# Venn Diagrams: Level 5

## Indicator of progress

Students can correctly describe straightforward relationships between sets of everyday and mathematical objects verbally and can illustrate these with Venn diagrams and two-way tables as appropriate.

From Level 3, students have been able to place individual items correctly in a two-way table or simple Venn diagrams. At Level 3.5 they can focus on the sets and subsets of objects involved, and begin to describe relationships verbally as well as pictorially (Venn diagrams) and in two-way tables. At Level 3.75, they are able to describe more complicated relations between sets. At higher levels, they can use more complicated sets describing them verbally, pictorially and finally using set theory symbols.

Illustrating relationships between sets and attributes with Venn diagrams and two-way tables is an important cross-curricular thinking skill. Clear thinking about set relationships is important for logical thinking in all areas as well as for searching effectively using technology.

### Illustration 1: Showing and describing subset relationships

Pose this scenario: any number that divides into one number will also divide into any multiple of that number. This is because factors of the smaller number are also factors of the multiple.

For example, factors of 6 {1, 2, 3, 6} are a subset of the factors of 12 {1, 2, 3, 6, 12}.

At this level, students should be able to illustrate this relationship with a Venn diagram, and describe it verbally as "all factors of 6 are factors of 12" and "some factors of 12 are factors of 6".

### Illustration 2: Everyday concepts are generally easier than mathematical concepts

These faces differ in hair colour, eye colour and happiness. A set of faces varying on these attributes can be downloaded here (Word - 36Kb) for classroom activities, or teachers can easily make their own.

Students will find it easier to learn about classification and Venn diagrams by classifying according to everyday concepts such as these, than by classifying according to mathematical concepts.

Similarly, it is generally easier to classify everyday objects than mathematical objects.

### Illustration 3: A common error is identifying too many sets as disjoint

Sometimes sets are disjoint. For example the sets of odd and even numbers are disjoint; they have no members in common.

Students often treat classes of objects as if they are disjoint, when it is not necessarily appropriate. A likely cause for this is that students think that if things have different names, then they must be very different.

One example is that students often think that the sets of equilateral triangles and isosceles triangles are disjoint - that there are no equilateral triangles that are also isosceles. However, all equilateral triangles are isosceles - isosceles triangles need 2 sides the same length (not exactly 2 sides the same length)

Students might consider the set of square numbers and the set of cube numbers to be disjoint, but they overlap, since 1 and 64 belong to both sets. (NOTE: 64 = 43 and 82.) See Activity 1

### Illustration 4: Classifications depend on definitions

Sometimes sets have members in common. For example the set of rhombuses (the quadrilaterals with four equal sides) and the set of rectangles (the quadrilaterals with four right angles) both contain the set of squares.

Students who do not understand the distinction between different types of relationship between sets will probably wish to make the sets of rectangles, squares and rhombuses into separate, disjoint sets. This is not correct, because squares fit the definitions to be both rectangles and rhombuses.

## Teaching strategies

The aim is to show students the existence of a variety of different types of set relationships. There is a need for careful thought about definitions and some thought being given to the most suitable type of diagram: disjoint, subset or intersection. Venn diagrams and two-way tables are both useful because they are both visualisation tools. Students also need practice in using correct language.

Activity 1: Sets of square and cube numbers aims to develop the awareness of an intersection in the context of simple powers of numbers, by having pupils search two lists for common elements. It then requires students to use language to reinforce this idea.

Activity 2: Multiples and common multiples aims to develop awareness of set-subset and intersecting relationships, by using Venn diagrams and also two-way tables to help analyse multiples and common multiples. Again language reinforces this idea.

Activity 3: Two-way tables or Venn diagrams? shows when two-way tables are more appropriate than Venn diagrams. It uses equipment readily available in schools.

Activity 4: Making Venn diagrams with attribute blocks shows a simple Venn diagram activity, using a set of specially designed attribute blocks.

### Activity 1: Sets of square and cube numbers

These sets of numbers are also studied in Number, where sets of wooden cube blocks will have been used to show the derivation of the names.

1. List some square and cube numbers

Students make a list of some square numbers and a list of some cube numbers.

2. Identify whether or not there is an intersection of the two sets (ie, decide if some elements/numbers are common to both sets and overlap)

By making a list of some square and cube numbers they will quickly find that there is overlap; 1 and 64 arise quickly. Students should draw the overlapping sets and place at least 1 and 64 into the intersection.

3. Use of language

Students consider the truth or otherwise of these statements and explain their answers by reference to the numbers themselves and the Venn diagram. (Note: Only c and d are true.)

 a) All square numbers are cube numbers. b) All cube numbers are square numbers c) Some square numbers are cube numbers. d) Some cube numbers are square numbers e) No square numbers are cube numbers. f) No cube numbers are square numbers

4. Students should use calculators to find other numbers in the intersection.

You may need to demonstrate how to use the repeated multiplication key to find the squares (on many calculators this is done by pressing <number>, ×, =, =) and cubes (an extra =). A table is useful for recording the results.

 Number 1 2 3 4 5 6 7 8 9 10 11 … Square Cube

5. Extension

As an extension, some students might be challenged to identify the defining property of the set in the intersection: 1, 64, 729, ... They are the sixth powers eg 729 = 3 × 3 × 3 × 3 × 3 × 3 = 36

### Activity 2: Multiples and common multiples

The sets of multiples of 2 and multiples of 5 have an intersection: the multiples of 10. This is the set of common multiples. For example, for numbers up to 30:

However for sets where one number is a factor of the other (e.g. 2 and 4, 3 and 6) the intersection (the common multiples) comprises not just the product of the two numbers, but is the subset itself. For example the common multiples of 2 and 4 are multiples of 4.

Again, when two numbers have a common factor (e.g. 4 and 6, 6 and 9) the intersection (the common multiples) comprises the product divided by the common factor. For example the common multiples of 4 and 6 are the multiples of 12, and 4 × 6 ÷ 2 = 12. For example, for numbers up to 30:

The resource sheet explores this idea, using the two-way table as a tool for visualising the relationship. All students should be expected to explain their reasoning, and they should challenge each other so that only correct reasoning survives the process.

Link to student resource sheet 1 (Word - 25Kb)

Two-way tables

Two-way tables can also illustrate these relationships.

Example 1: Universal Set - Numbers up to 24. Subset 1 - Factors of 12. Subset 2 - Factors of 18

 Factors of 12 Not factors of 12 Factors of 18 1,2,3,6, 9,18 Not factors of 18 4,12, 5,7,8,10,11,13,14,15,16,17,19,20,21,22,23,24

Are SOME factors of 12 also factors of 18? Yes. These numbers are 1, 2, 3 and 6.
Are ALL factors of 12 also factors of 18? No. Both 4 and 12 are factors of 12 that are not factors of 18.

### Activity 3: Two-way tables or Venn diagrams?

A two-way table is able to record information about 2 attributes (eg size and colour) simultaneously. Two-way tables are particularly suited to displaying multi-valued attributes (e.g. 3 sizes, 4 colours). This can be very complicated with a Venn diagram.

As with Venn diagrams, introductory activities can use classroom materials with clear attributes, or features of the students themselves.

These little teddies have 4 colours and are in 3 sizes. A two-way table can show all the types of teddies. The table can be introduced by placing teddies in the appropriate cells of the table.

 Number of teddies Small Medium Large Green Yellow Blue Red

Jill has some teddies. She has recorded which teddies she has in a two-way table.
(a) Draw (or make) Jill's collection of teddies.
(b) How many yellow teddies does Jill have? Can you work this out just from the table? ( 1 yellow teddy - add 1+0+0 across the 'yellow' row)
(c) How many big teddies does Jill have? Can you work this out just from the table? ( 4 big teddies - add 3+0+0 +1 down the 'big' column)
(d) How many teddies does Jill have that are not red? Can you work this out just from the table? (9 - Add across all the rows other than 'red')
(e) How many teddies does Jill have altogether? Can you work this out just from the table? Can you work this out from the table in two different ways? (12 - Add the totals of all the rows, OR add the totals of all the columns)

 NUMBERS OF JILL'S TEDDIES Small Medium Big Green 2 0 3 Yellow 1 0 0 Blue 0 3 0 Red 0 2 1

Students can make their own collections and record their collections in a two-way table. If they hide their collections from their friends, they can pose questions for their friends to answer using the two-way table.

### Activity 4: Making Venn diagrams with attribute blocks

The first Venn diagrams that students work with, will be for classifying objects or people according to two attributes.

Although many sets of materials can be used, the commercially produced sets of 'attribute blocks' are a very rich resource for classification activities. The set contains 60 pieces, involving combinations of 5 shapes (triangles, squares, rectangles, hexagons, circles) in 3 different colours and 2 sizes. Some are shown in the picture below. Although not evident in the photo, some pieces are thick and others thin, making a fourth attribute.

Students can make Venn diagrams using sets of attribute blocks to show many characteristics. The Venn diagram below shows big pieces in the green-bounded set, circles in the red-bounded set. The big circles are in the intersection. The pieces that are not big and also not circles are outside both. Students should describe each of these sets using appropriate words as in the first two dot points, and be working towards understanding compound expressions as in the third dot point for Level 3.75:

• all the big pieces are in the green (large 7-sided) region
• some of the circles are big and some of the big pieces are circles
• all the pieces that are not big and are not circles are outside the regions.

As well as making Venn diagrams to specifications, students can create Venn diagrams and give them to other students to describe the criteria used.