out the form.

Recall our sample GeoGebra dynamic worksheet on Ceva’s Theorem. When you open the worksheet, it automatically takes you to slide 29/29 in the slideshow, but you can jump to slide 1 and scroll through them one by one. What you’re seeing is actually the construction of a single dynamic worksheet, one objectÂ at a time.Â (In this case, I think it is extremelyÂ helpful to see the dynamic worksheet built up like this, rather than just seeing the end result.) Actually, if I had selected slide 1 instead of 29 when I saved the .ggb file to upload, I wouldn’t have to jump to slide 1 in the dynamic worksheet.

If you’d like to add this feature to your own dynamic worksheet, all you have to do is turn on the Navigation Bar for Construction Steps.

In the desktop app:

1. Click on the View menu
2. Select Layout
3. Select Preferences – Graphics from the top of the window (the icon is the overlapping green circle and blue triangle)
4. Under Navigation Bar for Construction Steps, select Show (you can include the play button too if you’d like to automate the slideshow)

Then, when you upload your GeoGebra applet to the online GeoGeobra worksheet that you’re constructing, the navigation bar will appear as it does in the Ceva sample linked above.

I hope this helps with your GeoGebra projects. I can’t wait to see them!

Instructions for your Geogebra project can be found here.

Select one of the following topics and comment here with your selection to claim your topic. Check to make sure nobody else has chosen your topic.

• The theorem of Menelaus (Venema Chapter 9)
• Simpsonâ€™s theorem (Venema 11.6)
• Ptolemyâ€™s theorem (Venema 11.7)
• Napoleonâ€™s theorem and the Napoleon point (Venema 12.1)
• Morleyâ€™s theorem (Venema 12.6)
• Circumscribed circle and circumcenter (Venema 4.1)
• Extended law of sines (Venema 4.1)
• Angle bisector concurrence theorem (Venema 4.2)
• The medial triangle (Venema Exercises 5.1.1 to 5.1.4)
• Desargueâ€™s theorem (Venema 11.2)

Due date: Tuesday, October 24, 2:30pm

Individual topics will be assigned later. More details will be added later.

In addition to a presentation in class, your assignment is to includeÂ the following items in your ePortfolio:

1. The statement of the theorem/result that you have been assigned, written inÂ $LaTeX$, in the body of the post. You may copy this statement word-for-word from the text, or paraphrase it. Either way, it must be complete and precise.
2. A link to a GeoGebra dynamic worksheet (uploaded to GeoGebra Tube) that helps students understand the statement of your theorem. The dynamic worksheet should be completely self contained. Think of the worksheet as playing the following role:Â You are teaching a geometry courseÂ and will be absent for one class. The lesson for that day is the topic you have been assigned for this project.Â The substitute teacher assigned to cover your class does not have a background in geometry, so your students will have to learn the topic exclusively from your dynamic worksheet.Â Your worksheet must take advantage of the benefits GeoGebra has over traditional paper worksheets (for example, you should make use of the drag test).

You may also include extra details either inÂ the body of your post or in the dynamic worksheet,Â if you think they will be helpful. For example, you mayÂ include hints for the proof of your statement (why is the statement true?) or you mayÂ include helpful applications. These are optional and should only be included if they help students understand the statement.

There are many resources available online for help creating dynamic worksheets.Â Hereâ€™s one. Read Chapter 3 of the Venema text for other helpful tips. As a sample,Â hereÂ is the dynamic worksheet on Cevaâ€™s theorem that we will explore in class.

Once again, your classmates willÂ be asked to score your worksheet and offer detailed feedback. This will be similar to the rubric and feedback form for the Desmos mini-project.

Topic assignments

For the first project, I want to teach how students to interpret and create bar graphs and line plots by utilizing the Desmos program as a visual display. I made an example of both types of graphs by plotting points in a table for the line point graph. I spaced the points on the graph so that they represent the data correctly. I also used inequalities to fill certain spaces in order to fill the bar graphs.

https://www.desmos.com/calculator/gjjuusjxwv

I will teach lesson explaining how to solve inequality and how you graph it using the graph line and also useÂ  desmosÂ  in order to benefit from using the technology in order to let the student understand the topic more clear, I have to mention to my student the meaning of the inequality and to make sure that the students have the ability to distinguishÂ  between theÂ Â  inequality

signs <, >Â  and also the signs bigger than or equal and smaller than and equal . I will teach my student how to solve those equation having the inequality signs and let them graph it at the board also I will make sure they knows how to use desomsÂ  in order to show their solution for the inequality equation

Project #1 consists of two components: an oral presentations and an OpenLab post.Â The project must be completed individually, but you are encouraged to discuss withÂ your fellow classmates.

Instructions:

1. Select a standardÂ from theÂ New York State Common Core Learning StandardsÂ for math or another math topic of your choosing. (Some sample topics are listed below.) Each studentâ€™s topic must be different; inform the instructor of your choice.
2. UseÂ DesmosÂ to create an introductory activity or motivating example for this topic. (Create a free account so you can save your work.)
3. Prepare a 10-minute lesson that you will present to the class.
4. Create a post on your own EPortfolio briefly summarizing the lesson. (See the sample here.) ExplainÂ how you used Desmos and what features you used. Include links and/or screenshots. Let us know who the intended audience is for your lesson and what math content knowledge you are assuming they have. Title your post: â€śProject #1 â€“ [your topic].â€ť

Due date: Thursday, September 7
OpenLab post dueÂ beforeÂ class. Presentations in class.

SampleÂ topic ideas:

• The definition of the derivative: the tangent as a limit of secant lines
• Riemann sums and definite integrals
• Transformations of functions and their graphs
• Inverse functions and their graphs
• The definition of trigonometric functions using the unit circle
• Solving inequalities
• Regression: lines and curves of best fit
• Graphs of linear functions: changing slopes andÂ $y$-intercepts

Sample Desmos activity

Feedback:
On September 7, there will be 2-3 minutes between presentations. During this time, you will submit feedback for your classmatesâ€™ presentations using this form.

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