One of my pushes this year is to get my Algebra II students to write math better. Last year I put “explain this” problems on a few exams and wasn’t so impressed with their responses. This year I am teaching my kids to write responses.
On their first assessment, I put a question similar to one we talked about in class:
Explain to someone who doesn’t know a lot about math why you can never find an which would make .
The responses were disappointing across the board. There were bits and pieces of gems, but nothing complete. Not a single student was able to construct a well-written response. Things I received included:
- The other side of the equation is negative, leaving no possible solution to the problem.
- You can never find x because the answer is negative and an absolute value problem with a negative after the equal sign is not possible.
So what I did was type up the following document and passed it out a few days after the assessment:
We talked about the vagueness of the responses, the use of pronouns like “it” and making references to “the other side of the equation,” and most crucial, the lack of reference in almost every solution to the original equation. How can you answer a question about an equation without even talking about the equation?
My favorite moment of the discussion this generated was when one student raised her hand and critiqued her own solution, and then said: “I wrote this and don’t even know what I meant.”
On the next assessment, without telling them I was going to do this, I threw the exact same question down. It was on. I saw my kids reread their responses after they wrote them, and really pay attention to their writing. Let me tell you: it all paid off. On this second round, most students got full marks. (On the first assessment, almost no one got full marks, or close to it, for that matter.)
Here are some random smatterings of their thoughtful answers:
- You could never find an to make the absolute value equation above true because you would have to subtract -5 from -6, which still gives you a negative number. . An absolute value equation cannot equal a negative number because absolute value is the distance from zero and is always positive [my correction: or zero].
- In this absolute value equation there is no solution because any number in the absolute value has to be 0 or a positive number. And if you subtract 5 from 0 or a positive number, there is no possible way that can equal -6. So there is no solution to this equation.
- An absolute value of anything can never be equal to a negative number, since it expresses a distance. When this equation is simplified, it becomes . If the ‘-1’ were replaced with a positive number, you could find the answer [for] . But since it is a negative, you already know that is impossible.
I am continuing to ask them to express themselves through writing. On that same assessment where I asked them to repeat the absolute value problem, I also asked the following two questions, to which I got some really nice writups.
The following two questions build upon each other. The solution to part (a) will very much help you explain part (b).
(a) Explain why without using your exponent rules. Explain it to someone so they can understand it simply!
(b) Explain why is true. You can assume and are positive integers. Explain it to someone so they can understand it simply!
I still have to do more work with this, but I just wanted to say: it is worth it to talk with your kids about writing. One 15/20 minute conversation has already yielded great dividends for me.