How many balls?

Personal Reflection:
I saw this image recently and, I believe, the question was, "How many balls are there?"  There were a ton of responses, most of which made little to no sense, but two of which I felt made a lot of sense.  And no one seemed to acknowledge that the way this image was created...was it possible to really know the answer?

Grade Level: 6-7

Course: Algebra

Standards:  6.G.4, 7.G.3, 7.G.6, G-GMD.4
SMP: SMP.2, SMP.3
Skills: Geometry, spatial reasoning, number sense, analysis


How to use this as a mad minute:
You have 60 seconds.

Oliver thinks that this stack of balls has 20 balls in it.  Olive is sure that there are 30.  Who do you think is right?  Why?  Justify your solution.

How to use this as a warm up:
You could ask the students to consider one of the following:
1.  Name all the shapes you can see in this picture.
2.  How many balls do you think there are in this stack?  Why?
3.  Can you see a different possible solution?  Where?  How?
4.  How many faces does this 3-dimensional shape have?

How to use this as a mini-lesson:
With 20 minutes:
0:00  Show the photo.  Give students 2 minutes to work independently on trying to figure out how many balls there are in the stack.
2:00  Ask students how they would name the stack of balls.  Is it a pyramid?  What do they know about pyramids?  Have them discuss with a partner and brainstorm what they know.
3:00  Have students share out.  Record their thinking.  Do all pyramids have square bases?  What would other pyramids look like?
6:00  Have students work in partners.  Do they agree on the number of balls in the pyramid?  If so, can they prove it?  If not, can they explain their thinking to the other?
10:00 Select students with different solution strategies to present.  Ask them to explain their thinking to the class.  Ask the class members to prepare either an observation or a question for the presenter.  Select a student who "guessed", a student who "counted" in a somewhat random way, a student who saw the base of the pyramid as a triangle, and a student who saw the base of a pyramid as a square to share out.
15:00  Ask students to dicsuss whose method made the most sense and why.  Focus on the fact that we cannot see the shape of the base, so we have to assume, so both answers could be right.
17:00  Ask students to add a layer to the pyramid.  How many would there be if it was a triangular base?  How many if it were square?  Give them work time before sharing out.

How to use this as a full lesson?
I'm not sure this has enough opportunity for depth to be a full lesson.  However, you could definitely extend the last 5 minutes of the "mini lesson" to have students explore the patterns in the "layers" of the two different pyramids and then challenge them to find the number of balls in a 21 layer pyramid, etc.  You could also ask students to try to write a rule for the number of balls in any sized pyramid.

How to use this as an assessment?
I would not use this as an assessment.

Please feel free to use any of these ideas and modify them to meet your needs.  However, please acknowledge the original source of the items and my own lesson outlines.  ©NatalieRSprigg 2013