Topic outline
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Video transcript
This unit of work is on set notation and Venn diagrams
Students often struggle to recall the notation of Venn diagrams and to link it to common English terms as well as to represent this notation diagrammatically.
We are going to look at how to draw and use Venn diagrams to represent information for increasingly complex problems. Set notation will be explored and learners will use Venn diagrams for finding probabilities.
This unit of work is just one of several approaches that you could take when teaching this topic and you should aim to adapt the resources to match ability level of your learners as well as your school context.
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Lesson 1 video
Video transcript
Suppose we want to sort these people into two groups:
Those with black hair and those wearing a tie.
In maths we call groups of this nature sets and the items within them, elements.
In this case the people we are sorting are the elements.
A Venn diagram can help us.
The Venn diagram consists of a rectangle that acts as a boundary into which all the elements must be placed.
This rectangle is referred to as the Universal Set and it contains the two further sets, each representing the characteristics with which we wish to sort our people.
We can take each element in turn and add them to the Venn diagram by placing them in the correct position.
The first person has both black hair and a tie so we place them in the Venn diagram where the two sets overlap.
This section of the Venn diagram is called the intersection of the two sets.
The second person has a tie, but does not have black hair, so we place them in the set for ‘having a tie’, making certain that they are not in the part of set that overlaps with having black hair.
The third person has black hair but no tie so is placed in the ‘black hair’ region of the Venn diagram, outside the region that overlaps with having a tie.
The fourth person has neither a tie or black hair, but every object needs to be placed somewhere inside the universal Set on a Venn diagram so this person is placed inside the universal set, but outside of the indicated sets.
Rather than placing the objects physically into the Venn diagram it is often more useful to simply represent the number of items in each region with a number.
In this case the number of people wearing a tie is equal to 4.
The number of people with black hair is equal to 5.
You will notice that if you add together the number of people with black hair to the number of people wearing a tie, you get a total of 9, which is more than the number of people placed in the two sets.
The reason is that people who have both black hair and a tie - have been counted twice.
We call this overlap the intersection.
And it represents the elements that are in both of the sets because they have both black hair and a tie.
The intersection of two sets is written as shown, and is an important concept in Venn diagrams.
We can see that there are 2 people who have both black hair and a tie.
If you want to know how many elements are in at least one of the two sets - In other words - the number of people who satisfy at least one of the conditions of having black hair or of wearing a tie, then we need to count everyone in each of the two sets.
We call this set the Union of the two sets and it is written as shown.
We can see that there are 7 people altogether who have black hair or who wear tie.
The union of two sets is often described by using the word OR, but this also includes both possibilities happening and at the same time.
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