Category: Algebra

My “Fake World” Task

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.

Algebra

Student-Created Polynomial Digital Notebooks

Post author By Bob Lochel
Post date November 25, 2013
2 Comments on Student-Created Polynomial Digital Notebooks

Over half-way done my first time teaching Algebra 2 in over 8 years (under block scheduling), and it’s amazing how much technology has changed many of my former approaches. Nearing a chapter on polynomial functions, I was somewhat dreading the experience. This is a pretty dry chapter…synthetic division, rational root theorem, complex conjugate roots…there’s a lot of rules and regulations here, and not much room for engagement. Here’s what I settled upon to avoid dry lectures, promote student ownership, and encourage a serious review of resources:

DIGITAL NOTEBOOKS FOR POLYNOMIALS

On day 1 of the unit, I gave students their assignment for the chapter, which you are welcome to download: Polynomial Notebooks. The link describes the assignment:

This chapter contains many landmark theorems and ideas for analyzing polynomial functions. Your job is to create a digital notebook of the ideas below, using a Google Document to house your information, and eventually share with the class. You may use online resources, or examples you create, to serve as examples for each idea. The goal of this document is to serve as a resource which demonstrates an understanding of the ideas, and which helps you study their meaning and usage.

To be honest, this did not start off as I expected, as my students were simply not accustomed to not being fed material. The idea of seeking outside resources, being able to weigh their merits, and summarize them for use was foreign. But our second day in the computer lab was fruitful, as students had great questions about the Intermediate Value Theorem and Descartes’ Rule of Signs; what was really fasicnating were the student attempts to think about the ideas in language which made sense to them: “Does this mean that…?”, “Does this example work here…”.

Not all class time was spent doing research. We started class with sythetic division problems done on boards around the room. Also, I developed 3 “food for thought” questions each day during the unit, which helped drive discussions.

Name a polynomial with roots 2i and -3 (multiplicity 2).
How many possible negative real zeroes does the following polynomial have?
Show that polynomial given must have a real zero between 2 and 3.

Enjoy all of these Food for Thought questions: food_for_thought_daily_qs2

It’s also interesting how different groups have chosen to format their notebooks. While some are going through my list in a linear fashion, others have developed connections and examples to help make sense of the rules. One ambitious young man has developed an outline of the topics, rearranging the ideas, providing examples, and organizing links.

Tomorrow we will have a group quiz on the material, and I feel like this experience has helped students personalize their needs, and think about their gaps. I also feel more confident in the aiblity of the class to think about these polynomial rules as a whole, rathern than as a set of disconnected ideas. Hoping for a good day!

Tags algebra 2, polynomials

Algebra High School

Applications of Quadratic Functions

Post author By Bob Lochel
Post date November 4, 2013
2 Comments on Applications of Quadratic Functions

My class just completed its unit on quadratic functions, where we looked at all of the old favorites: completing-the-square, quadratic formula, -b/2a. We also looked at “sideways” parabolas (those of the form x=…), and the formal definition of a parabola, including the focus and directrix.

But beyond the important algebraic processes to be mastered, I wanted students to appreciate the many vital applications of parabolas. Towards the end of the unit, I posted the following task on Edmodo:

Research an application of parabolas, and explain how the properties of parabolas make them an effective shape for your chosen application, including specific vocabulary. Find a picture of your parabola in action, and use an application like Geogebra to find its equation. Turn in electronically as a single slide or page which could be posted.

Many of the questions students had about the assignment dealt with my intentionally non-specific instructions: Do I have to use the focus? What if I can’t find anything about the directrix? How do you use geogebra? How much do I need to write? This was one assignment where I needed to “play dumb”; I wanted to students to think about what was essential, and craft explanations carefully.

HOW THE CLASS DEVELOPED THE RUBRIC:

On the day the assignment was due, I printed out all student papers. I told students that the assignment would be worth 20 points, but I wanted their advice on how I should grade it. What should I look for? What evidence of understanding should I see? After a rough start, where we argued the benefits of “creativity”, the class settled on a nice list of 4 ideas:

Proper use of vocab: while not all vocabulary words are required, those that are mentioned should be used properly.
Understanding of the application: pretty self-explanatory
Equation: is it correct? does it model the situation?
Structure / Flow: this was the compromise for creativity. A good paper should have a logical structure which a reader should be able to follow.

The next step was to assign points to each category. I told students that the assignment was worth 20 points. With their group, I told students to come up with a method for allocating the 20 points. After a minute or two, all groups had contributed their point-allocation ideas, which were recorded on the board:

points

I then took a pseudo-average from the group results, and a 7-7-3-3 point structure was agreed upon.

NEXT STOP – SPEED-DATING

Armed with this rubric, I placed students in groups of 3, and used a version of Kate Nowak’s great speed-dating method to have students peer-assess their work. With the clock set for 4 minutes, students shared their parabola discoveries, and discussed ideas for improving their paper. After 4 minutes, the students (labeled A, B and C) moved to a new group.

A’s moved one group to their left
B’s moved one group to their right
C’s stayed at their current group

So, in 12 minutes, students had a chance to have their work evaluated by 6 peers, and see how their work stacked up to the class standard. At the end of the “speed dates” I gave students 2 more days to revise their papers, based on their peer reflections, and this revised paper would be what was graded.

The class found some great applications of parabolas, and the chance to reflect and revise not only made the papers so much sharper, but also allowed students to share each other’s applications. Some examples were:

Bridge Design:

Bridge