At a glance. One lesson of about 60 minutes. It assumes the symbols from Lesson 1. Students use n(A ∪ B) = n(A) + n(B) − n(A ∩ B), find the region counts of a two-set diagram, and fill a three-set diagram from the centre outwards.
Calculator: no calculator. None is needed. The arithmetic is addition and subtraction of small numbers, and the structure of the diagram is the reasoning.
What students will learn
Know
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
- n(U) − n(A ∪ B) is the number outside both sets.
- The eight regions of a three-set diagram.
Understand
- A total counts every region inside its circle, including the overlaps.
- Working from the centre outwards means each subtraction removes only counts that are already placed.
- n(A ∪ B) is never more than n(A) + n(B).
Do
- Find any region of a two-set diagram from totals.
- Fill in a three-set diagram from totals that include the overlaps.
- Say how many are in at least one set, exactly two sets, or none.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the union formula and the regions of two- and three-set diagrams (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to predict the union before moving the overlap, then to say how the union changes when the overlap grows.
- Criterion C. Communicating: labelling every region with its count and setting out the subtractions in order (the worked example and question 7).
- Criterion D. Applying mathematics in real-life contexts: surveys of clubs and transport, and why a total can exceed the number of people.
The one thing to do with the figure
Make students commit before they slide. In the first figure, set n(A) = 14 and n(B) = 11, hide the overlap, and ask “If 4 are in both, what is n(A ∪ B)?” Many will say 25. Then reveal.
In the second figure, cover the totals list and ask students to say which region they would fill first and why. The order is forced: each total includes the regions inside it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the two lists that add to more than the year group. Collect predictions about the overlap. |
| 10 min | First figure and key idea. Settle the hook. |
| 12 min | Second figure, the warning card and the worked example; pairs finish the faded example. |
| 5 min | The check. If many choose A, return to the overlap in the first figure. |
| 16 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 9. |
The answers
| Warm-up | {2, 4}; 16; 13. |
|---|---|
| Hook | Some students must be on both lists. If all 60 are on at least one list, at least 38 + 29 − 60 = 7 are on both. |
| Worked example | 6 own neither. |
| Faded example | A only = 7. |
| 1. The union formula | B. A adds without removing the overlap; C removes it twice; D adds it again. |
| 2. Neither, from two totals and an overlap | 16. union 30; 46 − 30. |
| 3. One set only | 8. 15 − 7. |
| 4. A missing total | 26. 45 − 28 + 9. |
| 5. Exactly two of three sets | 18. 7 + 6 + 5. |
| 6. Complement from regions | 15. 11 + 4. |
| 7. Total from a three-set diagram | 74. 5 + 16 + 47 + 6, where 16 is the pairs-only regions and 47 the singles. |
| 8. Bounds on a union | C. A forgets the overlap; B reverses the inequality; D reverses the other one. |
| 9. Both, from a survey | A. B gives the union; C repeats the given 5; D gives the sum. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Adds without removing the overlap, removes it twice, or adds it again. | Draw two circles, write 8 in the middle, then ask how many of the 25 are not in the middle. |
| 7, 25 or 30 in question 2 | Counted the overlap twice, removed it twice, or gave the union. | Ask: how many play at least one sport? Then what is left over? |
| 15, 7 or 10 in question 3 | Gave the whole set, the overlap, or the number in neither. | Shade only the part of the music circle outside the art circle. |
| 17, 8 or 64 in question 4 | Stopped at B only, took the overlap off B only, or reversed the formula. | Substitute the answer back into the formula as a check. |
| 27, 24 or 21 in question 5 | Did not remove the centre from each pair total. | Fill in the centre of a three-set diagram first, then the pair-only regions. |
| 16, 11 or 27 in question 6 | Gave n(A), only B only, or the union. | Shade A′ on the diagram; read off the regions that are shaded. |
| 100, 69 or 68 in question 7 | Added the totals, forgot to add the centre back, or forgot the none region. | Fill the diagram, label all eight regions, and add them at the end. |
| A, B or D in question 8 | Thinks the union is always the sum, or reverses an inequality. | Draw a case with a big overlap and a case with none. |
| B, C or D in question 9 | Gave the union, repeated the neither, or gave the sum. | Write each given number on the diagram and mark the region still missing. |
Questions worth asking
- Why does the sum n(A) + n(B) count the overlap twice? Listen for “the elements in both sets are in A's total and also in B's total”.
- In a three-set diagram, why fill the centre first? Every total and every pair total includes the centre, so nothing else can be found without it.
- When does n(A ∪ B) = n(A) + n(B)? Exactly when A ∩ B is empty: the sets have nothing in common.
- Can n(A ∪ B) be smaller than n(A)? No: A is part of A ∪ B.
Differentiation
A way in for anyone stuck
- Give a printed three-circle diagram with the eight regions numbered, and have students write which region each given total covers.
- Use a different colour for each total, and colour the regions it covers.
- Write the order on the board: centre, pairs, singles, outside.
Stretch
- Derive the three-set formula n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C) from the regions, and say why the last term is added.
- Among 100 people, 70 own a phone and 65 own a laptop. What are the smallest and largest numbers who could own both?
Students learning in English as an additional language
“Overlap” means one thing partly lying over another. “At least one” means one or more, and “exactly two” means two and not three. Read each sentence aloud and underline words such as “only”, “both”, “neither” and “at least” before drawing anything.
| Word | Plain meaning | Example |
|---|---|---|
| overlap | the elements that are in two sets at the same time | students on both the drama and the debating list |
| only | in this set and not in the others | in drama only |
| neither | in none of the sets being discussed | on no list at all |
| at least one | in one set or more than one | on one list or on both |
| total | the number of elements in a set, including any overlap | n(A) |
| region | one of the separate parts of a Venn diagram | the part where three circles meet |
Sentence starters for explaining aloud or in writing:
- The overlap is counted twice because …
- To find A only, I take … away from ….
- I filled the centre first because ….
Two things not to say
“Just subtract the overlap.” Subtract it from what, and why? Students need to see that the sum counted it twice.
“Put the given number in the region.” A given total covers several regions, so it is a sum of regions, not one region.
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.
- Two setsn(A) = 20, n(B) = 12 and n(A ∩ B) = 4. Find n(A ∪ B).
28. - OnlyIn the same diagram, how many are in A only?
16.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextOf 50 visitors to a museum in Lima, 30 saw the gold room, 25 saw the textiles and 8 saw neither. How many saw both?
50 − 8 = 42 saw at least one, and 30 + 25 − 42 = 13 saw both. - Earlier work (Lesson 1)U = {1, …, 12}, A = {x | x is even}. List A′.
{1, 3, 5, 7, 9, 11}.
3Challenge
Reasoning, generalizing and working backwards.
- Reasoningn(A ∩ B) = 0, n(A) = 12 and n(B) = 9. What are the smallest and largest values n(U) could have if n(U) must hold both sets?
At least 21; there is no largest, because U can have any number of elements outside A and B. - Working backwardsn(A ∪ B) = 40, n(A only) = 18 and n(B only) = 15. Find n(A ∩ B).
40 − 18 − 15 = 7.
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? Draw the three-circle diagram on the board and have each pair fill in a different layer.