Activity: Drawing SquaresFor this activity all you need is a grid of dots, a pencil and your brain.Let us discover how many squares you can make on a:

Note:"1 by 1" means how manysides(not how many dots).

So, let's try drawing in some squares and count how many:1by 1Well, that's easy, there's just one:

2by 2That seems to be easy too. There are four of them, aren't there?

But wait, that's not the complete answer.There's also this bigger one:

That makes five squares altogether - four 1 by 1 squares and one 2 by 2 squareYour Turn !3by 3Over to you now.Here's the grid:

Hint:For the 3 by 3 case, you will expect to get 1 by 1 squares, 2 by 2 squares and 3 by 3 squares. How many of each?Now you can start to fill in a table:How Many1 by 1 squaresHow Many2by 2 squaresHow Many3by 3 squaresHow Many4by 4 squaresHow Many5by 5 squaresTotal

1by 1 Grid:11

2by 2 Grid:415

3by 3 Grid:

4by 4 Grid:

5by 5 Grid:

Did you notice anything about the numbers in the table?They are allsquare numbers: 12=1, 22=4, 32=9, etc ...and the totals are found by adding together square numbers.Formula to The Rescue ... !There is actually aformulafor adding the firstnsquare numbers:Sn= n(n+1)(2n+1) / 6Example: The number of squares in the5 by 5caseTrysubstitutingn = 5into the formula:Sn= n(n+1)(2n+1) / 6S5= 5 (5+1) (25+1) / 6S5= 5 6 11 / 6S5= 55So, we seem to have solved the question. Yipee!But wait ... there's more!I said you would need to use your brains. Let's go back to the 2 by 2 case:2by 2There is another square too, this one:

Why is it asquare?It has four equal sides and four right angles, so that's a square.So, that makes six squares altogether.Four 1 by 1 squares, one 2 by 2 square and onex by xsquare.What is the value ofx? We can usePythagoras' Theoremto find it:x2= 12+ 12= 1 + 1 = 2So x = 2So, we havefour 1 by 1 squares,one 2 by 2 squareandone2 by2 square.Your Turn !3 by 3 Are there any more squares?YES! Can you find them?4 by 4 and 5 by 5Also try the 4 by 4 grid, and the 5 by 5 gridAs you proceed, you will find squares like these:

What are the lengths of the sides of these squares?You can usePythagoras' Theoremto work that out yourselfIn each case, how many do you get of each one?Here is a table to help you:How Many1 by 1How Many2 by 2How Many3 by 3How Many4 by 4How Many5 by 5How Many2by2How Many5 by5How Many8by 8How Many10 by10How Many13 by13How Many17 by17Total

1by1Grid:11

2by 2 Grid:4116

3by 3 Grid:

4by 4 Grid:

5by 5 Grid:

AdvancedCan you find a formula to calculate the number of squares that have lengths that are square roots?Can you then find a formula for the total number of squares in each case?Are there any more squares lurking in there that we've missed?ConclusionWhat started off as seeming to be a simple exercise turned out to be quite complex. You reallydoneed to use your brains to think this one through, but it's a challenging and rewarding exercise.Activity: Garden AreaHave you ever wondered what the area of your garden is?Let us try and find out!

You will need a garden, a tape measure, pen and paper ... and your brains.I don't have a garden, so what can I do?If you don't have a garden, I'm sure a friend has one, or your relatives have one, so use theirs.

How accurate should my measurements be?Try to measure to the nearest centimeter (or half-inch), so the error will be as small as possible.You should get a good estimate, as long as you are careful with your measuring.Is there a simple way to find the area of my garden?If your garden is a rectangle, then you have a simple calculation. You just have to measure its width and length and multiply them together:Rectangle: Area = W L W = width L = length

But that makes this activity just too easy ...... so go find another garden with a more interesting shape!My garden is a difficult shape, so how can I find its area?Good! This activity just gotinteresting...It may be one of the shapes on the pageArea of Plane Shapes, then you just have to decide which shape, make the measurements, and use the formula.But you could also break up your difficult shape intotriangles:

Then measure the base (b) and height (h) of each triangle:

Write down each measurement carefully so you know which triangle it belongs to.Now go inside and calculate each area (usingArea = b h) and add them all up.But my garden is different ...... in fact it's not any shape at all ... it has some straight edges and some curved parts. What should I do?Maybe it looks something like this:

You could try covering your garden with a grid of squares these could be 1 meter squares or 1 foot squares, something like this:

How do I make a grid?Try using pegs in the ground and join them up with string. Make sure they are the right distance apart and all angles are right angles.How does this help? The grid and the outline of the garden don't match. There are lots of corners and curved parts.Count the squares!There are special methods talked about on theAreapage. The simplest method is: more than half a square counts as 1 less than half a square counts as 0

An estimate for this area is41 m2. This is just an example. Your garden will be different.(If your grid was 1 foot, then the area will be in square feet)

Why should I want to know the area of my garden?There might be lots of reasons: You want to re-turf the garden. How much grass should you order? How much will it cost? You want to plant the garden with tomato plants. These have to be planted a certain distance apart. How many plants could you plant? What will be your expected yield of tomatoes? You want to hold a barbeque party. How many people could comfortably fit into your garden?You can now doActivity: Grass for the GardenI am finished ... what have I learned?You have learned about measuring, recording data, drawing, and calculating area, well done!

Activity: A Walk in the Desert

Crash!Jade has crash-landed in the desert.There is a village somewhere nearby, direction unknown.So Jade comes up with a cunning plan: Fill up a water bottle from the plane, and take a compass, Then walk 1 km north, change direction and walk 2 km east, then 3 km south, 4 km west, 5 km north, 6 km east, and so on, like this:

This way he will find the village no matter what direction it is in, and can (hopefully) find his way back to the plane for fresh water and shade when he needs it.But he needs to know, at the end of each stage: How far he has walked altogether How far (in a straight line) back to the planeOK, Let's Do the Calcs ...After one stage of the journey, Jade has reached point A: Jade has walked 1 km altogether. He is 1 km (in a straight line) from the plane.After two stages, he has reached point B: Jade has walked 3 km altogether. To answer the second question, we make a right-angled triangle OAB:

We can calculate the length of OB usingPythagoras' Theorem, as follows:OB2= OA2+ AB2= 12+ 22= 1 + 4 = 5OB = 5So the answer in this case is:At Point B, the distance back to the plane (in a straight line) is5 kmAfter three stages, he has reached point C:Hey, it's your turn Now!Fill in all the other values ... if it gets hard, read below for some helpPointDistancewalked altogetherDistance (in astraight line) from O

O00

A11

B35

C6

D

E

F

G

H

I

J

How To Make It EasierDistance walked altogetherAt the end of each stage, the total distance is the sum of the series 1 + 2 + 3 + 4 + 5 + 6 + ...So just add the new distance each time.ORyou can calculate each value using:n(n + 1)/2wherenis the number of stages.Like this:Number ofstages (n)Total distance walked=n(n + 1)/2

11 2 / 2 =1

22 3 / 2 =3

33 4 / 2 =6

4etc ...

This works because it is the "Triangular Number Sequence":

Try doing the calculations both ways, for fun.Distance (in a straight line) from OTo work out the distance back to the plane, we can map out the journey on aCoordinate Grid, like this:

Now it is just a matter of finding thedistance between two pointsThe distance between the points (xA,yA) and (xB,yB) is given by the formula:

and one of those points is always the origin, which is at (0,0), so whenxBandyBare zero we get:

Example, for the pointE (-2, 3), x = -2 and y = 3, and so:

Hopefully that will help make your job easier.

Direction?There is one more thing Jade must know: to get back to the plane, what compass bearing must he travel?This will be covered inActivity: A Walk in the Desert 2