11 plus Venn Diagrams questions

What a Venn Diagrams question looks like

A Venn Diagrams question in the 11 plus is a drawn figure, not a wall of text. Two or three circles overlap, each carrying a label that is a rule, and some numbers are already written into the regions. Almost every rule here is a number property: multiples, factors, primes, square numbers, odd, even.

That is what separates it from its neighbours. Classification and logic questions ask the same sorting in words alone, and number properties questions test one number for one property, while here the figure is already part filled and the marks come from reading its regions, counting or totalling what sits in them and spotting what is missing. It is a good place to start eleven plus data work, because there is no arithmetic beyond the times tables.

Example one

Two circles overlap. The left one is labelled Multiples of 7 and holds 7 and 49. The right one is labelled Even numbers and holds 2 and 18. The number 15 sits outside both. The overlapping part in the middle is empty.

Which of these numbers belongs in the overlapping part: 21, 16, 28, 35 or 26?

A number in the middle has to obey both labels at once, so test each one twice. 21 and 35 are in the 7 times table but are odd. 16 and 26 are even but do not divide by 7. 28 is even, and 28 divided by 7 is 4.

The answer is 28.

The tempting wrong choices are 21 and 35. They are in the 7 times table, and a child who has found one rule that works often stops there. Being inside one circle is what keeps a number out of the middle.

Example two

Two circles overlap. The left one is labelled Multiples of 5, the right one Multiples of 6, and the numbers from 1 to 30 have been sorted into them. The number 30 sits in the overlapping part.

How many numbers from 1 to 30 belong in the left circle only, meaning inside it but not in the overlap?

Write the multiples of 5 up to 30 first: 5, 10, 15, 20, 25, 30. That is six numbers. Now cross off any that also divide by 6. Only 30 does, because 30 divided by 6 is 5, so it has moved into the overlapping part. What is left is 5, 10, 15, 20 and 25.

The answer is 5 numbers.

The tempting wrong choice is 6, from counting every multiple of 5 and forgetting that the shared number has moved out of that region. Six answers a different question, “how many are in the left circle altogether”, and that is the trap.

Example three

Three circles overlap, labelled Odd numbers, Multiples of 5 and Less than 50. Where the first two overlap sits 75, where the first and third overlap sit 9 and 33, and where the last two overlap sit 20 and 30. The region in the very centre, inside all three circles, holds a number called x.

Which of these could x be: 55, 40, 27, 35 or 10?

The centre needs all three rules to hold, so check every option against all three. 55 is odd and a multiple of 5, but it is not less than 50. 40 and 10 are multiples of 5 and less than 50, but they are even. 27 is odd and less than 50, but it does not divide by 5. Now 35: it is odd, 35 divided by 5 is 7, and it is less than 50.

The answer is 35.

The tempting wrong choice is 55, because two of the three rules work and two feels like enough. Passing two rules puts a number in an outer overlap, not the centre, and that is the difference a three-circle question tests.

How to answer it

1. Read the labels first and say each one as a rule: “left circle, divides by 5; right circle, divides by 6”.
2. Work out what the region named in the question demands: the middle means both rules, a circle “only” means one rule holds and the other does not, and outside both means neither holds.
3. Test each option against every rule in turn and give it a yes or a no. Never stop at the first yes.
4. If the question asks how many, or what the total is, write them out in a list and then count or add. Counting straight off a picture is how numbers get missed or counted twice.
5. Check your answer back against both labels. It takes five seconds and it catches the near miss that passed one rule.

Common mistakes

• Reading “in the left circle” when the question says “left circle only”. The overlap sits inside the left circle, so it is easy to include, but “only” excludes it. More marks go on that one word than on any calculation.
• Stopping as soon as one rule passes. Nearly every wrong option is built from this: 16 is a multiple of 4 but not of 3, 21 is a multiple of 3 but not of 4, 27 is a multiple of 9 but not of 6.
• Forgetting the space outside the circles. A question asking how many numbers there are in total counts the ones written outside, and “outside both” means a number obeying neither rule.
• Treating 2 like the other primes. Where one circle is primes and the other is odd numbers, 3, 5 and 7 sit in the overlap, but 2 sits in the prime circle alone, because it is even.
• Mixing up factors and multiples. Factors of 30 are the numbers that divide into it, 1, 2, 3, 5, 6, 10, 15 and 30, and nothing larger. Multiples of 30 come from counting up in thirties.

How to practise 11 plus Venn Diagrams questions

The useful practice here is not more arithmetic, it is slower reading. When your child gets one wrong, ask them to point at the region the question named and say its rules out loud before they give an answer. Nine times in ten the mistake was in the region, not the numbers. At home, give them two rules and the numbers 1 to 20 and ask them to sort all twenty into the four regions. 11+ Daily has a Venn Diagrams project alongside the rest of the data and statistics work, on laptop, tablet or phone, and the full set is on our 11 plus maths practice page. Short and regular beats long and occasional, the pattern we set out in how much 11 plus practice a child needs each day. The bank is re-checked in rounds, every question is original to 11+ Daily, and nothing is a past paper.

Questions parents ask