Where did I go wrong?

Today, I made a bunch of huge math mistakes in front of my classes. Intentionally.

Where did I go wrong?

At the board, I simplified, added, and multiplied a bunch of rational expressions. The hook was that I’d make an error in every solution. They just had to spot it. This pedagogy is a riff on the joy kids have at telling me I’ve done something wrong.

[Edit: Here’s a quick video of the technique in action and the PDF of several problems I used today.]

I incorporated ActivExpressions, because I have a set (though this activity hardly required the tech). We held a short voting period after each solution. After a quick peek at the results, I asked a volunteer to show the class my mistake.

Some implementation tips:

  • No spoilsports: tell the kids to hold their enthusiasm & corrections until the end
  • label every step so kids can reference exactly where you made your mistake
  • talk out loud as you solve (“I need to get a common denominator before I can add these rational expressions”)
  • pause about 10 seconds between steps so kids have a chance to absorb what you just did

This lesson was a blast! My kids were hanging on every word. Seriously.

Row game: simplify rational expressions

1 March 2015 Update: get the Word version of this file without the Scribd membership from my Google Drive.

Here’s a contribution to Kate Nowak’s row games collection. I played with the page layout so a pair of students could collaborate on a single sheet of paper. For the uninitiated, a row game is played by two students. They solve separate problems that have the same answer. It’s a collaborative event because when their answers don’t match, the kids have to work together to find the error in each other’s work. (After a quick look from Player B’s POV, I realize the questions are numbered right to left. Wonder how many kids will be bothered by that?)

Standards alignment: Georgia Performance Standard MM1A2d Students will simplify and operate with radical expressions, polynomials, and rational expressions. Add, subtract, multiply, and divide rational expressions.

Alice in Wonderland Logic

I’ll be teaching logical statements (inverse, converse, contrapositive…) this week. There’s a great little logic exchange in the scene below. For the sake of my deaf students and English Learners, I added subtitles to this clip from the 1999 made-for-tv movie, “Alice in Wonderland”. It works with the Georgia Math 1 task “From Wonderland to Functionland Learning Task“.

The part you want to see starts at 1:49 and goes for about a minute.

Relevant state standard:

MM1G2. Students will understand and use the language of mathematical argument and justification.

a. Use conjecture, inductive reasoning, deductive reasoning, counterexamples, and indirect proof as appropriate.
b. Understand and use the relationships among a statement and its converse, inverse, and contrapositive.

Function Family Scrapbook Project

This project was totally cribbed from someone else. Unfortunately, I cannot find the original link. I rewrote it and am kinda proud of the standards-based rubric at the end.

Using the metaphor of functions grouped into families, I ask students to create a scrapbook showing all the family members (function transformations) and describing them (characteristics). I will be assigning this project to be completed by students on their own time but you could also use it as an in-class assignment. Format of the completed scrapbook is unimportant.

Download it from scribd

Georgia Performance Standards:

MM1A1. (a-e) Students will explore and interpret the characteristics of functions, using graphs, tables, and simple algebraic techniques.

Carpenters do the best geometry

A carpenter’s measuring tricks fascinate me. These guys have figured out ways to do math without all that pesky math getting in the way.

The most recent trick I learned is a great geometry class problem: how to divide anything into equally-sized sections. For example, suppose I want to place the handles exactly 1/4 the way from each end on this drawer front.

I’ve also read that cabinetmakers use this trick to set equally-spaced dovetail joints.

You can prove the “trick” using a triangle congruence theorem1.

Better yet, there’s a tool to do the same division. Again, you can prove the tool works with a triangle congruence theorem2.

Edit: I took this in to my 9th grade students who are repeating Georgia’s Math 1 class (it’s a combination of algebra and geometry) and they crushed my excitement. Perhaps it’s just the crowd but those guys shot me down in flames. Zero interest in a cool trick. Zero interest in why it works. Honestly, I had higher hopes for “hey guys, you can divide anything — any size — into equal segments with only mental math.”

Georgia Performance Standard: MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons. c. Understand and use congruence postulates and theorems for triangles (SSS, SAS, ASA, AAS, HL).

1 ASA, as shown here
2 SSS

How long to fill the sink?

Next up with my ninth grade math class: solving rational equations. I’ve been having fun with combined rate problems such as, “How long does it take Timmy and Marsha to mow the grass if it takes Timmy 2 hours on his own and Marsha 1.5 hours on her own?”

I, therefore, with great pride present you my first Vimeo video (Edit: Filling the Sink):

How I’m using it:

  1. Put it out there as a warmup — how long do you think it’ll take to fill this sink? Show about 10 seconds of the video.
  2. Put it out there in the lesson — what about when I combine the faucet and pitcher?  Teach them combined rate problems as a specific case of rational equations.
  3. Watch “Trial 1 (faucet only)” and “Trial 2 (pitcher only)”, then compute time to fill the sink with both working together.
  4. Play “Trial 3” and compare your experimental results to my actual results.
  5. Close class with this: Assuming your computed time to fill the sink is different from the actual time, discuss what you think is the most significant source of error and how I could fix it in “Filling the Sink, Part II”.

Side note that’s aggravating but true: I’ve struggled with finding great applications of the concepts I teach because they’re rarely as complex as the stuff we’re expected to work with. With rational equations, I’m still looking for something where I’ll need to factor the numerator and denominator, cancel some terms out, and move along. My gut tells me this thing doesn’t exist.

Georgia Performance Standards: MM1A3d Solve simple rational equations.