PrepShorts · Study sheet · Class 8 Mathematics · Chapter 3, Proportional Reasoning-2
Chapter 3 · Proportional Reasoning-2
Building a pie chart: turning counts into angles
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Forty students across five grades, drawn as a circle cut into five slices. That is not really a picture of counts — it is a picture of angles.
The idea
A pie chart is the previous section's division of a whole with one substitution: the whole is 360°. Nothing else is new. And that is exactly why the chapter's first move — reducing the count ratio to its simplest form before computing any angle — cannot change the picture: dividing every count by the same number divides their total by it as well, so each count's share of the total, and therefore each angle, comes out untouched. The chart is a drawing of a set of fractions, and fractions are what survive cancelling.
What you should be able to do
- Turn a table of counts into a ratio, and that ratio into a set of central angles summing to 360°
- Reduce the count ratio by its HCF first, and show that the resulting angles are identical either way
- Compute the degrees-per-item rate and the degrees-per-part rate, and state how the two are related
- Construct a pie chart with compasses and protractor: base radius first, then each successive angle measured from the radius just drawn
- Check a finished chart by confirming the angles close at 360°
- Read a pie chart in reverse: from a sector angle to a fraction of the whole, and from a fraction plus one known count to the total
- Recover an unlabelled sector's angle from the labelled ones
- Say what a pie chart cannot be asked — a category it does not contain has no reading on it
- Read a two-sector chart against a whole of 24 hours
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| pie chart | a circle cut into sectors whose angles are proportional to the quantities they stand for | printed in this chapter, §3.5 (Part II p.60); the printed section title is "A Slice of the Pie" |
| slice | the chapter's word for one sector of the chart | printed in this chapter, §3.5 (Part II pp.60–62) |
| proportions of a whole | the shares a pie chart is drawn to display | printed in this chapter, §3.5 (Part II p.60) |
| HCF | the highest common factor, divided out of every term to reach simplest form | printed in this chapter, §3.5 (Part II p.61) |
| simplest form | the equivalent ratio left after dividing out the HCF | printed in this chapter, §3.5 (Part II p.61) |
| radius | the segment from the circle's centre to its edge, the line each new angle is measured from | printed in this chapter, in the construction steps (Part II p.61) |
| anti-clockwise | the turning direction the construction steps specify | printed in this chapter, Step 2 of the construction (Part II p.61) |
| degrees per item | 360 divided by the total count, the rate that converts one count into one angle | an added term; the chapter computes the equivalent per-part rate and names neither |
Where people slip up
- "You can only draw a pie chart if the numbers add to 100, or to 360." Forty students work fine. The total is whatever it is, and 360 is divided in its ratio. Item 1 of the exercise set happens to have 360 people, which makes the arithmetic look like a rule; it is a coincidence.
- "Reducing the ratio changes the chart." It cannot. Dividing every count by 2 halves the total too, so every share is unchanged. Show both routes to 108° and the objection dies.
- "The angles are the percentages." Fifty percent is 180°, not 50°. The conversion rate is 3.6° per percent, and it should be computed once.
- "A bigger circle means more students." The radius carries no information at all. Two charts of different sizes with the same angles say exactly the same thing.
- "Every sector has its angle printed on it." The car sector on Part II p.68 does not, and recovering it from 360 is the whole task.
- "If the question names a category, the chart must have it." Taxis are asked about and are not on the chart. The honest answer is that the chart shows none, and saying so is a skill.
- "Sectors go in order of size." Nothing forces it, and the two printed charts make the point between them: the finished chart on Part II p.60 is in descending order (12, 10, 8, 6, 4), while the transport chart on Part II p.68 is not. So the p.60 chart is not the counter-example — say plainly that its ordering is a choice its author made, not a rule the object carries. Note also that the chapter's two charts disagree about starting radius but not about direction — both run anti-clockwise; see section 6.
- "Reading a chart is a different skill from drawing one." It is the same proportion run backwards: angle over 360 is the share, and share times total is the count.
- "A chart with only two arcs is not a pie chart." The sleep rings on Part II p.67 are two-sector charts against a whole of 24 hours. Same object, fewer cuts.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 3.5 Q1, Figure it Out · 3.5 Q2, Figure it Out · 3.5 Q3, Figure it Out · 4 Q4, Figure it Out · 4 Q5
Transcript1,421 words
Forty students, five grades. Twelve got an A, ten a B, eight a C, six a D, four an E. You could leave that as five numbers in a row. Or you could draw it. A circle, cut into five slices, one slice per grade, the biggest slice for the grade the most students got. And here is the thing about that picture. It is not really a picture of the counts at all.
It is a picture of the SHARES - what fraction of the class each grade took. That distinction is going to do all the work today. Start with what makes a slice big. Not its length round the edge, and certainly not how far out it reaches. The angle at the centre. That is the only thing a slice really has. So the rule has to be: the angle follows the count. Twice as many students, twice the angle. Equal counts, equal angles.
And the circle gives you exactly three hundred and sixty degrees to hand out. No more, no less. Which means this is the same problem as before, wearing a hat. Divide a whole in a given ratio - and the whole is three hundred and sixty. So do it. The five counts add to forty. That is the total the class actually has. Three hundred and sixty degrees, forty students. Divide, and you get nine degrees per student.
Nine degrees is the price of one student. Everything else is multiplication. Twelve students at nine degrees each is a hundred and eight. Ten is ninety. Eight is seventy-two. Six is fifty-four. Four is thirty-six. Notice there was no rule about the total having to be a hundred, or three hundred and sixty. Forty works. Any number works. Now, most people do not do it that way. They do something else first, and it is worth understanding why it is safe.
Look at twelve, ten, eight, six and four. Every one of them is even. So divide the lot by two and you get six, five, four, three and two. Simpler numbers, same ratio. But watch what happened to the total while you were not looking. Forty students became twenty parts. The total halved as well. It had to - it is made out of those five numbers, so whatever you do to all of them, you do to it.
Which changes the rate. Three hundred and sixty degrees over twenty parts is eighteen degrees a part. Eighteen, not nine. So did the chart just change? Six parts at eighteen degrees is a hundred and eight. The same hundred and eight. And ninety, seventy-two, fifty-four, thirty-six - every angle identical. The rate doubled and the count halved, and the two cancel exactly. And be careful how you say that. Eighteen is TWICE nine. It is not nine more than nine, even though it happens to also be nine more. Say times, and you will never be wrong; say differ by, and one day you will subtract.
Here is why that had to work, and it is the whole idea of the video. A chart draws shares. Each slice is one count over the total. Divide every count by two and you have divided the total by two as well - so every fraction has a two on top and a two underneath, and they cancel. The shares come out untouched. And the angles are the shares, so the angles come out untouched.
Test it on every chart in this video, six of them, and the angles never move once. But it only works if the factor reaches EVERY count. Halve grade A alone and the counts read six, ten, eight, six, four - a chart nobody drew, in which the slice you halved has shrunk and every single other slice has grown. Reducing is safe. Reducing one thing is not. So let us actually draw it, because there is a technique here and it is easy to get wrong.
Circle first, with compasses. Mark the centre. Draw one radius out to the edge - any direction you like, that choice is free. Now put the protractor down with its middle exactly on the centre and its baseline exactly along that radius. Measure a hundred and eight degrees round, and draw the second radius. That slice is grade A. And here is the step people miss. The next angle, ninety degrees, is measured from the radius you just drew - not from where you started.
Each new slice starts where the last one stopped. That is what stops the angles piling up on top of each other. Seventy-two more for grade C. Fifty-four for D. And thirty-six for E, which should bring you exactly back to the radius you began with. Add them up. A hundred and eight, plus ninety, plus seventy-two, plus fifty-four, plus thirty-six. Three hundred and sixty. The circle closes. That check is not decoration. If your last radius misses the first one, something is wrong upstream, and you know it before you have coloured anything in.
One more free choice, while we are here. Nothing forces the slices into order of size. This one happens to run biggest to smallest, but that was a decision somebody made, not a rule the object carries. Now turn the whole thing round. Somebody hands you a finished chart. What can you get out of it? The same proportion, run backwards. A slice's angle over three hundred and sixty is its share of the whole.
And its share times the total is its count. So a ninety degree slice is a quarter, and on a class of forty that is ten students. It works in the other direction too, and this is the useful one. If you know a slice's share AND how many things are in that slice, you can recover the total you were never told. Watch that earn its keep. A survey of how children get to school: walking, bus, two-wheeler, car and cycle.
Four of the five slices have their angles written on them. Walking ninety. Bus a hundred and twenty. Two-wheeler sixty. Cycle sixty. The car slice has no number on it at all. Which is the whole point of the question. Because you already know what it is. Those four add to three hundred and thirty, and a circle is three hundred and sixty. The car slice is thirty degrees. Thirty over three hundred and sixty is one twelfth. So if eighteen children came by car, and eighteen is a twelfth of everybody, then eighteen times twelve is two hundred and sixteen children surveyed.
From one blank slice and one count, the size of the whole survey. Two more readings off that chart, and the second one is the one to remember. The bus is the commonest, at a hundred and twenty degrees - that is a third of two hundred and sixteen, so seventy-two children. The two-wheeler and the cycle tie, at sixty degrees each. Thirty-six children apiece. And now: how many children come by taxi?
Stop. There is no taxi slice. The survey asked about five ways of getting to school, and taxi was not one of them. The honest answer is that this chart has nothing to say about taxis. Not zero - nothing. A chart answers about the categories it carries, and refusing to invent a number is a skill, not a failure. Last thing. The whole does not have to be a count of people.
Give me percentages instead - fifty, twenty-five, fifteen and ten. They add to a hundred, and a hundred has to spread over three hundred and sixty degrees. So one percentage point is three and three fifths of a degree. Fifty per cent is a hundred and eighty degrees, not fifty. Twenty-five is ninety, fifteen is fifty-four, ten is thirty-six, and they close. Or make the whole a day. Twenty-four hours round a circle is fifteen degrees an hour, and a two-slice ring showing eight hours asleep turns a hundred and twenty degrees of it.
One warning. Ask three hundred and sixty people a question and the counts and the angles come out equal, because the rate is exactly one. That is an accident of the number three hundred and sixty. Of six different totals in this video, exactly one behaves that way. A pie chart is a division of a whole in a given ratio, and the whole is a full turn. Find the rate, multiply, and check the circle closes.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Dividing a quantity in a given ratioClass 8 · Ch 3, Proportional Reasoning-2
- Simplest form, and using it to test whether two ratios are proportionalClass 8 · Ch 7, Proportional Reasoning-1
Comes up again in
- Telling a story with data, and letting it raise the next questionClass 8 · Ch 5, Tales by Dots and Lines
Either side of this one
- When one quantity rises and the other falls by the inverse factorClass 8 · Ch 3, Proportional Reasoning-2