﻿ Circus towers: square stacks
Square stacks
Square stacks
Welcome to the circus! I'm the ringmaster.

Your job is to help me direct the acrobats to form square-shaped towers.

How do you make a square tower?
Square stacks
This is a square tower.
Square stacks
The bottom layer of the square tower is the base.

In this base there are four acrobats.
There are also four layers in this square tower.
To make a square tower, the number in the base and the number of layers must be the same.
Square stacks
Enter a number: Go
Your first task is to create a square tower with a two-person base.

How many acrobats are needed to do this? Enter the total number of acrobats you need in the space.
That doesn't look right, does it? Try again
There are not enough acrobats in your tower.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
There are too many acrobats on stage.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
That's still not right.

To create a square shape based on two acrobats, you need two layers.

Two acrobats in the base times two layers is four acrobats: 2 x 2 = 4.
Square stacks
That doesn't look right, does it? Try again
There are not enough acrobats in your tower.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
There are too many acrobats on stage.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
That's still not right.

To create a square shape based on two acrobats, you need two layers.

Two acrobats in the base times two layers is four acrobats: 2 x 2 = 4.
OK
Square stacks
Now use the table to enter the number of acrobats in the base and the total number in the tower.
Acrobats in base Acrobats in tower
That's not right. Count again.
That's still not right.

Count the number of acrobats in the base and enter this number in the top row.

Enter the number of acrobats in the whole tower in the bottom row.
That's still not correct. I'll show you the right numbers.
The numbers in the table can also be shown by a point on a graph.
Number of acrobats in square tower Number of acrobats in base Acrobats in tower
Square stacks
Enter a number: Go
Now create a square tower with a three-person base.

How many acrobats are needed to do this? Enter the total number you need in the space.
That doesn't look right, does it? Try again
There are not enough acrobats in your tower.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
There are too many acrobats on stage.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
That's still not right.

To create a square shape based on three acrobats, you need three layers.

Three acrobats in the base times three layers is nine acrobats: 3 x 3 = 9.
Now use the table to enter the number of acrobats in the base and the total number in the tower.
That's not right. Count again.
That's still not right.

Count the number of acrobats in the base and enter this number in the top row.

Enter the number of acrobats in the whole tower in the bottom row.
That's still not correct. I'll show you the right numbers.
Acrobats in base Acrobats in tower
Acrobats in tower Number of acrobats in square tower Number of acrobats in base
Good. Can you see a pattern beginning to form?

The pattern relates to square numbers. These are numbers that are made by multiplying a number by itself, such as 2 x 2 = 4 and 3 x 3 = 9. We can write 3 x 3 as 32 and we call it '3 squared'.
Square stacks
Enter a number: Go
Acrobats in base Acrobats in tower
Acrobats in tower Number of acrobats in square tower Number of acrobats in base
Now create a square tower with a four-person base.
That doesn't look right, does it? Try again
There are not enough acrobats in your tower.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
There are too many acrobats on stage.

You need to create a square shape with two acrobats in the base layer. So how many do you need in total?
Try again
That's still not right.

To create a square shape based on three acrobats, you need four layers.

Four acrobats in the base times four layers is sixteen acrobats: 4 x 4 = 16.
Now use the table to enter the number of acrobats in the base and the total number in the tower.
That's not right. Count again.
That's still not right.

Count the number of acrobats in the base and enter this number in the top row.

Enter the number of acrobats in the whole tower in the bottom row.
That's still not correct. I'll show you the right numbers.
That's still not correct.

There are five acrobats in the base. To create a square shape, the number in the base and the number of layers must be the same. So you need five layers in the tower.

To find the answer, multiply 5 x 5: you need 25 acrobats.
That's still not correct.

There are seven acrobats in the base. To create a square shape, the number in the base and the number of layers must be the same. So you need seven layers in the tower.

To find the answer, multiply 7 x 7: you need 49 acrobats.
This one is trickier.

There are 2 500 acrobats in the tower. To create a square shape, the number in the base and the number of layers must be the same. So you need to find the number you can multiply by itself to make 2 500.

Since 50 x 50 = 2 500, there must be 50 acrobats in the base and 50 layers in the tower.
The numbers in the table can also be shown by a point on a graph.
Can you see the pattern now? You can also see it on the graph.
This time, instead of making a tower and recording the numbers in the table, just find the missing numbers in the table.

For help, look at the pattern in the table and on the graph.
Now look at the graph. Notice that it's not a straight line. It forms a curved shape called a parabola.

This curve shape occurs because the total number of acrobats does not increase by an equal amount each time an acrobat is added to the base.
Now we're going to jump ahead. This time you need to look at the pattern and enter the number required to build a tower with seven acrobats in the base.
Take a look at the graph.

Is it easier now to see why the points don't make a straight line?
Now we're going to jump ahead even further. This time you need to enter the number of acrobats needed for base when the total number of acrobats is 2500.
This number is too big to show on this graph.

Notice that when you plot squared numbers in a sequence, the graph curves sharply upwards.
Now let's look at how this pattern can be described in words.

Choose the correct answer for each of the questions that follow.
Question 1 In any square tower, the number in the base is: double the number of layers larger than the number of layers the same as the number of layers smaller than the number of layers
That's not right. Try again
That's not right. Think about the shape of the square towers you made. Try again
That's still not right.

The answer is (C). In a square tower, the number in the base is always the same as the number of layers, no matter how large or small the tower.
That's still not correct.

The answer is (D). We work out the total by multiplying the number in the base by the number of layers.

In a square tower, these numbers are always the same, so you multiply the number by itself.
Well done!

In a square tower, the number in the base is always the same as the number of layers, no matter how large or small the tower.
Question 2 Which of the below best describes the pattern in the table and on the graph?

The total number of acrobats in any square tower is:
the number in the base + the number of layers the number in the base x 2 the number in the base ÷ the number of layers the number in the base x the number of layers
That's correct!

A mathematician would write the rule like this:

n x n or n2
Good work!

Thank you for your help making square towers.