Math Detours · Grade 5 · Around the corner · 17

Sorting with circles

Multiples of 2 in one circle, multiples of 3 in the other. What lives in the overlap, and what is left outside?

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Tonight's idea

Take the numbers 1 to 12 and two rules. Rule one: the number is a multiple of 2. Rule two: it is a multiple of 3.

Draw a circle for each rule and let the circles overlap. This picture is called a Venn diagram.

multiples of 2 multiples of 3 2 4 8 10 6 12 3 9 1 5 7 11
The numbers 1 to 12, sorted. Every number has exactly one home.

The picture has four regions. A number can pass rule one only, rule two only, both, or neither.

The overlap holds the numbers that pass both rules. Here that is 6 and 12, the multiples of 6. The numbers outside both circles pass neither rule.

Now count. There are 6 multiples of 2 and 4 multiples of 3. But only 8 numbers are inside the circles, not \(6 + 4 = 10\). Adding counted 6 and 12 twice.

Inside the circles = first circle + second circle − overlap. Here \(6 + 4 - 2 = 8\).

Watch one: football and chess

A class has 24 children. 14 play football, 9 play chess, and 4 play both. How many play neither?

Fill in the overlap first: 4.

Football only: \(14 - 4 = 10\).
Chess only: \(9 - 4 = 5\).
Inside the circles: \(10 + 4 + 5 = 19\).
Outside: \(24 - 19 = 5\).

Five children play neither. Check with the rule: \(14 + 9 - 4 = 19\). Starting from the overlap stops you counting anyone twice.

Your turn

  1. Draw two overlapping circles in a box. Label them "multiples of 2" and "multiples of 5". Put each number from 1 to 20 in its home. Which numbers are in the overlap?
  2. Now use the numbers 1 to 30, with circles for "multiples of 3" and "multiples of 5". Do not draw it. (a) Which numbers are in the overlap? (b) What one rule describes them? (c) How many numbers are outside both circles?
  3. A club has 25 children. 16 have a brother and 14 have a sister. Every child has at least one of the two. How many have both a brother and a sister?
  4. Sort 1 to 12 again. This time the circles are "multiples of 2" and "multiples of 4". (a) One of the four regions is empty. Which one, and why? (b) Draw a better picture for these two rules, with no empty region.

Stretch: three circles

Draw three circles that all overlap each other: multiples of 2, of 3 and of 5. Sort the numbers 1 to 30. Which number sits in the middle, inside all three? How many numbers are outside all three?

Answers

  1. Multiples of 2 only: 2, 4, 6, 8, 12, 14, 16, 18. Multiples of 5 only: 5, 15. Overlap: 10 and 20. Outside: 1, 3, 7, 9, 11, 13, 17, 19.
  2. (a) 15 and 30. (b) They are the multiples of 15. (c) There are 10 multiples of 3 and 6 multiples of 5. Inside the circles: \(10 + 6 - 2 = 14\). Outside: \(30 - 14 = 16\).
  3. \(16 + 14 = 30\), but there are only 25 children. The extra 5 were counted twice, so 5 children have both. Check: 11 brother only, 9 sister only, 5 both, and \(11 + 9 + 5 = 25\).
  4. (a) "Multiple of 4 only" is empty. Every multiple of 4 is also a multiple of 2, so 4, 8 and 12 all sit in the overlap. (b) Draw the "multiples of 4" circle inside the "multiples of 2" circle. Then 4, 8, 12 go in the inner circle and 2, 6, 10 in the ring around it. The odd numbers stay outside.

Stretch. Only 30 is in the middle: it is the first number that 2, 3 and 5 all divide. Eight numbers are outside all three: 1, 7, 11, 13, 17, 19, 23, 29.

Next year's version counts with three circles: grade 6, sheet b26.

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