I love the Gold Rush Problem because it is similar to a problem that I had used in my class before; the version I used involved maximizing the area of a garden given a limited amount of available fencing to surround the garden. What I liked about the gold rush problem is that it took the garden problem a step further by asking students to analyze what happens to area when perimeter is doubled, tripled, etc, and I like how it invites different strategies for solving.
I also like how this problem encourages creative thinking about basic shapes. It invites students to construct several different rectangles, which ideally will lead them to a better understanding of how dimensions relate to perimeter. Doing this also reinforces computation skills, and so it works well in a multilevel classroom. Once students arrive at their answers, there is a lot to talk about. For example, is a square a rectangle? Is a rectangle a square? Assuming that I did share a plot of land with another prospector, what would be the most equitable way to split it? Do we split the land, or do we split the profits? At what point in joining the ropes together does the workload become too much for one prospector? And so on.
And finally, there’s a bit of historical context here, and so it fits well into a class that also has a history component. It acts as a good springboard into a discussion about the pre-Civil War period and the waves of westward migration that were occurring around the time.
How I Solved It
This is an optimization problem. Let x and y be the dimensions of the rectangular plot. Given the constraint of only having 100 meters of rope, the perimeter of my plot would be 2x + 2y = 100. The area would be A = xy. I started by solving the perimeter equation for y so that I could substitute it into the equation for area.
Substituting this into the equation for area, I have A = x(50 – x), or A = 50x – x2. The graph of this equation will be a parabola with a single critical point, and that critical point will give me the x-value that will maximize area. To find that point, I need to know the derivative of A with respect to x; this will be the equation for slope of the tangent line to the graph. The critical point I’m looking for will have a tangent line with slope 0.
The derivative of my area formula is A’ = 50 – 2x, where A’ represents the slope of the tangent line at a chosen point. Since I’m trying to find the point where the slope is zero, I substitute 0 for A’.
Now I have 0 = 50 – 2x. When I solve this for x, I get the solution x = 25 meters. So this is the optimal length, which means that my optimal width is also 25 meters. The shape that will maximize area is a square that is 25 meters by 25 meters.
To answer the second part of the question, I applied the same rationale to a rope that is now 200 meters in length. If the optimal shape is a square, then it would be 50 meters by 50 meters, and it would have an area of 2500 square meters. This means that each prospector would get 1250 square meters of land, which is twice as much as they would have before. So it does make sense to “join the ropes.”
Anticipating Student Approaches
The method I outlined above is impractical for teaching, and I only tried it to challenge myself and to see if I could remember how optimization problems worked. So after solving it algebraically, I wanted to examine the relationship between area and perimeter just so that I could see how much the area changed when I made slight modifications to the dimensions. I drew a few different rectangles and ended up at the square that was my final answer from before.
I also wrote it out in table form, just so that I could have an organized chart showing the areas given by different dimensions. I started the table at 40 by 10, as shown below, and worked my way up.
The table is interesting because it provides the opportunity to see the consecutive difference in area each time that the dimensions are adjusted by 1 meter. I noticed a pattern, which is added in the updated table below:
My Goal for Student Learning
One goal in presenting this problem is that my students will be able to apply the basic concepts of area and perimeter in a setting that is a little different from what they might be used to. I also hope to see that they’re able to think creatively about the problem and make adjustments to the shape of their plots in order to see how the shape of the rectangle has a significant effect on its area. In other words, I want them to be able to create a possible plot but then—without my intervention—try drawing other rectangles as a way of checking to see if their answer is correct. Another goal is for students to verbalize the relationship between the shape of a rectangle and its area. What the students should notice is that, as the rectangles become more square-like, the area increases. I also would like to see an organized approach to solving this problem, although I realize that the way students organize their work will differ greatly.
Supporting Productive Struggle
I anticipate a number of students drawing one rectangle and thinking that they’ve answered the question after they’ve successfully calculated its area. Moreover, I anticipate some resistance when I prompt them to try drawing other rectangles so that they can compare the areas of each. I also anticipate some issues with understanding the situation. Even though the prompt specifically mentions rectangular plots, I have a feeling that some students will miss this part. They’ll understand that they’re getting four stakes and a rope, but they won’t really know where to go from there. So I might have to intervene a bit just to clarify exactly what the question is asking them today. I also foresee students jumping to a quick conclusion about the second part of the question. That is, I think that some will gloss over the part about joining two ropes together and just assume that because you’re sharing with another prospector, you would get less land.
To support students who are struggling, I will first ask them to tell me what is happening in the problem. I would want to make sure that they understand exactly what they’re getting from Billy and why they are getting those materials. If they are unable to make a rectangle, I might ask them to draw one, and then I would ask what the length of the rectangle could be. They could then try a few things and check their work. For students who try to stop after drawing one rectangle, I’ll ask how they know that the one they drew provides the most land to work with. So after they try one more, I’ll ask that they try another. And so on. I have some students in my class who really struggle to do long multiplication, and so I may allow them to use calculators. The goal of this activity is to encourage reasoning about shapes; it’s not about crunching numbers.
If some students finish early, I would ask what would happen if three, four, or five people joined their ropes together. How much land would each person get in these cases? And is there a pattern to the increase in land you get by working together with other prospectors? How could you organize the data to see what the pattern might be? Could this be viewed as an input/output table, or a function? If so, what would be the rule of the function? How do you know? How many ropes would you need to join together so that you could get 7500 square meters to work with?
One of my favorite professional development activities is to talk about student work with other teachers. Download all the samples of student work on the Gold Rush Problem.
Elisa and Belen’s Approach
Belen is one of the brightest students in the class. Elisa struggles and has missed several classes because of her work schedule and other issues. This group had a hard time getting started, but once they figured out a pattern, they were able to make progress. What I like about their representation is how organized it is. They begin with a rectangle that is 30 meters by 20 meters; it has an area of 600. The next rectangle they drew had dimensions of 28 by 22, with an area of 616. When I talked to B. and E. about this, they said that they were surprised about what happened to the area. They explained that they noticed how, when they decreased the length and increased the width, the area got bigger. So they kept doing this until they arrived at the dimensions 26 by 24, for an area of 624. This was the greatest area possible, they said.
When I asked why they didn’t go a step further and try 25 by 25, they reasoned that it wasn’t allowed: The plot had to be a rectangle, and 25 by 25 would be a square. I was interested in this solution because I predicted that students would get hung up on the square/rectangle issue, and these two were adamant that the plot could not be a square. So we talked about this. Also, notice their reasoning at the bottom. It says, “After a while we figure it out that if you increase the width, then you have to decrease the length in order to have the same perimeter, but bigger Area.” I understood what they meant, but we talked about it for a while to get some clarification. Is there a point at which decreasing length and increasing width doesn’t increase area anymore? What is that point? Why does it work this way? This group’s graphical approach was very typical of what other students tried.
Travis and Latoya’s Approach
What interested me about this group’s approach was that the first rectangle they drew was actually correct. But they didn’t know that. So I prompted them to try drawing a few others. Travis was sure that he could find one with a greater area, because he reasoned that as the length value got bigger and bigger, the area would too. He wasn’t really thinking multiplicatively yet. So he tried some other rectangles: 40 by 10, 20 by 30, and 45 by 5. He told me that he was really surprised to find that the 45 meter by 5 meter rectangle had the smallest area. So he, Latoya, and I went into a hallway that was about 5 feet wide and looked at how narrow this would actually be.
Travis and Latoya were able to complete the second part of the question pretty quickly. Latoya said that she knew the shape would need to be a square again, since the square from part 1 had a bigger area than the rectangles. They did some calculations and concluded: “It would be better to join the ropes because you can make your width and length wider by each side. By doing this you increase your profit. There is also more land for you and your partner to dig.” I was really interested in the comment about profit, and so we talked about it with the whole group. We wondered whether having more land would necessarily guarantee more profit. So in talking about this, we touched on probability, and we also began thinking about what the most equitable way of sharing the plot would be. Is it more fair to split the land, or is it more fair to split the total profit? Most of the students concluded that it would be the most fair to split the total profit, or weight in gold, equally. Though some said they would prefer to take a gamble and have half of the land all to themselves. This was interesting, I thought.
Rodolfo, Clemente, and Julio’s Approach
Rodolfo and Julio have been with me for a while, but Clemente is pretty new. All have limited English proficiency, and so it was fascinating to me that their description of how they solved this problem was the most verbal of any that I saw. What isn’t clear from this photo is that, before they submitted this poster, they had done another that contained pictures and nothing else. I told them that I would like to know more about what they had to say about the problem, meaning that I would like them to talk about it to the group. They decided to start over and take the approach you see above.
After finishing part 1, Rodolfo was sure that there was no way it would be beneficial to work with another prospector. So I asked him to prove it to me, and he started working. When I checked back with their table only three or four minutes later, Rodolfo told me that he was wrong: If he worked with another prospector, he would get twice as much land. Because they answered so quickly, I asked: “What if all four of us decided to join our ropes together? How much land would we get then?” And they produced the explanation on the right. Their drawing is interesting. It suggests that the four small squares could be put together to form the big square with area of 10,000 square meters. I asked them about this. They explained to the group that they didn’t mean it that way, and they realized how their drawing didn’t accurately represent their thinking. This approach to presenting their solution was great and not at all what I was expecting.
Crystal and Steven’s Approach
I liked Crystal’s approach because of its clarity and simplicity. But it’s also worth noting that Crystal needs almost constant support in the classroom. She has a hard time struggling on her own, and her hand shoots up to ask for my help once every five minutes or so. When she first looked at this problem, she gave up right away and said that she didn’t have any idea where to start. So we first just talked about what was happening in the problem. Once Crystal figured out that she needed to make a rectangular plot, she was able to produce the four rectangles above. And she worked independently for the next fifteen minutes without asking a single question. After a few tries, she arrived at the correct answer.
Steven was really struggling. Despite some help and some tips from me, he wasn’t able to find any rectangles that had a perimeter of 100 meters. The only one he could come up with was 25 by 25. This was good, but I wanted to see some flexibility in how he was thinking about this, so I pushed him to keep trying. Crystal was sitting next to him, and when I stepped away to talk to another student, she started showing Steven what she had been working on. Steven followed along with what she was saying and asked her questions. I thought this was a really great moment for Crystal. Here was a student who had no confidence in her own abilities, teaching another student how to create rectangles. I also noticed that Steven was really listening. So I stayed out of the way, and they finished the project together, with Crystal doing most of the heavy lifting and Steven asking good questions along the way.
I really enjoyed doing this problem with my classes, and it’s one that I would highly recommend using with any class level. I wasn’t sure whether or not to have small groups present their strategies to the class using posters, but I’m really glad I did. In some cases, I was explicit with students in asking them to represent all of the steps they took to get to their answers—meaning, I wanted to see the mistakes as well as the successes. But with other groups, I just let them go. I found this to be an effective way of structuring the discussion about student responses. By doing this, we got to talk about different ways of structuring and illustrating our thinking, but we also got to talk about the choices that the students made in terms of what to include and what to take out when creating their posters. Over the past few years, a big part of my teaching has involved talking about student work, and this activity only reinforced it for me. Time spent talking about thinking and talking about strategy is just as valuable as time spent solving equations or graphing lines. I also learned a lot about my students’ ability to persevere and struggle from doing this activity. I do at least one of these long- form problems every week, and at the beginning of the cycle, my students tended to give up, get frustrated, and ask me why I was making them do problems like this. But now that we’ve done ten or twelve of them, my students have become real problem-solvers. It was affirming to see that we can teach persistence and that our students do benefit from it.
What I Might Change
When I did the problem this time, I asked my students to work independently for about twenty minutes, but then I allowed them to work with the other people at their table for the next forty minutes. I think that this improved the “presentation” element of working on this problem, but I would be interested in seeing what would happen if students just work independently the entire time. I plan to try this next time around, just to see what I get from them. My sense is that the small- group work facilitated some good discussion, and it helped keep struggling students engaged. Even if they weren’t able to completely solve the problem on their own, they were able to provide input and feedback as the group worked together. I’m also interested in trying this activity over a period of a week. Students could submit something on the first day. I would then provide some feedback and ask them to clarify their thinking in places, and I would ask them to resubmit their work. I’d like to see how their explanations and processes would change if they were given several days, rather than just an hour, to think and elaborate.
I used this problem with two groups of students who didn’t have a lot of experience with geometry. Most of them were able to pick up on area and perimeter quickly—in large part because it wasn’t completely new—but some had a very hard time. I can think of two or three students who just couldn’t figure out how to make a rectangle have a perimeter of 100 meters. Or, if they were able to find one, then they couldn’t find one with different dimensions. In these cases, I just asked the students to focus on creating rectangles, not finding the one with the biggest area: “Calculating the area can wait; for now, let’s just see how many different rectangles we can find that have the perimeter we’re looking for.” Next time I’ll be better prepared to help students with this part of the problem.
My students really liked this problem, and they liked getting the opportunity to explain how they solved it. The students did learn some important mathematical concepts, but I think that the most important thing they got out of it was thinking about how they would create their posters so that they could talk about their thinking. They learned that they needed to show the beginning and intermediary steps before just getting to the answer because this would help their classmates understand where the approach came from and how they worked with it. Through the course of the activity, I saw my students start to think like teachers. When talking about their strategies, they explained all their steps and they fielded questions, both from me and from their peers.
Advice for Teachers
This is a great problem that gives students a lot of material to talk about. On the surface, it just seems like another word problem, but there are lots of extension questions you could pose to encourage further thinking, and there are good discussions that can arise after the students have already found the solution. So don’t feel like you have to rush through it. Take your time, talk to your students about their thinking, and then ask them to show their thinking to their peers. You’ll also get a sense of what your students are interested in. Mine, for example, were really interested in turning this into a function (because we had just covered functions). Each student will find something interesting about this problem. So take a little bit of time to let the class go where they want it to go. Your students will appreciate it.