Friday, August 12, 2011

Virtual Math Manipulatives for Algebra

When I last posted about the NROC text resources, I focused on the use of the written text itself, but did you notice the manipulatives? Take a look at Unit 4, Lesson 1, Topic 3 of NROC’s Algebra 1—An Open Course. There, you’ll see a manipulative that relates the slope “m” and constant “b” of a linear equation in slope intercept form to its graph. Below is a snapshot of the manipulative.

Like physical manipulatives, virtual manipulatives are awesome tools for students trying to really understand and internalize (I’d use the term “grok,” but I doubt that word is commonly understood, these days) how a mathematical concept works. Here is a way to use this tool in your classroom.

Learning Objective(s)
• Understand how graphing is used to represent solutions to a linear equation
• Recognize how changing coefficients and constants change the graphed solution.

Assessment Type
The formative assessment mentioned here should be employed as part of an introduction or reinforcement activity before students are thoroughly familiar with the concepts referred to.

Assignment Details
In a classroom with only one internet enabled computer, one can have one of NROC’s manipulatives running on through an overhead projector. Explain the basics of what they're seeing in the manipulative, and then ask them to predict what will happen when you toggle one of the variables.

For example, to use the manipulative mentioned, remind students of the y=mx+b form of linear equations, and possibly run them through how to graph one using an X,Y table. Hint that you’re going to be showing them a shortcut soon. Read the introduction to the manipulative aloud to the class and field any questions, but do not use any of the slider bars to change the manipulative. Then, ask “What will happen to the graph if we increase the value of b in the manipulative below? How about if we make B negative? ” Have them record their prediction and share their prediction with one partner. Then, “What will happen if we make ‘m’ greater? If ‘m’ is a fraction? Negative?”

If at all possible, though, these manipulatives should be given to students directly to explore on their own. You’ll need to give them basic direction on how to make the manipulative go, but once you have, allow them to have fun exploring the manipulative. Eventually, structure their investigation, asking them “What happens if…?” questions and “Why does…?” questions. I recommend having them write down their responses for you. (See the Example Questions below)

So, that’s a great way to use virtual manipulatives in your classroom. I hope you have fun with them! Yet, what do you do if you want a manipulative, but NROC doesn’t have it in its text?

I have two go-to sources, Wolfram Math Demonstrations and Geogebra, both of which have large libraries of manipulatives and a build-your-own option if you need something new. Both are absolutely free, although user accounts are required for Geogebra. Both can also be embedded in your webpage. In fact, Geogebra is what was used to make the demonstrations made in the NROC text.

See my hints and tips below for more information on getting started with these two great manipulative resources. If you build your own, give me a link to it in the comments section. I’d love to see it!

In summary, manipulatives are great because they instantly provide a playful math experience while allowing the student to explore and internalize “If I change one thing, what happens to another?” You can access manipulatives through NROC’s text, by visiting the Wolfram Math Demonstrations page, or making your own with Geogebra!

Example Questions
1. What do you think will happen to the graph if we increase the value of ‘b’ in the manipulative? How about if we make b negative? What really happened when ‘b’ changed?
2. What do you think will happen if we make ‘m’ greater? What if ‘m’ is a fraction? Negative? What really happened when ‘m’ changed?
3. Overall, describe what ‘m’ changes in a graph of y=mx+b? What about b?
4. *Extra Credit Challenge!* If you have two y=mx+b equations graphed on the same page and ‘m’ is the same in both, but 'b' is different, what is the special name for how these two lines’ relationship?

4 Points for participating well in the class activity and discussion.
2 Points per question (6 total). The first point is for any honest attempt at a prediction given in a complete sentence. 2nd point is for noting the correct action of the graph when transformed. No penalty for skipping the Extra Credit, but take the opportunity to discuss the answer later as it introduces parallel lines.
Total: 10 points

Instructor Notes

Hints and Tips for Geogebra:
Download Geogebra and register with them to access their materials.

Once you register for Geogebra, go back and log in. Browse the wiki-library of resources, but be patient as things take a while to load. Geogebra's library is divided by language—you won’t see a list of manipulatives on the “Main” page. Look instead under the link that says “English.

If you want information on how to create or embed a manipulative in your website, look in their "Introductory Materials" for orientation. Their format is do-it-yourself and share. Again, if you make a manipulative, please post me a link of your creation! I’d love to see it!

Hints and Tips for Wolfram Math:

You’ll need to download a CDF player (free) to use the demonstrations, but wow! It’s worth it.--CDF Player

To navigate, try looking within the topics list in the drop down menu linked to here.--
Wolfram Math Demonstrations Topics List. You can also do a keyword search.

If you get stuck or find yourself with questions on how to make your own demonstration--Wolfram FAQ.

Also, if you see a demo that’s almost what you want, but not quite, emailing them is very effective. I usually get a response within a week. Or, if you’re at all familiar with Mathematica and/or programming you can make a DIY manipulative here too! Have a great time!