2.  Think of a number (single digits are easiest)  Multiply it by 10  Add 6  Divide by 2  Add 2  Divide by one more than your original number  The answer is......

3.  Patterns are evidence of underlying mathematical structure  Patterns of same-ness; patterns of difference

4.  Different starting numbers same relationship

5. 60 ÷ 1 = 60 60 ÷ 2 = 30 60 ÷ 3 = 20 60 ÷ 4 = 15 60 ÷ 5 = 12 60 ÷ 6 = 10 60 ÷ 7 = ?? = 0.1 = 10% = 0.333... = 33%

6.  Teacher – generated (deliberate or accidental)

7. 1 ÷ 9 = 0.111 … 2 ÷ 9 = 0.222 … 3 ÷ 9 = 0.333 … 4 ÷ 9 = 0.444 … 5 ÷ 9 = 0.555 … 6 ÷ 9 = 0.666 … 7 ÷ 9 = 0.777 … 8 ÷ 9 = 0.888 … 9 ÷ 9 = 0.999 …

8.  23 ÷ 10 = 2.3  23 ÷ 20 = 0.23

9.  222222 ÷ 13 =  444444 ÷ 13 =  666666 ÷ 13 =  777777 ÷ 13 =  101010101010 ÷ 13 =  131313131313 ÷ 13  121212121212 ÷ 13 17094 34188 51282 59829 7770007770 of course ???

10.  Generated from past experience

11.  Patterns in the effects of actions can be evidence of mathematical meaning

12. ~~~ ~~ ~ = = =

13. XXX XX X = = =

14. +++ ++ + = = =

18.  Reflection on a whole piece of work can reveal similarities and differences with other objects

19.  Think of a number › Multiply it by 12 and divide by 3 › Now multiply the output by 3 and divide by 12  Think of a number  Multiply it by 15 and divide by 5  Now multiply the output by 5 and divide by 15  Think of a number › Multiply it by a and divide by b › Now multiply the output by b and divide by a

20.  Two triangles enlarged additively and multiplicatively

21. › What do I know now that I did not know an hour ago? › What could I do now that I would not have thought of an hour ago? › How did I come to know these new things? Reflection can trigger awareness of change and growth in knowledge

22.  reflect on what they have produced, to look for patterns  reflect on the effects of their actions, so they can make predictions, check their work, invent short cuts  know that reasoning from experience can lead to false conjectures so we need logical reasoning too

23.  Provide sequences of tasks that reveal mathematical patterns: different/same  Encourage learners to reflect on the effects of their actions, so they can make predictions, check their work, invent short cuts  Provide tasks that show how reasoning from experience can sometimes be misleading

24. well-structured tasks and prompts can make reflection a habit methods and concepts of the multiplicative relationship interact reflection can transform knowledge about calculation engaging the reflective mind can enrich understanding