Jane started the meeting by telling us that her class has been studying signed numbers recently. She has been looking for creative ways for them to understand adding and subtracting signed numbers. One example was to imagine taking away negativity as the same thing as making someone happier.
Jane’s students have been using number pyramids to practice adding signed numbers. Each number in the base is added to get the number in the box above. Jane handed out the worksheet and we did the first one together and everyone got 9. Then we worked independently to fill in the other pyramids. Jane then asked us to check our totals with our elbow partners. I made a mistake that Mike found. Parvoneh found one of his.
Then, Jane gave us a 10-base pyramid and asked us to complete it. When I checked my top number with Mike, I found that we had different numbers again. After some double-checking of different rows, we came to a number we agreed on.
Jane asked if anyone had made mistakes while they were doing this computation, then told us about looking over student work on a recent flight. Students had made mistakes and she was making mistakes when looking over the work. Shouldn’t there be a way to figure out what the top number is going to be, without having to add up the totals of every single box? (I love this example of a headache in search of aspirin. I quickly became frustrated with doing all the computation and wanted a faster way.)
So, that was our task, starting with the smaller pyramids and then moving on the 10-base pyramid. Is there a way to predict the number of the top box without calculating each foundation box individually?
A few related questions came up:
- How many computations are students doing to complete each triangle (3-base, 4-base, 10-base, n-base)?
- What is the best strategy for balancing out negative and positive numbers in the base so that addition problems with different signs continue up into the higher levels of the pyramid?
- Is there a pattern in the number of As, Bs, Cs… in each row as you climb the pyramid? The following image shows the number of each variable in the 1st, 2nd and 3rd rows. For example, there is one A, one B, one C… in the first row. In the second row, there is one A, two Bs, two Cs, etc. Is there a pattern as the rows are added? The B column seems to add one B each row.
We spent some time working on the main question and sharing strategies:
Jane then asked to look what could be called a powers of 2 triangle. It’s similar to Pascal’s triangle but you put 2’s on the outside rather than 1’s and multiply instead of add. Here’s an example. What do you notice?
We finished the meeting by watching a Vi Hart video on sick number games.
Oh yeah, and Jane gave us this lovely holiday problem for homefun!
Thank you to the Fortune Society for hosting and providing snacks! It was lovely to be in Parvoneh’s classroom!
Respectfully submitted by Eric
In attendance: Eric, Jane, Linda, Maggie, Michael S., Parvoneh, Solange
Programs represented: CUNY, NYCDOE, York College, BMCC, The Fortune Society
Location: The Fortune Society, 29-76 Northern Blvd., LIC, NY
Here’s a similar puzzle from Henri Piccioto:
http://www.collectedny.org/frameworkposts/number-pyramids/
And here’s a similar lesson by Dan Finkel, from Math for Love:
https://mathforlove.com/lesson/pyramid-puzzles/