Surface Area with TricubesTask 193 ... Years 4  10SummaryA task designed to clarify the meanings of area, surface area and volume while at the same time exploring the spatial skill of representing 3D objects using isometric paper. The basic shape is one Tricube. Its volume is constant at three unit cubes. Its surface area is constant at 14 unit squares. But its base area will depend on how it is placed on the table. One possibility produces a remarkable base area. All challenges are solved using exactly four tricubes. 
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IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 
If the students can interpret the first diagram on the card. They are on the way. To test their comprehension, and prepare them for the skill of isometric drawing, you might ask them to stand one Tricube on the table in a different way and draw it on isometric paper. Once the doughnut in Question 1 is created there is an opportunity to ask the mathematician's question Can I check it another way?.
Modelling these question often enough for the students to ask them for themselves leads to growing confidence through 'knowing' you are correct. (And much less of Is that right Miss?)

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. 
This is a great task to turn into a whole class investigation, but you must have a class set of Tricubes. Basically the lesson follows the structure of the card, although you might explore 2Tricube objects before tackling the Challenges. Doing so provides extended opportunity for discussion and development of concepts, developing awareness of counting strategies, refinement of isometric drawing skills and activation of the spatial perception required to tackle the Challenges.
Use the lesson to develop the 'check it another way' approach above, isometric drawing skills and measurement concepts. Then in the next lesson students will tackle the 'surface area' chapter exercises with vigour and comment on how easy they are.
These work samples are from Year 8, Wade High School, Griffith. Addtional ideas are:

Is it in Maths With Attitude?Maths With Attitude is a set of handson learning kits available from Years 310 which structure the use of tasks and whole class investigations into a week by week planner. 
The Surface Area with Tricubes task is an integral part of:
The Surface Area with Tricubes lesson is not in any Maths With Attitude kit but it can be used to enhance either of the kits which include the task. 