Time For Tiling

Task 140 ... Years 4 - 8

Summary

Sue is planning to build a square courtyard with dark and light tiles so that the main diagonals are dark. She realises that the patterns will be subtly different depending on whether she uses an even or odd side square. The odd ones always have one central tile. Her challenge becomes:

If I tell you any size square courtyard, can you tell me the number of dark and light tiles needed to build it.

Materials

2 x 20 counters in two colours

Content

properties of odd and even numbers
basic number skills
seeking & seeing patterns
generalisation
equivalent algebraic expressions
symbolic representation
substituting into equations
solving equations
graphing ordered pairs

Iceberg

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

The answers to the questions on the card can be found by making all or part of the model courtyard and are:

7 x 7 needs 13 dark tiles and 36 light tiles.
9 x 9 needs 17 dark tiles and 64 light tiles.
11 x 11 needs 21 dark tiles and 100 light tiles.

From the number patterns it could be predicted that if n is odd:

n x n needs [n + (n - 1)] dark tiles and (n - 1)² light tiles.

This prediction would only be correct if [n + (n - 1) + (n - 1)²] simplifies to n².

Looking at even squares in a similar way, ie: by construction, gives:

6 x 6 needs 12 dark tiles and 24 light tiles.
8 x 8 needs 16 dark tiles and 48 light tiles.
10 x 10 needs 20 dark tiles and 80 light tiles.

From the number patterns it could be predicted that if n is even:

n x n needs (n + n) dark tiles and [(n - 1)² - 1] light tiles (or n[n - 2]).

This prediction would only be correct if [n + n + (n - 1)² - 1] simplifies to n².

Extensions

So far this is only playing with numbers and patterns. Can we reconcile these generalisations with a visual explanation? The hint is in the dark tiles in each case.

The diagonal always has the same number of tiles as the side of the square - it's not the same length, but it does have the same number of tiles.

In an odd square this number can be counted by doubling the diagonal and subtracting 1 because the centre tile is counted twice. So, it is counted as either (2n - 1) or n + (n - 1).
In an even square this number can be counted by simply doubling the diagonal because there is no centre tile. So, it is counted as either 2n or n + n.

In either case the number of light tiles can be calculated by subtracting the diagonals from the square number representing the total number of tiles.

But the mathematics asks: Can I check it another way?. In this context, can we see the problem another way? The answer is always yes and one way to see this problem if from the point of view of the light tiles. They are always in four clusters.

The number of tiles in the outside line of a cluster (vertical line in the case of the yellow clusters) is always two less than n because the end ones are part of the dark diagonal. Each row as you move toward the centre is then 2 less than the previous one, also because of the diagonals.

In an odd square the light tiles can then be counted by calculating:
4 x (1 + 3 + 5 + ... + [n - 2]).
In an even square the light tiles can then be counted by calculating:
4 x (2 + 4 + 6 + ... + [n - 2]).

In either case the number of dark tiles can be calculated by subtracting the number of light tiles from the square number representing the total number of tiles.

Note: If you need to explore ways of summing series such as those above, then Task 51, Staircase, explores that challenge by making towers of blocks to represent each number and placing them side by side to make a staircase.

Further extensions to this problem include:

Substitution
If I tell you the size of a square, can you tell me the number of dark and light tiles?
Solving
If I tell you the number of dark tiles (or light tiles) can you tell me the size of the square?
Graphing
Size of square against dark tiles (or light tiles).
Designing
A different courtyard tiling pattern and exploring it.

These extensions could be the basis of an Investigation Guide.

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.

In preparation for this investigation you will need to make up sandwich bags of around 20 tiles or blocks in each of two colours. Alternatively, Poly Plug can be used, as indicated in the link below. You will also need some graph paper. Establish the initial story shell and encourage exploration. As students construct and count particular squares, gather the data on the whiteboard. Follow through the investigation guided by the information above.

At this stage, Time For Tiling does not have a matching lesson on Maths300.

Visit Time For Tiling on Poly Plug & Tasks.

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 Time For Tiling task is an integral part of:

MWA Pattern & Algebra Years 5 & 6

MWA Pattern & Algebra Years 7 & 8, which contains an extensive Investigation Guide

Follow this link to Task Centre Home page.