I was really excited about using this problem in my class because I felt that students with little knowledge about functions or algebra would still be able to answer it. Students would be able to use pictures and charts to get to the answers, and once different students had the answers, we could use that to lead into a discussion about functions. Often times students ask the question, “When will I ever use this?” By using different methods to solve these functions before we talk about functions, students will already know at least one scenario where they would use it.
I also like this problem because the answer is not readily discoverable for most
students. They need to investigate, explore, and work as teams to find the different answers. Also, the problem can be answered using different methods. I veered towards using a function early on, yet students relied more on graphics to solve the problem. As an added bonus to the problem, there are many different follow up questions or criteria that can be added for our students that finish early.
How I Solved It
I started by drawing diagrams to find a pattern. Once I found the pattern, I developed a function based on the pattern.
The function that I developed was y = 2x + 2, where x represented the number of tables, and y represented the number of seats.
Using this function, I was able to input the y values of 10 and 20 and solve for x to answer part 1.
10 people: 2x + 2 = 10, and so x = 4 tables
20 people: 2x + 2 = 20, and so x = 9 tables
Using the same function, I was able to input the x values of 10 and 15 and solve for y to answer part 2.
10 tables: 2(10) + 2 = 22 people
15 tables: 2(15) + 2 = 32 people
Anticipating Student Thinking
This problem could be solved in many different ways. Students could use pictures of the tables to solve the problem, though this will get challenging when they need to draw, say, 15 tables. Still, I think that drawing pictures will help them to see that there’s a pattern, which could also lead them to the solution. I think that some students might construct a table or chart to solve the problem. Especially if they recognize the pattern, they might be able to come up with something like this, which they could extend as far as needed:
I also suspect that some students will try a combination of these methods, or they might feel frustrated by drawing lots of tables and just try to take a guess. I hope that this does happen, because it will give me an opportunity to help these students take small steps and find entry points into the problem using one of the other solution strategies.
My Goal for Student Learning
My two favorite words: Productive Struggle. I want my students to really test themselves. When they feel ready to give up, me or another student will give them a little boost in the right direction and let them struggle on. In the end, their struggle will be worth it as they solve the problem and develop some foundational skills with algebraic thinking and functions.
Supporting Productive Struggle
The biggest challenge for students when they take on a problem like this one is usually “Where do I start?” I will ask the students to think about similar problems that we have done and ask them how they started those. I will also remind the students of our problem-solving strategies and ask them if one of those may be helpful. Finally, if some students continue to struggle, I will ask another student who has found a good starting point to talk to the struggling student about how they got started. The trick, I think, is to nudge the student in the right direction, as opposed to dragging them in the right direction.
This question extends itself nicely for follow up questions. I could give the students different numbers of tables or people to work with. I could change how many people fit at each individual table. I could add room dimensions and table dimensions to see the maximum number of tables and people that would fit. If my students are really up for a challenge, I might ask them to find a rule that they could use to determine how many people could be seated at any number of tables. Even though it’s not a real-world scenario, I might ask something like: “Let’s take a look at the answers you came up with for the first two questions. Well, what if there were 200 tables? A thousand? How many people could be seated?” This might be too big of a leap for some students, but it could be a nice challenge for others.
Heather, Leann, and Carol’s Approach
Heather, Leann and Carol began by immediately drawing pictures. They saw that as each table was added, there were two more people that were added. Once they found the answer for 10 people, they began a new picture for 20 people. They also used new pictures for finding how many people would fit at 10 tables and 15 tables. Their process worked well for them because they were able to easily add tables/people until the criteria of the problem was met.
Desiree had the same thought process as Heather, Leann and Carol. She used pictures to find the answers to the problem. Like the previous group, she made a new picture for every group. Desiree seemed to pick up on the pattern more quickly than other students, however, she continued using pictures as it was easiest for her.
Jack and Renee’s Approach
Jack and Renee did their work similar to the other groups, but there was a little difference. While the previous groups used a new picture for every component of the question, Jack and Renee drew one graphic, and as they answered one part of the question, they would add to their graphic to go onto the next part. With this method, they were using their graphic as a reference.
Kayla and Nicole’s Approach
Kayla and Nicole were not too sure how to start. As other groups began to gain momentum on the question, they would give Kayla and Nicole direction. When their graphics became a little confusing to me–notice how their tables aren’t lined up, as the problem specifies–I asked them to write out what they were thinking. It was apparent that they were relying more on what they saw and heard other groups doing, than what they were getting on their own. Although they were adamant that their way made perfect sense, I feel that this exercise may have gone past their point of productive struggle.
After the students finished the assignment, I noticed that the parameters of the question made it possible to answer all of the parts with just pictures. I wanted to see if any of the students could turn these pictures into an algebraic process. I turned the answers to their questions into a chart and then asked them how many people could be seated at 100 tables. Here is the chart that I gave them:
This proved a little trickier as some of the students tried to use proportions to solve the problem. Heather, Leann, and Carol said that:
I asked them if the ratio of 4 tables to 10 people was equal to the ratio of 9 tables to 20 people. They saw that it wasn’t, but then were not sure what to do. I asked them if they remembered doing input-output tables, which we had discussed earlier in the cycle. They did remember doing input-output tables and used that as a starting point for finding the correct answer.
Desiree began looking for the rule right away and quickly found the rule: y = 2x + 2. Using that rule, she found that 100 tables would seat 202 people. She then helped lead the other groups to that answer.
What I Might Change
I think I would give different students different sets of parameters so that not every
group is using pictures. Maybe ask one group to make a chart, ask one group to draw pictures, and if there is a higher-level group, ask them to find an algebraic expression or function.
My students’ biggest challenge was turning the information that they gained from their pictures and diagrams into algebraic expressions. A little productive struggle never hurt anybody, though, and I think that by processing our work as a whole group after everyone had finished working, the students were able to see more clearly what they were working toward. The strategy that worked best with my struggling students was guided questioning. If people were having a hard time getting started, I asked them to talk to me a little bit about what they were thinking about. I asked questions like, “Can you show me?” and “What could you do to keep this organized?” I wasn’t, however, quite prepared for students who had trouble understanding the situation and drawing a picture accordingly, as some of my students did. It was challenging to get them back on track.
From doing the activity, I learned that students who are intimidated by algebra in general were not intimidated by this problem. I think that since it was not introduced as algebra at the beginning, they were thinking of it in problem-solving terms as opposed to in algebraic terms. I also find that the students tended to view the problem more as a brain teaser than a math problem. As such, they were extremely engaged in the problem. The students took great pride in finding the answer. When it was then turned into algebra to show them that they were doing algebra all along, they had a great deal of pride and confidence. It also helps that each time we do a problem like this there are a few students in the room who have done similar problems with me in the past. This gives them the opportunity to become group leaders. All in all, students seemed to get what they wanted because they were engaged throughout the process, and when one group or individual stumbled, another group would lift them back up.
Advice for Teachers
Have some parameters later in the question that will encourage the students to jump from charts and diagrams to algebra. Some students may still use diagrams or charts to see how many people can sit at 100 tables, and that is okay. But the higher parameters really challenge the students to make the connection to algebra, after they have answered
the questions. Also, encourage the students to ask their peers questions before they ask you. It will help to focus authority with the students, and it allows for students to take the lead and become teachers themselves.