A couple months ago, one of my favorite math Tweeters and bloggers, Fawn Nguyen, posted this, which I promptly liked and retweeted:
REMOVE the ? from the problem initially as Ss will always come up with ? you had in mind! Do “I notice/wonder” first pic.twitter.com/C6XbD1ScTT
— Fawn Nguyen (@fawnpnguyen) April 23, 2016
The structure of this activity really appealed to me. In our NYC Community of Adult Math Instructors meetings, we often begin our explorations with two questions:
What do you notice?
What questions do you have?
From here, we come up with interesting ways to approach a mathematical situation, and we each tackle different questions that are interesting to us. This model has become a routine in my math class. When we look at charts and graphs, for example, we always start with these questions. This practice helps students anticipate the kind of question that they could be asked, and it gets them talking to one another.
So of course I was super excited about the problem Fawn posted, and I couldn’t wait to teach it. There are so many different things I like about the problem. I like that it connects spatial reasoning with proportional reasoning; I like that it’s so immediately accessible but ultimately challenging to solve; and most of all I like the way it builds student interest by giving them time to talk to one another about what they notice and what math questions they might be able to ask and solve. I decided to teach this problem to two very different groups of students: one, an HSE-level class in Carroll Gardens, and two, an ABE class in Red Hook. I also changed the names of the people in the problem; I like to use the same names in my classes, as my students start to remember things like “Eric is the guy who sold the fish tanks, right?” Or, “I remember Solange from that problem about the taxi.” It’s fun.
Students Asking Questions
These two classes had very different levels of mathematical ability. The first class was composed primarily of HSE-level students who had been with me for a while. This problem was challenging for them, though, because we had spent so much more time on algebraic reasoning than we had on fractional and proportional reasoning. Here are some of the things they noticed and the questions that they had:
This group asked the question that I hoped they would ask: What is the area of the yard? The second class also asked this question, and they “noticed”–as I hoped they would–that this problem didn’t actually ask a question!
They asked some great questions, and they were interested in figuring out who mowed the most. Some of the questions they asked were:
- How big is the lawn?
- How much is 1/3 of the lawn?
- How much is 1/4 of the lawn?
- Why use three different words for lawn?
- Who mowed the most?
Altogether, this part of the activity took about a half hour. Students took five minutes to jot down their questions, and then they talked about them in small groups for another five to ten minutes. After that, we talked about the problem as a whole class, and I noted all of their questions on the board. Both classes decided that they wanted to figure out the area of the lawn, which is exactly what I was hoping for.
How I Solved It
First I figured out what fraction of the lawn had already been mowed by Solange and Jane by adding 1/3 and 1/4.
1/3 + 1/4 = 4/12 + 3/12 = 7/12
I knew that 7/12 of the lawn had already been mowed, so the remaining 5/12 was the part left for Denise. At this point I set up a proportion to find exactly how many square feet the entire lawn was.
I cross-multiplied and divided to find the total area of the yard.
The yard is 1680 square feet in total.
Anticipating Student Approaches
I knew that my students were going to struggle with this problem, especially because they aren’t always comfortable setting up proportions correctly. They are, however, very good at drawing pictures to help them clarify their thinking. So I assumed that several students would correctly add the fractions of the lawn that Jane and Solange mowed, and then they would be able to draw a picture of 7/12 to help them find how much Denise has left to mow. We worked with strip diagrams before, so I expected to see something like this, where the green part represents the amount that has already been mowed, the white part represents what Denise still needs to do, and the entire strip represents the lawn:
I thought that if students were able to do this, then they would be able to see that Denise’s five parts are equal to 700 square feet. From here they would be able to divide by 5 to figure out the number of square feet in each “piece” of the strip diagram.
700 ÷ 5 = 140
Now, to find the total area of the lawn, they could multiply 140 × 12 = 1680 square feet.
Supporting Productive Struggle
Once they had established that they want to find the area of the yard, many of my students struggled to figure out what to do next. Several quickly figured out that they needed to add 1/3 + 1/4 to get 7/12 but then got stuck. So to support productive struggle, I asked a lot questions about what the 7/12 meant in the context of the question.
- What is the 7/12? Who mowed 7/12 of the lawn?
- So if 7/12 has already been mowed, what fraction of the lawn still needs work?
- Will Denise need to mow more than half of the lawn, or less than half?
- Talk to me about 7/12. What does 7/12 look like to you? Could you draw a picture?
A few students tried using a proportion to solve the problem, but they forgot to label each item in the proportion, which got them into a little trouble. To help them get back on track, I asked a lot of questions about what each element in the proportion represented.
- I noticed you have 5/12 on the left side of the proportion. Can you talk to me about the 12? What does it represent? What about the 5? If you wanted to label each of those, what labels would you use?
- It looks like you have 700 on the bottom on the right side of your proportion. Can you talk to me about that? What is the 700? How would you label it?
My goal was to get students talking and to support whichever strategy they were working with.
I was interested in Glenn’s approach to the problem because he started out by drawing a picture, as he has done in class before, but it didn’t seem to get him very far. I thought it was interesting that in his first drawing, 1/4 is drawn larger than 1/3. Glenn had also completely forgotten how to add fractions. At first, he added 700 + 1/3 + 1/4 and got an result of 4207/6. When I asked him about this, he couldn’t really explain what he did. He was sitting next to Priscilla, who I noticed had already added the fractions correctly, and so I asked Glenn to look at what she did and try to reproduce it. He was able to raise the fractions properly, though I’m not sure much of the work was his own. It did, however, give Priscilla a teaching moment, which I liked.
Glenn still needed a lot of help, so I asked him to draw the fraction. His drawing looked good, and so I asked him some questions about what the shaded and unshaded parts meant. Once he was able to tell me that the 5 unshaded pieces represented the part that Denise still needed to mow, I stepped away. He was able to do the rest on his calculator, and when I came back over, he and Priscilla were comparing answers.
Priscilla is a sharp student who always solves problems in creative ways, and so I was pleased to see that she produced this:
Priscilla correctly added her fractions, and labeled them “Jane, Solange.” When I had checked in with her and Glenn, she had already drawn a picture, and then she took a break from her solution strategy to help Glenn add his fractions.
In her drawing, she circled two separate groups of five, knowing that each represented 700 square feet. This meant that 10/12 of the strip diagram represented 1400 square feet. I loved this! What she did next was interesting too. She knew that 5 of the pieces needed to be 700 square feet, so she looked for a number that she could multiply by 5 to produce 700. Instead of dividing, she did this through guess and check. Once she found that the number she was looking for was 140, she multiplied it by 2, knowing that these were the 2 parts that weren’t yet accounted for. She added 140 + 140 = 280, and then added 1400 + 280 to get the correct answer of 1680.
In this photo, it’s hard to see just how much work Edith had done and then erased. She knew that she could use a proportion, but at first she just couldn’t make it work, no matter how hard she tried. Near the top right of her work, she creates the proportion
This led her to think that the total area must be 1200 square feet. I asked Edith several times to check that and see if it would work. For example, “If the total area is 1200 square feet, then how many square feet did Solange actually mow? What about Jane? Does that leave 700 square feet for Denise? Hmm….Maybe we need to rethink your proportion.” So then Edith set it up a little differently. This time she wrote out the proportion:
On the left side of her proportion, she compares the 5 parts that Denise needs to mow to the 7 parts that have already been completed. She then correctly put Denise’s 700 square feet in the numerator of the ratio on the right and solved. She got an answer of 980. This confused her for just a minute, but then we talked about how the 980 square feet corresponds with the 7 parts that have already been mowed, and she realized that she still needed to add the two: 700 + 980 = 1680. I really liked how Edith not only used a proportion, but also that she did it in a way different from what I was anticipating.
I really love this problem and will definitely keep it as a part of my teaching toolkit. It worked well–and was challenging for–both groups of students because the “notice/wonder” structure of the problem really helps to engage learners at all levels and give them time to think about the situation posed before they start crunching numbers. As I mentioned before, this structure has become routine in my math classes. Try it for yourself! Start problems with these two questions: “What do you notice?” and “What do you wonder?” It will allow all students an entry point into the problem. Some may ask more challenging questions and will want to struggle to answer them, while beginning students might work on other questions more appropriate to their level. I’ve used the notice/wonder structure with visual patterns, geometry, tables, graphs, among other things. It’s been fun helping my students get into the rhythm of anticipating questions and making mental notes as soon as they start interacting with a problem. Because the students come up with the questions themselves, they also feel more invested in answering them. This is just another way for the teacher to step back, focus authority with the students, and act as a facilitator in their learning.
A final note: I can’t emphasize enough how much the math Twitterverse has impacted my teaching. There is a truly robust, supportive, and creative group of math educators on Twitter, and they’re always posting new ideas about lessons and good math pedagogy, as well as linking to great problems and teaching resources. For example, to find samples of activities that are structured like this one, check out the hashtag #noticewonder. To find blog posts by Fawn Nguyen, as well as other “Math Twitter Blogosphere” folks, search #MTBoS. Are there other hashtags that you follow? Leave them in the comments below. Oh, and follow me at @rezloh!