Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

Areas of Parallelograms

Stage: 3 Challenge Level:

Here are two parallelograms, defined by the vectors $\mathbf{p}$ and $\mathbf{q}$. Can you find their areas?

a) $\mathbf{p}=\left(\begin{array}{c}3\\ 0\end{array}\right), \mathbf{q}=\left(\begin{array}{c}5 \\ 2\end{array}\right)$
b) $\mathbf{p}=\left(\begin{array}{c}3 \\ 2\end{array}\right), \mathbf{q}=\left(\begin{array}{c}0 \\ 4\end{array}\right)$

Select different vectors $\mathbf{p}$ and $\mathbf{q}$, where one vector is along a co-ordinate axis,and find the areas of the corresponding parallelograms. Can you discover a quick way of doing this?

Here are two more parallelograms, again defined by vectors $\mathbf{p}$ and $\mathbf{q}$. This time, neither $\mathbf{p}$ nor $\mathbf{q}$ lies along an axis. Can you find the areas of these parallelograms?

c) $\mathbf{p}=\left(\begin{array}{c}4 \\ 1\end{array}\right), \mathbf{q}=\left(\begin{array}{c}3 \\ 3\end{array}\right)$

d) $\mathbf{p}=\left(\begin{array}{c}2 \\ 4\end{array}\right), \mathbf{q}=\left(\begin{array}{c}-1 \\ 3\end{array}\right)$

Try some others.

Now try to generalise this.

Can you find the areas of a family of parallelograms, e.g. $\mathbf{p}=\left(\begin{array}{c}a \\ 2\end{array}\right)$ and $\mathbf{q}=\left(\begin{array}{c}4 \\ 5\end{array}\right)$?

Can you find the area of the parallelogram defined by the vectors $\mathbf{p}=\left(\begin{array}{c}a \\ b\end{array}\right)$ and $\mathbf{q}=\left(\begin{array}{c}c \\ d\end{array}\right)$?

Why do some areas turn out to be negative? Can you predict which vector pairs have this effect?