Dotty Shapes

Introduction

Take a look at this activity on nRich. Can you think of a quick way of working out the area of any square drawn on a dotty grid?

Now what if the area is not a square? Try this activity. Play with the dotted grid, make your own shapes and find the areas of those shapes. You may wish to challenge your friends to finding the areas of shapes you have created.

Pick's Theorem

Let's now look at how at how a shape’s area can be determined from dots. With any luck, we will (hopefully!) discover Pick’s Theorem, discovered by George Alexander Pick in 1899.

Take a look at the shapes on the right and for each shape, work out the following:

p - how many dots are there on the perimeter?
i - how many dots are there inside the shape?
a - what is the area of the shape?

How would you organise the information in a table?

Now draw a few more shapes either on dotty paper (PRINT ME) or this pin board (you can change the grid size by clicking on the cog wheel on the right).

You may wish to check that your answers are right on geogebra

Add your findings in the table. How can you make this systematic?

Can you find a relationship between p, i and a?

Support

if you need some prompts, take a look at these:

In the task there were 2 variables you could influence – the perimeter dots (p) and the inside dots (i). The area (a) will be affected by both of these.
So try to alter only 1 variable at a time to monitor its impact on the area (if we change both at once it is hard to see how much each of them individually affects the area).
Keep the inside dots (i) fixed and only change the perimeter dots (p).
- Hint: make i = 0.
- Organise your findings systematically in a table.
- How does area change as the perimeter increase?
- Can you link "a" and "p" in a formula?
Keep the perimeter dots (p) fixed and only change the inside dots (i).
- Hint: make p = 16.
- Organise your findings systematically in a table.
- How does area change as the number of inside dots increase?
- Can you link "a" and "i" in a formula?
Can you now put the last two points together and link "a", "p" and "i" in a formula?

Further Questions and Challenges

How many different shapes can you make such that:

p = 6, i = 0 and a = 3?
p = 8, i = 1 and a = 4?
p= 10, i = 2 and a = 6?

Look at the worksheet below:

Can you find the area of each shape?
Can you find the perimeter of each shape? Note: this is NOT the same as the number of dots on the perimeter. To do this, you will need to apply something called Pythagoras Theorem which you will learn more about in Year 9.

Dotty Polygons.pdf

Further Practice

Some of the essential skills introduced in this lesson are areas of shapes such as square, triangle, parallelogram, trapezium, kite, L-shape, and composite shape. Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.

Transum

Area and Perimeter of a Rectangle - try this self-marked exercise. There are altogether 2 levels plus an investigation at the end.
Area of a Triangle - try this self-marked exercise. There are altogether 3 levels.
Area of a Trapezium - try this self-marked exercise. There are altogether 2 levels.
Area and Perimeter of a Parallelogram - try this self-marked exercise. There are altogether 6 levels.
Kite Maths Exercise - try this self-marked exercise.
Areas of Composite Shapes - try this self-marked exercise. There are altogether 5 levels.
Area and Perimeter - try this self-marked exercise. Name the shapes then find formulas to calculate the area and perimeters from the given lengths.
For more goodies on area on transum, click on the hyperlink.

Extension

For those of you who have superior algebra skill, you may wish to look at this proof for the Pick's Theorem. This is way beyond what is required of you in Year 8!

nRich

There are so many good activities on exploring mathematics on dotty grids on nRich. Below are a few:

Exploring Area - the starting activities are based on ideas here.
Appear Act - It seems we can use the same pieces to make two shapes with different areas! Can you explain why? Is this magic?
- A similar idea is this super famous missing square puzzle. Scroll down to to see videos and explanations of this.