Algebra's Tiling

I don't think algebra tiles or blocks are a panacea for what ails student algebraists, but I do think they are powerful.  Even though as a mathematician I am pretty comfortable with symbolic reasoning, at heart I am a visual thinker.  Having a way to visualize algebra opens up many possibilities for learners, even when they had a good symbolic understanding beforehand.  The goal for me is not to replace symbolic manipulation, but support the concepts.

I designed the activity for students that may have not done a lot of investigation before.  So it starts with a lot of modeling, and then letting them try.  Students did an amazing job.

Even before doing the mental math (making the point that even number operations can be visualized, plus setting the context for the follow up activity), I asked the students to take a  look at the blocks to see what they noticed. Mr. Boeve had had the students play with the blocks the day before, which is an excellent idea.  They noticed corresponding dimensions, colors, different designs.  These were the tiles from Algebra Lab Gear, so there's 1/2 x blocks, 1/4 x blocks, 5 sticks and 25 sticks.

None of the students had seen the visual multiplication before, but they were willing to give it a try.  They made connections as to why the pieces represented what they did.  The verbal connection, x squared, was biggest, but then a few students recognized the x times x relation.  They picked up the symbol to picture representing quickly, and that gave an opportunity to talk about how there are many different ways to write things in math.  They had 2x+3+1x+2 and 2x +3 +x+2 and 3x+5.  We introduced what mathematicians call simplifying, which they connected to fractions.

I raised the problem of negatives - how could we show negatives, because algebra has a lot of those.  They thought we could have two color blocks, or use some kind of design.  How could we do it with the blocks we have?  Maybe we could separate the positives and negatives.  Nice thinking!

Sometimes I think of mini-lessons like these as equipping the students for problems.  The problem list offered practice, light extension and serious problems.  After 15 minutes to do their choice, which they loved having, we came back together and I asked if there were any they wanted to see me do?  They suggested problems, and if there was a student to explain them, they gave it a try.  One of the themes throughout was "give it a go."  I made sure to ask some students who had incomplete or incorrect thinking so we could talk about that, too.  Because the whole situation was different, it helped with making that safe.

In discussing the subtraction problems, they got to three separate ideas: taking away, zero pairs, and adding the opposite.  The students exploring these ideas were able to give reasons for why it made sense with the blocks.


We just worked with the document camera, but if you want virtual algebra tiles, there's the National Virtual Manipulative Library tiles, a two color version from the Michigan Virtual University, and NCTM's Illuminations Algebra Tiles.  None of them are ideal, but all are serviceable.  It's very possible to make homemade algebra tiles, and there was a good article about that in the Mathematics Teacher: "Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," Annette Ricks Leitze and Nancy A. Kitt, September 2000, Volume 93, Issue 6.

Thanks to Mr. Boeve and his classes for the nice opportunity!  Jill Beauchamp came along for experience with the algebra blocks, and she was a great support to the kids, so thanks to her, too.




Playing With Blocks

Photo credits: Eamonn @ Flickr

Variable and a Problem

The Geogebra post on variable was gathering resources for a middle school inservice with math teachers and special education teachers.  (I've added the geogebra files for my sketches.)  Unfortunately, everything that worked smoothly the night before caused hangups on the day.  Sigh.

I'm working on this project with Esther Billings and Pam Wells, who both have a great grasp on teaching this subtle concept and are amazing teachers.  Pam is a wonder at working from student/participant work towards her goals, and Esther has this great integration of research understanding with what it means for teaching.

Objective:  TPW understand variables can represent changing quantities or represent an unknown; experience doing algebra with meaning and understanding of underlying concepts.

The day before they had spent a lot of time working on and watching student work for finding the patterns from pictoral sequences and expressing the relationship symbolically.  Including the very nice Modeling Middle School Math video of the Beams and Rods problem, from the Math in Context curriculum Building Formulas lesson (videos 8-13).  

We wanted to keep the element of having the teachers have a chance to do mathematics instead of just talk about it, and we also wanted it to connect to the Connected Mathematics Project Bags of Gold problem from the Moving Straight Ahead unit.  Swapping problems help develop the idea of equality, which is central to the idea of variable as unknown.  The bags of gold gets at it by putting the same amount in each bag, but you don't know how much that is.  So if 2 bags and 3 coins is the same as 15 coins...

I hate contrived situations, so my first thought about swapping was Magic cards (or Pokemon or Yu-Gi-Oh) but that seemed irrelevant to the audience.  Pam had the nice Stuart Murphy book Dinosaur Deals about that sort of thing.  My next inspiration was currency trading.  Found the current rates, jiggled it around, and realized it was entirely proportional.  Save it for later.  I wanted an exchange plus.  I thought about a classroom rewards situation (these were teachers) that a friend uses, and then complicated it.  So I wrote this problem:

Classroom Rewards

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board).  For more significant achievements  or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip.  For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip.  If you can get to 6 blue chips, you get a jolly rancher on Fridays for free.

Two white chips can be traded in for a jolly rancher.
Six white chips can be traded in for three red chips and a jolly rancher.
Five red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips?  At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers?  What other questions does this raise?  

Connections:  Does the situation make sense?  Do you need more information?  What does an answer to the problem look like?

Focus:  How will you answer the question?  Have you ever solved a problem like this one?  Is there a representation that would be helpful?

Activity:  Solve the problem.  Try to record your thinking.
Extension:  If you were going to let the class trade in chips as a whole group for a pizza party or doughnut day, what would make a reasonable goal?  Why?

Reflection: How would you check your work?  Now that you know the answer would you solve it another way?

Teacher Work
This was my first time with the group, so I was worried:  was it too messy? Too easy?  Too contrived?  (I hate contrived problems - especially mine!)

But they were amazing.  They dived right into the problem, asking great making sense questions.  What is this?  How does that work?  Why wouldn't they just...?

People worked with equations (because that seemed more mathy), tables (because that seemed helpful) and making pictures with blocks (which we had put out on the table beforehand without comment).  The blocks group had the most rapid progress and worked things out in multiple ways.  Nobody's answer matched mine.  But that wasn't the point.  They did want to be told the answer, but were okay with 'later.'

The free Jolly Rancher caused the most problems.  People thought you were trading the 6 blue for 1 Jolly Rancher, or that it was in addition or a one time thing.  The other confusing thing was that the red and blue chips come out surprisingly (to people there) close in value.  Given that, I reworked the problem a bit.  This retains the messiness and elements of non-proportionality.  If you want to make the problem considerably cleaner, make it 11 white chips for three red chips and a Jolly Rancher


Classroom Rewards, v2

For small achievements or solid whole group work, students in Ms. Smittyck’s class earn a white chip (like presenting a problem at the board).  For more significant achievements  or an action that benefits others (like raising your grade in the class or helping another student meet a standard), a student can earn a red chip.  For very notable work or effort or action on behalf of another student (like figuring out a way to increase recycling in the classroom), students can earn a blue chip.  If you get into the 6 blue chips club, you get a jolly rancher every Friday without having to trade anything in.

Two white chips can be traded in for a jolly rancher.
Twelve white chips can be traded in for three red chips and a jolly rancher.
Ten red chips can be traded in for two blue chips and a jolly rancher.

What would be a fair trade involving white chips and blue chips?  At the end of the year, what would be a fair trade for blue chips in terms of jolly ranchers?  What other questions does this raise? 

Questions
How would you work on this problem?  How would students?  Is it too messy for students?  If you're interested, I'd be curious about your comments.