This week’s problem makeover from Dan Meyer features a room in need of carpeting:

There’s a lot of stuff going on in this problem:

- Overlaying rectangular carpet over an irregular shape. We’re going to need to worry about waste at some point, perhaps.
- The double-sided tape. We’ll have to look at the perimeters of the individual carpet pieces and add them up later.
- Costs. We definitely don’t want to waste carpet or tape, or at least minimize the amount of waste.
- Meters vs Yards vs Feet. Yeah, this aspect of the problem is just mean Maybe if Lesa had her fill of coffee before measuring the room, she would have used feet.

I think the piece of this problem which may be easy for kids to overlook, and least comprehend, are those seams. Seams are bad, and add to cost. This problem needs something tactile to get kids thinking about the seams and coming up with their own ways to think about them and minimize them.

**HERE’S WHAT I AM GOING TO DO**

Groups of students will be given a bunch of 4×4 inch sqaures and a bigger shape to tile.

The instructions are simple:

Use the squares to cover the triangle in the most efficient manner possible.

As groups complete the task, they will then meet with other groups and defend why they feel their coverage method is the most efficient.

I’ve left the phrase “most efficient” open for interpretation here. I’m hoping that a strong definition of efficiency will emerge from group discussions. As groups continue discussions and start to reach agreement on the characteristics of efficiency, have members of teams approach the front board, and start a class-wide list of these characteristics. Here’s what I am hoping will emerge:

- Using the least number of squares as possible is important.
- Having squares overlap, if it can be avoided, is preferable.
- Seams are ugly, but necessary. When we are able, we want to minimize these seams.

This last bullet may be the toughest to coax from kids. But this is a good time to bring in carpeting as a nice analogy. Does the carpet at your house have seams? Where are they? If you were laying carpet, where would you try to put them?

Once we have a stronger definition of efficiency, it’s time to try a second tiling:

This time have students write up their solution and defend their efficiency. There’s a nice opportunity for differentiation here. Students with a strong geometry background could use rules regarding secants and chords to generalize the optimal solution, or even use software to model the scenario. Middle school students, or those with less exposure to geometry, could simply measure the seams.

Now for the main event. Students will be asked to tile one last shape:

This time, we add a new wrinkle to the problem. To get students thinking harder about efficiency, we add some costs to the mix.

- Each square used costs $10.
- Every inch of seam in the tiling will cost $2 per inch.

It’s Saturday morning during the summer, so I’m not sure if these are “good” costs for the task, but we can adjust them. Teams will calculate their costs, defend their solutions, and be assessed on how well they minimized the costs.

Geometry isn’t usually my “thing”, but this challenge appealed to me, because the original task was so filled with stuff that I know many students would shut down. And I feel there is a natural opporunity here for students to assess their solutions and communicate. In the end, I want my students to:

- Communicate their problem-solving process
- Defend their process, and evaluate the process of other teams
- Utilize their findings in different situations.

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