Math Question Banks from New York

Recently googled: JMAP ExamView Question Banks of NY Regents math exams…going back to 1890!

Oh, and I go back to work on Tuesday, to a building with 50% more students than in May, to the year my classes’ English Learner population should tip 50%, probably to “float” into other teachers’ classrooms, to teach physics!, to my 8th year in the classroom, and with the best math department in the world. Am I ready? Heck no. Am I excited? Heck yeah!

Polar Clocks

This is the Polar Clock (apparently, it’s soooo 2009). I recommend grabbing the screensaver or smartphone app (Win/Mac/Android/iPhone versions all available). In a pinch, you can watch a video of someone else running the app here.

Polar Clock isn’t precisely a Meyerian[1] What Can You Do With This? creature.  But I do think the Polar Clock falls in the same genus as WCYDWT because it could inspire some pretty cool mathematical investigations.

If you’re a Georgia math teacher, check out standards from Math II, specifically the properties-of-circles stuff under MM2G3.

I ask y’all, what mathy questions does the Polar Clock inspire? Leave ’em in the comments.

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[1] As I understand it, Dan Meyer’s WCYDWT requires that the problem have a hook anyone can guess at, that the math scaffolding can slowly be lowered, and that the photo/video/hologram/whatever look good. I got this based on my reading of http://blog.mrmeyer.com/?cat=70.

Treasure Hunt!

Thanks for the inspiration, Kate Nowak. Your Circumcenters was an amazing lesson that my colleagues and I turned into a full-fledged project.

The day before Thanksgiving break, my students searched for approximately 25 treasures that were hidden inside and out of my school. We secured permission to hide treasures in offices of the most feared administrators, on the doors of teachers the kids love to hate, and on the walls of our halls.

The kids used Geometer’s Sketchpad with an embedded blueprint of our school (upstairs and down) to locate vertices of a triangle as given in a clue, then constructed all 4 points of concurrency. Upon showing me their 4 points, I unlocked a second part of their clue: hints that told them which point of concurrency marks the spot. In a mad dash, the kids grabbed the hall pass, a camera, and embarked on finding the flag. If a teacher or administrator busted them breaking rules or removing the treasure, they forfeited it. Students returned with photographic evidence of them at the site of the treasure flag.

Ignore the Man Behind the Curtain

Or, how this huge project came together

The numbers: Approximately 300 students participating. Three teachers @ 5 hours each to write the project. $400 in treasures and treasure flags.  Seven donors bought some of our supplies through DonorsChoose and we 3 teachers bought the rest.

Here’s how we set the project up:

  1. Get a blueprint for your school (I scanned the fire escape map from my wall) and paste it into Geometer’s Sketchpad or Geogebra.
  2. (In GSP), Right click the image, choose Properties, then uncheck the “Arrow Selectable” checkbox. This way, you and the students can’t move the picture around.
  3. Find a place to hide your treasure! This needs to be at one of the points of concurrency of a triangle. Here’s how we did it: Construct a triangle and all 4 points of concurrency. Get a little GSP help starting on page 22 of “Meet Geometer’s Sketchpad”. Manipulate the triangle by moving vertices until one of the points of concurrency falls in an interesting spot. Here’s one example:Screenshot from GSP showing a triangle and its 4 points of concurrency.
  4. Write a clue to tell students how to place the 3 vertices. Add a second step to the clue that tells them which point of concurrency the need to search out. (That last bit was important as we want students to construct all 4 points of concurrency but only hide treasure beneath one of them.)  In hindsight, I’d spend more time making the clues easy to read and decipher. Clue example:
  5. Repeat steps 3 & 4 until you have a whole bunch of these triangles.
  6. Name each of the treasures, associate prizes with them. Our treasures included: foam airplane toys, playing cards, hand sanitizer, candy, doughnuts for breakfast with a math teacher of your choice, and teacher buys you ice cream with lunch.
  7. Package the clues in interesting envelopes. I found colorful envelopes at a craft supply store.

The project was an amazing success. Kids loved it and were all excited about playing the game — even though we ran it the day before Thanksgiving Break. Teachers: this is completely worth the time to set up for your school. Can’t say enough good about the wonderful donors who helped with $300 worth of goodies, either.

Georgia Performance Standards Alignment:
MM1G3. Students will discover, prove, and apply properties of triangles, quadrilaterals, and other polygons.
e. Find and use points of concurrency in triangles: incenter, orthocenter, circumcenter, and centroid.

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