This is my third (and most exciting) post about my new found love for Visual Patterns. My enthusiasm stems from a growing appreciation of how these patterns can be used in such a wide range of grade-levels, including advanced algebra. The use in an elementary or lower-level secondary classroom is easy to see. However, the teacher and student need to dig a bit deeper into the make-up of these patterns in order to generalize them.
It’s not immediately apparent what step 4 should be. But even more so, the quadratic nature of this pattern is not necessarily simple to comprehend. From yesterday’s pattern #5, the students had a method for finding the number of penguins in the nth step by converting the penguins into a table and creating a system of equations. I didn’t want to encourage this method, as it is very procedural and tedious. However, it was a good place for students who liked to work in a more algebraic way to feel successful.
Also, the table allowed them to explore what the difference of differences really told them. I had a student, let’s call her Kay, ask “I wonder what the constant difference of differences represents in our equation for the nth step.” She came up with a conjecture by comparing it to our problem from pattern #5. Kay concluded that the “a” value in the equation ax^2 + bx + c = y is half of the constant difference of differences. I challenged Kay to continue to examine these values in future problems to see if her conjecture holds true. I had another student, Em, wonder if that meant that the “a” value in a cubic function is equal to one third of the difference of difference of differences. This she will investigate as well. What is very exciting about these questions is that they were non-existent 5 weeks ago. It wasn’t that the students didn’t WANT to be mathematically curious, they just didn’t know HOW. It was a huge thrill for me as a teacher to see these kids move from looking at a math problem with a single solution to being able to ask new questions. A nod to Christopher Danielson for helping me realize that learning is having new questions to ask.
Back to the problem at hand: How many penguins are in step n? A few of the students were able to get the answer without using a table. These were mostly the students who like to do things in their head. The ones who want to fully process the problem in their brain, but not write any of it down. [Side note: these are usually the ones who are brilliant with numbers but get lower grades in traditional math classes because they don’t want to “show their work.”] Anyway, I wanted to challenge those who used the table method and set up a system of equations to relate their model back to the picture. Spoiler alert! The answer is 1/2n^2 + 1/2n + 1, but I wanted my students to be able to relate that back to the picture. What do the individual pieces of the expression represent in penguins? This way, the students were able to make that connection of a picture or pattern that didn’t seem quadratic to begin with and flesh out its quadratic properties.
When the students figured this out, it was a magical moment. I had to capture it:
Another cool experience with this problem: the same evening that I did this problem, our school hosted parent-teacher conferences. One of my students came into conferences with her parents and her three little sisters, ranging in age from about 5 to 10. One of the little sisters sat down and wanted to be part of the conference. I pulled up the visual pattern and asked her how many penguins would be in the next step. It was a validation of my initial thoughts of how open and accessible these problems are to all levels of mathematics. Here was an 8? year-old looking at the same pattern that her 17 year-old sister explored earlier that day. And it was mathematically applicable to them both. Beautiful.