Building a pie chart: turning counts into angles

What to assume they know

• Dividing a whole in a given ratio, and the fraction form of the rule — Dividing a quantity in a given ratio
• Reducing a ratio to simplest form by its HCF — Simplest form, and using it to test whether two ratios are proportional
• Highest common factor of a small set of numbers
• That a full turn is 360°, and measuring and drawing an angle with a protractor
• Drawing a circle with compasses and drawing a radius
• Reading a percentage as a fraction of a whole

What they 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

Where it usually goes wrong

• "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.

Questions to check understanding

• Convert a table of counts into central angles and draw the chart
• Convert a table of percentages into central angles and draw the chart
• Given a chart with one sector's angle missing, recover it
• Given a sector's angle and one known count, find the total
• Given a total and a sector's angle, find that sector's count
• Identify the largest and smallest categories from a chart without computing anything
• Identify two categories with equal counts from a chart
• Answer, correctly, a question about a category the chart does not carry
• Reduce a count ratio before computing, and justify that the answer is unaffected

Examples worth working on the board

Values marked printed are worked out on the page. Values marked not in the book are worked out here or an added reading of a figure. The chapter prints every angle for the grade example and no answer for any Figure it Out item.

• The grade data (Part II §3.5, p.60). A two-row table: the grade row reads A, B, C, D, E; the student row reads 12, 10, 8, 6, 4. Printed, and set in a pink-ruled table.
• The chart already drawn (Part II p.60, artwork right of the text). A finished five-sector pie chart titled Grade Distribution (40 students), with the count printed inside each sector — 12, 10, 8, 6, 4 — and the grade letter printed just outside it. Read off the printed page: starting from the twelve-o'clock radius the sectors run A, B, C, D, E anti-clockwise, and they are coloured green, yellow, blue, magenta and orange in that order. The title carries the total, 40 students, which the table itself does not state.
• The angle argument (Part II §3.5, p.61). Printed: each grade's angle has to be proportional to the number of students holding that grade, the circle offers 360° in all, so 360 is what gets divided in the ratio 12 : 10 : 8 : 6 : 4.
• The reduction (Part II p.61). Printed: the HCF of the five counts is 2, and dividing through gives 6 : 5 : 4 : 3 : 2.
• The five angles (Part II p.61). Printed: grade A is 6 over the sum 6 + 5 + 4 + 3 + 2, that is 6 over 20, times 360°, shown as 6 × 18 = 108°; grade B is 5 × 18 = 90°; then 4 × 18 = 72°, 3 × 18 = 54° and 2 × 18 = 36°. The page shows the first two as full fractions and the last three in the compact multiply-by-18 form.
• The two rates (not in the book, and the argument of sections 4 and 5). Before reducing, the total count is 40 and the rate is 360 ÷ 40 = 9° per student, so grade A is 12 × 9 = 108°. After reducing, the total is 20 parts and the rate is 360 ÷ 20 = 18° per part, so grade A is 6 × 18 = 108°. One rate is exactly the HCF times the other — 18 is 2 × 9 — because one part is two students. Say "times", not "differ by": the two rates differ by 9, and a student who hears "differ by the HCF" will subtract. Both routes give every angle identically, and that is the concrete form of the thesis. The chapter takes only the second route and never says why the first would agree.
• Closure check (not in the book): 108 + 90 + 72 + 54 + 36 = 360.
• The construction, seven steps (Part II pp.61–62). Part II p.61 carries Steps 1 and 2 as a one-by-two row of boxed panels; Part II p.62 carries Steps 3, 4, the combined "Steps 5,6" and Step 7 as a two-by-two grid. Printed: Step 1 draws a circle and a radius, lettering the centre A and the edge point B. Step 2 measures 108° from AB, turning anti-clockwise about A, and draws the new radius AC. Step 3 measures 90° from AC and draws AD. Step 4 measures 72° from AD and draws AE. Steps 5 and 6 are given together as "finish the rest the same way". Step 7 colours and labels the slices. Read off the printed pages: in the completed line drawing the five radii are lettered AB, AC, AD, AE and AF, with the marked angles 108°, 90°, 72°, 54° and 36° reading round from AB, and in the coloured panel only two slices are labelled — Grade A in green and Grade B in yellow — while the other three are left white and unlabelled. That half-finished last panel is the chapter inviting the student to complete it.
• Figure it Out, Part II p.62, three items. All inputs, no printed answers:
  • 360 people each name one favourite among rainy, winter and summer; 90 pick summer, 120 pick rainy, and the remainder pick winter. Draw the chart.
  • Students' favourite channel types, given as percentages: entertainment 50, sports 25, news 15, information 10. Draw the chart.
  • Collect the class's own favourite-subject counts, one subject per student, into a printed blank table and then chart them. The table's seven subject columns read Language, Arts Education, Vocational Education, Social Science, Physical Education, Maths, Science, and every data cell is blank — confirmed on the printed page, not inferred from extraction. The table sits wholly on Part II p.63; what breaks across the page boundary is only the sentence of item 3, which runs off the foot of Part II p.62 and resumes at the top of Part II p.63. Nothing of the table is on p.62.
• Worked answers to those three (not in the book): item 1 — winter takes 360 − 90 − 120 = 150 people, and with 360 people the rate is exactly 1° per person, so the angles are 90°, 120° and 150°; the coincidence between the head count and the degree count is worth pointing out and then warning against, since it holds only for this number. Item 2 — percentages convert at 3.6° per percent, giving 180°, 90°, 54° and 36°, which close at 360°. Item 3 has no answer to have.
• The transport pie chart (Part II p.68, exercise item 5 — see Notes for why it is here). A five-sector chart with the sectors labelled Walk, Bus, Two-wheeler, Car and Cycle. Printed angles, read off the printed page: Walk 90°, Bus 120°, Two-wheeler 60°, Cycle 60°. The Car sector carries no printed angle — that is the point of the item, and it is a claim. The four questions asked: which mode is commonest; what fraction travel by car; if 18 children travel by car, how many were surveyed in all, and how many use taxis; and which two modes carry equal numbers.
• Worked answers to that item (not in the book, none printed): Car must be 360 − (90 + 120 + 60 + 60) = 30°, so the fraction by car is 30/360 = 1/12; 18 children being one twelfth means 18 × 12 = 216 children surveyed; the number using taxis is zero as far as this chart can say — taxi is not one of its five categories, and the question is a check on whether a student will invent a reading; Bus is the commonest at 120°; and Two-wheeler and Cycle tie at 60° each, which is 36 children apiece on a survey of 216.
• The sleep rings (Part II p.67, exercise item 4 — again, see Notes). Eight ring charts laid out four and four, each split into just two arcs, one indigo and one orange, with a drawing of a sleeping creature in the middle: a giraffe, an elephant, a human child, a tan four-legged animal that reads as a puppy, a cat, a squirrel, a coiled green snake and a hanging bat. The student is told to assign each one an average daily sleep from a printed list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18. Not in the book: the whole here is a day, so 24 hours spans 360° and one hour is 15°; the indigo arc is the sleeping share. Measuring the eight indigo arcs on the printed page gives giraffe 2.5, elephant 3.5, human child 8, puppy 10.5, cat 13, squirrel 15, snake 18, bat 20 — each list value used exactly once. That assignment is an added measurement and is flagged as such in Notes.

Figures to have open

• The chapter's finished grade chart (Part II p.60) with counts inside the sectors and letters outside. Redraw as a schematic, keeping the anti-clockwise A-to-E order and the five distinct colours; the counts inside the sectors are what make sections 1 and 8 possible.
• The seven construction panels (Part II pp.61–62). Redraw as a step-by-step sequence rather than seven stills — the point a still cannot make is that each angle is measured from the radius drawn immediately before, not from the base radius. Keep the lettering A, B, C, D, E, F as printed, including the jump to F on the last radius.
• A side-by-side of the two rate computations, 9° per student and 18° per part, with the shared answer 108° ringed. Standard schematic. This is the topic's central figure and the chapter has nothing like it.
• The transport chart (Part II p.68) with four angles printed and one sector blank. Redraw; the blank sector must genuinely be blank.
• One sleep ring beside a 24-hour dial. Standard schematic. The full set of eight is optional decoration; one worked ring carries the idea.
• A protractor overlay on a circle, correctly seated on the centre with its baseline on the existing radius. Standard schematic, and worth its own moment — seating the protractor is where classroom charts actually go wrong.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 3, "Proportional Reasoning-2", §3.5 "A Slice of the Pie", Part II pp.60–63. §3.5 begins in the lower half of Part II p.60 with the grade table and the finished chart; the angle computation and construction Steps 1 and 2 are on Part II p.61; Steps 3 to 7 and the three-item Figure it Out are on Part II p.62; and the blank subject table belonging to item 3 sits wholly on Part II p.63, immediately above §3.6, with only item 3's sentence carrying over the page boundary.
• Two chart items that sit inside §3.6's exercise sets and belong to this topic mathematically: the eight sleep rings, Part II p.67 item 4, and the transport pie chart with its four questions, Part II p.68 item 5. When one quantity rises and the other falls by the inverse factor notes the handover.
• Backward pointer: Dividing a quantity in a given ratio for the division rule this section applies to a whole of 360°, and Part I printed Chapter 7, §7.3, for simplest form.