I have this written on the board:
“We can stack anything to represent a triangular number. Are the any of the numbers below a triangular number?”
I try and avoid problems that are always neat and perfect. When I search for something that we can connect to triangular numbers, I accept that an exact match is neither always needed or preferred. Here we have examples of number that are almost triangular. This is perfect because it helps students define why they are not triangular numbers and what we would need to make them exactly triangular numbers. The small can of coke is missing its top cube, but the extra cube provided is not the same size. As a class we talk about why its important that a model be consistent throughout. If we distort some of the cubes or dots, it could skew our pattern in a misleading way.
This discussion is usually a lot of fun since students spend time defining triangular numbers in order to describe why these piles are not exactly triangular numbers. This has a much higher return rate than just asking students to write a definition and example of triangular numbers.
I hand out building materials (cubes, pennies and anything else that works) and give students 10 minutes of independent work on this task. This is unlike the square numbers because it is much more difficult. Students are now working with a less uniform shape and pattern. They need some independent time to experiment with their ideas, to be inventive and conjecture and test their ideas. Too often a group will cling to one idea and only develop that idea. Meanwhile other ideas are left by the wayside. I like to give each idea the highest chance of growing. In an ideal situation a group will combine , accept and explore several approaches.
After the independent time, groups have 10 minutes to develop their 3D sculpture pattern and 10 minutes to record and reflect on their work. They try and imagine how others will interpret their 3D sculptures and if those sculptures will help others predict the nth triangular number. This is fun for them and natural to us as teachers. Our job is to always imagine the ways in which students will process the information we produce.
This summary is essentially another show and tell. Students have carefully constructed a 3D triangular number sequence and we do a gallery walk and visit another tables sculptures. The idea is to review their work, figure out how the pattern is growing (not just in total cubes but in its structural layout), think about how the shape lends itself to an algorithm and then share their thoughts with the class. This is difficult because everyone already “knows” the answer. Our goal is to focus on how the pattern will encourage others to find an answer and how the pattern might lead us to new insight. What makes even trickier is that you can create these patterns and only have a sense of the structural growth. Seldom will students design a pattern thinking, “if I place the cubes this way, that will help students come up with a specific algorithm that is different from anything we have tried so far.” The class conversation focuses on mathematics as a deep and meaningful activity, not just something that has a specific right or wrong answer. Its an attempt to bring more meaning to the process and focus of mathematics.