Task 101 ... Years 2 - 10
SummaryThe 3D spatial challenge that is the beginning of the task can be tried by students of any age. Solving it suggests a number pattern in the number of spheres at each level and that too can be accessed by quite young children. Seeing the pattern leads to predicting based on it and that's where things start to get more difficult. It's not so much predicting the number of balls in any given layer of a growing pyramid that is challenging, rather, it is predicting the total number of balls needed to make a pyramid of that size. Further, asking the question:
This cameo has a From The Classroom section which derives from a teacher connecting two pieces of mathematics, one of which includes the work of two Year 10 students, and coming up with a new hypothesis. It also confirms how deeply embedded the process of working like a mathematician is in this school.
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
Hint: Give a paper plate to each student along with the puzzle. This will (a) provide a non-slip surface and (b) make it almost impossible for the longer pieces to roll onto a hard floor and perhaps snap.
If these are the pieces:
The solution can be derived from these diagrams:
Or, for those who prefer an oral explanation, Side View A gives a clue:
Students can work out the answers to Question 2 either by counting as they manipulate their pyramid, or by using the clues in the Side View on the card.
All faces of a regular tetrahedron are identical, so the Side View is also the Base View. This realisation helps work out a 5 layer tetrahedron. Continuing from above gives:
And continuing the pattern to a 10 layer tetrahedron gives:
Extension AOne direction for the iceberg of this task begins with the question:
There are several ways to calculate:
A more challenging iceberg question is:
The clue to achieving this calculation is in recognising that each pair of consecutive Triangle Numbers makes a square:
It seems that to find the sum of the Triangle Numbers it will first be necessary to sum the Square Numbers. So another new challenge arises. Not necessarily Year 12 stuff since Greek mathematicians were able to derive a formula using spatial reasoning, but definitely open again to checking the solution by Mathematical Induction.
Extension BAnother direction for the iceberg begins with the question:
Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
It is sometimes remarkable how much more mathematics students see when their hands are holding the object of mathematical inquiry. You may be tempted to generate a whole class investigation of this problem based on it being used by various groups in previous task sessions, or perhaps even from drawings, but there is no real substitute for a class set of Pyramid Puzzles. Unless, perhaps, you have access to a snooker triangle and enough balls to build a pyramid above the balls usually supplied.
The general outline of the whole class investigation is above. More detail, including formulas for the sums of Natural Numbers, Square Numbers and Triangle Numbers, and proofs by Mathematical Induction can be found in the companion Maths300 lesson.
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 138, Pyramid Puzzle & Other Algebra Investigations.
Is it in Maths With Attitude?Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Pyramid Puzzle task is an integral part of:
The Pyramid Puzzle lesson is an integral part of::
From The Classroom
MacKillop College, Swan HillDamian Howison
For example, the actual pieces (arrays of spheres) that come with Pyramid Puzzle (representing the first four days of the song) are:Now Damian, it's one thing to see the pattern, and even recognise that the total of spheres is the number needed to build a Size 5 pyramid, but do you mean that these five pieces will actually fit together to make a Size 5 pyramid?!?
So could the 5-layer pyramid be made of the 5 pieces based on the song?
- 4 partridges - 4 days x 1 partridge
- 6 turtle doves - 3 days x 2 doves
- 6 french hens - 2 days x 3 hens
- 4 colly birds - 1 day x 4 birds
If so, in this way you can see the pyramid as:
- 5 partridges - 5 days x 1 partridge
- 8 turtle doves - 4 days x 2 doves
- 9 french hens - 3 days x 3 hens
- 8 colly birds - 2 days x 4 birds
- 5 gold rings - 1 day x 5 rings
I've known those patterns for some time but never connected the second pattern to the song itself. The Pyramid Puzzle always seems to have another surprise in store!
- an accumulation of days with triangular numbers of gifts, and
- an accumulation of gifts with rectangular numbers for each type of gift.
Actually that's exactly what he did mean, as explained in a following email with photos attached:
Morning Doug,So the hypothesis is that for a Size N pyramid (tetrahedron) a construction puzzle can be created by choosing pieces using the 12 Days of Christmas song up to the Nth day. The pieces are then also placed according to the song beginning with the Nth day, ie: 1 x N spheres (x = 'rows of'). The proposition makes sense because:
Actually I had a pair of year 10 students try out the 5-layer pyramid last year as part of their investigation in our algebra replacement unit. So I already have the polystyrene pieces! I've photographed them just now.
I had intended to send some photos to you and was hoping that some students pick up on it again this year because the task was a very popular choice in the kit this year. But none of the groups wanted to go that way with their work. Anyway, here are some photos from some pieces made by a pair of Hannahs. Actually I just traced back to last year's reports and found their work. One of the reports also contains some good photos.