One of the most enjoyable aspects of mathematics teaching is working with pupils on Puzzles and Investigations. Witnessing young minds grappling with puzzles and teasers and seeing the eventual joy at solving a problem is very rewarding for us teachers.

Mathematical problem-solving skills are recognised as fundamental to a solid mathematics education and is one of the three core aims of the 2014 National Curriculum for Mathematics. It requires that pupils ‘can solve problems by applying their mathematics to a variety or routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions,’

Here at Cargilfield we think that problem solving should be a core focus on mathematical teaching.

Furthermore, the wider benefits of teaching children problem solving in maths can be seen when considering how problem-solving skills in maths are actually thinking skills that can be widely applied to other areas of learning, working and living. It is a widely held opinion by experts in this field, that not only does mathematics develop logical, deductive reasoning but - somewhat surprisingly - engagement with this subject can also foster creativity. Therefore, mathematics is an important context for developing problem-solving strategies that potentially have significance in all areas of human activity.

There are roughly 5 different types of problem in maths and these cover the range of skills that children need to practise repeatedly and explicitly in order to become excellent problem solvers.

· Trial and improvement

· Working systematically

· Pattern spotting

· Working backwards

· Reasoning logically

· Visualising

· Conjecturing

All very jargony!!

**Some simple problems to explain some of the above strands**

**Problem 1 - Rows of Coins**

*Take five coins: 1p, 2p, 5p, 10p and 20p.*

**Put them in a row using these clues.**

*The total of the first three coins is 27p.*

*The total of the last three coins is 31p.*

*The last coin is double the value of the first coin.*

(This example can be seen as a simple __Trial and improvement__ and __Reasoning logically__ problem)

Younger children, especially, can look at a problem like this and panic a little. They tend to want to write the correct answer in one go and therefore not be comfortable having an attempt at the solution. First attempts can sometimes even be rubbed out leaving the correct solution!

**Three important skills teachers should be assessing their pupils on and striving to improve are :-**

1) Can the child make an attempt at the start of an investigation?

2) Can the child keep trying for a prolonged period of time, despite not having success?

3) When stuck, can the child be able to try different approaches?

__Problem 1__

** First attempt **-

Doesn’t work for other clues.

__ Second attempt__ – 2p, 5p,

Doesn’t work for the third clue.

** Third, fourth, fifth attempts **– at this stage the 20p is fixed in the middle and combinations of 2p, 5p on the left, can be written down with combinations of 1p and 10p on the right to satisfy the last clue.

Of course, the pupils can by-pass a few Trial and improvement attempts by noticing that with __2p and 5p__ on the left and __10p and 1p__ on the right, that only __10p is double 5p__ will work.

Interestingly, at the start, pupils who have a “feel” for numbers would recognise that the totals for the first three numbers and last three numbers would mean that __20p is “ double counted”__ so has to be in the centre.

Total for first three = 27p Total for last three = 31p Total for five coins = 38p

Total for first three added to total for last three = 27p + 31p = 58p

58 – 38p = __20p__ The 20p has been ** “double counted”** and therefore must be in the centre.

(This is a skill widely used for Venn diagram problems with intersections.

__For example:__ Of 50 pupils surveyed, 36 pupils play football and 29 play basketball.

How many play both? (36 + 29) – 50 = 15

Solution: 21 play only football, 15 play both and 14 play only basketball)

**This simple coin problem can be seen to have tested Trail and improvement and Reasoning logically from the list above.**

__(Solution - 5p, 2p, 20p, 1p, 10p)__

Problem 3 – Roly Poly

*The dots on opposite faces of a die add up to 7.*

**Part 1**

*Imagine rolling one die.*

*The score is the total number of dots you can see.*

*You score 17.*

*Which number is face down?*

*How did you work out your answer?*

**Part 2**

*Imagine rolling two dice.*

*The dice do not touch each other.*

*The score is the total of dots you can see.*

*Which numbers are face down to score 30?*

(This example can be seen as a simple __Trial and improvement__ but possibly better to use __Working backwards__)

Playing board games with dice is maybe a dying activity for some youngsters.

The knowledge that opposite faces add to 7 and that the probability of scoring a double with two dice is the same as the probability of scoring a six with one die, comes more easily to children playing lots of board games. (Backgammon is another blog, maybe!)

Looking at this problem, the possibilities for the six faces can be written out:

1, 2, 3, 4, 5 and 6

and five numbers can be randomly added to try to get to 17.

This is Trial and improvement but not too elegant!

Better to look at the total of the six faces.

**This is of course 21** 3 x 7 from three pairs of opposite sides or if they are mindful of their triangular numbers, the sum of the first six consecutive numbers.

__Working backwards__ from 21 we have lost a 4 which makes the total of 17.

__Therefore 4 is face down.__

__Part 2__ is just an extension with 2 x 21 – 30 = 12.

The only two faces adding to 12 are the 6 and the other 6.

These problems can be extended to use more dice where the Trial and improvement would become too cumbersome and maybe using __dominoes__ instead of __dice__.

With practice the pupils become braver, more resilient and less concerned about making mistakes and writing down “wrong” answers – these “wrong” answers are of course just stepping stones to the final solution!

**Problem for Parents**

**Money Bags**

Sam divided 15 pennies among four small bags.

He could then pay any sum of money from 1p to 15p, by handing over one or more bags, without opening any bag.

How many pennies did Sam put in each bag?

**** Repeat the question but with 1023 pennies and 10 bags!! ****

RF

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of what life is like for the girls and boys at our school. We would love to welcome you in person to tour Cargilfield and explain more fully exactly what makes a Cargilfield education so special and so different.
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** [email protected] ** or you can telephone her on 0131 336 2207.