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

Who Are the New Teachers?

Guest post today from one of our student teachers from this past semester.  Sarah Cavazos will be student teaching in Fennville, MI this fall and is - to steal a phrase from Sir Ken Robinson - exceptional but not an exception.  She is bright, dedicated and passionate about teaching.  I find that many of our novice teachers have much in common with her.  This past semester I got to see her try a game of her own design in a classroom that had not done much of that, and adjust it on the fly to improve both gameplay and better address learning outcomes.  Then she gave a terrific Little Big at the end of the semester about teaching all students.



Triangle Rummy

Object of Game: Obtain a set containing the picture, the name and the definition of the same triangle.

Triangle Rummy is a game made for two to four players that contains 24 cards (7 pictures, 7 names, 7 definitions and 3 Free cards). There are 7 different triangles in the deck of cards, each demonstrated by a picture, a name and a definition. There are also three Free cards (Free Picture, Free Name and Free Definition).

To start the game, the dealer deals out three cards to each player and places the rest of the deck face down between the tables. The youngest play starts by taking the top card from the deck. In order to keep three cards in their deck, the player must discard one of the cards in their hand face up next to the facedown pile. The player to the left can chose to take the face up card in the discard pile or choose to take a chance and take the next card in the discard pile. In order to win, a player must obtain the same triangle demonstrated by three different cards.

If any player obtains a Free card, they can choose to use that card in place of a triangle name, picture or definition depending on the name of the card. For example, say a player had the picture of an isosceles right triangle and the name of the isosceles right triangle and then obtained the Free Definition card. In order to win the game, the player must say the definition of an isosceles right triangle. If the Free Picture card was played, the player must draw the correct picture as well as have the matching name and definition cards. If the Free Name card was drawn, the player must name the correct triangle and have the matching definition and picture cards.


The Unreachables






Photocredit: qthomasbower @ Flickr

Power Up

A math game for exponents.

A colleague gave me a game created by one of her students Steven Reynolds.  The idea was having the power as an exponent connected with the power of a superhero.  I liked the premise, as I'm a comic fan myself and have a son who's obsessed.  The timing of playing this with the 5th graders was perfect, the day before Free Comic Book Day, the closest thing the superhero community has to a holiday.

The fifth graders had had no experience with exponents before, so this was the introduction.  The game was good ground for getting them talking to each other about things like 3^4, how to calculate it, and noticing some of the properties.  The main confusion was typical: 2^3 = 2x3.  But they went from me reminding them to reminding each other to the vast majority being comfortable with the notation.

The most exciting thing about the game was how much people got into the context and then wanted to play.  Some students that had never gotten this much into a math game were engaged; not just in the creative side of it, but in playing the game, also.  The students asked if they could choose to work more on their origin stories for their writing time later and Mr. Schiller gave them that option.  Two students gave me permission to post their stories, which are after the revised game below. (As usual, good suggestions from the students.)

I launched the game by creating my character on the overhead, at the same time as they were creating theirs.  While people were still working on different steps, I had a few people share what they had so far.  My super hero was the Time Keeper, who got his powers by finding a watch that turned out to belong to Father Time.  He got super strength (base 4), super speed (base 3), and the magic watch (random).  Then I played a game vs Shadow the Hedgehog in front of the whole class in which he pounded me.  The game play is pretty quick, so kids were all over the room playing each other.  Overall, they rated this game a definite keeper.

Download from Google as pdf  I'm grateful to my son, budding comic artist at almost 11 yr.s, for the following illustration of the example from the game sheet. (Click for full size.)

Here's two of the origin stories.

Shadow the Hedgehog by Jim (created by science, doesn't know if he's a hero or villain)
At a space complex ark, he was looking out the window. "Shadow, you've been looking out that window all day," Maria said. Shadow replied, "I just want to know what it's like down there. I've seen pictures, but... I just want to know what it's like."

(3 years later) "Doc, why am I nervous?" asked Shadow curiously. "You will see," Doctor Robotnik said.

A blaring alarm sounded. Doc cried, "Intruders! Shadow, guard Maria with your life!"

Shadow and Maria were running away from the GVN soldiers when one yelled, "Stop or I'll shoot!" Shadow told Maria, "run! I'll try to stop them." One of the soldiers pulled out a pistol, fires, and Shadow yelled, "Maria!"  Shadow causes chaos control with Maria, and she says, "Shadow, before I die... Be friends with the people on earth." Shadow yelled, "NO!!"

The Doc says, "take out the gun soldiers and their leader and see how they feel when their loved one is lost." Shadow said, "yes, sir."

When the Doc was arrested, they did not know about Project Shadow.

The Changer, by Katrina
"Ah, the sweet smell of safari!" I said to the driver.  "Now don't go too far from the jeep, 'cause you'll get bit," the driver said.

I walked for hours.  I could not find the jeep, when all of a sudden a cheetah was face to face with me.  I ran and ran.  I could already see the blood coming out of me.

The cheetah bit me.  I screamed for my life.  A couple of minutes later, I was home in an instant.  I went to my lab, and mixed some potions together to get the cheetah out of me.  But instead, I was on fire.

I could feel the flames.  I felt like I was off the ground.  I looked ..."What the heck," I said, " What is this, a joke?" Something in my head told me it wasn't.  "Now I know what to do," I said with an evil laugh.  I lifted up my car with no trouble.  Then I knew that I was now a super hero with super-strength, super-speed and firepower.  I help fight crime.  I help police catch villains.

And my name is the CHANGER.

Photo credits: The Tick I have no rights to, but he is my favorite. Legal use, Fern R and Spencer77 @ Flickr.

Triangle Mosaic

Holy cow, have I been busy.  Sorry for the lack of new posts.  What makes it worse is that I have had several guest posts to get up that students were kind enough to send me weeks ago.  In addition to scads of things that I want to write up for myself!


The first is from a preservice secondary teacher named Jill Beauchamp.  She is active in coaching cheer, and in our local Dutch culture.  (And it's almost tulip time.)  I'm pretty sure she's a licensed wooden shoe dancer.

On an assignment that gave a choice of follow up options after playing with Pierre Van Hiele's mosaic puzzle in class (from “Begin with Play,” by Pierre van Hiele, Teaching Children Mathematics, Feb 1999), Jill chose to make a activity based on my triangle puzzle.  And she was willing to share it!  I like how she really captured Van Hiele's idea of beginning with play, and uses the puzzles to get at the triangle properties.  She makes the most of what I was designing the puzzle to do, have one triangle of each type.



The assignment:

Teaching Math – Mosaic Making

Choose one or more of the following to do for this:

1. Analyze Van Hiele’s mosaic. What geometric properties of the pieces permit all the combinations we saw in class?
2. Create your own mosaic puzzle and document your design process.
3. Create a new lesson using PvH’s mosaic or my 7 triangle mosaic at http://mathhombre.blogspot.com/2010/12/triangle-puzzle.html

Document your work, and be sure to include a reflection.

Schema: I decided to take a look at your 7 triangle mosaic – nice work! This would be difficult for me to create on the computer, so I am very impressed. When first thinking about a lesson in regards to the mosaic, I could only consider it being a fun puzzle. With our exposure in class to different workshops regarding the original mosaic, I began to think about the properties each triangle in your mosaic possessed. You have:

• -Two right triangles
• -One isosceles triangles
• -One right isosceles triangle
• -One equilateral triangle
• -Two scalene triangles (One acute and one obtuse)

Fabulous! You have one example of everything.

Focus: With my class, I would want to explore why these triangles fit together the way they do. Assuming the students have not yet learned about triangles, this could be used as an introduction. Let’s say I have this class for 60 min. Here’s how my day would go.

Lesson: Properties of Triangles

Introduction: (5 min) Talk about puzzles!

• What kinds of puzzles do students like to do?
• What makes a puzzle puzzling?
• What are some mathematical properties of puzzles?

Introduce Mosaic

Mosaic Play and Record: (15 min) Allow students to play with the pieces and try to create the mosaic. As they do this, I would like them to document their actions:

1. What they tried
2. What pieces worked together?
3. What didn’t work together?
4. Qualities they notice about the triangles

**If students solve the mosaic, they should focus on:

1. Is there another way to solve it?
2. Why do some types of triangles fit together and others don’t?

Discussion: (20 min) I would ask all students to pull apart their mosaics and separate the individual triangles. Then I would ask them if they saw any similarities between any of the triangles?

*As this is happening I will write up student ideas on the board. If need be, they may come up to the board and illustrate their thinking.

Assuming they already know terminology for a line, angle, point etc. I will have students pull out the rulers and protractors to assist them in drawing more comparisons. Once we have a pool of properties, we can begin to group the triangles accordingly. Once we are able to do this accordingly by the deduced properties, I will write the names of the triangles on the board (but not yet with their corresponding group). Instead I will ask students what they think goes with each.

Properties: (With any luck, we get some or all of the following, although I’m sure I’ll get some other interesting thoughts!)

• 3 equal sides
• 2 equal sides
• No equal sides
• 3 equal angles
• 2 equal angles
• No equal angles
• Right angle
• Obtuse angles
• Acute angles

Hopefully, they will see comparisons between the word “Equilateral” and the same angle and side measures, “Right” and the triangles with 90 degree, or right angles, “Scalene” and the triangles that depend on their individual scale/measure, although “Isosceles” doesn’t work too well, but it can be the odd guy out.

I will want to pay special attention to that sneaky little purple “Right Isosceles Triangle.” This guy is important because he shows that two properties can hold for one triangle. Maybe we could explore which properties can hold together and which ones don’t (As I’m writing this these ideas are just kind of coming…)

• A scalene can be a right triangle. Why? Because one angle may be 90 degrees, the other two differing, and all sides of different lengths. A scalene cannot be isosceles or equilateral because it goes against the definition of scalene.
• An isosceles triangle can also be right, but can an equilateral triangle also be isosceles? No, the definition of isosceles is EXACTLY two sides of equal length. Although it can be either acute or obtuse depending on the size of the angles
• A right triangle can then be isosceles or scalene. It cannot be equilateral because one angle must be 90 degrees, thus going against the fact that all angles in an equilateral triangle must be 60 degrees.

Teacher Question: So then, are triangles actually right triangles? Or does the word “right” just classify a specific type of isosceles or scalene triangle? A right triangle cannot exist outside of one of the two classifications.

Sorry for my tangent. The above discussion over the “right isosceles triangle” may be something for another day! My hope would be to get to the last part of my lesson…

Discovery: (15 min) The students would then need to reassemble the mosaic (I will show them the put together puzzle if they need it). With their protractors and rulers I would like them to work on:

Measuring the divided angles in the corners of the square. What is the sum of these angles? What type of triangles have an angle like this?

Measuring the divided angles along a straight line within a puzzle. What is the sum of these angles? What do they notice about all of these sums along a straight line? How does this compare to the sum of the angles within a triangle?

Lastly, I would like them to paste their mosaic together on a piece of paper and write out the angle measures, side lengths, and classification for each triangle. Students should make a note of anything else they notice.

Reflection: (last 5 min of class) What is one realization that surprised them today? Can they put anything they’ve seen into another context? How might it relate to something else?

MY Reflection: Wow, This was wonderful. I had the initial idea for the lesson because I thought it was so cool how the angle measures across a straight line will add up to 180 degrees. A simple concept, but it helped a lot of things make more sense when I recognized it. I think a lot of times we have this subconscious knowledge that we utilize everyday but don’t fully recognize. Once I started planning out how I would eventually get to a measuring activity, ideas just lead into one another, making this a lot longer lesson that I intended. There is no way I would get through the discovery part in 15 min! For me, this order of events seemed to make the concept clear. Perhaps it should be a day and a half sort of lesson?




Do you have any feedback for Jill or I about the lesson?  What would you try?


Photo credits: Jill Beauchamp, quinn.anya and bjornmeansbear @ Flickr

Division into Decimals - Undone

This is a pretty focused game and it is undone.  Even the fifth graders couldn't help me finish it... maybe a reader can help?  We were playing Manga High games with my preservice middle school teachers this week, and they noticed something.  The context can be completely silly.  They liked "the penguin game" in particular.  They thought it was mathematically worthwhile, teaching estimation.  They found the problems worth doing in the game and felt like it could help students improve their estimation and computation.  But we never really have to chuck penguins to safety.

There's something appropriate to the context.  We're approximating, acting quickly, there's the idea of what fraction of the way across are we... but it doesn't bear much scrutiny.  It's just silly and a bit of fun.  I think that's an element missing from some of my practice games.

The Game
The game premise is complete simplicity: race from one end of the paper to the other (25 cm) by drawing rectangles with area = 10 sq.cm.  Each turn your rectangle is determined by rolling the width with 2 dice.

That's it.

The Lesson
I shared that we were playing a game that had no name and no context.  I needed their help.  Immediately they began shouting names and ideas.  If I were a game designer, I would do this.  Just get a bunch of kids and let them throw ideas at you.  Holy cow were they enthused and specific.  Sadly, I had to tell them I wanted ideas for this game, and we'd have to play it first.  Me versus the entire class, like we usually do.

I showed the graph paper and explained that the goal was to get to the top by drawing rectangles with an area of 10 sq. cm.  I drew a 1x10 rectangle.  But that wouldn't be very interesting. So instead we're going to roll the width of the rectangle.  I rolled a 2 and asked, "how tall should it be?"  A couple students quickly jumped to 5, so I drew it and we checked - there were 10 squares.  We rolled for them.

Aside: that worked well enough that I would fake a 5 if doing this again.  (Faking a 2 is greedy.)

Their roll - a 7!  That's how wide it should be, but how tall? "3!" kids are quick to say. Hmm - that makes for an area of 21.  "1!"  That's only 7.  "1 and a half!" Let's check.  They talk me through 1.5x7. (Not a few couldn't recall.) Hmm, 10.5, too much.  "1.4!" They walk me through that... so close! 9.8. "1.45?" We try it and get 10.15.  Wait, I say, aren't we really doing division?  What times 7 equals 10? We started an saw that 7 goes into ten once.  "It doesn't work," someone said.  Another student said, "add a zero!" I used the notation my son has been using, which I like.  We got to 1.428, and then realized that was close enough for drawing the rectangle.  "Is it point-4-2-8 repeating?" "Will it ever repeat?"

We finished our practice game, and recorded our calculations on the white board.  When they played for themselves (mostly 2 on 2, with a few 1 on 1's), they referred to the ones we had calculated, but had also several others that needed to figured out.

In the summary, they thought the game was fun enough to play, and had good math.  25 cm made for about the right length of game.  (Hah!) They really got into trying to come up with a context. REALLY.  "Monkeys climbing!" "Monkeys escaping from sharks!" "Tiger sharks!" "People escaping from sharks!"  "You're escaping evil aliens."  "It's an alien trying to climb to the moon!" "It's ..."

OK... it should connect somehow to what we're doing.  I see why the climbing.  But why sharks?  Where is the shark chasing?  Could it connect to what we're stacking? What would you stack that has different sizes? "Students escaping an evil teacher by stacking homework!" "Books!" 

OK... maybe the game remains undone.  Maybe it should be unfinished?

Reflection
The game made a good context for modeling the division on which they needed practice.  The representation of the rectangles was supportive.  We talked about the connection between 10/2 and 10/4, for example.  They got the indirect variation aspect of it, and were rooting for small numbers for themselves and large numbers for me.  The familiarity with common division computations was good.  For such a vanilla game, it might be a nice aspect to keep it vanilla, and keep the game design aspect to the lesson.

If I was developing this into a video game, the subsequent levels would introduce some variability of the dividend also.  I also think you could have kids estimate the division.  If they overestimate, no block.  Increased decimal places would get you farther faster.  One variation that I ruled out was to have the game be a race to 2.5, so that the kids were dividing 1 instead of 10.  But then the area is .1 sq. units - not nearly as intuitive.  My next lesson would be how you can use these results to get at other computations, and to strengthen that multiplication connection.  I'm convinced that 99.9% of kids do not see the partial product connection with the division algorithm.

My 1cm graphpaper (pdf) is at Scribd if you'd like it.

Twitter Conditions

I had the good fortune to win a bet recently (well, best 2 out of 3) by the performance of the Yukon Huskies (jk) in the 2011 NCAA Division I men's Basketball Tournament.  My prize? A guestpost from Dave Coffey, @delta_dc.  (I was actually rooting for Butler, but that's a quality consolation prize!) This is Dave with Juneau.  Juneau is asking, "How could you bet on bulldogs?  Have I taught you nothing?  Haw!"

A few weeks back, one of our teacher assistants said, “You just started on Twitter this semester. I never would have guessed.” I wasn’t sure if she was talking about my quantity or quality. I chose quality and explained that it could be traced to Cambourne’s Conditions of Learning (a Foundational Framework of our Teacher Assisting Seminar).

This reminded me that John, my co-teacher, had asked me about blogging about the Conditions. I turned to him and said, “I’m thinking about writing about how the Conditions of Learning helped me to communicate using Twitter.” I thought this would be a good example of authentic learning in action.

The teacher assistant chimed in, “Maybe you could describe each condition in a Tweet.” John laughed, understanding that she had issued me a challenge without knowing it. Well, “challenge” accepted…


[Note from John - t was very tempting to put this in twitter-typical reverse order... but that would make it less readable.  Please forgive the lack of verisimilitude.]

Thanks, Dave and Jim Calhoun! And Brian Cambourne, of course.  The Reading Teacher has put the article introducing the Conditions online for download.  Or you can read the whole story in his book The Whole Story.  Also, I put the date on the cartoon at '95, the date of the RT article, but 1988 would be more accurate.

Photo Credit: Kathy Coffey, Rosaura Ochoa @ Flickr

Sites, Gadgets and Widgets

Warning: entirely a novice.  Trying to give that perspective on using these tools.

My university, Grand Valley State University, is in Michigan, a famously economically hard-pressed state.  Much like Wisconsin, any kind of education funding is suspect and under inspection.  We have think-tanks requesting faculty emails, and newspapers and TV stations soliciting financials.  (The financials are public info - no worries.  The emails - creepy.)

We got an email from the administration with some information to contest the spin being put on the data, and I started wondering about making it a webpage.  So I used it as a chance to practice with Google sites, embed a Google spreadsheet, and a Wolfram|Alpha widget.  Unfortunately, you can't embed a W|A widget directly, you have to wrap it in a Google gadget, then embed it in the site.  This site was very helpful, and the whole process boiled down to pasting the W|A javascript code in the Google gadget builder. I was using the builder at this page, just adapting their most basic example.

The W|A gadget was pretty easy to manufacture, following their steps.  After entering a request, you'll see options for sharing.  The far right W|A sigil starts the widget builder.

The thing that took me the longest to get was just me being dense.  To get at component properties, select a component in your proto-widget.  And then the settings will show in the bottom left.

Then to get the embed code to paste in your blog (will work directly) or to put in the gadget to put in a google site, click on the code button in the embed box on the finished widget.

The (sort of) finished project is here: Facts and More: How does GVSU stack up?

And, of course, I'd happily take any suggestions for improving that page.