Lesson 17 Sets and subsets

# Episode 3

Reasoning Resources: Worksheet 3 (for extra activity), Scissors
Extending understanding into a three-set diagram appreciating the different nature of the overlaps - some are an intersection of two sets; the subset in the centre is the intersection of all three sets. Whole class preparation

With some classes you may be able to cover most of the agenda of this episode in a whole class mode, selecting the challenge that is most appropriate for the class to work on independently. With some classes you may be able to cover most of the agenda of this episode in a whole class mode, selecting the challenge that is most appropriate for the class to work on independently.

What if there are 3 sets which overlap?
Pair and small group work
To overlap both the sets for multiples of 2 and 3, the third set could be: multiples of 5, 7, 11 or 13, or 'square numbers'. Using blank sheets of paper and the numbers 1-30, pupils explore what happens when three sets for multiples are put into a set diagram. Ask pupils to write the numbers 1-30 in the correct sets as follows:‘multiples of 2’, 'multiples of 3!' and then to describe the numbers in the intersection. Think of another set and tick all the numbers that should be in it. Draw a ring and move all these numbers inside it. How many numbers in the new set altogether? How many numbers outside all the three sets? The new ring adds four new subsets to the diagram. Describe each set.

There are advantages in allowing pupils to choose any criterion for their third set. If it is included, excluded, or overlapping with only one of the sets this will make a profitable focus for discussion.
Whole class sharing and discussion
Consolidating the understanding of the totals of the numbers in the subsets and the overall total being different. Share pupils’ sets and their descriptions of these and the subsets. Count the number of elements in each subset and add to get a total. Compare with the total number of elements in the sets. Why are they different? More able pupils may be able to explain the difference of total exactly as twice the count of the double overlap (central) subset plus the counts in the single overlaps.
Whole class reflection
The nil set.

Discussing the properties of the shapes and where on the diagram they fit.

Rehearsing the language needed to describe the shapes should not overpower the thinking agenda.
Bring the class's attention back to the basics of having an overall set, a rule for selecting one, two or three sets, and representing that in set diagrams. Some classes may enjoy considering themselves as the overall (universal set) and looking into overlap of three sets in it. A creative energy can be harnessed where the class invents sets for ‘sports-players at lunch-time', ‘dark haired‘, ’computer club‘, 'wearing something black’ etc, and attempt to draw a set diagram for three sets with overlaps, and consider the numbers in each subset. The set of pupils over 20 years old is a nil set.
Extra activity
Thinking about the differences between the intersections in the diagram in terms of a shape’s properties. Using Worksheet 3, ask pupils to cut out the shapes and place them in a set diagram with three overlapping sets:‘quadrilateral’; ‘equal sides’;‘with a right angle’.

• Choose a shape in one of the intersections. Describe one change you need to make to it so that it fits into a different subset,
• Choose one shape not in the intersections. Describe the changes you need to make to it so that it fits into another set, passing through an intersection.
• Start with the square. What shapes can you make with one change? What changes did you make?
• Recognising the similarity of overlap in shape and number, with the difference that the criteria for shape can be visual, while criteria for number are related to properties that must be thought about deeply. End of Lesson Reflection

• What is similar about all the set diagrams you have drawn?
• is different?