Inspired by Brian’s post involving numberless word problem’s, I threw a “mostly numberless word problem” at my students for our warm-up today. When students came in the following statement was on the board: “Mr. M went out to dinner and paid for his food, tax, and tip. His total bill was $26.” I asked students, like I always do, to just silently think about the question on their own for a minute. During this time, several students tried raising their hands, looking confused and uneasy. I knew what they wanted to ask me, but I told them to just think about it. Then I told them to turn and talk with their group members and share their thoughts. It was great. Some groups immediately began trying to assign dollar amounts to dinner, tax, and tip, some lamented the fact that there wasn’t a question, and some were just confused about what was going on.

After a couple minutes I began randomly calling on students to share their thinking with the class. The responses were incredibly interesting. One student responded that he thought that the food would cost around $18. I asked why and he said because that seemed like a reasonable amount considering that I also had to pay tax and tip. Others gave specific amounts; one student said she thought the food cost $15, the tax was $3, and the tip was $8. Then I called on one boy and his response was, “Pasta.” The class of 8th graders obviously got a kick out of that and giggles ensued. I asked what he meant by that and he replied that considering the total bill was $26 including tax and tip, he thinks I ordered a pasta dish because the cost of pasta dishes that he’s seen in restaurants are usually a little less than $20. It was a really cool insight and thought that I never expected.

Finally a few accused me of not giving them enough info, and I played it up saying, “Wait, you mean I DIDN’T give you a question?! Whoops. We all make mistakes sometimes.” But instead of giving them a question, I asked them what they wondered. I immediately got questions involving how much the food cost, how much tip I left, what percent tax there was, etc. I told them I couldn’t remember all of the details, but I did remember that the tax was 10% and that I left a 20% tip (I made the percents very reasonable because this was all mental math). We decided to figure out how much my food cost. Students thought for a minute on their own and then began discussing with their groups, which again led to some really interesting insights. Some wanted to subtract 30% of 26 from itself and came up with a cost of $18.20. I asked them to prove it by working “the other way” from 18.20 and confirm their solution was accurate. they realized it wasn’t and went back to the drawing board. Other groups used guess and check and began guessing numbers and trying them. In the end, many of the groups were able to come up with an answer of $20, which we then discussed as a class and made sure that it worked.

Overall I thought this was a really cool setup for a question. It definitely got students thinking about what information they wanted to know from the problem, and let them essentially choose the question to be asked. I look forward to using more of these types of problems throughout the year. Hopefully I’ll get more answers as good as “Pasta.”