All you need is lots of counters!! Answering the question ‘Why?’

An approach to proportional thinking and then on into Mathematical explanation... aka Algebra! An approach to mathematical explanation involving modelling a range of problems aka Algebra! some of which are proportional problems

What’s the answer?

How did you get the answer?

does it work ? always Why

When are we doing Mathematics? Jack has 81 fruit smoothies to sell at a stall in the school gala. He sells 27. How many are left? How would you work this out? = = 54 So 54 are left “Turn it into an addition. Start at the lower number and work up......jump to the next ten number, then jump to the ten number just below the big number...see how many more are needed...then add up all the jumps!” 81 is 9 x9, 27 is 3x9 So we end up with 9 – 3 lots of 9 6 x 9 =

So what is algebra for? communication formulas to allow others to do a procedure description the mathematical structure of a problem derivation transforming from one expression to another solution finding values for quantities

Algebra is Generalised arithmetic Manipulating generalised quantities The system of symbols and rules that we use to work with variables Mathematics with the context that originated it stripped away. Laying out stuff in a particular way so that we can see what’s going on.

Maths is......explaining organising stuff so that you can understand what’s going on. When we answer the question ‘why?’ we are doing maths need to come up with a ‘picture’.....a model so that we can see the structure....

= 3 x = 4 x 16

Problem Materials Model Re-organise to gain insight Practise with different cases Generalise Algebraic Definition Refine ideas until structure appears Informal Definition

Using counters show me what each of the basic operations mean: Addition Multiplication Subtraction Division

If we add two odd numbers we get an even number. If we add two even numbers we get an even number. If we add an odd and an even number we get an odd number. Why?

If we multiply two odd numbers we get an odd number. If we multiply two even numbers we get an even number. If we multiply an odd and an even number we get an even number. Why?

Choose any three different digits (eg 7,5,8). Add them up. Form all the 2 digit numbers you can using them (6 of them). Add all these 2 digit numbers up. Divide this result by the total of the three digits. What happens? Why?

+3x2 +3 1,2,3,4,5,6 8,10,12,14,16,18 5,7,9,11,13, Why? +x x+

My family is very mathematical and food is always distributed to the children in proportion to their ages. Mike is 14, Bridey 10 and Joe 7 and it’s pizza night! There are several pizzas. Joe gets a quarter of a pizza. What fraction of a pizza should the others get?

Rule for divisibility by 9 is Why does this work? Rule for divisibility by 11 is Why does this work?

Consecutive Sums What numbers... are the result of adding two consecutive numbers? are the result of adding three consecutive numbers? are the result of adding four consecutive numbers? Why is this?

Choose any 4 different digits and write them down in any order to form a 4-digit number Now use the same 4 digits, jumble them up in any order to make another 4 digit number Subtract the smaller form the larger – 1825 = the result is always a multiple of 9! Why is this?

1/3 as a decimal is Why? What about 1/6?.....

A familiar problem ! Choose(any?) three digit number.328 Reverse it, subtract the smaller from the larger 823 – 328 = 495 Take the answer, reverse it and add = 1089 Why do you always get 1089?

Arithmagons

cups and counters...equations

super subtraction

Number cells

When we do algebra what happens? Is it a linear, step by step process as is often portrayed in textbooks? Or does it happen by insight (haha! Moments)...when you see the structure of a problem and how to solve it....must be like this? Otherwise you are blindly going step by step with no idea of an endpoint?