Venn diagrams and two-way tables

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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• Worksheet Download Copy address for Prepare Print the student copy; the Teacher edition has the answers.

Students will be able to

• I can sort a group into one, the other, both and neither.
• I can build a Venn diagram or a two-way table from a story.
• I can change a Venn diagram into a two-way table, and back.

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 sample spaces and fractions as probabilities.

The one idea: two properties make four groups, and every person is in exactly one. The common mistake is writing a whole total (22 like tea) in the 'only' part, counting the overlap twice (Lia's mistake).

The board shows the diagram and the table together; choosing a region highlights the matching cell. In Modify, students fill one from the other (the board ticks when all four are right).

Probabilities from these (and, or, not) come in two lessons. Tree diagrams and grids (the other representations in 8P02) are next.

Running the lesson

1. Predict: ask who might have been counted twice.
2. Run: ask which two groups make the whole Sport circle.
3. Investigate: ask Lia how many like tea but not coffee.
4. Modify: ask where 'both' goes in the table.
5. Make: ask which picture answers 'how many ride a bike in all?' faster.

The hard part (on purpose)

The overlap. Ask students to write the 'both' number first, every time, and take it from each total.

When students get stuck

Answers to the checks

1. more than the class: In a class of 30, 20 play sport and 13 play music. 20 + 13 = 33. How can that be?
   Some play both. They were counted once with sport and again with music. Here 8 play both.
2. find the groups: Choose everyone who plays sport: click the parts of the diagram (or the table, or the buttons). Does it match the table's total? Who was counted twice in your prediction?
   
   Sport only (12) and both (8) make 20. The 8 who play both are also in the music circle: 20 + 13 counts them twice.
3. Neither: 40 people: 22 like tea, 18 like coffee and 7 like both. How many like neither?
   7
   Tea only 22 − 7 = 15, coffee only 18 − 7 = 11. 15 + 7 + 11 = 33, so 40 − 33 = 7 like neither.
4. Find the mistake: Lia draws the Venn diagram for the same 40 people. Click the line where she goes wrong.
   line 3
   Line 3: 22 like tea, but 7 of them are already in the overlap. Tea only is 15 and coffee only is 11. A negative count shows something went wrong.
5. Tables or diagrams: Which is easier to read from a two-way table than from a Venn diagram?
   The totals, such as how many play sport in all
   The table has total rows and columns; in a Venn diagram you add the parts yourself.
6. Diagram to table: Fill in the two-way table from the Venn diagram.
   
   Dog and cat 4, dog not cat 9, cat not dog 6, neither 11. The totals check it: 13 have a dog, 10 a cat, 30 in all.
7. Table to diagram: Now fill in the Venn diagram's four numbers from the table.
   
   Glasses only 7, both 2, left-handed only 3, neither 16.
8. Exactly one: In the Sport and Music class (12 sport only, 8 both, 5 music only, 5 neither), how many students do exactly one of the two?
   17
   Sport only and music only: 12 + 5 = 17.
9. Outside the circles: Which cell of the two-way table matches the part of the Venn diagram outside both circles?
   Not sport and not music
   Outside both circles means in neither group.
10. Row total: A two-way table: 18 students walk and own a bike, 7 walk and don't own a bike. How many students walk in all?
    25
    The Walk row: 18 + 7 = 25.
11. Case only: 60 people: 35 have a phone case, 20 have a screen protector, 12 have both. How many have a case but no screen protector?
    23
    35 − 12 = 23.
12. Neither: In that group of 60, how many have neither?
    17
    Case only 23, both 12, protector only 8: 43. So 60 − 43 = 17 have neither.
13. Review: complement: Review: a bag has 7 counters, 2 of them green. What is P(not green)?
    5/7
    1 − 2/7 = 5/7.

Solutions

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

from one to the other › Diagram to table

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

from one to the other › Table to diagram

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

from one to the other › Exactly one

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

from one to the other › Outside the circles

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

your own survey

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

Stretch and support

Stretch: Three properties need three circles: how many regions does that make? Try it for sport, music and art.

Support: A blank Venn diagram and a blank two-way table on one sheet, with 'both' marked in each.

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 "Venn diagrams and two-way tables".

The idea: Representing two-event combinations: I can Students represent all possible combinations for two events using two-way tables, tree diagrams and Venn diagrams.

What understanding looks like:
- Correctly lists all combinations for a two-event situation.
- Represents combinations accurately using at least one method.
- Identifies the advantage of each representation for different questions.

Mistakes students often make here:
- A student writes the whole total in the 'only' part.
- A student forgets the 'neither' group.
- A student mixes up rows and columns in the table.

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.
- 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 "Venn diagrams and two-way tables".

Exit ticket

50 students: 30 study French, 25 study Japanese, 10 study both. How many study neither?

French only 20, Japanese only 15, both 10: 45, so 5 study neither.

What each step is for

• more than the class: Committing to an explanation for 20 + 13 > 30 shows whether they think of people who are in both groups.
• find the groups: Choosing regions shows each person is in exactly one of four groups, and a whole circle is two of them.
• build them from a story: Start with the overlap and subtract it from each circle's total; putting a whole total in the 'only' part counts the overlap twice (Lia).
• from one to the other: Filling a table from a diagram and a diagram from a table shows the four cells are the four regions.
• tables and diagrams: Reading and completing tables and diagrams, mixed; the overlap counted twice is the planted slip; complements are the review.
• your own survey: Planning a two-question survey and its table and diagram shows they can set up both representations.

Where it leads

This starts AC9M8P02 node 1.

Where it leads: grids and tree diagrams (next lesson), and 'and', 'or' and 'not' probabilities.