At a glance. One lesson of about 60 minutes. Students find sector angles from frequencies and percentages, check that they add to 360 degrees, and compare shares between pies.
Calculator: no calculator. Nothing here needs one. Every total divides 360 or is a friendly percentage.
What students will learn
Know
- The angle of a sector is its frequency divided by the total, times 360 degrees.
- Sector angles add to 360 degrees.
- A percentage of a pie is that percentage of 360 degrees.
Understand
- A pie chart shows proportion of its own whole, not the amount.
- Finding the angle for one item first makes every sector a single multiplication.
- Two pies with equal angles can stand for different numbers.
Do
- Calculate sector angles from a table.
- Convert between percentages, angles and frequencies.
- Check a pie chart by adding the angles.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: sector angles from frequencies and percentages (questions 1 to 7).
- Criterion B. Investigating patterns: the table figure. Ask students to find the rule linking the total to the degrees for one student, and to say why all six totals divide 360.
- Criterion C. Communicating: explaining why equal angles do not mean equal numbers (question 7 and the what-if).
- Criterion D. Applying mathematics in real-life contexts: breakfasts, clubs, pets and ticket sales (the hook and worked examples).
The one thing to do with the figure
Ask for a prediction before the first slide. “24 students share 360 degrees. How many degrees for one student?” Then change the table and ask for the new value before each change shows it.
In the second figure, set 50% and then 25% and ask for 75% before moving. Name the pattern: every 5% adds 18 degrees.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure on tables and sectors. |
| 10 min | Key idea; the second figure on percentages. |
| 10 min | Worked example on clubs; students finish the faded pets example in pairs. |
| 5 min | Drawing: each student draws the pets pie with a protractor, each angle measured from the last line. The last sector is the check. |
| 5 min | Question 1, the check. If many choose A or B, go back to the first figure. |
| 10 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 130; 75%; the number of values. |
|---|---|
| Worked example | 144°, 108°, 72°, 36°. |
| Faded example | 144°, 120°, 72°, 24°. |
| 1. Sector angle from a frequency | C, 72°. A uses the count; B uses the total; D is the other students' sector. |
| 2. Degrees for one item | 18. 360 ÷ 20. |
| 3. Sector angle from a frequency | 100. 5 × 20. |
| 4. Angle from a percentage | 108. 3 × 36. |
| 5. Frequency from an angle | 5. 25 ÷ 5. |
| 6. Missing sector angle | 60. 360 − 300. |
| 7. A pie shows proportion | B. A treats equal angles as equal counts; C misreads the angles; D rejects a fair comparison of shares. |
| 8. Frequency from a pie chart | 40. 120° is 1/3, and 1/3 of 120 = 40. |
| 9. Angles of one pie chart | B. A gives the total; C reads the angle as a count; D forgets that the pie is only its own 60 students. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used the count, the total, or the other group's angle. | Ask what one student is worth in degrees before counting. |
| 20 or 340 in question 2 | Gave the number of students, or took it from 360. | Say aloud: share 360 equally into 20 parts. |
| 5 or 260 in question 3 | Gave the frequency, or the angle for everyone else. | Ask which group the question is about. |
| 30, 90 or 1080 in question 4 | Gave the percentage, used 3 for each per cent, or 36 for each per cent. | Write 10% = 36° and count the tens. |
| 72, 0.2 or 14.4 in question 5 | Gave the angle, left the share as a decimal, or stopped at one student’s angle. | Ask what fraction of 360 is 72, then take that fraction of 25. |
| 300 in question 6 | Gave the sum so far. | Draw the circle with the missing sector blank. |
| A, C or D in question 7 | Treated equal angles as equal counts. | Ask what fraction each is and what the whole is. |
| 120 or 30 in question 8 | Gave the angle, or the child tickets. | Ask what the question is about before the arithmetic. |
| A, C or D in question 9 | Gave the total, read the angle as a count, or forgot the pie is only its own group. | Ask: what fraction of the circle is 120 degrees, and a third of which number? |
| Drawing the pie (worked example, step 4) | Measured every angle from the first radius, so the sectors overlap. | Say: each angle starts at the line you drew last. Check that the last sector comes out at 36 degrees. |
Questions worth asking
- Why must the sector angles add to 360? They are shares of one whole circle. Listen for “the whole circle”.
- Which is quicker, one student's angle first, or a fraction of 360 each time? Both work; one-student first is quicker when totals divide 360.
- A pie chart has a sector of 180. What can you say? Half the group. Listen for “half of them”, and that the number depends on the total.
- When is a pie chart a poor choice? When there are many groups or you need exact counts. Listen for a reason.
Differentiation
A way in for anyone stuck
- Give a pre-drawn circle marked in 10 degree steps and start with totals of 20, 36 and 40.
- Write the angle for one student in a box before any sector is calculated.
- Use 10% = 36 degrees as the anchor for percentages.
Stretch
- Question 8, then: 120 tickets, and the Child sector is 60 degrees. How many child tickets? (20.)
- A pie chart has total 48. Why is each student a decimal number of degrees? (360 ÷ 48 = 7.5.) Which totals give whole numbers?
Students learning in English as an additional language
A “sector” is a slice of a circle, like a slice of pizza. A “pie chart” shows how a whole is shared. Say “out of the whole” each time you say “share”.
| Word | Plain meaning | Example |
|---|---|---|
| pie chart | a circle cut into sectors that show shares of a whole | how 24 students have breakfast |
| sector | a slice of a circle between two radii | the sector for Rice |
| angle | the amount of turn of a sector at the centre, in degrees | 150 degrees |
| proportion | a share of a whole | a quarter of the group |
| total | the whole amount | 24 students |
| degrees | the units for angles; a whole circle is 360 | 90 degrees is a quarter turn |
Sentence starters for explaining aloud or in writing:
- One … is worth … degrees because 360 ÷ … = …
- The sector for … is … × … = … degrees.
- The angles add to …, so …
Two things not to say
“Divide by 360.” It is the other way: the angle is the fraction of 360.
“The biggest slice is the best.” The biggest slice is the largest share, not the best.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- One itemA pie chart is for 12 students. How many degrees for one student, and for the 4 who chose tea?
30°; 120°. 360 ÷ 12 = 30, and 4 × 30 = 120. - Add themFour sector angles are 150°, 120°, 80° and 20°. Could they be the sectors of one pie chart?
No. They add to 370°, which is more than a whole circle of 360°.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA shop sold 90 items: 45 books, 27 pens and 18 rulers. Find the three sector angles.
180°, 108°, 72°. One item is 4°. - Earlier work (Unit 2, lesson 3)Find 40% of 360 and say what it is on a pie chart.
144°; the angle for 40%.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA pie chart of 60 students has a sector of 54°. How many students? Check it as a percentage.
9 students; 54 ÷ 360 = 15%. - ComparePie A has 80 students and a 45° sector. Pie B has 160 students and a 45° sector. How many in each sector?
10 and 20: equal shares of different wholes.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Use a protractor and a circle template, or a paper plate folded into quarters and eighths.