# Smooth Edge Tiles

### Task 220 ... Years 4 - 8

#### Summary

The story shell is about creating tiled rectangular patios. Only the tiles on the border have smoothed (curved) edges. Some of these tiles have one smooth edge. The corner tiles have two. The challenge is to calculate the number of each tile - 0, 1, or 2 smooth edges - needed for a given size patio. This task is a perfect partner for Garden Beds, 4 Arm Shapes, Making Monuments and several others.

#### Materials

• 25 tiles
• 9 with zero marked edges
• 12 with one marked edge
• 4 with two L-shaped marked edges

#### Content

• algebra, concept of a variable / function
• algebra, generalisation in words & symbols
• arithmetic, multiplication / division
• equations, substitution & solution
• factors, multiples & primes
• mental arithmetic
• multiplication, array concept
• multiplication, calculations / times tables
• patterns, number
• patterns, visual #### Iceberg

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

The thinking in this task involves realising that:

• The L-shape tiles are always in the corner, so there will always be four of them.
• Further, because the L-shape tiles take up the corners there will be two less single smooth edge tiles on each length and width.
• The tiles with zero smooth edges will always form a rectangle in the middle that is 2 less than each of the length and the width of the patio.
So, for an M by N patio:
• 2 smooth edges: 4
• 1 smooth edge: 2(M - 2) + 2(N - 2)
• 0 smooth edges: (M - 2) x (N - 2)
However, the development of this symbolic notation is not the objective of the task. The objective is to learn to:
• 'see' how the problem is constructed
• 'see' the construction in alternative ways if possible
• describe the general principles of construction to someone else in natural language
Then, when symbolic notation is used, to be able to give meaning to the symbols and operations in terms of the physical problem.

#### Extensions

All of Kareema's patios are rectangles.
1. If Kareema had only 14 single smooth edge tiles and all the other tiles she needed, what size patios could she build?

2. Which number(s) of single smooth edge tiles (less than or equal to 20) gives the largest number of possible patios that could be built if Kareema has all the other tiles she needs.

3. Kareema tells you she has built a patio using 36 tiles altogether. What dimensions might her patio be and, for each possibility, how many of each type of tile?

4. Which total number(s) of tiles (less than or equal to 100) can make the most different patio sizes?
Extensions 3 & 4, involve three discussion points:
• Can Kareema build a patio just one tile wide with the tiles she uses?
• Can Kareema build a patio just two tiles wide with the tiles she uses?
• Is an A x B patio different to a B x A patio?

#### 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.

With a good stock of square tiles it is possible to make bags of tiles like the ones in the task quite quickly, especially if a faculty works together. The edges can be drawn by hand with a stroke of a permanent marking pen. Alternatively students can use graph paper to explore the various patios. The focus challenge of the lesson is:

If I tell you any dimensions for a rectangular patio, can you tell me how many of each type of tile are needed to build it?

At this stage, Smooth Edge Tiles does not have a matching lesson on Maths300.

#### 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 Smooth Edge Tiles task is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 7 & 8 