Six Ways to See Visual Pattern #324

I know you weren't asking, but I see n overlapping squares.

— Hunter P. (@professorpatton) November 12, 2019

What about you? How do you see the pattern in the tweet above growing? Please take a look before I completely ruin it for you.

I loved that Hunter saw overlapping squares in pattern #307 from Caden Glover. I saw the pattern like this instead — with the constant four-circle square, and groups of three circles wrapping the upper right corner, and the number of groups is one less than the step number.

Some 30 hours later, I was at my desk creating another pattern, and I started out with a diagonal of increasing cubes (highlighted in yellow), and above and below this diagonal are more cubes (marked in purple). Therefore, for any step n, I see the middle as (n + 1) and the purple ones are two equal groups of n.

It was not until I finished creating step 3 that I realized I'd made the same pattern as the one from Caden. I didn't want to scrap it because I didn't mean to copy his pattern and really had built it from scratch, and now it is pattern #324 on visualpatterns.org.

It might be my favorite one thus far because I can see it in different ways. Here are the overlapping squares that Hunter saw:

What's fun is sometimes I don't see the pattern in its "simplified" form until I've simplified the equation. (I have to do this for the answer key). The number of cubes C is related to the step number n, such that C = 3n + 1.

And I wouldn't be me if I didn't always try to see a rectangle in any pattern. (The green rectangles have dimensions of 0 by 3, 1 by 3, and 2 by 3 for steps 1, 2, and 3, respectively.)

Many of my students will try to see if the entire step can be enclosed in one rectangle, then minus the negative space. The negative space (missing cubes) of this pattern is fun to discover too!

Do you see another way?

Six years ago when I created the site I had hoped to have 180 patterns to match the number of school days. We are now at 324! I'm so grateful for all the pattern submissions and for all the ways that the site gets shared.

Jelly Beans or No Jelly Beans

What's not to love about Would You Rather; I use it with my students and always recommend it as one of the great warm-up routines. This one caught my attention last week.

Each entry always includes this statement:

Whichever option you choose, justify your reasoning with mathematics.

This statement is important, especially the word "mathematics," because you might have a student who says [truthfully] that she doesn't like jelly beans or is allergic to them and will prefer to give them all away. Then, you might have another student in the same class who [untruthfully] makes the same claim to avoid having to do any maths. There's also that thoughtful student who wants to give more to his friends and keep fewer for himself. Or a student might consider giving more away to make friends. If I were to pose this question to my students, I'd mention the above possible reasonings, but then I'd add, "This question assumes that you love jelly beans and want to keep more of it for yourself. For now, it's the mathematics we're after." I didn't ask my students this question though, I asked them to choose between this question and the one on the right.

As a student, would you rather be given the problem on the LEFT (jelly beans) or the one on the RIGHT?

Using the same image above, I asked on Twitter and of my colleagues.

As a teacher, would you give your students the problem on the LEFT (jelly beans) or the one on the RIGHT?

*****

Out of my 68 sixth graders, 71% of them chose the left problem. The words they used for their reasoning:

colorful, vivid imagination, visual, more pleasant to the eyes, interesting, engaging, not boring, not basic, better reason because you like jelly beans

My fellow educators, meanwhile, overwhelmingly chose the left problem — of the 781 people who took the survey, a whopping 90% chose the left problem.

Well, I prefer the one on the right that I’d typed up. How did I get it so wrong? I'm normally not this lame. But, truth be told, I don't love the jelly beans question. At all. Maybe the one on the right is the wrong "fix" for the left one. If I could retype the problem on the right, I'd remove the equal signs since the question is just asking which one yields a larger difference, not caring exactly what each difference is.

I want to believe that anyone who spends 5 minutes with me learns that I love mathematics. I love numbers, I love math problems and don't give two shits if they are real-world either. I'm the one who loves the problem about carrying 3,000 bananas across the desert and the one about the emperor pouring oats on every other guest's head. One of my 247 all-time favorite problems is Noah's Ark, even if I grew up in a house with a Bible in every corner and found it hard to wrap my head around this story. (However, Eddie Izzard's take on the Ark — language caution — makes me laugh.) I love problems that are simply stated, yet they beg you to savor your perseverance as you think deeply and creatively. There's great joy in solving a good problem, especially the ones that at first blush, you weren't even sure how to begin.

The jelly beans problem above is not one of these problem-solving tasks. It asks for a mathematical justification, so I'm going to assume that the mathematics is finding the difference or another arithmetic operation. I see it as a number talks problem. Students get to share their strategies. Mine might include:

364 minus 188... I'd need 12 more to go from 188 to 200, then 164 more to get to 364, so the difference is 176. Similarly, to do 281 minus 137, I'd need 63 more and 81 more, or 144. Problem A has a bigger difference of 176. Comparing the two totals, 364 and 281, the difference is 19 plus 64, or 83. While the difference between the give-away quantities 188 and 137 is 51. I start out with 83 more in problem A, but I only have to give away 51 more, so problem A leaves me with more to keep.

That's why I prefer the one on the right. Numbers are beautiful. I want students to focus on the numbers and play with them, learn to regroup, try massaging them and making them flexible, be comfortable with numbers. Math is badass, so let's do maths for maths' sake. I feel protective of numbers and don't see why they need to be dressed up in colors or dunked in forced contexts. (I suddenly think of little dogs in ridiculous outfits that I doubt if anyone asked for the dogs' permission.)

Nine out of ten of you disagreed with me. That's okay because I can make phở better anybody. But guess what though? My own 23 and 25-year-old kids chose the one on the right. This fact was comforting! Sabrina (23):

If you were one of the survey respondents, I thank you thank you thank you.