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Variation: A n Introduction claire.morse@winchester.ac.uk jane.jones@winchester.ac.uk

claire.morse@winchester.ac.uk j ane.jones@winchester.ac.uk. Variation: A n Introduction claire.morse@winchester.ac.uk jane.jones@winchester.ac.uk. Friday 16 th June 2017. Today we will explore together…….

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Variation: A n Introduction claire.morse@winchester.ac.uk jane.jones@winchester.ac.uk

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  1. claire.morse@winchester.ac.uk jane.jones@winchester.ac.uk Variation: An Introductionclaire.morse@winchester.ac.ukjane.jones@winchester.ac.uk Friday 16th June 2017

  2. Today we will explore together…… • How mathematics is about noticing and discerning and how this serves as the precursor to generalisation • The role of relational thinking

  3. 9999 + 999 + 99 + 9 + 5 = 2 x 37.5 + 3.75 x 20 = ( 4 + 1 ) + ( 5 + 1 ) + ( 6 + 1) + ( 7 + 1 ) + (8+ 2 ) = 5 6 6 7 7 8 8 9 9 10 What are you attending to? What can you see? What can you notice?

  4. Use of relational thinking 7 - = 6 – 4

  5. Relational thinking with true/false number sentences 8 x 3 = 8 x 2 +8 6 x 7 = 3 x 7 + 21 8 x 6 = 8 x 5 + 6 7 x 6 = 7 x 5 + 7 9 x 7 = 10 x 7 - 7

  6. EXHAUSTIVE LISTS – to reason you need to be able to notice thingsSometimes there are an infinite number of answers and sometimes we can create an exhaustive list. Children can look for what is ‘general’ about the examples and what they all have in common. Pairs of whole numbers which have a product which ends in zero The numbers that give a remainder of 2 when you divide by 5 Numbers with exactly 3 factors

  7. Noticing leads to generalisationhttp://nrich.maths.org/6814

  8. The solution we found was that you always put an odd number at the bottom when it was 1-5.There were three odd numbers and two even numbers. To make it balance you've got to get rid of an odd number so you put an odd number at the bottom, because then you have two odds and two evens.You have to put the biggest remaining odd number with the smallest even number on one arm, and the smallest remaining odd number with the biggest even number on the other arm to make it work. • If n is the first number then the other numbers are n+1 n+2 n+3 n+4, the three solutions for every number are:

  9. Variation & Reasoning A 9 litre vat of jam is used to fill some 3 litre jars. How many jars can be filled? A 9 litre vat of jam is used to fill some 1 litre jars. How many jars can be filled? A 9 litre vat of jam is used to fill some 1/2 litre jars. How many jars can be filled? A 9 litre vat of jam is used to fill some 1/3 litre jars. How many jars can be filled?

  10. Designing purposeful learning for mathematics Variation is an approach to teaching Art of sequencing similar but increasingly complex examples to “generate disturbance of some sort for the learner” Festinger(1957)

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