Lesson 17 Sets and subsets

# Episode 1

Reasoning Resources: Cardboard rings (optional), Worksheet 1
Whole class preparation
Describing properties of different sets of numbers. Classifying using overlapping criteria.

Finding precise descriptions for the different subsets produced.

Understanding that numbers
can be in two different sets at the same time and how this can be represented in a set diagram.
On the board list the numbers 1 to 20. Ask: What types of numbers do we have within this set of numbers? For example, some are even numbers. Write suggested descriptions on the board. Emphasise that you want a description of subsets, not just lists.

Choose two of the subsets which overlap, to demonstrate how to draw a set diagram. (Odd numbers and prime numbers make a good combination.)

Draw a simple rectangle for the whole set and a ring for the first subset. Then involve the class on how to draw gy the second subset and what to put inside it. You may want to use a cardboard ring. Emphasise that the placing of the second ring is flexible but if this subset includes primes any elements of the first subset then it has to overlap it (or be entirely within it).

Emphasise also that the numbers can be written into the diagram in any order, as long as they are in the correct place, and that each number appears only once.

Ask for descriptions of the different sets, including the overlap and the numbers outside the overlapping sets. Push for precise language, e.g.‘Odd numbers which are prime’ ‘Even numbers which are not prime’ etc.
Pair and small group work
Puzzling out why when the numbers in each set are added together the result is more than the total number in the combined sets. Recognising that whenever we talk about two sets, we can actually talk about many more sets through relationships between them. For example for two sets A and B there are : the overall set (often assumed or ignored); the two sets A and B, their complements, A ‘{i.e. Not A) and B ‘; their overlap ANB and its complement; their union AUB and it's complement (AUB) ‘ and more. The numeric relationships between them are precise and can be written in Boolean algebra: n(AUB) = nA +B -H(ANB) or n(AUB) = 11 E- n(AUB) Introduce Worksheet 1 and explain that they are to write all the multiples of 3 from the list inside the ring, using pencil. For the second subset give some of the pupils ‘multiples of 2, some ‘multiples of 5; some ‘more than 10° Can you use a set diagram to show these sets of numbers? The pupils need to identify where to place their second set - this may involve trial and improvement. Encourage pupils to draw loops for sets, without unnecessary worry about the loops being perfect circles or equal sizes.
Whole class sharing and discussion
Share pupils’ results for the different sets, either on the board or using an OHT. Then ask pupils for their precise descriptions of the sets. Count the numbers in each set separately. Add to get a total. Then count the numbers in both sets together (only counting numbers in the intersection once). What do you notice? How can we explain this?
Whole class reflection
Most pupils will be able to put in words the idea that the overlap is included twice when numbers in two overlapping sets are counted separately. Therefore the number in the overlap has to be known and subtracted from the total. Some may recognise that it is easier to take off the ‘outside set’ from the overall set.

A real-life comparison is possible. Ask the class how many ‘come to school by bus, ‘have a brother at home’ and ‘neither come by bus nor have a brother. List on board the total
number and each of the sets, and look at the relationships possible. Do not give any hints here — leave them time to puzzle this out. There is little use in most cases to formalise this work on sets to Boolean algebra.