g10-01-04 · Core topics
Can these class counts coexist?
How many pupils choose neither option?
1 MIN · PREDICT ON PAPER
Predict four disjoint category counts before changing the overlap.
Discuss or write first. No typed answer is required to continue.
Feasible counts sum to 40.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
The overlap is counted twice in a+b, so subtract it once. Invalid negative categories signal inconsistent data. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: Counts are integers; 0≤overlap≤min(a,b), and a+b−overlap≤N.
Mathematical justification: The categories are a−o, b−o, o and N−a−b+o. Nonnegativity gives max(0,a+b−N)≤o≤min(a,b).
1 MIN · EXIT CHECK
N=30,a=20,b=18: can the overlap be 5?
Check after answering
No; the union would be 33, exceeding 30. The overlap must be at least 8.
Teacher guidance and extension
Before: Set operations and integer counts
Ask: Keep an impossible case on screen and ask which assumption it violates.
Watch for: An overlap cannot be chosen independently of the class total and margins.
Challenge: Derive the smallest and largest possible overlap from N,a,b.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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