### 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.

# Linear Area

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

Sketch diagram
$y=-2$ means that the $y$ coordinate will always be $-2$, regardless of what $x$ is. So this is the line through $(3,-2)$, $(-6,-2)$, $(0,-2)$ and $(1000000,-2)$ - that is, a horizontal line 'along' $-2$:

Next, $y=4x+6$ has gradient $4$ and $y$-intercept $6$ - so it slopes up steeply and crosses the $y$ axis at $(0,6)$.

Next, $x+y=6$ has gradient $-1$ and $y$ intercept $6$ ($x+y=6$ is the same as $y=-x+6$). So it slopes downwards, much more gently than the blue line, and also crosses the $y$ axis at $(0,6)$:

You can already see that the height of the triangle is $8$ units, from $-2$ to $6$ on the $y$ axis.

For the base of the triangle...
Finding the coordinates of the points of intersection
Suppose $y=-2$ and $y=4x+6$ intersect at some point $(a,b)$.
$(a,b)$ is on the line $y=-2$, so $b=-2$.
$(a,b)$ is on the line $y=4x+6$, so $b=4a+6$. But $b=-2$, so $-2=4a+6\Rightarrow -8=4a\Rightarrow -2=a$.

So $y=-2$ and $y=4x+6$ intersect at $(-2,-2)$.

Suppose $y=-2$ and $x+y=6$ intersect at some point $(c,d)$.
$(c,d)$ is on the line $y=-2$, so $d=-2$.
$(c,d)$ is on the line $x+y=6$, so $c+d=6$. But $d=-2$, so $c-2=6\Rightarrow c=8$.

So $y=-2$ and $x+y=6$ intersect at $(8,-2)$. This is shown below.

So the base of the triangle is from $x=-2$ to $x=8$, which is $10$ units.

So the area of the triangle is $\frac12\times8\times10=40$ square units.

The gradient of the line $x+y=6$ is $-1$, so when going down $8$ units from the top to the bottom of the triangle, it must also go along $8$ units, as shown on the right.

The line $y=4x+6$ has gradient $4$, so it is $4$ times steeper. This means that going down by the same amount corresponds to going $4$ times less far along.

So the base of the blue triangle shown below is only $8\div4=2$ units.

We could find the areas of the green and blue triangles separately, or of the whole triangle, whose base is $10$ units in total.

Green triangle: $\frac12\times8\times8=32$ square units

Blue triangle: $\frac12\times8\times2=8$ square units

Whole triangle: $32+8=40$ square units, or $\frac12\times8\times10=40$ square units.

Plotting the lines on a grid
Plotting the lines carefully, you can find the dimensions of the triangle.

Now, you can count that the base is $10$ units and the height is $8$ units, so the area is $\frac12\times8\times10=40$ square units.
You can find more short problems, arranged by curriculum topic, in our short problems collection.