The shape of the lesson
By the end of the lesson, learners can
- Set language: In the lab, set the universal set \(\xi = \{1, \dots, n\}\), choose a rule for A and for B, and watch each number drop into its region of the Venn diagram.
- Venn diagrams from counts: Often you are given numbers of elements, not the elements themselves.
- Three sets and subsets: Extended.
- List each set in full before combining them.
- Union \(\cup\): either or both, each element once. Intersection \(\cap\): both.
- Complement \(A'\): everything in \(\xi\) not in A.
- From counts: \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).
- Three sets: centre first, then the pairs, then the 'only' regions, then outside. Check the total.
One picture every idea returns to
Fill a Venn diagram from the middle outwards: the intersection first, then the 'only' regions, then the outside. \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\), because the overlap is counted twice otherwise.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Draw the overlap.. A set is a collection of things, and a Venn diagram shows how sets overlap. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Set language | In the lab, set the universal set \(\xi = \{1, \dots, n\}\), choose a rule for A and for B, and watch each number drop into its region of the Venn diagram. Project tab 3 of the lab (Number lab, sets and Venn diagrams) and let a student drive it while the class predicts. Model the worked example Listing sets on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Venn diagrams from counts | Often you are given numbers of elements, not the elements themselves. Model the worked example Finding the overlap on the board, one line at a time. Your turn: \(n(\xi) = 40\), \(n(A) = 22\), \(n(B) = 19\) and \(n(A \cap B) = 8\). Find \(n\big((A \cup B)'\big)\). | Pairs |
| 21 to 30 min | Teach and check: Three sets and subsets | Extended. Model the worked example Three sets (Extended) on the board, one line at a time. | Anticipation / Reaction Guide |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 18 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Three sets and subsets, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Anticipation / Reaction Guide
Before the lesson, students mark five statements true or false. After the teaching they mark them again and justify any change of mind in one sentence each. The statements come from the examiner traps below.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| In a union, list a shared element only ONCE. | Writing \(A \cup B = \{2, 4, 6, 8, 10, 12, 3, 6, 9, 12\}\) double-counts 6 and 12 and gives the wrong \(n(A \cup B)\). |
| Do not write \(n(A) = 18\) in the 'A only' region. 18 is the whole circle; the 'A only' region is \(18 - 7 = 11\). | Writing 18 there makes the diagram add up to more than 30. |
| \(\varnothing\) means the empty set; \(\{0\}\) is NOT empty, it has one element (the number 0). | And \(n(\varnothing) = 0\). |
Sets, set notation and Venn diagrams: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
(a) \(\xi = \{1, 2, \dots, 15\}\) and \(A = \{x : x \text{ is a prime number}\}\). List the elements of A and write down \(n(A)\).
(b) List the elements of \(\{x : x \text{ is an integer},\ -2 < x \le 3\}\).
\(\xi = \{1, 2, \dots, 16\}\), \(A = \{\text{odd numbers}\}\) and \(B = \{\text{square numbers}\}\).
(a) List \(A \cap B\). (b) Find \(n(A \cup B)\). (c) List \(A' \cap B\).
50 people were asked whether they drink tea (T) or coffee (C). 28 drink tea, 23 drink coffee and 9 drink neither.
(a) Find how many drink both. (b) Find \(n(T \cap C')\) and say what it means.
Describe in words the elements of (a) \(A \cap B\) (b) \((A \cup B)'\).
(Extended) \(\xi = \{x : x \text{ is an integer},\ 1 \le x \le 10\}\), \(E = \{\text{even numbers}\}\) and \(F = \{x : x > 6\}\).
(a) List \(E \cap F\). (b) Is \(\{8, 10\} \subseteq E\)? Give a reason. (c) Complete with \(\in\) or \(\notin\): \(7 \ \_\_\ E\). (d) Find \(n(E' \cap F')\).
(Extended) 50 students were asked which of Art (A), Biology (B) and Chemistry (C) they study. 3 study all three. 5 study Art and Biology only, 4 study Art and Chemistry only, and 6 study Biology and Chemistry only. \(n(A) = 20\), \(n(B) = 22\) and \(n(C) = 18\).
Find (a) the number who study exactly one of the subjects (b) the number who study none of them.
Answer key and marking notes
- (a) \(A = \{2, 3, 5, 7, 11, 13\}\), \(n(A) = 6\)
- (b) \(\{-1, 0, 1, 2, 3\}\)
- \(A = \{1, 3, 5, \dots, 15\}\) (8 elements), \(B = \{1, 4, 9, 16\}\).
- (a) \(A \cap B = \{1, 9\}\)
- (b) \(n(A \cup B) = 8 + 4 - 2 = 10\)
- (c) square numbers that are not odd: \(A' \cap B = \{4, 16\}\)
- (a) \(n(T \cup C) = 50 - 9 = 41\); both \(= 28 + 23 - 41 = 10\)
- (b) \(n(T \cap C') = 28 - 10 = 18\): the number of people who drink tea but not coffee.
- (a) elements in both A and B
- (b) elements of \(\xi\) that are in neither A nor B
- \(E = \{2, 4, 6, 8, 10\}\), \(F = \{7, 8, 9, 10\}\)
- (a) \(E \cap F = \{8, 10\}\)
- (b) Yes, because 8 and 10 are both even, so both are in E.
- (c) \(7 \notin E\)
- (d) not even and not more than 6: \(\{1, 3, 5\}\), so \(n(E' \cap F') = 3\)
- Art only \(= 20 - 3 - 5 - 4 = 8\); Biology only \(= 22 - 3 - 5 - 6 = 8\); Chemistry only \(= 18 - 3 - 4 - 6 = 5\).
- (a) \(8 + 8 + 5 = 21\)
- In at least one subject: \(21 + 5 + 4 + 6 + 3 = 39\).
- (b) \(50 - 39 = 11\)
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
\(n(A) = 15\), \(n(B) = 11\) and \(n(A \cap B) = 4\). Find \(n(A \cup B)\).
- A22
- B26
- C30
- D18
In a class of 30 students, 18 play football, 12 play tennis and 5 play neither. How many students play both football and tennis?
- A5
- B0
- C25
- D13
\(\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) and \(P = \{\text{prime numbers}\}\). List the elements of \(P'\).
- A\(\{1, 4, 6, 8, 9, 10\}\)
- B\(\{4, 6, 8, 9, 10\}\)
- C\(\{2, 3, 5, 7\}\)
- D\(\{1, 4, 6, 8, 10\}\)
The Venn diagram shows the number of elements in each region. Find \(n(A \cap B')\).
- A8
- B11
- C3
- D12
40 students each study some, all or none of French, German and Spanish. 2 study all three, 3 study French and German only, 4 study French and Spanish only and 5 study German and Spanish only. In total 15 study French, 14 study German and 16 study Spanish. How many study none of the three languages?
- A11
- B5
- C9
- D0
\(\xi = \{1, 2, 3, \dots, 12\}\), \(M = \{\text{multiples of } 4\}\) and \(E = \{\text{even numbers}\}\). Which statement is true?
- A\(M \subseteq E\)
- B\(E \subseteq M\)
- C\(M \cap E = \varnothing\)
- D\(6 \in M\)
Quiz key
\(n(A \cup B) = n(A) + n(B) - n(A \cap B) = 15 + 11 - 4 = 22\). The tempting 26 is \(15 + 11\), which counts the 4 elements in both sets twice. 30 adds the overlap a third time, and 18 counts only the 'A only' and 'B only' regions (\(11 + 7\)), leaving out the overlap.
\(30 - 5 = 25\) students play at least one sport, so \(n(F \cup T) = 25\). Then \(18 + 12 - 25 = 5\) play both. The answer 0 comes from \(30 - 18 - 12\), forgetting the 5 who play neither. 25 is the number who play at least one sport, not both, and 13 is \(18 - 5\).
\(P = \{2, 3, 5, 7\}\), so \(P'\) is every other element of \(\xi\): \(\{1, 4, 6, 8, 9, 10\}\). Leaving out 1 treats 1 as prime, but 1 has only one factor so it belongs in \(P'\). \(\{2, 3, 5, 7\}\) is \(P\) itself, and \(\{1, 4, 6, 8, 10\}\) misses 9, which is \(3 \times 3\) and not prime.
\(A \cap B'\) is the part of A that is not in B: the 'A only' region, which holds 8. The tempting 11 is \(n(A) = 8 + 3\), which wrongly includes the overlap (that part IS in B). 3 is \(n(A \cap B)\), and 12 is \(n(B') = 8 + 4\).
Centre: 2. Two-language 'only' regions: 3, 4, 5. French only \(= 15 - 2 - 3 - 4 = 6\), German only \(= 14 - 2 - 3 - 5 = 4\), Spanish only \(= 16 - 2 - 4 - 5 = 5\). Inside the circles: \(2 + 3 + 4 + 5 + 6 + 4 + 5 = 29\), so none \(= 40 - 29 = 11\). The tempting 5 comes from forgetting to subtract the centre 2 from each 'only' region (giving 8, 6 and 7, a total of 35). 9 comes from \(40 - (45 - 12 - 2)\), and 0 from treating \(15 + 14 + 16 = 45\) as more than everyone.
\(M = \{4, 8, 12\}\) and \(E = \{2, 4, 6, 8, 10, 12\}\). Every element of \(M\) is in \(E\), so \(M \subseteq E\) is true. \(E \subseteq M\) is false because 2 is in \(E\) but not in \(M\). \(M \cap E = \{4, 8, 12\}\), which is not empty, and 6 is not a multiple of 4, so \(6 \notin M\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.