Batch 3 - Class 110 - Venn Diagrams

• (Dudeney - 120) Farmer Longmore had a curious aptitude for arithmetic, and was known in his district as the "mathematical farmer." The new vicar was not aware of this fact when, meeting is worth parishioner one day in the lane, he asked him in the course of a short conversation, "Now, how many sheep have you altogether?" He was therefore rather surprised at Longmore's answer, which was as follows: "You can divide my sheep into two different parts, so that the difference between the two numbers is the same as the difference between their squares. Maybe, Mr. Parson, you will like to work out the little sum for yourself." How many sheep did Longmore have?
• The farmer had one sheep only! If he divided this sheep into two parts, making one part 2/3 and the other 1/3, then the difference between these two numbers is the same as difference between their squares - that is, 1/3. Any two fractions will do if the denominator equals sum of the two numerators.
• An alternate answer can be that it is any even number, so when you divide it into two equal parts, the difference between these two numbers is the same as difference between their squares - that is, 0.

• One property diagram: "Has brother" - Ask kids to fit themselves into this venn diagram - kids should understand that there are two regions in this diagram
• How do we add another property to the venn diagram, lets say "Has Sister"?
• Ask kids to draw a diagram that allows them to place all possible options. Test for it
• Show them the correct one and ask them to place themselves in the Venn diagram
• How do we add another property to the venn diagram, lets say "Has Cat"?
• Ask kids to draw a diagram that allows them to place all possible options. Test for it
• Show them the correct one and ask them to place themselves in the Venn diagram
• Of course, all combinations might not be permissible! Lets do some exercises to fit sets to Venn diagrams!
• Instructor Notes: Go around in circle, and ask individual kids to try and fit 
• Are there any that are left? Can we find examples for those?
• How do we add another property to the venn diagram, lets say "Has Dog"?
• The squarish venn diagram is the most likely to pop up. Ask kids to place various combinations in it

  Has Brother, Has Sister, Has Cat and Has Dog?Has Brother, Has Cat, but doesn't have dog or sister?Has Brother only?Has Sister, Has Cat, Has Dog, but does't have brother?Has Brother, Has Sister, but doesn't have a Dog or a Cat? - This diagram fails for this case!
• Can you come up with a Venn diagram that covers all options?

  Let kids try for a while, and then put up the test diagram. Test it for various combinations as above

  Is this even a Venn Diagram - what properties should the Venn Diagram have?

  Cover all casesExclusivity - no two region should represent the same case?

  What happens on Has Brother, Has Dog in the case above?

  Two separate areas!
• Can you come up with a Venn diagram that has exclusivity property?

  Let kids try for a while, and then put up the test diagram. Test it for various combinations as aboveSeems to workHow many regions does this Venn Diagram have?
• Do Venn diagrams have to be circles and ovals? Lets this this new pattern

  Is this a Venn diagram - Why or why not?

  Lets split this apart and test for various combinations

  Does it check out? Yes!Does it avoid duplicate areas? No! How do we know?
• Can we keep adding more properties - 5? 6? 7? 

  Yes, the Cog wheel exampleAnd, some more beautiful ones too - which don't seem possible for all numbers

• There is a zoo where there are 32 cages. The Zookeeper wants to classify the animals by 5 properties. Can you help classify them!
• The problem set is available in accompanying pdf file - the last slide has the answers - don't look at that before you solve!