Monday, February 27, 2012

Some Sum to One

Another quick activity. Put this version of it together for the 5th graders, but didn't get a chance to do it with them. This would be suitable 5-8, and my preservice teachers got a lot out of it also. I added some more support (in structure) on the square for middle school age kids. The grid divides well into thirds, fourths, twelfths and the like.

The idea was probably Mondrian inspired, mixed with a geometry activity that I like: divide a geoboard into four non-congruent equal parts.

The task for this one is to make different fractions that add, or fit together, to one, or a whole square. Had very interesting conversation with Dave Coffey today about fractions and introducing operations, and representation... and then that conversation spread to Lisa Kasmer.  Nice to work with such interesting colleagues.

EDIT: oops! had the permission set incorrectly on this. If it's not appearing below, here's the link to the pdf.

Two examples from my preservice teachers. The second one was made with flaps that lifted up to reveal the value of the fraction shown. Nice!

You can have your students simplify or not - both create interesting situations.

A nice extension challenge is to do this Egyptian style, with all unit fractions.

Friday, February 24, 2012


Had just a quick time with the elementary students today. On short notice, I needed a game suitable for 3rd to 5th graders. This obviously isn't super-original, but a spin on the fabulous Yahtzee.

The game was popular with the kids ("5 dice?!") and many had played Yahtzee so there was  not a lot of explaining necessary. I wanted to experiment with 2 rolls instead of 3 rolls.

To introduce the game, I talked about how mathematicians like to notice. We rolled five dice and talked about what they noticed. The scoring rolls were chosen to provide more relationships to notice as well as some computation practice. The dice pips are a good support to the third graders who were novices at the multiplication, and the 5th graders were really seriously considering which scoring slots were related to what they had rolled. One team of girls played cooperatively instead of competitively, and that worked well.
God bless you, Professor Yaht!
image Josh Kenzer @ Flickr

I think the game has some replay value - due to the brilliance of the Yahtzee designers. Dice randomness plus getting better with more familiarity with the scoring is good for repeat business. The math is reasonable for looking for and increasing automaticity with computation.  Let me know what works for you and what variations you might try if you give it a go.

Thursday, February 16, 2012

Math Teachers at Play 47

Doesn't this look like that door from the story problem?
Welcome to the 47th edition of the
Math Teachers at Play Blog Carnival

by mag3737
The Number Dictionary reveals two particularly interesting facts about 47.
  • 47 is a prime and a Gaussian prime.
  • 47 is the difference between two squares.
Don't these make you wonder?

I mean 7s and 3s show up in the ones places of many primes, of course.  (What is the frequency of one's place digits in prime numbers anyway?  Are 7s and 3s more common than 1s and 9s?)  But 47 is so nicely nestled between 43 and 53. What's the next 7-prime that's the only prime between two adjacent 3-primes? Jim Loy has a nice page on Gaussian primes - that's extending the idea of prime numbers to complex numbers. Surprising things happen then - like 2 and 5 are not Gaussian primes. In fact there are only 12 integers that are Gaussian primes less than 100. Aren't there 24 primes less than 100? Is there a reason that it's half the number? There's some nice complexity.

Being the difference of two squares is also interesting. Is there a triangle connected to that? (Bet I could illustrate that in GeoGebra...) The difference of two adjacent squares - is there a pattern to those numbers?  How would you quickly estimate which squares? Can you tell precisely just from knowing they're adjacent? Could the difference of adjacent squares pattern help you find more Pythagorean triples?

I don't think I've appreciated 47 nearly enough before this carnival. But we should move on since there are a lot of neat entries this month.

by Eva the Weaver
About teaching 
David Coffey sent in Whose problem is it? from Delta Scape, which is good because it saved me poaching it for the curator's corner below. He's thinking about supporting students in being problem finders not just problem solvers.

Bon Crowder instructs How to Create an Inquiry Zone for Math Learning from Math is Not A Four Letter Word.  Making learning safe for students - even in math class.

Peter Price blows Are Wind Farms The Solution? Do the Math! our way from Classroom Professor, How math should be used to inform students' learning about the environment and how to protect it, using video about wind farms and the challenging questions they raise.  

Peter Rowlett asks Have you used maths in the news in school? posted at Travels in a Mathematical World Blog. He would be interested in hearing from people who have used maths in the news to enrich their teaching.

Becky Johnston shares a young learner's Metamorphosis pictures Wide Open Campus, and argues for a broader view of mathematical thinking.

Rodi Steinig explains What We Did in Math Circle, AND Why We Did It posted at Talking Stick Math Circle Blog, presenting Math Circles' pedagogy via a session with 8-10 year olds.

Natasha Chen from IMACS persuades us to avoid teaching tricks and instead urges us to make sense of word problems.

Balazs Koren (kobak) presents digitális bennszülöttszelidítés (digital teaching tools) posted at kobak pont org. (The Google translation; there's a slideshare which doesn't translate but which you can read most of.)
by Leo Reynolds

Algebra and Geometry 

Alexander Bogomolny presents Finding a Parallelogram in 3D posted at CTK Insights, with three solutions based on the definition and properties of parallelogram.He de "Almost a classic problem of finding a parallelogram cross-section of an irregular pyramid. " Plenty of his insightful and excellent visualizations.

at Let's Play Math!, the founder of this here carnival here,  writes Understanding Algebra: How Many Roots? because "I wish my algebra teacher had explained it like James Tanton does. It makes so much sense!"

Fawn Nguyen presents Playing with Barbies posted at her self-titled blog, which uses Barbie dolls to teach proportions. "The kids really get into the activity and are proud of their work."

Cassie Becker is developing 16 algebra activities based on the Common Core to promote student engagement. The first 5 are available already.

Luis R. Guzman, Jr. presents A Complex Problem looking at roots of Complex numbers at Guzman's Mathematics Weblog, using standard and Euler notation. But he's not afraid to get negative, too.

Guillermo Bautista also speaks C, and investigates Complex Conjugates at Math and Multimedia.

Maria Miller shares a triangle area problem (with her own solution) from at her Homeschool Math Blog.

by Leo Reynolds
Mainul Maksud posts an algorithm for Extracting Square Roots with several examples. From Mental Math with Tricks and Shortcuts.  Does his method apply to Luis' method for complex numbers?

Yan Kow Cheong shares Social Media Math posted at Singapore Math, 8 exercises using social media as a context.

Santo D'Agostino has been writing a series, How Much Mathematics Should A Student Memorize? Part 5, The Multiplication Table is up at QED Insight Most of the post is in an 18 page pdf file that quite thoroughly investigates multiplication facts and properties. It even addresses one of the 47 problems from up above!

Christy presents A few more math books and activities from Just another step to take.... Instead of following their regular math curriculum, they've been reading some math books and doing math play.

Peter Rowlett has Favourite popular mathematics books posted at Travels in a Mathematical World Blog, saying, "I collected this list of favourite popular mathematics books from people on Twitter and Google+ and there's more in the comments." (Love these kinds of lists, and there's lots of overlap with my list, but definitely new stuff, too.)

Chintamani Gadre further develops some divisibility rules in this post.

I recently wrote up one of my favorite games, Fraction Catch. Pretty good game, and some constructive mathematics.
by Ruth Hill
Early Learners  
Karyn Tripp presents What Equals What? With Math Fables posted at Teach Beside Me, adding some quick representation to Greg Tang's book.

Teaching My Baby Math from Jennifer Bardsley's Teaching My Baby To Read. She wonders: can two year olds identify and name the quantity three?  Even after all of my trials and errors with my own daughter, I’m still not sure. Can your two year old do this?

Christy has Math and Graph Games. posted at Just another step to take.... With bonus connections to flow charts and Rock, Paper, Scissors, Lizard, Spock.

Amy Bowers paints numbers with art grids posted at mamascout. I'd be curious to see the math side of this more explicitly. The art is beautiful.

Lilac shares Homemade Math Manipulatives - Sorting Hearts for teaching grouping and sorting skills to preschoolers, at Learners in Bloom. Also a game, Blocks in Socks - Early Math Game.

by mafo

Colleen Young presents World Maths Day 2012 posted at Mathematics, Learning and Web 2.0, saying, "With World Maths Day approaching, it seems appropriate to have a post on the topic as part of the carnival." Some info on the day and links to lots of resources.

Dan MacKinnon programs better late than never: Mandelbrot Set  at mathrecreation. He's using the open source programming language called Processing. 

Sunday at the Park with Guillermo results in a post from his Mathematical Palette blog on Points, Pixels and  Pointillism.

Opening a whole can of worms, Misty presents Free High School Math and everything else you ever wanted to learn posted at Homeschool Bytes, saying, "We've been using Khan Academy almost exclusively for our math work lately and the kids love it."  I'll share a counterpoint here from Frank Noschese.  That said, Khan is a huge resource that there must be good ways to use.

by Eva the Weaver
Curator's Corner
Sam Shah started a conversation on planning. Read the comments.

Jason Buell lays out some nuts and bolts of complex instruction. Strong stuff.

Derek Bruff has a nice probability lesson, but moreso builds a case for using student prediction in your teaching.


...and out!
by pieplate
That concludes this edition. Submit your blog article to the next edition of math teachers at play using our carnival submission form. Past posts and future hosts can be found on our blog carnival index page. Next time around Bon is hosting at Math Four. Could get crazy.

47 lb.s? We do like this wine in my house... but supposedly this was a real rooster from a turn of the 20th century Texas carnival. A typical rooster weighs about 10-18 pounds and about a foot tall... so how big was HRM Rex Goliath?

 Image credits: mostly captioned under photos, all from Flickr. But I want to especially point out Eva the Weaver at Flickr who has many math themed photos, geometry and counting as well as numerals. The title graphic is from DizzyGirl, altered with Skitch.

by Eva the Weaver
of course

Thursday, February 9, 2012

Teacher Evaluation

The Diane Rehm show 2nd hour today was about teacher evaluation, centering mostly around the research of economists Raj Chetty and John N. Friedman of Harvard and Jonah E. Rockoff (show guest) from Columbia. They measure how much teachers add value by comparing teacher's students' test scores with their incoming test scores. Their conclusion is that the top 5% of teachers in this value added measure have dramatic impacts on their students' lives. They have a website with much more info about their research.

Good teachers are good for students. Huh. Who knew?

Seriously, though, of course what they are claiming that they can now identify these teachers using much-maligned test scores.  I am not familiar enough with their research to rip it to shreds professionally, but critical thinking would seem to raise several problems quickly.  Where these discussions always seem to go is that now we know whom to fire. Finally! I love (sarcasm) how in a discussion about dramatic student improvement, there is no thought given to teacher improvement. Of course, it's not really a growth mindset in terms of the students, either, but rather an idea that they are in the control of these good or bad teachers. It inspired this cartoon, based on my favorite Sydney Harris cartoon.

I went immediately from there to a mentoring meeting, where after a brief discussion of international humor, we began talking about how we self-evaluate our teaching. Here's what a half-hour led to:

We noticed that the majority of these were student outcomes, instead of teacher behavior. The question arose: do we assess these characteristics that we care the most about?

Furthermore we came up with some (also student-centered) monitoring questions for ourselves:
  • Are the students equipped for the activity?
  • Do the students know what they are hoping to achieve/where they are hoping to get to?
  • Are students getting more independent?
  • What can the students do?
The big question was raised: how do we know if these things are really about effective teaching or are they just what we want?

I'm not suggesting that this, finally, is the teacher evaluation system for which we have been looking. Instead, I want to suggest that this kind of teacher conversation is what will lead to better teaching.  Follow it with a chance to see each other teach, and talk about these criteria based on common experience. In other words, improvement in teaching is much like learning in any area.