Making shapes
Problem
I have three counters. I arrange them on a grid to make a rectangle like this:
If I had six counters, I could make a rectangle like this:
Are there any other rectangles that I could make with six counters?
Imagine you have $18$ counters to put on a grid.
Arrange any number of counters on the grid to make a rectangle (not just its outline).
How many different rectangles can you make with each number of counters?
Getting Started
How about starting with a small number of counters and working up?
How will you know that you haven't missed any out?
Student Solutions
Sydney worked hard at this problem. He wrote:
I tried various combinations of numbers of dots to make rectangles. I discovered by factoring each number of dots I could figure out how many rectangles I could make out of each number of dots.
That is very well expressed. In other words, by finding pairs of numbers that multiply together to make each number of dots, you can find out how many rectangles there are. Sydney continued:
For example:
So, all numbers make a skinny rectangle.
6 also makes a 2x3
12 dots make three rectangles: 1x12, 2x6, and 3x4.
8 a 2x4
10 a 5x2
12 a 2x6 and 3x4
14 a 2x7
16 a 2x8
18 a 2x9 and 3x6.
Pippa from Newbald Primary School sent in the following;
If you have 3 counters, you can make 2 rectangles. 1 x 3 and 3 x 1
If you have 6 counters, you can make 4 rectangles 1 x 6, 2 x 3, 3 x 2 and 6 x 1
If you have 18 counters, you can make 6 rectangles, 1 x 18, 2 x 9, 3 x 6, 6 x 3 , 9 x 2 and 18 x 1.
It's basically the times tables.
When you work out one answer e.g 3 x 6 =18 just do the opposite to the numbers you are multiplying e.g 6 x 3 = 18.
And with other numbers of counters think of the times tables they are in.
Michael from Cloverdale Catholic in Canada wrote;
You can make about 6 rectangles:
1x18
2x9
3x6
6x3
9x2
18x1
Imagine these O'S are counters:
18x1 - OOOOOOOOOOOOOOOOOO
9x2 - OOOOOOOOO
OOOOOOOOO
6x3 - OOOOOO
OOOOOO
OOOOOO
3x6 - OOO
OOO
OOO
OOO
OOO
OOO
2x9 -
OO
OO
OO
OO
OO
OO
OO
OO
OO
1x18 -
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
O
Thank you for these you certainly got good answers.
Teachers' Resources
Making Shapes
I have three counters. I arrange them on a grid to make a rectangle like this:
If I had six counters, I could make a rectangle like this:
Are there any other rectangles that I could make with six counters?
Imagine you have $18$ counters to put on a grid.
Arrange any number of counters on the grid to make a rectangle (not just its outline).
How many different rectangles can you make with each number of counters?
Why do this problem?
This problem links shape with factors and multiples, and is a great way to introduce children to the idea of visualising numbers - in this case as rectangles or arrays.Possible approach
Once the group has grasped the idea, they can explore in pairs using counters and $2$cm squared paper. Allowing them time to investigate in this way is very valuable, but you might like to bring the whole class together at various stages in order to discuss what they have found so far.