I want to give a little breathing room to a thoughtful debate going on in the comments. I know it requires an extra click to get to those comments so maybe we’ll get the attention of a few more readers by dedicating a post to it. Additionally, if Common Core spurs thoughtful debate about relationships between math content and students’ minds, we’ll all be smarter for it.
John is in a 10 mile walkathon for breast cancer. He looked at his watch when he started walking- it was 7:02. After a half mile, he saw that it was 7:17.
- We discuss what’s happening. Hopefully a few students are curious about either a) how fast he’s going or b) a reasonable approximation for when he’ll finish.
- We decide on a way to measure how fast he’s going. We discuss measuring by minutes or by hours, and hopefully come to the conclusion that miles per hour serve our purposes best.
- The students work in pairs. They use 1/2 mile per 15 minutes and get to 2 miles per hour (the desired unit rate) in any number of ways. Formal proportional reasoning, informal proportional reasoning, number sense, a graph, a table- whatever is thrown out.
- We discuss how these strategies are related by placing them side by side.
- We debate which are the most efficient of these strategies.
- Now the teacher has a decision to make. Personally, I feel that while it may be unholy and it’s definitely arranged, this is not a terrible transition into a discussion about complex fractions.
- We write (1/2)/(1/4) on the board, and discuss its relationship to 1/2 per 1/4. This may be our major line of disagreement, as I don’t think this a terribly sophisticated jump. Assuming the students have some experience with slope and rate of change, this feels like fair game.
- A number of strategies, again side-by-side, are used to solve this expression. We break it apart visually with manipulatives. We convert both the numerator and the denominator into decimals. We show the algorithm. The students notice that “2″ as a solution is the same “2″ that they saw before.
- The teacher states that it may not be the most efficient way to find the unit rate in this particular problem, but that it may be in future ones. We speculate about when.
- In subsequent days, the teacher can veer towards abstraction with complex fractions. When misconceptions arise, there is the race analogy to give it footing.
Obviously this isn’t perfect. But if complex fractions are a necessary component to a middle-school curriculum, where else do they land outside of unit rates and proportions?
Really thoughtful stuff. Much appreciated. I have more to say but I’ll hang back and let Sean’s ideas simmer for a few days.