Shear Magic
Shape C is a parallelogram with a base of 2 and a height of 3.
Can you use the two pictures below to work out the area of the parallelogram?
Here are two more parallelograms made by shearing a rectangle with a base of 2 and a height of 3.
Can you draw similar diagrams to work out their areas?
Use the GeoGebra applet below to explore other parallelograms with a base of 2 and a height of 3.
What do you notice? Can you explain it?
Explore other families of parallelograms with a particular base and height, by moving the red and green dots.
Can you come up with a general rule for working out the area of a parallelogram if you know its base and height? Can you explain why your rule works?
I wonder what happens when we shear triangles...
Here is a family of four sheared triangles with a base of 1 and a height of 2.
Can you work out the area of the triangles?
You might like to use the GeoGebra applet below to help.
Explore other families of triangles with a particular base and height.
Can you come up with a general rule for working out the area of a triangle if you know its base and height?
Can you explain why your rule works?
Shearing is a transformation of a shape in which a particular line (in this case the base of the triangle or parallelogram) remains fixed and all other points in the shape are translated parallel to that line by an amount proportional to the distance from that line.
How does the area of the triangle compare to the area of the parallelogram? How do you know? How can you use this to create a formula for the area of a triangle in terms of the area of a parallelogram?
Thank you to everyone who sent in their solutions and ideas for this problem. We received correct answers, or partial answers, from many students around the world including: Aarush, Ronik, Seyon and Yuchen from Wilson's School; Harriet from Romsey School; Helena, Mia, An and Laura from Loughborough High School; Camille from the German European School in Singapore; Emily at The Garden International School in Malaysia; Tessa, Alysia, Hailey, Penny, Matilda, Vihaga and Leia at the Frederick Irwin Anglican School; Tanna and Jazz at Stephen's School Carramar in Australia; Ahan in Singapore; Mahdi from the Mahatma Gandhi International School in India; Sanika at PSBBMS in India; Rose and Pavan (who did not leave their schoools' names) and, Gia and Ken at the British Vietnam International School in Hanoi. Well done to you all.
We began by asking you to share your thoughts about working out the area of Shape C, a parallelogram with base 2 and height 3, and other paralellograms with the same base and height as Shape C.
There are many ways to explain our mathematical thinking to others. This is how Yuchen explained her thinking:
Firstly, I know that the base of picture 1 is 6 and the height is 3 because if we know that base of C is 2 and it takes up 2 squares then we can do the same to the base and height to get 6 and 3.
Now we need to work out the area which is 18 because to work out the area of a rectangle you need to do the length of the base multiplied by the height, which is this case is 6 x 3.
Knowing that the area is 18, we now need to work out the area of A and B. The base for A is 4 and the height is 3, and it is the same for B, which also means the area for both of them is going to be the same, as they are both triangles. Now we do 4 x 3 = 12, however working out the area for a triangle the formula is (base x height) x 1/2. So we do 12 divided by 2, which gives us 6, which means the area for A and B is 6.
Now we do 6 + 6 = 12 and then 18 - 12 = 6 which gives us 6 which means the area for C is also 6.
Well done, Yuchen.
Seyon explored the second part of the question using the following visual approach:
Seyon also noticed a pattern in his answers after investigating the areas of several more parallelograms created by shearing rectangles:
I noticed that all the parallelograms that have a base of 2 and a height of 3 have an area of 6. From this I can infer that finding the area of a parallelogram has something to do with both its base and height.
Vihaga and Leia also noticed the importance of knowing a parallelogram's base and height. They generalised their findings as follows:
No matter how stretched out your parallelogram is, the area is always the base multiplied by the height.
Sanika shared the following thoughts:
After I worked out the area of the parallelograms above, I figured that a rectangle was just a unique parallelogram i.e., a parallelogram could be moulded into a rectangle and the area would not change. The rectangle would then have a base that equals the parallelogram's base and a height which equals the height of the parallelogram. The area of the parallelogram would therefore be the area of the rectangle which is nothing but the base of the parallelogram times its height.
Using the GeoGebra applet, I figured that as long as the base and height of the parallelogram were kept constant, the area would not change.
We also encouraged you to investigate what you noticed about a family of sheared triangles
Here is a family of four sheared triangles with a base of 1 and a height of 2:
Tanna and Jazz shared their thoughts:
Parallelogram = base x height
Sheared Triangle = base x height x 0.5
This works because a triangle is half the area of a parallelogram.
Matilda wrote:
The rule is that the area of the triangle multiplied by 2 is the area of
the parallelogram.
Are they correct? How do you know?
Sanika explored her ideas further using GeoGebra:
I noticed that the area of a triangle was half the area of a parallelogram which can be formed by diagonally reflecting the triangle. The parallelogram would then have the same height and base as the triangle. Since the area of a parallelogram is base times height, I could say that the area of a triangle would be half its base times height.
Why do this problem?
Many students have only a superficial understanding of the formula for the area of a triangle, particularly when they move away from right angled triangles. This problem starts with an exploration of shearing parallelograms and then encourages students to think of a triangle as half a parallelogram, which should lead to a more secure understanding of where the triangle formula comes from, and a better appreciation of how areas change or remain the same as we transform shapes.
Possible approach
Start by showing students these two images and inviting them to jot down anything they notice:
You could pose the question "Which area is bigger, C or D? How do you know?" and give students time to come up with a justification for why we know C and D must have the same area.
Then share these two images:
Invite students to draw similar diagrams to find the area of these two parallelograms.
"I wonder if all parallelograms with base 2 and height 3 will have the same area..." Students could explore using pencil and squared paper. When they are ready, they can move on to explore other bases and heights, perhaps using the first interactivity in the problem.
Once students have had time to explore, bring the class together to develop a formula for the area of a parallelogram based on their findings.
If appropriate, introduce the word "shear" to describe this type of transformation - a suggested definition is included in the notes and background section at the bottom of the problem.
Next, share this image of a family of triangles:
"How can we use what we found out about parallelograms to help us find the area of these triangles?"
Again, students can explore using pencil and squared paper, or using the second interactivity in the problem.
Once students have had time to explore, bring the class together to develop a formula for the area of a triangle based on their findings. The interactivities could be used to test out the formula, by creating a triangle and then inviting students to predict what the area is before clicking "Show area".
Key questions
Which areas are the same?
How do you know?
How does the area of the triangle compare with the area of the parallelogram?
Possible support
Show some examples using the interactivities, and then encourage students to draw the diagrams for themselves on squared paper, paying particular attention to drawing the correct parallelogram for a given triangle.
Invite students to predict what will happen if the red dot moves one unit to the right, or if the green dot moves 1 unit up. Then take time to explore those patterns, recording them using squared paper.
For some students, cutting out the shapes in the first part of the problem may help them to see the conservation of area.
Possible extension
Students could use what they have learned about areas of triangles to explore Isosceles Triangles.