Amazing Venn Diagrams Years 4 - 6

Summary Given a Venn Diagram starting point like the one shown children are asked to investigate which numbers go in to which sections. At first sight this might seem a bit straightforward, even procedural, but the conversation snippets Lisa captured in her classroom show that the activity has a far more profound influence on student thinking. Asking the mathematician's question What happens if we change the labels on each circle? makes the activity suitable for threading. Materials Hula hoops and label cards Digit cards from (at least) 1 to 20 Poly Plug sets Butcher's paper and markers per group (optional)

Acknowledgement

This activity was contributed by Lisa King, Cygnet Primary School

I had been doing a lot of data collection and graphing with my Grade 6 class. They were really enjoying using and creating Venn Diagrams, but refused to believe they were actually maths!

I decided to give them a Venn Diagram involving concepts of number. Adapting an idea from Profiling Mathematics: Tasks for Assessing Learning (Griffiths and Clyne, 1995 published by Longman) I gave them the diagram above.

Working in pairs, they had to place the numbers 1 to 50 in the correct places on the diagram. For example, the number 6 is a multiple of both 2 and 3, so it is placed in the section of the diagram where those two circles overlap. Numbers that are not multiples of 2, 3, or 5 were written outside the diagram. Most could figure out where to place the numbers by using mental computation. A few chose to use a calculator. Occasionally students checked with their Poly Plug. Some of the more confident students expanded the task to include all the numbers to 100. Once finished we discussed their findings.

I could not believe the responses to this activity! A lot of these children have in the past had great difficulty with the concepts of multiplication, multiples, factors and times tables. Yet presented in this visual format and coupled with their sound understanding of the way Venn Diagrams work, it became one of those 'aha' moments about which Calculating Changes speaks so passionately!

• Oh, I get it!. This is like our times tables.
• I didn't know that 45 was in the 3 and 5 times tables.
• Multiple just means the numbers or answers to the times tables.
• Gee, 30 can be divided by a lot of numbers.

We discovered that the only number to appear in the middle of the diagram was 30, and those who took the investigation further, to 100, told us that 60 and 90 would also be in there (as Anna's diagram shows). The discussion led us into an extended conversation about the way numbers work and how they relate to each other. I heard ideas such as the following (from someone who has really struggled to grasp concepts in the past and finds number work, especially tables, very difficult): Question: What can you tell me about the numbers in the middle of the diagram? Daniel: They are 30 numbers away from each other. They are in the tens (ten times tables) and they can be divided by 2, 3, and 5. (The children have yet to realise why they are 30 numbers away from each other.)

I asked the class: What can you tell me about the numbers on the outside of the circle?. The children had affectionately labelled this the 'reject bin'. Here are some of their responses:

Tamara: They are all odd numbers and they are not numbers that 3, 5, 2 can go into evenly.
Charlene: They don't go in the 2 - 3 - 5s. They are all odd numbers. They don't fit inside the circles.
Wade: They are all odd and not even.
Kelsey: The numbers are all odd numbers that don't go into 2, 3, 5, at all.

Kelsey: My prediction is because this time there are different numbers than before, the numbers (in the circles) will be shared differently.
Dion: I predict the 2 and 4 will have the most numbers.
Elizabeth: I predict that 2 and 4 will have more numbers because they're even.
Jaimey: I predict it will be way different because there are two even numbers and only one odd whereas before there were 2 odd and one even.
Sally: I predict that the 2s will get more and the 7s will get least.

Extensions

1. Ask students to use their Poly Plugs to make equal rows for some of the numbers in one of the sets, then compare this with what is possible for numbers in another set.
   Can they, for example, discover that all the numbers in the Multiples of 2 set can be made into 2 rows of something, whereas all the numbers in the Multiples of 3 set can be made into 3 rows of something.
   Taking the time to do this, and record some in their journal, reinforces:
   
   • the brain picture of multiplication as a rectangle
   • the concept that if a problem displays a 'pattern' in numbers then there will be a corresponding spatial pattern and vice versa.
2. Factor Trees and Factorgrams (Factorgrams is Maths300 Lesson 104).

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