MEI Conference 2014
Algebra with cards and
paper
Kevin Lord
kevin.lord@mei.org.uk
Expressions for perimeter
One card A business card is a rectangle with length and width given as follows :7cm
b
7cm
5 cm
a
Write down the perimeter of each card.
Two cards: Two business cards can be arranged in different ways. For example :-
Write down the perimeter for each arrangement.
Use a and b for the width and length to write expressions for the perimeters.

What do you notice about the expressions? Can you explain what you have noticed?

Can you find any other arrangements for two cards with different perimeters?

What is the maximum and minimum perimeter for a shape made of two cards?

How would you write an expression for the perimeter of these arrangements?
Three or more cards:
How many different shapes and expressions for perimeter can
you find?
a
Perimeters of Pyramids (Ziggurats really!)
b
A business card is a rectangle with length b and width a
Write down expressions for the perimeter of these pyramid shapes.
Write down some expressions for the perimeters of larger pyramids

What do you notice about the expression for the perimeter of pyramids?

Can you explain what you have noticed?
Perimeters of Windmills
Write down an expression for the perimeter of this windmill arrangement.

What do you notice about the expressions for
the perimeter of different sized windmills?

Can you explain what you have noticed?
Other shapes
What other shapes made from cards could you investigate the perimeters of?
a
Perimeter of Rectangular Rings
b
A business card is a rectangle with length b and width a
Four Card Ring
Write down an expression for the perimeter of

the outer rectangle of the ring;

the inner rectangle of the ring;

the total perimeter.
Can you simplify your expression?
Six Card Rings
There are two possible arrangements of six cards in a rectangle with a hole.
For each arrangement, write down an expression for the perimeter of

the outer rectangle of the ring;

the inner rectangle of the ring;

the total perimeter.
Simplify your expressions where possible?

What do you notice about the expressions?

Can you explain what you have noticed?
Eight or More Card Rings
How many different rings can you make with 8 or more cards?
What are the expressions for total perimeter of each ring?
a
Area of Rectangular Rings
b
A business card is a rectangle with length b and width a
What is the area of one business card?
a
Four Card Ring
The shape produced is a large square with a square hole.
Write down an expression for

the length of one side of the large outer square;

the area of the outer square;

the area of the hole;

the total shaded area of the ring.
The total shaded area of the ring = area of the outer square - area of the hole
Do your expressions satisfy this equation?
Six Card Rings
There are two possible arrangements of six cards in a rectangle with a hole.
For each arrangement, write down an expression for the area of

the outer rectangle;

the inner rectangular hole;

the total shaded area of the ring.
Check your expressions satisfy the equation
The total shaded area of the ring = area of the outer rectangle - area of the hole

What do you notice about the expressions for the area of the rings?

Can you explain what you have noticed?
Overlapping Rectangles
One card A business card is a rectangle with length and width given as follows :7cm
b
7cm
5 cm
a
a
Write down the area of each card.
Two Cards Overlapping
Two cards are placed so that they overlap each other as shown.

In each case find expressions for the areas of the overlapping and other sections of
the shapes.

Find an expression for the total area of the resulting shapes.

Can you find different ways of finding the expressions for the total area?

If b > a, which of the arrangements has the greatest area?
Cards and Coordinators
b
A business card is a rectangle with length b and width a
a
Imagine four cards layed on a coordinate grid.
If a = 3 and b = 7 what is the coordinate of
 point P
 point Q?
Finding the length and width of each card
Write down equations in terms of a and b to find how long and wide the cards are in these
(23, 12)
arrangements?
(21, 9)
(2, 1)
(2, 1)
(13, 14)
(1, 4)
(16, 16)
(2, 1)
(-1, 0)

(-7, -28)
Make up some arrangements of your own and write down expressions of
change in x coordinates and the change in the y coordinates.
Problems involving Area of Rectangular Rings
b
A business card is a rectangle with length b and width a
The ratio of the dimensions can be written
a
or
Four Card Ring
Find the ratio of width to length of a card for which the
Area of the shaded ring = Area of the Hole
Six Card Rings
There are two possible arrangements of six cards in a rectangle with a hole.
For each arrangement, find the ratio of width to length of a card for which the
Area of the shaded ring = Area of the Hole
Eight or More Card Rings

Investigate the areas of larger rings to find the ratio of the width : length of cards for
which the area of the ring = area of the hole.

Is there any pattern or general rule?
Polygons from Business Cards
b
A business card is a rectangle with length b and width a
The ratio of the dimensions can be written
a
or
Three Cards
Three identical business cards are to be cut so that when laid over each other the six short
sides create a regular hexagon.

What is the ratio of the dimensions of the cards?

Cut three cards with dimensions in the correct ratio to check your solution.
Four Cards
Four identical business cards can be laid over each other in a similar way to make a regular
octagon. What is the ratio of the dimensions of the cards?
Five or More Cards
You can use increasing numbers of business cards to produce regular polygons.

Can you find a relationship between the number of sides of the polygon and the ratio
of the dimensions of the cards?

What happens to the ratio as the number of sides of the polygon increases?