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Lessons From the Classroom
Venturing Into the Sequel of Penny Pyramid
I didn't get around to doing Penny Pyramid when I first saw it last year. But Dan's 3-post series and Nathan's recent mention of it were the reminders I needed to make it happen.
Act 1
how many pennies
how much money is that
how long did it take
who in their right mind would do this/who has that patience
how much does it weigh/is the table gonna collapse
what is the volume/surface area/height
what is the ratio of pennies from one level to the level above it
(Student who gave the highest high guess did correctly say her written number as "one hundred quadrillion." It made me happy that she knew this.)
Acts 2 and 3
Lauren F: Is there a way to multiply consecutive numbers quickly? You showed us the addition one...
Maddie: Isn't that the exclamation point operation?
Gabe: But we're not multiplying consecutive numbers!
Mia: Doing 40 by 40 then by 13 gives all huge numbers, so we're doing a simpler problem, then find an equation.
Lauren P: Our group is finding a pattern and making a table.
Gwen: We're doing layer by layer. There are more of us (4 instead of 3), so it's pretty quick to divide up the work.
Gabe: I already have the answer because I was too eager to do the math, but I didn't say anything to the group. (He got the answer about 2 minutes after we formed groups.)
Julia: And I got it 3 or 4 minutes after Gabe.
Angela: And I got the answer after Julia. Without her help.
Me [to Gabe, Julia, and Angela who were in same group]: Aren't you guys special. You seriously just sat there and did nothing then while I walked around?
Julia: Well, yeah, we're kinda admiring our work.
Me: Geez Louise. What do you think I'd have asked you if I knew you'd found the answer to this pyramid?
Gabe: If it was 100 high?
Me: No. A million high. A billion high.
Gabe: Hehe. That's why we didn't want to say that we're done.
Two students figured out why each stack had 13 pennies.
Their other questions were answered to their satisfaction, except we didn't know exactly how long it took Mr. Bezos to build it, but we talked about how we might be able to estimate this.
Kids remembered from last week's lesson that a square pyramid has 1/3 the volume of a cube with same dimensions, but that our penny pyramid had jagged lateral edges.
While everything up to this point had gone as well as I'd expected. Kids immediately responded to the video with WOAHs and WOWs. They asked solid questions in both Acts 1 and 2. They worked well in groups.
However, the kids and I knew that no one really struggled with the task of just finding the number of pennies. The math was pretty basic and with a calculator, 40 layers of pennies didn't make anyone break a sweat.
What was meant as an "extension" or "sequel" really needed to now become the focus of our lesson — at least for this group of students who valued a good struggle. We needed to try to figure out the equation for this penny pyramid.
But I also realized that it would be very unlikely for my 8th graders to come up with the equation because it involved summation of a sequence. (You're right, Nathan, it is unlikely, even for Gabe.) But the process of getting there might be worth it. I wouldn't be their teacher if I didn't ask them to explore the patterns that they might see along the way.
I gave them small interlocking cubes and colored chips so they could build smaller models of the pyramid.
Their collective frustration arose from how "simple" the pyramid was built — nothing more than a sum of layers whose square dimensions were consecutive.
Incomplete Cube
We started with a smaller problem. We did a 5 x 5 square pyramid with a height of 5. We didn't like the "jagged" lateral edges of the pyramid either, hence we pushed the cubes into one corner like this so at least the cubes stacked squarely.
One way would be to imagine that we had a whole 5 x 5 x 5 cube, then subtract from this the small cubes that were missing. We noticed the missing pieces were these L-shapes.
We see a pattern in these missing L-shapes:
4 pieces of (2n-1) or (n-1)(2n-1) or (2n^2)-3n+1
3 pieces of [2(n-1)-1] or (n-2)(2n-3) or (2n^2)-7n+6
2 pieces of [2(n-2)-1] or (n-3)(2n-5) or (2n^2)-11n+15
1 piece of [2(n-3)-1] or (n-4)(2n-7) or (2n^2)-15n+28
Incomplete Rectangle
How else can we see this pyramid? Because my mind has a tendency to reshape things into rectangles, I flattened the pyramid into an incomplete rectangle like this:
The dimensions of the rectangle were straightforward enough, and unlike the missing L-shapes of the incomplete cube, the missing pieces here were rectangular and came in pairs. For example, in the above right sketch, the missing pieces were two 1 x 4 and two 2 x 3 rectangles. But if n were even, then the number of missing pieces would be pairs of rectangles plus 1 lone square piece.
I talked with them about the sigma notation, and since they knew how to add {1 + 2 +... + n} quickly — we refer to this as "Gauss addition" in class — they thought it was fun to learn the new symbol.
Then we went into WolframAlpha and typed in what we wanted. The equation came up with the "newly" learned summation notation.
The kids saw patterns. They learned a fancy new sign. They knew that the right math could help solve for any penny pyramid. But I really think they look forward to learning more math in high school.
I'd like to feature this comment from the old blog:
May 19, 2013 2:21 PM
l hodge wrote:
If you draw two copies of the rectangle sketch mirroring each other, with a 1 unit space between them, you have a nice sum of squares proof. The space between the two copies is easily seen as a re-arranged sum of squares. Divide the area of expanded rectangle by 3 and you have your formula.
May 20, 2013 2:41 PMfawnnguyen wrote:
Thank you, l hodge! Mind blown. So happy to know that we were on the right track of flattening out the pyramid into an incomplete triangle. We did make another copy of the flattened pyramid but turned it around (1800 rotation) to look at that double-pyramid-with-extra-spaces rectangle, but time ran out. So, we drew this together in class today. So #nguyening!!
Circles Galore
This lesson is from MARS. Using graphics from the original source, I created this recording sheet.
But instead of passing out the worksheet and starting right in like I had intended, I placed the stack in my desk drawer and decided to ask the kids to follow my instructions to create the diagrams.
I gave each kid one sheet of unlined paper and a compass. These were my instructions to them, and I followed along also.
Put a dot in middle of paper.
Draw a large circle using the dot as its center.
Draw in the diameter.
On this diameter, mark half the radius. (I could ask them to bisect the radius, but to save time, we just estimated "half" using a ruler.)
Now draw a circle, centered at this half-radius mark, using a radius half that of the original circle, draw two adjacent circles along the diameter.
Shade in the outside of the two small circles — their drawings should now look like Stage 1 on worksheet.
We stopped here, and I asked Question 1) What fraction of the large circle is shaded? They quietly did their thing.
After seeing about half of the kids finished answering this question. I said, "I know not everyone is done with #1, which is perfectly okay, but I'm going on with my instructions because some people are done, and I need you to follow along. Then if you need to, you can go back to #1."
I repeated directions 4 and 5 above, and before I asked Question 2, Josh said, "What fraction is shaded now?" They went to work again.
I then repeated the instructions to get their drawings to Stage 3. However, our compass can't make a circle smaller than 0.5-inch radius, so we freehanded them in. Everyone knew next came Question 3) Now, what fraction is shaded?
And Josh immediately asked, "Are we doing this again, make even smaller circles?!" We drew in the next 16 shaded circles for Stage 4.
You guessed it. Question 4) What fraction of the large circle is shaded? I asked them to work alone on the questions, but that they would have a chance to discuss with other classmates during the last 15 minutes of class. They plugged away.
Then the talking began; and there was a lot of it, so I started recording their conversations. I was more interested in those who did not get 1/2 for Question 1. I happened upon Zach telling his two classmates that the answer was 2/pi:
Did you catch what Zach said? He reasoned that (16*pi) + (16*pi) = 2*pi*32 by saying "You can't get rid of stuff because we haven't done any mathematical operation to get rid of pi." I was so glad this came up! Sometimes I feel a wrong answer is just as important as the correct one.
(But I thought it was funny too how Slater reacted, "That's wrong." Then there was peer pressure, "He made me change my answer." And there was alleged copying. Then there was, "Taj showed me the way." I will miss this class.)
Daniel started the discussion with this group.
The "formula" that Daniel had trouble articulating was never written on his paper, it was all in his head. Then I noticed that he'd carried his calculations to stage 8! I was so pleasantly surprised by this that I was speechless.
Class ended and everyone agreed that the answer to question 1 was 1/2.
I passed out the worksheet above for homework, asking them to try to figure out the answers for stages 2 through 4 if they hadn't already, then to give stage n a try if they had time. Daniel never saw the worksheet to know that I'd wanted him to keep thinking of the pattern, but he was already heading there on his own.
Tomorrow we'll finish out this lesson, but I wanted to capture what we've had so far before my amnesia sets in.
Some thoughts:
When I search for lessons/activities for my kids to do, I always search for lessons "recommended" for kids at least one grade level above mine. Meaning, I look under 9th and 10th grade activities for my 8th graders. Why not? I could always work back from there.
I remember adding "stage n" to the worksheet just because there was some left over space at the bottom. This made Daniel happy.
I'm glad I had them draw the circles. This was a last minute decision that paid off — they made pretty sketches!
I learn that my kids are smarter than I am. (But let's not tell them this.) Daniel's "equation" was so elegant that he could do all of it in his head.
Stage n, if we get it or not tomorrow, is icing on the cake. I feel pretty full.
Speaking of stage n. What do you think? Is there a limit that we are approaching?