I wanted to share some feedback on this problem because it’s so similar to the Goats and Chickens Problem that Daphne Carter-McKnight wrote about recently. You can read her fantastic writeup of the problem here. I like to use the Movie Theater Problem to assess how well students are able to make connections between the two problems and to see if they’re able to try some new approaches that maybe they didn’t try the first time around.
I like the problem because even though the situation is easy to understand, solving the problem is difficult. Even if my students are somewhat familiar with algebra, they usually aren’t able to make a solve a system of equations, and so they’ll have to puzzle it out using a bit of guessing. They also need to exercise precision and control over basic operations, which is always nice to reinforce. Often, I’ll have a group of students that does very well with calculation, and that doesn’t have too much trouble working out rather complex problems when the methods required are clear to them. However, they struggle when they have to puzzle things out for themselves, especially when there are no given answers to choose from or clear pathways toward the solution. One other thing I really like is that students can benefit from stepping back and thinking about how much a movie ticket could cost. This often helps them to find an entry point when they’re feeling stuck.
How I Solved It
I solved this problem algebraically. Let x be the price of each child’s ticket. An adult’s ticket costs twice as much as a child’s ticket, so the price of each adult’s ticket would be 2x. There are two adults, and so the total price for their tickets would be 2x + 2x, or 4x. There are three children, and so their total ticket price would be 3x. The total for all tickets would therefore be 4x + 3x, or 7x. This should be equal to the amount paid, $43.75. So:
7x = 43.75
x = 6.25
A child’s ticket costs $6.25, which means that an adult’s ticket would cost $12.50.
Anticipating Student Approaches
One possibility would be guess and check. Let’s assume that a child’s ticket is $5.00. If this were true, then the total for the children would be 3 times $5.00, or $15.00. Since the adult tickets cost twice as much and there are two adults, the total for their tickets would be $10.00 times 2, or $20.00. The total for this scenario would be $35.00, which is too low. I could eventually get to the correct total cost by adjusting my guess until I get the correct result.
I could also put this information into a table, which would help me to organize my guesses. Let’s say I started by guessing that a child’s ticket is $4.00, and then I recorded each subsequent guess into the table. It might look something like this:
In both of these, I started with a prediction about the price of a child’s ticket, because that’s what the problem asks me to find. But I think that students will often start by thinking about what the price of an adult’s ticket would be. This shouldn’t make a huge difference in the way that they make and check guesses, but it will be interesting to see how it plays out.
Supporting Productive Struggle
Students often have more difficulty with understanding the question than with performing calculations. I anticipate that some will incorrectly assume all the ticket prices are the same; others might misinterpret the number of people seeing the movie; and others might return the price of an adult ticket instead of a child’s ticket. As we know, students can have a hard time digging into really difficult problems when the answer choices aren’t given to them. I think that this problem will make them try a few different methods or guesses to arrive at the answer. Some might get frustrated in doing this and might try to give up. I think that students will also struggle to understand and meet the two criteria set forth by the problem. That is, not only does the total cost need to be $43.75; the price of a child’s ticket has to be exactly half the price of an adult’s ticket. So I think that students will submit answers in which the total cost would be correct but the relationship between the ticket prices will not. I’ll need to make sure I ask students to talk through the thinking that led them to their answers.
To support students while they struggle, I will emphasize careful reading and ask good clarifying questions. If a student is struggling, I’ll read through the question with them and make sure that they understand both the criteria set forth in the problem and exactly what it is that the question is asking them. This will help to set them on the right track in terms of figuring out how to solve the problem. “I don’t know where to start” is something that I hear a lot, and when this happens during this activity, I’ll ask what the student thinks the answer could be. So, if someone is really struggling with this problem, I’ll ask them about how much they usually pay for movie tickets, and then let them take things from there. I also keep a running list of problem-solving strategies in the classroom, and so if a student can’t get started, I might suggest one.
Natasha has been in my HSE class for about three months. When she first joined the class, she would work diligently but quietly and would often talk to me about how lost she felt. In a short period of time, though, she has gained a lot of confidence and has started helping other students in the class when they get stuck. It has been a really amazing transformation to watch.
Natasha got her first answer, the one circled at the bottom of the page, very quickly. She didn’t really explain where the $2.75 came from, but when she multiplied it by 3 and subtracted it from the total cost, she was left with $36.50, which should mean that each adult’s ticket was half of that: $18.25. She circled her answer and called me over. My response was, “Awesome! Now let’s check and make sure it works.” She proved it to me mathematically by talking through each of her steps, and so I reminded her that we had one other thing to check: “So this also says that the price of a child’s ticket is half the price of an adult’s ticket. Do we have that here?” She was able to identify that her answer didn’t work out, and she set back to trying something else.
On her next attempt at the problem, she really fixated on the relationship between the two ticket prices. This helped her to structure her guesses, which are at the top of the page. She guessed that an adult’s ticket would be $15.00, then $14.00, then $13.00, then $12.00, and then, finally, $12.50. This time, when she called me over and asked if her answer was right (and I gave my usual reply: “I don’t know–talk to me about it”), she didn’t miss a beat before saying, “I know it’s right this time.” This approach was pretty common for the students in this class, but Natasha got to the answer very quickly. Another confidence booster!
Delphine has been taking classes for almost two years now, and though progress has been slow, I’m seeing some significant improvements. Her computation skills are great, but her problem-solving skills still need some work, and she really struggles with the sense-making aspect of math. She really has a hard time approaching problems strategically, and often, when I ask her why she tried a particular strategy or operation, she tells me that she just guessed.
I was really interested in the work that Delphine did, because even though I sat down and talked to her for a while about how the prices of an adult’s ticket and a child’s ticket need to be different, I didn’t see this reflected in her work.
Delphine seems to have assumed that the ticket prices were the same for everyone because her guesses show her adding the same number five times and then checking the total. She tried $8.00, $18.00, $13.00, $8.50, and so on. What’s interesting is that she didn’t try to divide the total by five! Once she noticed that $8.00 didn’t work, and neither did $8.50, she started to give up. At this point, Natasha, who was finished, stepped in to talk with her. Natasha is really great about asking questions when she helps her classmates, so I just let her go. Natasha helped Delphine to understand that the ticket prices needed to be different, and so her guesses started to reflect this. Still, she wasn’t able to grasp the idea that their is a specific relationship between ticket prices, and I didn’t see evidence of much structure in the way that she determined the ticket prices. She does have the correct answer written at the top, but this is because Natasha revealed it to her at the very end of the activity.
I think this was a little discouraging for Delphine. Almost every other student in the class made it to the correct answer, but she just couldn’t get there. I also felt discouraged that I couldn’t lead her to a successful solution. What I was happy to see, though, is that Delphine stuck with it and tried not to give up.
I saw a lot of interesting things happen in Carla’s work. First, she divided the total amount by five and then tried to work things out from there, but she hit a dead end pretty quickly. What she noticed is that she ended up, in a very roundabout way, finding that Nick and Katie’s tickets would cost the same amount as their children’s tickets.
At this point, she was stuck and asked for help. We talked for a little while about problem-solving, and she mentioned trying some different guesses. What I really like about her work is how she immediately made a table to organize her guesses. She was able to do some of the calculations in her head, but the rest she did at the bottom of the page. I was actually surprised by how atypical this was. Very few other students tried using a table, and none were able to create one as organized and effective as Carla’s.
Francisca worked through the problem by guessing and checking, like most students did. Interestingly, at one point she identified the correct $6.25 for a child’s ticket but had that price paired up with a $12.25 adult’s ticket, and so she couldn’t get it to work out. Only after a lot of struggling and playing around with numbers did she get the correct numbers for both the adults and the children.
But what was most interesting is something Francisca brought up after she solved the problem by guessing and checking. In passing, while talking to the class about her solution, she said that it helped her to think of two children as being the same as one adult. I asked her to talk about this a little more, and she asked if she could come up to the board and draw it out. English isn’t her first language and she felt more comfortable showing than telling. Francisca drew the family on the board, and then she divided the total cost by 7. See what she did there?
The other students didn’t understand why she was dividing by 7. Francisca again explained that she was seeing what would happen if she thought of each adult as having the same “cost” as two children. I asked her to draw it out so that everyone could see it.
She drew seven children and arrows showing the relationship. The other students were amazed. “This was so much easier!” “I can’t believe I didn’t notice that!” It was a great moment.
I have taught this problem a number of times before, and the approaches my students used this time around were very similar to what I have seen before. I was a little bit surprised by how much unstructured guessing and checking I saw in the room, but I was pleased to see that it was fruitful and that, for the most part, my students got to see their struggle pay off. We had done the Goats and Chickens problem about a month earlier, and during the whole-class discussion about solution strategies, a few people actually referenced the similarities in this problem. However, they weren’t able to draw a picture as easily this time around, which explains why so many students resorted to guessing and checking. It was also interesting to see how, even though we have done a lot of work on algebra, my students didn’t try to use variables in their approaches. I like to think that this is because we have spent so much time talking about how all solution methods are valuable, as long as they make sense to the student.
What I Might Change
I actually thought that too many of my students finished the problem too quickly, and so maybe I could adjust the language a bit to introduce another degree of challenge. Next time, I might try saying that a child’s ticket costs 50% of the cost of an adult’s ticket. Would students interact with the problem differently if I used a percent instead of “half?” I’d be interested in seeing what came out. Also, I didn’t have any good extension questions planned other than asking students to try solving it another way. While it’s always good to ask students to try other solution strategies, I feel like I should have had something else prepared. Maybe another theater that they could compare prices to? If you have any ideas for good extension questions, please feel free to add them in the comments. I’d love to read them.
I really didn’t think that so many students would have quite as much trouble working with and understanding the constraint built into the problem—that children’s tickets cost half as much as adult tickets. Several of them were able to come up with answers, but I don’t think that any students showed me a correct answer on their first try. In the past when I’ve taught this problem, students often say that guessing and checking isn’t “real math.” I thought it was interesting that this didn’t come up this time around. I’m interested in finding ways to make explicit connections between the strategies that the students try and the algebra underlying those strategies. I think that doing this will be a challenge, though.
I think that my students got a lot out of solving this problem. They got some practice doing computation involving decimals on the fly, which was good since we usually don’t do a lengthy unit on decimals in my HSE class. I think that the students also got a lot out of talking to each other. For example, Natasha got the opportunity to teach Delphine and a few other students, and Francisca got to explain her elegant method involving drawings.
Also, this class had been spending so much time on graphing lines and evaluating functions that it was nice to pull back and look at a very different nonroutine problem. The class really rose to the challenge, and it was fun to see how they were able to look at the list of problem-solving strategies and get to work.
Advice for Teachers
Like the Goats and Chickens problem, this problem is great for structuring a discussion of problem-solving strategies. It can help students see the connections between guess-and-check, making a chart, drawing a picture, and even solving the problem using algebra. It works really well for students of all levels, and students have a real sense of accomplishment in puzzling it out and arriving at a correct answer. Just keep asking questions and encouraging perseverance!