And, or and not

Classroom resources

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Students will be able to

• I can find 'and', 'or', 'not' and 'neither' on a Venn diagram, a table or a grid.
• I can work out P(A or B) without counting the overlap twice.
• I can explain that 'or' in maths includes both.

Timing

Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min

Before class

Do the Run board yourself first. Students need Venn diagrams and two-way tables, grids, and complements (the last three lessons).

The one idea: 'and' is the overlap, 'or' is everything inside the circles (one, the other or both), 'not' is outside a circle, and 'neither' is outside both. The common mistake is adding the totals for 'or' (Zoe's mistake), counting the overlap twice.

Inclusive 'or' is the maths meaning; everyday 'or' is often exclusive ("tea or coffee?"). Name the difference once. The addition rule P(A or B) = P(A) + P(B) − P(A and B) appears as a routine, not a formula to memorise.

Wider: Venn diagrams with three circles, and checking 'or' with a complement (neither).

Running the lesson

1. Predict: ask whether a probability can be more than 1.
2. Run: ask how many times the 8 are counted now.
3. Investigate: ask Zoe where the 4 with both are.
4. Modify: ask which outcomes are both over 9 and doubles.
5. Make: ask how many outcomes are in A, in B and in both.

The hard part (on purpose)

The overlap in 'or'. Ask students to point to the overlap and say how many times they have counted it.

When students get stuck

Answers to the checks

1. sport or music: 30 students: 20 play sport, 13 play music, 8 play both. What is P(a student plays sport or music)?
   25/30. 12 sport only, 8 both and 5 music only: each counted once. 33/30 counts the 8 twice (and a probability can't be more than 1).
2. choose the regions: Choose the students who play sport or music. How many are there? Compare with your prediction.
   
   Sport only, both and music only: 12 + 8 + 5 = 25, so P = 25/30 = 5/6.
3. And: 30 students: 13 have a dog, 10 have a cat and 4 have both. What is P(a dog and a cat)?
   4/30
   Both is the overlap: 4 of 30 = 2/15.
4. Find the mistake: Zoe works out P(dog or cat) for the same class. Click the line where she goes wrong.
   line 3
   Line 3: the 4 with both are in the 13 and in the 10, so they are counted twice. 13 + 10 − 4 = 19, so P = 19/30.
5. Or on a dice: A dice is rolled. A is 'an even number' and B is 'more than 4'. Which outcomes are in A or B?
   2, 4, 5, 6
   Even: 2, 4, 6. More than 4: 5, 6. Either or both: 2, 4, 5, 6, so P(A or B) = 4/6.
6. Not music: Choose the students who do not play music.
   
   Sport only and neither: 12 + 5 = 17, so P(not music) = 17/30 = 1 − 13/30.
7. Over 9 or a double: Two dice: choose every outcome with a sum over 9 or a double.
   
   6 sums over 9 and 6 doubles, but (5, 5) and (6, 6) are both: 6 + 6 − 2 = 10 outcomes, 10/36.
8. Neither: 30 students: 13 have a dog, 10 have a cat and 4 have both. What is P(neither)?
   11/30
   Neither is the complement of 'dog or cat': 1 − 19/30 = 11/30.
9. At least one: Which means the same as 'at least one of A and B'?
   A or B
   At least one of them: A, B or both. That is the maths 'or'.
10. Glasses or left-handed: A two-way table of 50 people: 12 wear glasses and are left-handed, 18 wear glasses and are right-handed, 5 are left-handed without glasses, 15 neither. What is P(glasses or left-handed)?
    35/50
    12 + 18 + 5 = 35, so 35/50 = 7/10.
11. Left-handed, no glasses: For the same 50 people, what is P(left-handed and no glasses)?
    5/50
    One cell: 5 of 50 = 1/10.
12. Using the rule: P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. What is P(A or B)?
    0.7
    0.5 + 0.4 − 0.2 = 0.7.
13. Exactly one: In the Sport and Music class (12 sport only, 8 both, 5 music only, 5 neither), what is P(exactly one of sport and music)?
    17/30
    Sport only and music only: 12 + 5 = 17, so 17/30. This is the everyday 'one or the other, not both'.
14. an 'or' event: Two dice: make an event of the form 'A or B', where A and B overlap, with exactly 15 outcomes.
    
    One way: the sum is 9 or more (10 outcomes) or the first dice is 3 (6 outcomes). (3, 6) is in both, so 10 + 6 − 1 = 15.

Solutions

One possible solution for each task. Other correct programs are fine.

not, or and neither › Not music

Open task: any program that does what the step asks.

not, or and neither › Over 9 or a double

Open task: any program that does what the step asks.

not, or and neither › Neither

Open task: any program that does what the step asks.

not, or and neither › At least one

Open task: any program that does what the step asks.

an 'or' event

Open task: any program that does what the step asks.

Stretch and support

Stretch: Find P(exactly one of A and B) for the Sport and Music class, and write it using 'or', 'and' and 'not'.

Support: Four mini Venn diagrams with 'and', 'or', 'not A' and 'neither' shaded, as a reference card.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "And, or and not".

The idea: Calculating compound event probabilities: I can Students use representations to calculate probabilities of compound events using 'and', 'or' (inclusive and exclusive), and 'at least' language.

What understanding looks like:
- Correctly calculates P(A and B) and P(A or B) from a representation.
- Distinguishes inclusive and exclusive 'or' in context.
- Uses the representation systematically rather than guessing.

Mistakes students often make here:
- A student adds the two totals for 'or'.
- A student gives 'and' as everyone in either circle.
- A student leaves out the 'both' people from 'or'.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Representing two-event combinations), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "And, or and not".

Exit ticket

40 students: 25 like maths, 18 like art, 10 like both. Find P(maths or art) and P(neither).

25 + 18 − 10 = 33, so 33/40. Neither: 7/40.

What each step is for

• sport or music: Committing to P(sport or music) shows whether they add the totals and count the overlap twice.
• choose the regions: Choosing 'or' on the diagram shows it is three regions, with the overlap counted once.
• counting once: 'And' is one cell; 'or' adds the totals then takes the overlap once; adding the totals alone counts both twice (Zoe).
• not, or and neither: 'Not' and 'or' on the diagram and on a two-dice grid, and 'neither' as the complement of 'or'.
• and, or, not: Reading and, or, not and exactly one from tables and totals, mixed; counting the overlap twice is the planted slip; exactly one separates the maths or from the everyday one.
• an 'or' event: Building an 'or' event to a given size makes them handle the overlap on purpose.

Where it leads

This completes AC9M8P02.

Where it leads: simulations of compound events (next lesson), and independent events in Year 10.