Too Puzzling

The content objective: adding and subtracting fractions with unlike denominators, for 5th grade students.

What game? On short notice... I have a couple race games and Mr. Schiller has pattern blocks - but those use the fraction cards which I like, but were also at my office.I was thinking about a game where students start at one, and one team tries to race to 2, while the other team takes away and races to zero.  That would be good for building intuition, and learning to record fraction number sentences, but this is their last day with fraction operations before moving on to something else. What could serve as a review?

I thought about a game you could play with a small number of cards (one sheet), or a game where they would be doing review problems and checking each other as they play. Somehow I was reminded of a tarsia puzzle. (Here's a bunch at tes, among other places. Not sure who tes is, really.) Mostly those are triangular, but I decided on a square pattern, more familiar to the students. Here's what I wound up with. Made the grid in GeoGebra, of course.


A lot of thought went into the puzzle. I repeated a couple values to make it so that a match did not conclusively mean that two pieces went together (harder). I made all of them doable with 24ths (easier).  I decided on a 3x3 vs a 4x4 (easier). I made almost patterns on borders (harder and easier). Made sure some of those were nonmatches (easier). Related problems, like 1/4+1/2 and 3/4-1/2 (easier). Some operations where the common denominator is one of the present denominators, 2/3-1/6. And one mistake. (Crazy harder. Fixed version above.) I tried to put in some common characteristics on adjacent tiles. (Hmmm?) If they've been doing these problems in general, I thought that this would be enough support for everybody, especially working in teams.

To differentiate, then, I was mostly thinking upward. On the original puzzle they could make 3 more squares to make it a 12 piece puzzle. I made one with some triangles blank for the students to work out sums, differences or make up a problem, and then an entirely blank one for them to make up their own Tarsia.

I launched the puzle by showing cut out pieces, telling them it was a puzzle and asking how it might go together. Through whole group discussion they figured out that the sums and differences matched some of the fractions. I compared it to the puzzles that are all squares that divide up pictures, which are pretty tough. After finding a few of the matches together, they formed pairs and came up to pick up their choice of puzzle. Only one group took an option on the partially filled in puzzle.

They did not like it. Found it too hard, or didn't know how to start. Mr. Schiller and I circulated and helped people get started. They found lots of matches, but before our time was up were moving on to other pursuits. Not interested in making their own.

To wrap up, we came back together and did some together to get a firmer idea of how to do it. They confirmed that it was beyond them. When I asked for words of wisdom, one student volunteered: "You might want to try it yourself, first. If it's too easy, add stuff to make it tougher. If it's too hard, make it easier."

Wise words, indeed.

To use this in fifth grade again, I think I might concentrate on first getting a square of four made, and then try to grow it. Mr. Schiller recommended either a fraction equivalence puzzle or a fraction-decimal equivalence puzzle to get it launched. I still love these kinds of puzzles, but fee like I learned something about introducing and using them with younger learners.

Post Script:
Jeff says: The equivalent fraction version of that worked a lot better.  I sort of sneakily encouraged them to also incorporate equivalent decimals too.  In the version we played in the afternoon,  they created their game board in pairs and then they matched up with another team and exchanged puzzles and it became a race to finish the puzzle first.  I sort of mentioned in an offhanded way - oh yeah, and if you wanted to make it a little more challenging for your opponent, you might include some equivalent decimals, too.  (That did the trick)