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Can you explain the strategy for winning this game with any target?

Counting Factors

Is there an efficient way to work out how many factors a large number has?

Is There a Theorem?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

Shear Magic

Age 11 to 14 Challenge Level:

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?

A 3 by 6 rectangle, the top left corner has a 3 by 4 right angled triangle shaded and marked A, the bottom right corner has a 3 by 4 right angled triangle shaded and marked B, and the remaining parallelogram area is shaded and marked C.       A 6 by 3 rectangle, the 4 by 3 rectangle on the left is divided into two right angled triangles A and B, and the 2 by 3 rectangle on the right is labelled D.

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?

parallelogram with base 2 and height 3 (top moved 1 unit to the right relative to the base)  parallelogram with base 2 and height 3 (top moved 7 units to the right relative to the base)

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.
 

Four triangles with base of 1 and height of 2, successively more sheared.

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?

 

Notes and Background

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.