A pictorial representation that will have you running naked through the streets

The sum of the first consecutive odd numbers is a square number.

Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.

Some of us (okay, me) may have presented something like this:

1     +     3     +     5 + … + (2n – 1)
(2n – 1) + … + 5     +     3     +     1

The sum of each column is 2n. We have n columns. The total is then × 2= 2n². We added the sum twice so 2n² ÷ 2 = n².

Can you see what perfect squares have to do with odd numbers? Me neither.

Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.

Inspired by this pictorial representation, I created this poster below.

 ¹ Lockhart might say it’s not the fact that perfect squares are made up of odd numbers which can be represented as L-shapes. What matters is the idea of chopping the square into these nested shapes.

The more sides you have, the smarter you are.

“How does shape affect your place in society?”
“The more sides you have, the greater your angles. So, the smarter you are.”

Two years ago, I created a lesson on Angles in a Polygon. The ‘hook’ was the opening minutes of the animated film Flatland: The Movie. In the story, Arthur Square asks his curious granddaughter if she has memorized her ‘laws of inheritance’.

Hex replies “Isosceles triangles have baby equilateral triangles. Equilateral triangles have baby squares. Squares have pentagons. Pentagons have hexagons, like me! And each new generation gets one new side until they get so many sides they look like a circle and become a priest.”

This film interestingly addresses many mathematical concepts, such as points, lines, and shapes in zero, one, and two dimensions as well as larger themes such as critical thinking.

Here it is:

I think it’s a pretty good lesson, but I decided to tinker with it. Here’s the new and improved version:

Yep. That’s it. Blank space.

I learned that from Sandra Ball when planning together for elementary school demonstration or team-teaching lessons. Just one of the many things I have learned from Sandra since joining the team a year ago.

The first activity is overly scaffolded. In the second version of the activity, the scaffolding is removed. Students will ask “How can I solve the problem?” versus “How does Mr. Hunter want me to solve the problem?”. Some students may need scaffolding, but I can better support these students by listening to and observing them. In the first assignment, I assumed all students would need scaffolding. And, really, if my students can’t think of using a table to organize information, what does that say about how numeracy is taught in my classroom?

Here are the documents as well as the three-part lesson plan:

Flatland Assignment
Flatland Assignment 2.0
Flatland Three-Part Lesson Plan

Math Manipulative of the Month – Pattern Blocks

MMM September 2011 Pattern Blocks (colour printer, double-sided)

Last year, a group of Surrey teachers suggested having a “Math Manipulative of the Month” at their school. Instantly, I thought this was a great idea. After this conversation, I created the brochure above. My hope is that this series of brochures can be used to generate conversations between teachers (and students, of course!).

Before trying the problems, I would ask teachers to get to know each MMM and list all they know about them. For example,

  1. “Two reds cover 1 yellow”, “Three triangles make 1 trapezoid”, etc.
  2. “All sides are the same length, except the base of the red trapezoid. It’s twice as long.”
  3. “The orange square and tan rhombus do not cover the other tiles.”

The symmetry problem ended up on the cutting room floor. Here it is: Pattern Blocks Symmetry.

Also, please see how the question “How many ways can you make 360 degrees?” becomes a problem-based lesson in Grade 6. Here’s the three-part lesson plan: Angles (format from Van de Walle).

I attempted to have a balance of primary and intermediate problems. How can each problem be adapted for the grade level that you teach?

Next month… Base Ten Blocks.

Never let them see you smile.

At least ’til November.

Anyone else remember being given this advice by veteran educators at the start of your teaching career? The thinking here was that it would prove too difficult to get students back on track once you loosened the reins. If you must, loosen up at the end of the semester. I could never pull this off. My true self, or at least my true teaching self, would make a special guest appearance by the end of the first class.

I often struggled with planning for the first day of classes. I’m just not able to lecture students for 75 minutes about consequences of unexcused absences, procedures for handing in homework, and lists of food & drink items that are acceptable to have in the classroom. Imagine sitting through this four times on Day 1. Welcome back!

“And one more thing… here’s a review worksheet that covers everything you should know from Math 9. See me or a counsellor if you’re having difficulties with it.”

I was also uncomfortable with the let’s-get-to-know-all-about-each-other approach. No “Find someone who…” searches for me.

When students left my classroom for the first time, I wanted them to believe that

  1. We were going to get to know each other as people, and
  2. We were going to do this while learning mathematics.

Here’s a PMa 10 1st Day Jigsaw activity that, although not perfect, attempts to convey this message.

I cut the squares and placed them in envelopes. In small groups, students pieced the puzzle back together so that questions and answers shared a common edge. An answer key is not provided, but the jigsaw puzzle part of the activity does provide students with some feedback.

These are not rich problems – they are review questions of important concepts & procedures from Math 9. However, I did listen to some interesting conversations. For example, in many groups, there were debates about which power (3^-2, -3^2, or (-3)^2) was equal to -9. One student said he remembered that a negative means flip (his words, not mine) and matched 3^-2 with 1/9. His group members asked him to explain why this works.

Please let me know what you think of this activity. Also, do you have a Day 1 lesson to share?

As a new school year begins, are you looking for posters to decorate your classroom? Learn how to create a gigantic math poster of your own.

Revisiting Pictorial Representations of Functions

The K-7 word walls were developed by my Numeracy Helping Teacher colleagues to help students and teachers communicate mathematically. They were not meant to ‘teach’ concepts but to help make visual and conceptual connections. The cards have been very popular with Surrey teachers. The Math 8 cards have been created and we will be sharing them with Surrey secondary teachers starting in September.

See the sample cards to the right. In an earlier post, I mentioned how concrete and pictorial representations of linear functions can enhance understanding. For example, in the expression 2n + 1, the coefficient of 2 can be interpreted as adding 2 tiles as the pattern continues.

The coefficient can also be visualized in another way. It may be easier to describe by looking at the card for constant. In the first figure, we can see two groups of one (one white square above the red square and one white square to the right of the red square). In the second figure, we can see two groups of two (one group of two white squares above the red square and one group of two white squares to the right of the red square). Similarly, in the third figure, we can see two groups of three. Finally, in the nth figure, there will be n groups of 2, or 2n, white tiles.

This can also be an interesting investigation when teaching quadratic functions (or a challenging extension when teaching linear functions). In the pattern to the right, the red squares in the first figure make a 2-by-3 rectangle. The red squares in the second figure make a 3-by-4 rectangle. We can see a 4-by-5 rectangle in the third figure. In the nth figure, there will be a rectangle with width n and length n + 1 . In each figure, there are also two white squares. Therefore, the expression is n(n + 1) + 2.

This pattern, too, can be be visualized in another way. For example, in each figure, the red tiles can be seen as being made up of a square and a rectangle. In the first figure, we can see 2 squares on top of a 2-by-2 square. In the second figure, we can see 3 squares on top of a 3-by-3 square, and so on. In the nth figure, there will be n squares on top of an n-by-n square. Remembering the 2 white squares, the expression is n^2 + n + 2.

The two expressions are equivalent but reflect different ideas.

How do you know that a relationship is linear? quadratic?
How are the pictorial representations of linear and quadratic functions the same? different?

To see more on this approach, visit I Hope This Old Train Breaks Down.

One more thing… I purposely did not circle the groups and shapes discussed above… I didn’t want to take away the fun of visualizing them for yourself.

iPad as Mathematical Communication Tool

When children think, respond, discuss, elaborate, write, read, listen, and inquire about mathematical concepts, they reap dual benefits: they communicate to learn mathematics and they learn to communicate mathematically(NCTM)

In general, I’ve been disappointed with many of the iPad apps categorized under Education. With new apps being added (270/day in June 2011), I’ve got to admit it’s getting better. A little better all the time.

As Orwell Kowalyshyn and/or Kevin Amboe mentioned last spring, apps from other categories such as Games or Photography may provide richer educational opportunities for students.

My daughter (6) is currently enjoying the game Slice It!. The goal is to slice shapes as evenly as possible. The number of slices you are allowed and the number of pieces the shape is to be sliced into is given. The challenges get increasing difficult. I can imagine using this app to explore mathematical concepts such as area, fractions, percents, and line symmetry. Perhaps students could take screenshots and explain their strategies to their classmates. Maybe they could explain how they know the pieces have approximately the same area. (The FAILED text that appears when not sliced into the correct number of pieces may turn off some educators. No noticeable signs of this affecting my daughter, at least so far.)

Students will benefit from iPads in the classroom not because there’s an app for practicing number operations, but because there’s an app for communicating their thinking. ShowMe, ScreenChomp, and Explain Everything have been listed/discussed in many math ed blogs. Students, not their teachers or Sal Khan, can create video explanations using these interactive whiteboard apps.

Meeting with Surrey & Vancouver secondary math teachers this summer, one teacher showed us a picture of two containers each filled with chocolate eggs. The number of eggs in the smaller container was given and we were asked to guess the number of eggs in the larger container (see Dan Meyer’s blog). Using her iPad 2, the teacher filmed us giving and justifying our estimates. In a classroom, teachers or, better yet, students could interview peers, administrators, parents, members of the community, etc. and then share and discuss these guesses and strategies.

One app that I had fun with this summer is iMotion HD. This app allows you to create and share stop motion movies from pictures you have taken. In the video below, I show why 1/2 + 1/3 = 5/6 using pattern blocks.

Using iMovie, I could have added narration but I chose not to. Why? Because I have no plans to share this with students*. I chose not to narrate my movie because students, not the teacher, should be doing the math. In this way, students communicate to learn and learn to communicate.

photo by pong0814

*Also, I have only one nephew. He is 18 months old and so far has been able to complete his algebra homework without asking his uncle to tutor him. The Khan Academy has already been widely and deservedly criticized by others. Please check out Karim Ani’s An Open Letter to Sal Khan on his Mathalicious blog.

How come we’re playing games today? I thought this was math class.

Inspired by reading the tweets & blogs of Surrey teachers over the summer, I thought I’d resurrect my blog.

In his blog, Richard deMerchant writes about how games, in addition to being fun, can help develop conceptual understanding of mathematics (http://rvdemerchant.wordpress.com/2011/08/31/count-down-part-two-games/). He also writes about the impact that debriefing strategies (“Why did you make that move?” etc.) has had on his son’s thinking.

I have seen this in my daughters as well. My 6-year-old loves the game/puzzle Camouflage. The challenge is to place polar bears on ice and fish in water while also having the game pieces fit on the board (see http://www.smartgamesandpuzzles.com/inventor/Camouflage.html for a better description). As she was playing, she went to place a piece down and then stopped herself saying “That can’t go there. It’ll make a square”. I asked her to explain this to me. She had figured out that if a move created a blank one-by-one square, then she would not be able to fit all the pieces on the board. (The game pieces are one-by-two dominoes or L-shaped triominoes). She developed this strategy on her own. As she completed the increasingly more difficult challenges, I could see her develop problem solving and reasoning skills (as well as spacial sense).

This year, I’m excited by the inclusion of the games learning outcomes in the Foundations and AWM pathways. This one comes from FoM 11:

But games/puzzles can also be used to address/enhance other learning outcomes in the math curriculum. For example,
  • rotations in Pentago
  • translations in Rush Hour
  • combinatorics in Mastermind
  • area in TopThis!
  • isometric drawings & volume in Block by Block
Each secondary school in Surrey will be receiving a games kit in the fall. Here’s the list: Secondary Games Kit
At a workshop in June, I asked teachers to play Blokus. Immediately, one teacher asked “What’s the point? Why are we doing this?”. It didn’t feel like math for him and it probably won’t feel like math for our students. However, aside from the strategy aspect, think of the possible connections to traditional math topics. For example,
  • transformations (when determining the number of game pieces, or ‘free polyominoes’)
  • area/ratios/percent (when determining the winner)
  • square roots (If the 4-player game board is 20-by-20, what should the dimensions of the 2-player game board be?)
In defining math, most of us math teachers will probably use words like ‘problem-solving’, ‘reasoning’, ‘patterns’, ‘estimation’, etc. (Would our students use these words or would they use words like ‘memorize’, ‘rules’, ‘formulas’…?) Compare a lesson in which students play (and discuss!) Blokus to one in which the teacher shows students how to divide rational expressions (1. factor numerators/denominators, 2. invert and multiply, 3. cancel) and students practice questions similar to the examples. In which lesson might you see the words listed above? In which classroom are students doing math?

Linear Functions – Concretely, Pictorially, Symbolically

Welcome to my blog!

I really enjoyed Marc’s Patterning the Blues activity (taken from Marian Small’s Big Ideas book that department heads received).

Teachers often talk about how manipulatives can help the struggling learner. I’m suggesting that having students solve problems concretely can assist all learners.

When I experienced this problem using the blue and yellow tiles, I gained a deeper understanding of the problem. The equation y = 3x + 2 now had meaning. I was able to find the pattern in the table to determine the number 3. By modelling the problem using tiles, I was able to see this as adding an extra 3 blue tiles every time the figure grew.

In the past, I had difficulty explaining to students where the 2 came from. I could convince them that it had to be there. For example, take the point (2, 8). Multiplying the 2 by 3 gives  6, so we need to add 2 more. Looking at this concretely & pictorially, the 2 now has meaning. For me, it is how many blue tiles there were before we start adding yellow & blue tiles. (See the photo below.)

Your students who used to get it symbolically will still get it if they approach it concretely. However, what it means to “get it” in your classroom will start to change.

Patterning the Blues
Patterning the Blues Concretely

I’d appreciate your comments. Maybe you have some thoughts on how this activity addresses the 7 processes?