In the last Geometry Problem of the Week, Finding a Supplement, students were told that two angles formed by intersecting segments had values that could be expressed as quadratics. A lot of students were successful finding the solution. And a great many of those students showed all the algebra that they used to reach their conclusion – it’s clear that many students understood how to use algebraic techniques to solve a problem like this. But there were differences in how much explanation those students included about why they were doing all that algebra.
All of the solutions started one of these ways, in order from most common to least common:
- 2x2 + 3x + 15 = 3x2 + x – 20
- Angle AED = Angle BEC
- I first drew a diagram as shown below, and saw that AED and CEB are vertical angles, which means their measures are equal.
The majority of submitters, by far, simply stated that the two given quadratics were equal, as in #1 above, then churned through some symbolic manipulations and ended up with one or two possible measures for the final answer. On the whole, their work was very well organized and easy to read. Typing all that algebra can be a lot of work, and it was clear that many of them put a lot of effort into their submissions. The next step for those students is to include some information about why they are doing the steps they’re doing, starting at the beginning with explaining why they can set the two quadratics equal to each other.
A few students started by stating that the two angles are equal, as in #2, and then launched into the algebra. But they didn’t say how they decided that the two angles were equal.
Then there’s #3. This is the beginning of a submission from Xingyao, from Conners Emerson School. She continues with, “If their measures are equal, then the quadratic equations expressing the measures must also be equal.” (Then she does all the algebra everyone else did, providing other useful bits of commentary along the way.)
I know that all the students noticed and decided the same thing Xingyao. Otherwise they wouldn’t have set those two quadratics equal to each other. But that’s exactly the information problem solvers need to write down. How did they start the problem? Did they decide to draw a picture? They should write that down. What did they notice about the picture that would let them take the next step? They should write that down. Then what did they decide to do? They should write that down. It’s important that they show the work that they did, but it’s equally important that they explain the decisions that they made along the way about what to do.
Maybe you could suggest to your students that they are telling a story about how they solved the problem. In this case, they didn’t start by setting the two quadratics equal to each other. There had to be something that happened before that, in the story, that helped them decide that they could set the two quadratics equal to each other. Encourage them to start the story from the beginning.
What if they imagine someone reading the beginning of their solution. If they think the reader will say, “Wait! How did they decide to do that?”, then they need to include some more information. You might support this in the classroom by having students trade explanations with a classmate. Each student should think about whether the other student started at the beginning of the story, or whether they left some details out. Have students give each other feedback, then reread their own explanation. Did they begin at the beginning?
What do you do in your classroom to support students in writing complete explanations? Are there strategies you use that are related to literacy practices at your school? Do you have peer review? Do students read their work out loud? Please share your ideas and your questions.
Some “Finding a Supplement” links in case you are interested:
- The problem [requires a Math Forum PoW Membership].
- Information about accessing “Finding a Supplement” (and a selection of all our PoWs) for 21 days with a free Math Forum trial account.
- Information about becoming a Math Forum Problems of the Week Member. Consider starting with a $25 membership, which gives you access to all of this year’s Current PoWs — and now you can create 36 student logins as well!