When everyone in a group is sorted by two yes/no questions, a Venn diagram and a two-way table show the same four groups: one, the other, both and neither. Each person is in exactly one of them.
In a class of 30, 20 play sport and 13 play music. That adds to 33, more than the class. Nobody miscounted. What is going on?
A Venn diagram shows two properties as overlapping circles inside a box (everyone). A two-way table shows the same people in rows and columns, with totals.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
More than the class
TryTry it on the board and compare the result with your prediction.
Find the groups
Click a part again to remove it. The table's inside cells match the four parts of the diagram:
Music
Not music
Sport
both
sport only
Not sport
music only
neither
Try it yourselfYour edits are saved in this browser.
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?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Build them from a story
Here is the routine. Start in the middle.
Worked exampleBefore each step, say what you think comes next.
25 students: 15 dog,10 cat,4 bothRead the story.
both=4Write the overlap first.
dog only=15−4=11,cat only=10−4=6Take the overlap from each total.
neither=25−(11+4+6)=4Everyone else.
Read Lia's working. Can a group have −7 people?
40 people: 22 like tea, 18 like coffee and 7 like both. How many like neither?
Lia draws the Venn diagram for the same 40 people. Click the line where she goes wrong.
ModifyChange the problem on purpose and see what happens.
From one to the other
Need a hint?
Both goes where the Dog row meets the Cat column.
Another hint
Dog only goes in the Dog row, Not cat column.
Need a hint?
The overlap is the cell where Glasses meets Left-handed.
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Fill in the two-way table from the Venn diagram.
Use the board. This ticks when you get there.
Now fill in the Venn diagram's four numbers from the table.
Use the board. This ticks when you get there.
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?
PractiseA few on your own, to check you have it.
Tables and diagrams
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?
60 people: 35 have a phone case, 20 have a screen protector, 12 have both. How many have a case but no screen protector?
In that group of 60, how many have neither?
Review: a bag has 7 counters, 2 of them green. What is P(not green)?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Your own survey
The board shows a finished example (bus and bike) to compare with. Ask your two questions to your class or family, then draw both pictures. Which is quicker to read, and for which question?
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Need a hint?
Pick two yes/no questions, such as 'Do you catch a bus?' and 'Do you ride a bike?'
Another hint
Count the overlap first.
For teachers
Students read, build and convert between Venn diagrams and two-way tables for two yes/no properties (AC9M8P02, node 1). About 40 minutes.
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.
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
Predict: ask who might have been counted twice.
Run: ask which two groups make the whole Sport circle.
Investigate: ask Lia how many like tea but not coffee.
Modify: ask where 'both' goes in the table.
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
A student writes the whole total in the 'only' part. Ask: Are some of those people in the overlap too? Fix: Write both first; only = total − both.
A student forgets the 'neither' group. Ask: Is everyone in a circle? Fix: The box is everyone; neither = total − the other three.
A student mixes up rows and columns in the table. Ask: Read the row name, then the column name: what do both say? Fix: Each inside cell is one row property and one column property.
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.
Where it leads
This starts AC9M8P02 node 1.
Where it leads: grids and tree diagrams (next lesson), and 'and', 'or' and 'not' probabilities.
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.
Now you can
Sort a group into one, the other, both and neither.
Build a Venn diagram or a two-way table from a story.
Change a Venn diagram into a two-way table, and back.