Why do this problem?
This low threshold high ceiling
task is designed to help students develop their estimation skills. Estimation is a hugely useful and important skill in all walks of life. This playful context provides an engaging hook in which to explore the big ideas of approximation and large numbers.
Show the images of the jar of teddies and the single purple teddy. (Before inviting students to estimate the number of teddies, you might simply ask "What do you see?" and allow some time to share thoughts.)
"Estimate how many teddies there are in the jar."
Give a few minutes of individual thinking time before encouraging students to share their thoughts with a partner, then with the wider group. Here are some examples of the sorts of methods that might emerge:
"There are six different colours of teddy. I think there are around 8 to 10 of each colour. So there's between 48 and 60 teddies altogether."
"There are about seven teddies visible on the top layer, and in the side view I counted around seven layers, so there must be about 50 teddies altogether."
"I can see about fourteen teddies in the side view. There are about four layers of teddies going back, so there must be around $14 \times 4 = 56$ teddies altogether."
"A teddy would fit in a 4cm by 4cm by 2cm space, which has a volume of 32 cubic centimetres. The jar is around 10 by 10 by 16 cm, which is 1600 cubic centimetres. 1600 divided by 32 is 50, so there are around 50 teddies."
Reveal that there are in fact 55 teddies in the jar and give students a few minutes to reflect on their estimations. You could reassure students that they are not expected to have got the answer exactly right and suggest that answers between 50 and 60 are reasonable estimates. Estimating from a photo is also quite tricky.
"The bottom of the jar is 10cm square, and the jar is filled to a height of 16cm.
If we wanted to design a jar that would hold 100 teddies, what dimensions could it have?"
Allow time for students to work in pairs, encouraging them to be prepared to share their reasoning, not just their answer, in due course.
In a mini plenary, invite students to share their ways of working and resulting answers. They are likely to have decided that, as there are approximately twice the number of bears now, a jar with twice the volume is needed. There is a variety of ways that this could be done. Watch out for the misconception that doubling all the dimensions will double the volume.
Show the picture of the larger teddy and share his dimensions: 16cm tall, 19cm wide and 12cm from front to back. Pose the question, "How would we work out how many teddies like this would fit inside our classroom?" (You may wish to divide this into the two subquestions "What do you need to know?" and "How will you work it out?".) Again, allow some time for thinking and discussion. This is
an opportunity for estimating the classroom dimensions and perhaps some practical measuring. Once the dimensions are known, students put their plan into action to work out their estimate, perhaps using calculators.
In a final plenary, discuss the assumptions that have been made and whether they are likely to lead to an overestimate or an underestimate.
Can you use the ruler in the picture to work out any useful measurements?
Does it help to work out the approximate volumes?
Can you tell me about your method?
The task could be tweaked to include items that students could physically investigate, rather than having to work from a photo.
Ask students, "What other mathematical questions could you explore?" and encouarge them to choose one or more to investigate.