April 2012 edit: interested in how I present this activity, and many others, to teachers? Check out my talk at the AMTNJ Tech Conference: 2014 Desmos Talk

Here on the blog, the conic sections project we use at my HS in Algebra 2 is one of the most popular posts, generating lots of hits and e-mails to me asking for more information. I am just now grading conic sections projects for this year, and want to share some new additions to the project, and a rubric you can use. The projects are all over my living room now, just waiting to be graded.

For the newbies to this project – the concept is simple: use equations you have used, specifically conic sections, to draw something. The Desmos calculator is perfect for this task, and students turn in their graph-based picture, then a completed, colored picture.

So, what’s new this year?

FOLDERS AND LINKS

In the past, students printed their equations and submitted them. This year, students instead shared a link to their Desmos product using Edmodo. Also, using folders has improved student organization, making it easier to locate and edit crucial equations.

VERIFYING SYSTEMS

In my conics unit, students learn to solve both linear-conic and conic-conic systems. This year, I asked students to choose two systems from their drawing and verify the intersection points. This served as a personal review of the chapter, and students had an investment in linking the algebra they had learned to their picture.

SCREENCASTS

Last year, I participated in a Desmos webinar where I explained the evolution of the conics project. For the webinar, one of our sophomore students recorded a screencast where she explained an aspect of her picture. Having a student comment and reflect on their work was so powerful that I made it a requirement for all students this year. Many students chose ScreenCastOMatic to record, and the reflections were excellent. Edmodo was used to share links, though some students had tech issues which I will work to head off earlier the next time I give this project. Below is a screencast from Nick, who was kind enough to allow me to share his work with you:

I have received many e-mails from folks asking for guidelines and a rubric for this project, and am happy to share with you a more detailed document. Feel free to use any part of it, and let me know how it works in your classroom!

We just completed our Experimental Design chapter in AP Statistics. It’s one of my favorites, as there are so many opportunities for fun, meaningful activities and class discussions. The activity in this post is an adaptation of a project I first heard of through my friends at Villanova University, where it is used in Introductory Statistics courses.

Think about these “old wives’ tales” and slogans. Which ones are accepted to be true? What ones seem dubious?

Carrots improve vision

Chicken soup can cure a cold

Toads cause warts

Choosy mothers choose JIF

The 5-second rule

In this project, teams select a “famous saying” to explore. Using the principles of experimental design learned in AP Stats, teams design an experiment which could be use to test the claim. Treatments must be described, randomization procedures explained, and comparisons suggested. Should the experiment be blocked? Will a placebo be employed? Should blinding be used? These all must be addressed by the team in a powerpoint submitted to describe the project. Finally, teams peer-assess other projects, and provide feedback. Below, you can download the project description, and the grading rubric.

On the day the project is assigned, I give 15 minutes for the teams to form. Teams look over the list of topics, and are given a playing card. Then the draft of topics begins! The team drawing an ace gets to pick their topic first, then two, then 3….so that no topic is repeated in the class.

Teams were given about two weeks to work on the project, as we continued through the unit. The actual creation of the project does not take that long, but we had the project simmering in the background as experimental design ideas were learned in class.

As teams turned in their files, my colleague Joel moved them to slideshare so that they could be house on his web page. Here are two examples:

After we had all the projects, all students were assigned 3 of the projects to assess. Using a Google Form, students provided comments, which I then sorted and gave back to the teams. Amazing how students are usually in target when asked to grade. And the comments provided were mostly consructive and on point.

This post will present solutions to two puzzles I have presented here on the blog, and some ideas for extending the learning with your students.

THE TAKE-AWAY GAME:

This is a game I proposed a while back on the blog, and it took my friend Anthony from Twitter’s gentle nudging to ask for a solution. Here’s a re-cap of the game:

23 marks are placed on a board

On each turn, a player must remove 1, 2 or 3 marks

The player who clears the board wins

I often challenge classes with this game, letting kids play me and try to figure out the secrets. The beauty here is that my students are often polite souls, and will let me go first…which leads them to doom. It will often take a few days before a student can conquer me.

Here’s a video example:

And here is the message I sent back to Anthony, who requested the secrets to the game.

Here’s the secret to the take-away game – if I can make it so that the number of dots remaining is a multiple of 4, then I will win the game – guaranteed. If they take away 3, I remove 1. When they remove 2, I remove 2. If they remove 1, I remove 3. The game is all about groups of 4.

With 23 dots to start the game, I will offer the student a choice; who plays first.

If they let me play first (which they often do because they want to be polite), then I erase 3 dots (leaving 20) and wait for their eventual demise. If they play first, then I need them to slip up and make it so that I can control the multiples of 4. If they erase 2 on their first move, then I remove 1, leaving 20.

It often takes a few days of playings at the end of class for kids to develop the strategy. They will first realize that being left with 4 is sure doom. Then 8….then eventually they get it.

Have fun

And here are some ways to tweak the game, and see if students can develop strategies:

Change the number of starting marks

Change the number of marks allowed to be removed

Play with 3 players (actually, I’ve never tried this…)

THE HOT SEAT

I posted about the Hot Seat game recently. Time to reveal the secrets to this interesting pattern game –

This table shows the winning chair in the game, based on the number of chairs which start the game. Note that if the number of chairs is a power of 2, the winning chair will equal the number of chairs. From any multiple of 2, the winning chair goes up by 2, until the next power of 2 is reached.

How would your students express this pattern? What vocabulary would they need to use in order to communicate the pattern? After a class develops the data table, challenge students to write a concise rule which will identify the winning chair. Then, have students trade their explanations and critique them.

Could an algebra student develop a formula which outputs the winning chair, if the number of chairs is given?

Take a look at the winning chairs, plotted vs the number of chairs. Can we write a function which follows the pattern? Share your function ideas in the comments, and enjoy the challenge.

Dan Meyer’s recent post on “fake world” math tasks has me thinking about many of the openers and games I have used in my classroom. I have written about The Take-Away Game before, and I still use it often…until the kids learn how to beat me and the strategy is revealed. This next one is not so much a game, but more of a task, similar in some ways to the Locker Problem.

THE HOT SEAT

In this task, chairs are placed in a circle. Chairs will be removed from the circle using the following rules:

Chair #1 is removed first.

The next remaining chair is skipped, and the next chair removed.

This continues, with chairs skipped and removed until only one chair remains.

Once a chair is removed, it is “out” of the circle

Whoever is sitting in the last remaining chair “wins”

Here’s a brief Doceri video which shows some game playings:

Like the “Take-Away Game”, I can’t recall where I first encountered this problem. They have both been sitting in my files for over a dozen years. If anyone can name a source, I’d be happy to award some credit.

Why I enjoy this problem:

It’s not intimidating. We have a chance to draw, get out blocks, magnets….whatever we want to use to model the problem. Great for working in teams.

I can let the problem marinate. On one day, I may ask the class “Where should I sit if there are 8 chairs?”, and come back the next day with “How about 24 chairs?” If it seems like discussion is flowing, I can put my foot on the gas.

I can use this problem with all levels of students. If we need to create a data table and look for a pattern as a class, that can happen. If my honors kids want to fly with it, that can also happen.

The answer is not obvious, but a clear pattern eventually emerges if you model enough circles. And there will be some nice vocabulary opportunities as the payoff.

There are a number of ways to express the solution. Later this week, I will post the “answer”. Until then, have fun moving around the furniture.