With this problem, I hope to observe how students will organize data and if they are able to recognize patterns. I also wanted to see what counting techniques the students might use to find all of the possible outcomes. Some of my students have experience developing harts and graphs, while others are new to these concepts. I was interested in seeing which students would figure out different solutions and help others to formulate solutions if they were having difficulty. My class is composed of students with math grade equivalents between 5.0 to 11.0 on the TABE.
I wanted to try this problem because I was interested in seeing the different ways that students would use to solve the problem. I have some students who attempt to solve problems in their heads and yell out answers before the rest of the class has time to plan, think, and solve the problem. I also have a wide variety of learning styles among my students. My classroom has round tables in it, and so I often encourage my students to work in groups of two or three.
The problem engages students in deep thinking, analyzing, patterning, and reasoning. I am looking to see if students are able to recognize and develop patterns. The tree diagram is just one tool for counting possible outcomes, but students have the opportunity to try different approaches.
How I Solved It
I solved the problem by making a visual tree diagram, like the one below.
Therefore, a cone with a choice of a flavor and a topping equals 9, and a cup with a choice of an ice cream flavor and a choice of a topping = 9. Since 9 + 9 = 18, there are a total of 18 combinations that you could have.
Anticipating Student Approaches
I think that students will try a few different things. Some of them will draw pictures, which I think will be helpful as a way to get started. However, I think that students will feel like this method takes too much work, and I have a feeling that they will miss some of the combinations.
Some students might also try organizing their combinations into a table or chart. I worry that students will come up with combinations but will have a difficult time figuring out how to keep everything organized. If this happens, I plan to ask some questions that will lead them toward a way of organizing all of their work.
My Goal for Student Learning
My goal is for my students to see and interpret patterns. Using a tree diagram is just one tool for counting all of the possible outcomes, but students can approach the problem in a number of ways. I hope to see students persevere in solving the problem, no matter which method that they choose, and I also hope that they are able to talk to me and to their peers about their chosen strategy.
While I know that I could teach the students to find the total number of combinations by multiplying 2 times 3 times 3 equals 18, I think it would be great if students could see this for themselves after they have already spent some time puzzling things out for themselves. I think that it will make the formula for finding combinations much more meaningful for them.
Supporting Productive Struggle
Some students will have difficulty figuring out where to begin: “I don’t understand this at all.” Some might give me a solution without showing it on paper–or they might not be able to articulate exactly why they got the answer that they did. I think that some students might give up early on and will just follow along with what their neighbors did.
In order to promote productive struggle, I plan to talk to the class about the different problem-solving strategies that we have tried so far, and I will encourage students through questioning.
- Can you show your approach to another student? How would you explain your approach to them?
- Do you see a pattern?
- Tell me more about that.
- Would it help if you created something visual? What might be one example of something you could order from the shop?
- Can you explain how you were thinking about this?
- Can you show us how you did that?
I plan to focus my line of questioning so that students can see connections among the different approaches that they tried.
If students finish with the problem and are confident that their answer makes sense and is correct, I will ask them to figure out what happens if the shop offers a fourth kind of ice cream: mint. Now how many combinations are there? If they really get on a roll, I might ask what would happen if they introduced more toppings as well.
Angela and Jaquia’s Approach
Angela and Jaquia sit next to each other in class. They like to bounce ideas off each other before putting something on paper. Angela started drawing each cup with one topping and flavor.
Meanwhile, Jacquia started by drawing ice cream cones showing each one with a flavor and a topping.
Angela and Jaquia then discussed their individual results and asked if they could combine their solutions because they knew the answer was a total of 18 combinations. I agreed to let their solution be written out together. Jacquia was eager to show her drawing on the Smartboard, which was great because it finally clicked for others when they saw this visual representation. At some point during class, Angela compared solutions with another student, and they were pleased and excited that their solutions agreed!
From the beginning, these two students worked together and with others to arrive at solutions. It is not uncommon for Jaquia to get up and go to another table to help students find a solution by offering an explanation or assistance. When this happens, I am careful to make sure that the student on the receiving end of her help is able to explain it.
Nicole was able to figure out her solution by using a tree diagram. Her visual helped her to process the problem very quickly.
While her written explanation had some gaps, her solution was accurate.
I have noticed that Nicole catches on quickly if she can make something visual. She is willing to assist others but always reluctant to come forward and show her work on the Smartboard. She did come forward to show her work and talk about it this time, and it was really helpful for other students. Even though she had difficulty writing about the solution, she was able to verbalize it for everyone else in the class.
Some students have difficulty in how to approach a math problem when it doesn’t just involve computation. I had a few students–in the beginning–who wanted me to give them the answer. I realize that I need to continue to allow my students to process, think, and solve problems, and not worry about making mistakes or being wrong.
What I Might Change
Most of the students understood the concept and were able to use their prior knowledge in problem solving but had difficulty in talking through their thinking or writing about their process. I asked them to solve similar problems a few days later to see if they understood the concept. I need to continue to encourage students to reveal their thinking process and to explain their methods to one another. Students need ample time to process, think, and work through math problems.
My favorite part about this activity was taking the time to let my students talk about their solutions. A few different people went up to the Smartboard and demonstrated their work, and the whole class had a really good discussion about the different strategies. The students were really energized by seeing that there are lots of different–and equally valid–ways of getting to the same solution.
Advice for Teachers
This problem is featured as a task on Jo Boaler’s Youcubed site. It’s a great side for quality problems, and so I knew this one would be good too. She recommends asking students to present their thinking, much like my class did, so that students can see different ways of organizing counting. This problem also opens the door to talking about other problems involving counting and probability, and it gives students some good background knowledge to rely on as they develop their skills. I love how it drew out a variety of interesting problem-solving strategies. I’ll definitely keep using it!