Building Views

Task 104 ... Years 4 - 8

Summary

Students are invited to build a set of rooms given the front and side views of how they are arranged. There is more than one answer and the challenge is to find the smallest number of rooms to suit the conditions.

Materials

25 wooden cubes

Content

2D/3D spatial ideas - plan views
spatial/visual reasoning
Developing a range of problem solving strategies
Combination - counting theory

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

There are many solutions to each challenge, but interest in each case is directed towards the minimum as shown in these solutions.

Questions 1 & 2

Questions 3 & 4

An extension of the problem is for Student A to secretly build a set of rooms and record their creation as a set of numbers on the grid as above. This 'top view' is given to Student B to recreate the building. Now the partners work on it together. Firstly they draw the grid and the front and side views in their journals. Then they investigate two questions:

How many more rooms can be added so that we still have the same front and side views?
How many rooms can be removed from this maximum so that we still have the same front and side views?

Another interesting question is:

How many variations are there on the minimum?

Whole Class Investigation
Tasks 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.
To run this as a whole class lesson you will need lots of cubes. You can purchase wooden cubes as a Task Centre support resource. Scroll down this link until you see 2cm wooden cubes. You will also need a 4 x 4 grid to match the size of your blocks.
Start with the problems on the card. They're easy to draw up on a computer and, for example, display on your Electronic Whiteboard. Each group first builds from the front and side views and is then challenged to find the minimum in each case. That will generate plenty of discussion.
Follow with a grid of numbers of your own.

I built one of these sets of rooms last night and recorded it like this. What do you think it means?

Groups then recreate your building and compare with the front and side views you also recorded. The challenge now is to find the maximum and minimum for your example. Again lots of discussion as groups try to convince each other. Extend from there as described above.
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 163, Building Views, which includes companion software.

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.
Building Views is task and lesson are only included in Points of View: Representing 3D Objects in 2D, a kit supporting the exploration of a Mixed Media teaching strategy. This kit also includes Building Views software extracted from Maths300. The kit is available to Members and non-Members of Maths300 alike.

From The Classroom

Maths300 members should note the following discovery sent to us by Michelle Patterson and answered by Charles Lovitt.

Dear Charles,
A class of 5/6 students have been very engaged in a Maths300 lesson 'Building Views'.
After a couple of lessons, the teacher began to correct what students had completed and we came across some answers that did not seem to match those answers in the lesson plan.
For No. 3 (Investigation Sheet 2) we got a maximum of 17, not 19 and also got more variations for No. 4. We were wondering if you had top view answers that we could have a look at to see where we went wrong?? Answers: Investigation Sheet 2 (as appeared in the lesson plan)
Max = 20, Min = 6, Variations = 4 Max = 21, Min = 11, Variations = 37 Max = 19, Min = 7, Variations = 5 Max = 22, Min = 8, Variations = 4
Any assistance you could provide would be greatly appreciated, as we have some pretty determined students and now a frustrated teacher!!
Thank you for your time.
Michelle

Hi Michelle,
Telegram to wonderful Grade 5/6 - stop!
Hiring your class as consultant mathematicians to our project! - stop!
Answer to Building Views Qu. 3 - definitely a maximum of 17 - not 19 - stop!
Answer to Variations in Qu. 4 - I figure it to be 42 - 7 ways of placing the two single towers - and with each of these are 6 ways to place the three towers of 2 - stop!
Thinking of excuses

put wrong answers in deliberately - stop!

'twas a typing error - stop!

Real conclusion - we are not as clever as we thought hence need your help!- stop!
Sincere congratulations for being excellent mathematicians! - and having the courage to point out our mistakes - you are the first to point out these particular mistakes to us.
I did hastily work out the 42 variations for Qu. 4 - I'd love you to double check my logic and let me know if you agree?
With best wishes,
Charles

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