When designing a task, I strive for a simple premise. Try this:
The two fraction bars below are mislabeled. Explain how you can tell they are mislabeled. Then drag the big black point until the fraction bars are correctly labeled. Test your solution by checking the “Reveal Solution” box.
Puzzle #1
Simple, right? But a simple premise isn’t enough. I also want my tasks to have rich connections between representations and concepts. Try the next three, and see what develops for you:
Puzzle #2
Puzzle #5
Puzzle #7
The solution to this puzzle does not fit on the page. Still solve it and find a way to communicate the answer. Then reveal.
If you were successful in solving some or all of them, how did you do it? What ideas came into play? I have asked a few of my colleauges to solve these puzzles over the last couple of days. The approaches we have developed vary in such wonderful ways.
Here is a video showing 3 methods that were not immediately obvious to us:
The connections that can be made here are some of the richest I’ve seen bridging the mathematics of elementary school with the mathematics of early high school. When engaging this task with the right set of skills, all of the following seem to be at play almost simultaneously in meaningful ways:
counting
whole number addition
whole number subtraction
whole number multiplication
whole number division
visualizing fractions based on their meaning
fraction addition
fraction subtraction
fraction division
variable expressions
equations
ratios
number patterns
functions
systems of functions
These puzzles are part of a series of tasks called Fraction Stretch! that I’ve developed over the past few days. Here is a link to the main page:
There you will find a link to a student page, which contains all seven puzzles, and a teacher page, which contains handouts for the puzzles as well as solution codes for each puzzle.
Given it’s richness and flexibility, I am looking forward to adding this to my repetoire this coming school year at multiple grade levels. If you try this with your students, I would love to hear how it went. In the meantime, I would love to hear your thoughts on the task based on your experience with it here.
Additional Mention
I started thinking about fraction tiles after seeing a pair of tweets by Graham Fletcher that wondered how labels on fraction bars affect our thinking about them. After I saw the tweets I thought “I wonder what would happen if you mislabeled fraction bars?”, and started exploring the idea. These tasks are what eventually resulted. Here are his tweets: