# The Arch Problem

I like the arch problem because it starts as a visual model, and then challenges students to develop their algebraic thinking skills. It’s also a problem that allows students of all ability levels to be successful. Lower-level students may be able to simply sketch some of the figures but might be truly challenged by some of the questions that show up further in the handout. Higher-level students will be challenged by the bonus questions, and I can even ask them to go a step further and create an equation that they could use to calculate the number of squares in any figure. The problem also asks for written explanation, so students will have to explain their mathematical thinking verbally–not just with numbers or pictures. This is something that students often struggle with, but being able to articulate mathematical ideas in writing is a skill that can help them to make their learning more concrete and to emphasize key vocabulary.

### How I Solved It

I looked for a pattern and noticed that each figure added two more squares to the end of each column of the previous figure. Figure 1 had three squares across the top and one down each side. Figure 2 had three across the top and two down each side. Figure 3 had three across the top and three down each side. This pattern will continue for any number of figures. In other words, whatever the figure number is, I would always have three across the top, and the number of squares down each side would be the same as the figure number.

### Anticipating Student Approaches

Some students might take question number 5 a little further and recognize that they could create a function to determine the number of squares in any figure:

y = 2x + 3

Here, x represents the figure number, and y represents the total number of squares in the figure. Two times the figure number plus three more equals the total number of squares. For example:

2(2) + 3 = 7        Figure 2 has seven squares.
2(3) + 3 = 9        Figure 3 has nine squares.

This would be an efficient way of figuring out the number of squares in, say, the 99th figure. I anticipate that some students will get frustrated and will just try to keep a table going, all the way up through the 99th figure. This would get them to the answer if they’re careful, but it would be very time-consuming and there would be lots of room for error.

One other thing that I’m anticipating is students “seeing” the pattern in different ways. When I solved it the first time, I saw three squares across the top, and then squares added onto the left and right sides. But I could see how some students might visualize the pattern in different ways, and rather than get them to see it my way, I want to support whatever seeing that they’re working with. In the end, all of them will produce the same result, which could lead to a pretty interesting conversation!

### My Goal for Student Learning

My students are not always comfortable explaining their thinking, especially not describing a mathematical process in writing. If I ask, “How did you come up with that answer?” or “How could you prove it to someone who wasn’t sure?” many respond by saying things like “I don’t know what I did–I just know that I have the right answer.” I want my students to gain confidence in their explanations of their thinking and the procedures that they took to produce an answer. For many of my students, English is not their first language, and I’d like to think that that’s the reason for avoiding verbal explanations. This is a great problem for generating a discussion on the different strategies that could be used; I want them to utilize different approaches to the problem, and I hope that we can have a rich, productive discussion about those approaches.

### Supporting Productive Struggle

I know my students will be able to complete Figures 4, 5, and 10 with little difficulty; however, some will be challenged by using words to describe the 10th figure. They will be able to draw the visual model, but they may not be able to write their thinking. I will encourage students to verbalize how they drew the three figures by asking them to talk about the pattern that they found: “Can you describe what you see here?” If they’re able to talk about it, then they should be able to write it. Some students may try to draw or make a table of all the figures up to the 99th. If they do, I will let them struggle with it for a while before stepping in to ask some questions about other ways they might approach the problem. Specifically, I will try to bring them back to the visual component of the pattern and ask them to tell me more about what they notice.

### Extension Questions

There will probably be a few students who finish early. If so, I will encourage them to come up with an equation that could be used to determine the number of blocks in any figure. Then, if they’re able to produce an equation, I would ask them to prove it by using the equation on a few figure numbers that I choose. If some students are really up for a challenge, I might ask them to figure out a way to determine the perimeter of any figure in the pattern–that would be a tricky one, but I think some students could handle it.

I also think it could be interesting to ask students to create their own visual patterns, and then ask the same line of questions of the pattern that they generated, just so that they could get extra practice in writing about visual patterns and seeing connections to algebra.

## Student Work

On all of the samples of work, I did not include the first side of the paper, on which I asked students to sketch the 4th and 5th figures. All of the students were able to complete that task.

### Taysha’s Approach

Taysha is a strong reader and problem solver. She arrived shortly after the rest of class began working on the problem, and she jumped right in.

Taysha didn’t answer questions 3 and 4 by describing in words; instead, she used her formula and a visual model. In reviewing her work, it appears she used the number 5 in her equation to represent the five blocks in the first pattern! She then took the figure number and multiplied that by 2 to represent the two columns, and finally subtracted 2 because each individual figure has two more than the previous one.

Taysha has a good understanding of order of operations and came up with an equation. She also showed algebraic thinking by recognizing a pattern. She used the same formula to show Figure 99. Taysha did not start using words to describe her thinking until the fifth question. She understood the pattern and was able to prove it by her description. I like how she eliminated the 5 – 2 from the original equation and just began to use the number 3 to represent the three boxes across the top. In question 6, Taysha was able to work backwards to find the figure number when the total number of squares was given. This showed her understanding of inverse operations.

What I really like about Taysha’s work is how she started out using an equation and a visual model to show her answer, but she didn’t use words to explain it. I would have liked her to write down more of her thinking so that it would have been clearer what the number 5 represented. When I questioned her about it, she was able to explain it to me–she just didn’t write it down. toward the end of the handout, she began writing down her thinking and proved it with an equation. Taysha likes using formulas and rules as a way of problem-solving. She immediately looked for an equation/rule to solve the arch problem. Although she relies on the calculator for computation, she can certainly recognize a pattern and run with it.

### Paula’s Approach

Paula completed the chart, so she was able to see that each figure had two more squares than the previous one, so she recognized a pattern. She didn’t describe what the 10th figure would look like; instead, she extended the table to figure out how many squares it had. Paula didn’t answer question number 4. She recognized the pattern but didn’t choose to continue the table any further because she said it would take too much time. When Paula did start to write down her thinking, it was brief and unclear.

I thought that Paula’s experience with the problem was interesting. Clearly she saw and understood the numerical pattern, but she could extend it beyond the 10th figure. Paula has good computational skills, but she really struggles with problem solving. She has been trying to apply different strategies, and she usually doesn’t give up easily. I think that if she had been given a little more time, she may have been successful. This is very typical of what Paula does in class. She comes from the “drill and kill” era of doing math. Doing basic computation is what she’s comfortable with, and problems like this one are new to her, so she really struggles with them. Paula comes to class every day and has begun to use some of our problem-solving strategies. She has, however, self-identified as learning disabled, and she is very open about how these kinds of nonroutine math problems are challenging for her. In any case, Paula always struggles productively.

### Robin

On the front side of the handout, Robin wrote the total number of squares above each of the first three figures: 5, 7, 9. She even wrote out the words “odd numbers by two” next to the figures. She recognized that the pattern–increasing by 2–would mean that every figure in the pattern would have an odd number of squares. Robin was one of the few students who wrote out a description for question 3, so she knew how to figure out the number of squares in the 10th figure. She seemed to get a little bit off track in question 4. She was trying to come up with 99 boxes, not figure 99. This is a mistake that a few other students made as well. Robin noticed her error, though, and used a visual model and equation to help. In question 5, she wrote down her explanation but again got off track, so she went back to the visual model in order to get back on track. I like how Robin was able to recognize her mistakes and apply a strategy to push through. She was also able to work backwards and use an equation to come up with an answer for question number 6.

I thought Robin’s work was interesting because she seemed to be all over the place with her calculations and descriptions. I like she was able to see her errors and get back on track. She persisted with the problem and used her calculations and visual aids to assist her. She didn’t give up, and in the end it paid off. This was pretty typical of Robin’s work in class. She often gets confused and doubts her abilities. But she is slowly gaining the confidence to continue the problem and not give up. If I had given Robin this problem a few months ago, I don’t think she would have been as persistent. She is learning to use different strategies in her approach to nonroutine problems, and it has been fun to watch.

## Final Thoughts

Through working on the arch problem, I learned that my students aren’t yet comfortable working on nonroutine problems. I have been incorporating these types of problems in my instruction for a few months now, but I have a long way to go before my students’ confidence levels increase. This problem can be used with students of all ability levels. Every student in my class was able to complete some of the questions, which gradually built up their confidence.

### What I Might Do Differently

I would encourage the higher-level students to write an algebraic equation instead of a numerical equation. I would also add additional questions between numbers 3 and 4. Going from the 10th figure to the 99th figure puzzled many of my students. I think that if I added the 25th figure and then the 48th, it might help them get to the 99th more easily. I would also give students the choice of working in a group or with a partner, and I would give each group a set of square tile manipulatives.

### Unexpected Challenges

I didn’t expect that some students would be so focused on the “two more squares” being added and not see the constant number of squares at the top (either three across or one in the middle). I found that it was challenging for me to come up with questions that would help them get back on track. Before teaching this problem again, I will spend a little more time thinking about focused questions I could ask to help struggling students.

When some of the students were stuck, I gave them square tiles to work with. This was helpful up until the 10th figure, when they realized that they would run out of tiles. I asked them to demonstrate how they started placing the tiles for figure 4. One students demonstrated that he started with “these 3” across the top. I asked him to continue placing the tiles, and placed 4 down each side. I repeated this with him for the 5th and 6th figure. He had an “ah-ha” moment and was able to verbally explain any figure from that point. The manipulatives really helped this student, and I’m sure they would have helped many others, but I just didn’t have enough of them on hand.

### Student Takeaways

What seems like a simple, routine task at first may actually be quite challenging. Students need to be comfortable taking the problem to the next level and being persistent. The students who started using a table soon realized that their strategy wouldn’t be the best approach when the figure numbers got larger. They will need to utilize many strategies when it comes to problem solving. Knowing multiple approaches can be a huge benefit.

Most of the students joined in the discussion on the different approaches. A few students went to the white board and drew or wrote out their thinking. One student even went to the board and wrote an equation for the 99th figure and explained how she got the number of squares. She said the equation would work on any figure number. A “doubting Thomas” asked if she could prove it with figure 5,281. I invited a different student to come up and prove it using the first student’s formula. I saw my class beginning to ask questions of each other, and I was happy to see the discussions beginning to bloom.