Areas by Audio: Podcast
W H Smith’s are selling audio revsion guides for GCSE Maths. The pack consists of 2 audio format CDs with about 25 short recordings plus a small booklet with diagrams and formulas. I found the speed brisk and the pack is for revision so it is assuming that students need to refresh ideas already covered. One of the GCSE students is now trying the pack out and I will have some feedback soon.
As I have at least one Access Maths student who is going to have to miss the areas lesson, here is an ‘audio lesson’ of my own in the form of a 14 minute MP3 with one sheet of diagrams. This audio lesson is not a revision lesson. I’m assuming that the user has not recently been taught basic area calculations. I have to assume that the students have recently studied perimeters including circles because that is how the Access Maths syllabus we use works. For the same reason, the area of a trapezium is not covered explicitly (that is a shame as the idea of averaging the parallel sides to get an equivalent rectangle usually goes down well – mature students begin to see the connections between topics when they grok the ‘average the two parallel sides’ leads to the (a + b)/2 bit in the formula).
- Download MP3 of the audio lesson [ 4.7 Mb, 48K mp3 sample]
- Download the PDF of the diagram sheet [ essential for use with lesson ]
The script for the audio lesson is copied below the fold of this blog post. I find scripting essential, and I usually need two takes. In this audio lesson, I had to use Audacity for editing out some bloops and misreads (about 4). I decided to splice in the instruction to turn over the diagram sheet before taking the summary test at the end of the lesson. I am still in two minds about the diagram sheet – perhaps I should leave spaces for the students to write in the formulas? Just a way of increasing interactivity?
The first take was made using the built in microphone on the iBook. Some background noise but very usable, and the recording level is high. The second take (the one finally used) was made using a cheap Logitech USB microphone with less background noise and a much lower recording level. The ‘normalise’ filter in Audacity brought the level up without any huge increase in background noise.
I’m linking this file to the Access Maths blog for students to try, and I will point out the BBC learning styles test as a possible indicator of a preference for auditory learning. We shall see…
Script of the areas audio lesson podcast
1600+ words in just under 14 minutes with spaces for answers to questions.
Areas
This audio lesson lasts for just under 15 minutes.
You will need a printout of the Diagrams for Area Podcast sheet, and you might want to make notes on the sheet as we go along. A calculator might be useful for the last two examples.
By the end of this recording, you will be able to state formulas for the area of rectangles, parallelograms and circles. You will also have worked through example calculations, and will be able to describe a good strategy for finding the areas of composite shapes.
What is area?
Area is the size of a two dimensional surface. You measure length with a ruler, and area is actually measured with small squares: commonly square centimetres for small areas and square metres for larger areas. When you paint a wall in your house, tins of paint often have the coverage written on the back of the label: usually a litre of paint will cover 12 square metres of wall with one coat.
Look around you: if you are in a room, car, bus, train or other engineered structure then the chances are that the surfaces around you are combinations of rectangles, parallelograms, triangles and circles or parts of a circle. These are the easiest shapes to make in concrete, wood and metal in industrial quantities. In the natural world, shapes are not simple but have a more complex often fuzzy structure. Maths has been calculating with the simple shapes for thousands of years, but is only just finding ways of describing natural shapes like clouds and lungs and the circulatory system.
The easiest shape to calculate the area for is the rectangle. Look at figure 1 on the sheet that shows a rectangle with a length of 4 centimetres and a width of 3 centimetres. As you can see by counting, there are 12 square centimetres in the rectangle. The area of the rectangle is calculated by multiplying the length by the width. A rectangle that is 20 centimetres long and 7 centimetres wide will have an area of 20 multiplied by 7 or 140 square centimetres.
A square is a special type of rectangle with four equal sides. A square of side 5 cm would have an area of 5 times 5 square centimetres. You can work that out by finding 5 squared – in fact the area of a square is where the idea of ‘square numbers’ came from.
The parallelogram is a shape that is made from two sets of parallel lines crossing at an angle – a bit like a pushed over rectangle. A rectangle has interior angles that are right angles, the interior angles of a parallelogram are not all equal; one pair of interior angles will be acute and the other pair obtuse. Look at figure 2 on the sheet. Imagine you cut a bit off the right hand side of the rectangle (see the triangle in dotted lines) and added the small area onto the left hand side of the rectangle. The result is a parallelogram but the area is the same as the rectangle we started with.
We name the sides of a parallelogram slightly differently compared with the rectangle. The base of the parallelogram is one measurement, and the other measurement is the perpendicular height of the parallelogram. The perpendicular height of the parallelogram is measured at right angles to the base: it is not the length of the slanting side. The area of the parallelogram is simply the base multiplied by the perpendicular height. The parallelogram shown in Figure 2 has a base of 4 cm and a perpendicular height of 3 cm. Again the area is 12 square centimetres.
Your turn: suppose I had a parallelogram with a base if 12.5 metres and a perpendicular height of 6 metres. What is the area of the parallelogram?
I have 75 square metres as the answer (I doubled and halved to make the multiplication easier to do: 25 times 3).
Look at Figure 3 – a triangle is a parallelogram sliced in half, and so the formula is simply multiplied by a half (or divided by two if you prefer). For example a triangle with a base of 10 centimetres and a perpendicular height of 5 centimetres would have an area of 0.5 times 10 times 5 or 0.5 times 50 or 25 square centimetres. The perpendicular height must be measured at right angles to the side you pick as the base for the formula to be valid.
Your turn: suppose then end of a roof had a triangular shape. If the base width of the roof was 8 metres and the height of the roof was 5 metres calculate the area of the roof. Answer: half of 40 is 20 square metres.
Circles have a slightly more complicated formula (look at figure 4). Like the square, the area of the circle is related to the square of one of the dimensions. The area of a circle is given by pi times the square of the radius. For instance, the area of a circle of radius 10 centimetres is 10 squared or 10 times 10 multiplied by 3.142 the usual value of pi. If you work out 3.142 times 10 squared, you should get 314.2 square centimetres. We usually round the areas of circles at the end of the question to three significant figures so you would give the answer to the question as 314 square centimetres.
Your turn: Find the area of a circle of diameter 10 centimetres. Answer – as you were given the diameter, you have to divide by 2 to find the radius of the circle, in this case 5 centimetres. 3.142 times 5 squared is 3.142 times 25, which I estimate to be a little more than 75 square centimetres. The full answer from the calculator is 78.55 square centimetres or 78.6 square centimetres to three significant figures. To get a quick check or estimate, just multiply the square of the radius by 3.
It is worth mentioning that you will sometimes need to find the area of half a circle or semi-circle and even the area of a quarter of a circle. The easiest way to do this is to find the area of the whole circle then divide by two or four as needed.
Composite shapes: As mentioned earlier, man made shapes are often formed from combinations of rectangles, triangles and parts of circles. Now you know how to calculate each part, you need some strategies for breaking down shapes into parts.
Look at figure 5a. The shape shown is made from a triangle and a rectangle joined together. The dotted line shows how to break the shape into a rectangle and triangle. To work out the area of the shape, calculate the area of the rectangle first, then work out the area of the triangle and finally add the two areas together.
The trick with these questions is to find the dimensions of the shapes you have decided to use. The rectangle has dimensions of 4 centimetres and 3 centimetres. The triangle has a perpendicular height of 4 centimetres and a base of 2 centimetres. The base length has to be calculated by subtracting the 3 centimetres from the total length of the shape.
I calculate the rectangle to have an area of 3 times 4 centimetres or 12 square centimetres. The triangle I find to have an area of half times 2 times 3 centimetres or 3 square centimetres.
Now look at a more complex example. Look at figure 5b. A flower bed is made from two semi-circles and a rectangular section. There is a square paddling pool in the centre of the flower bed. We need to find the area of the soil in the flower bed.
The easiest way to do this is to calculate the area of the outside shape, then the area of the pool, and finally to subtract the two areas. What is left will be the area of the soil.
The outside area is made from a rectangle of dimensions 40m by 20m, and the area of a whole circle: can you find the radius of the outer circle? Thats right, 10 metres. Don’t fall into the diameter trap.
I calculate the area of the rectangle in the outer shape as 800 square metres, and the circle has an area of 3.142 times 10 squared, or 314.5 square metres. The total area of the outer shape is the sum of these two figures, one thousand one hundred and fourteen square metres.
We calculate the dimensions of the paddling pool by squaring the side of 8 metres to give 64 square metres.
Subtracting these gives 1050 square metres for the area of soil.
Challenge: suppose the flower bed was surrounded by a path of width 1 metre. Can you find the dimensions of the path and calculate the area of the path? You will definitely need to draw a diagram!
Summary quiz.
You might want to turn your notes over now and try these 5 questions. Numerical answers are given later.
1) Can you say or write down a formula for the area of a rectangle?
2) What is the area of a parallelogram of base 10 metres and perpendicular height 12 metres?
3) Write down the formula for the area of a triangle.
4) Suppose the end wall of a tent is made from a square of material with a triangle on top. The square has side 2 metres and the triangle is 1.4 metres high. Can you calculate the area of material needed for the end wall of the tent?
5) A paddling pool has a diameter of 6 metres. What is the area of the pool?
Numerical answers: The parallelogram has an area of 120 square metres. The tent needs 5.4 square metres of material. The paddling pool has an area of 28.3 square metres.