### Spiral Snail

A snail slithers around on a coordinate grid. At what position does he finish?

### Bucket of Water

Lisa's bucket weighs 21 kg when full of water. After she pours out half the water it weighs 12 kg. What is the weight of the empty bucket?

### How Far Does it Move?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.

# Graph Area

##### Age 11 to 14 Short Challenge Level:

Plotting the graph using a table of values
If we plot the graph first, then we will be able to see the shape whose area we are looking for.

We can use a table of values to plot the graph, and we are interested in $x$ values between $3$ and $7$, so going from $2$ to $8$ will help us see the whole shape.

If $x=2$, then $y=2\times2-6=4-6=-2$. The rest of the values can be found in the same way.

 $x$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $y$ $-2$ $0$ $2$ $4$ $6$ $8$ $10$

The graph looks like this:

We need to find the area between $x=3$ and $x=7$, which is coloured green below.

The shape is a right-angled triangle, and its base is $4$ units and its height is $8$ units. So its area is $\frac{1}{2}\times4\times8=16$ square units.

Sketching the graph using relevant points
From the equation $y=2x-6$, we know that the graph will be a straight line.

Since we are interested in the area between $x=3$ and $x=7$, we could check where the graph is at those two points.

When $x=3$, $y=2\times3-6=6-6=0$.
When $x=7$, $y=2\times7-6=14-6=8$.

So $(3,0)$ and $(7,8)$ lie on the graph.

So the graph must be the straight line through those points, and the area required is the green area, as shown below.

The shape is a right-angled triangle, and its base is $4$ units and its height is $8$ units. So its area is $\frac{1}{2}\times4\times8=16$ square units.

You can find more short problems, arranged by curriculum topic, in our short problems collection.