Area Twice, not Thrice

This island explores creating as many shapes as possible within a square that have the same area at most twice. There are different possible variations using ‘L’ lines, straight lines and diagonal lines with fractional areas. Each presents a varying amount of challenge.

Starting with a 6×6 square, draw a vertical and a horizontal line that share a point with the other being on the side of the square. Essentially, draw an ‘L’.

Each time a line is added, new shapes are created inside the square. In the case of above, one has an area of six and the other, 30. How many lines can be added and how many shapes created whilst the same area appears at most twice?

‘L’ lines can share a point and intersect but cannot follow the same path. Above we have six shapes with the areas: 3, 3, 6, 6, 9 and 9.

We can still add another line though:

This line has created three new shapes to make nine shapes. We now have areas of 1, 2, 3, 4, 4, 5, 5, 6 and 6.

Is it possible to beat this score? Is it possible to create five shapes and then be required to stop? How many different scores are possible?

What if we used grids of different sizes?

Is there a pattern between the number of shapes/lines possible and the size of the grid?

What if our grids were not square?

When investigating, I would always use the same colour for the first line, then the same (but different) colour for the second line etc. This will ensure that solutions can be replicated.

I have created templates of blank grids if you wish to use them.

Templates of blank grids

What if we used straight lines?

This is a simpler alternative to using ‘L’ lines.

In the above example, I have managed four lines which makes nine shapes.

If I try and add any more vertical lines, I will have three shapes with an area of one.

If I try and any more horizontal lines, I will have three shapes with an area of two.

Is there a way of having more than four lines? What if we tried with grids of other sizes?

I’ve got an example of using a 10×10 square. Things get trickier here. I have come up with the below example which might seem like a good solution and that six lines are possible with 16 shapes.

However, there are three rectangles with an area of four as shown below. This is not allowed. These bigger grids create a bigger focus on how different shapes can have the same area.

Whilst the distributing the lines evenly horizontally and vertically seems aesthetically pleasing, it is not generally the optimal solution.

The children have to be more creative than that. Six lines is attainable with a ten-by-ten grid but using 4 horizontal lines and 2 vertical lines (or the other way around).

Can a pattern be found between the grid size and the number of lines and sections that are possible?

What if we used 45-degree straight lines?

A more complex variation of using straight lines that can have fractional areas. I have used a 3×3 square as the starting point here.

Children’s possible use of reasoning skills:

Search
Search inquiry reasoning skill in maths

The placement of the lines is the crucial use of Search skills. At first, it seems like you can just place them anywhere and it will be fine. It is surprising how quickly it becomes challenging. I tended to find that areas of one, two and three became difficult to avoid replicating. This is particularly true with smaller grids.

Organise

A high score table for different-sized grids for the class might allow inconsistent scores to be challenged and conjectures made.

It might be though that you just focus on one grid size and see how many scores are possible for it.

As mentioned above, to replicate results, colour coding the order that the lines go in will help you.

Discover

Lots of unknowns to explore and discover patterns for. With the different grid sizes easily investigable, conjectures can be made on what is possible before they are explored.

With these types of challenges, it is better to be focused on smaller grid sizes than spend ages on a massive grid.

They can then conjecture on what strategy they think will work best and investigate it.

Investigate
Investigate inquiry reasoning skill in maths

Children need to have a plan, a strategy to investigate. After they have had the opportunity to have a few goes, get them to investigate each others’ strategies for the placement of lines. Then they can be Investigated with different grid sizes.

Argue

There is a lot of potential here. The children can think carefully about why certain scores are possible and why certain ones might be challenging. This is particularly true when thinking about the maximum size.

Explore

This has so much potential for further exploration. I have given a couple of examples but changing the shape to no longer be a rectangle also holds lots of opportunities.

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