## First Class Maths: The Jewellery Case

Of all the maths tasks I have ever written (and there have been many!) this may well be my favourite. It’s The Jewellery Case task from First Class Maths, one of the logic puzzles from the resource.

The intrigue created by the context, uncovering the criminal from a robbery in a jewellers, hooks children immediately into the task. However, the children will need to show great perseverance and creativity to decipher how the five clues can be used together to find the final solution. Then, once the children have a solution, they have to find a way to articulate their reasoning – a real challenge!

It usually takes the children – even the most able – 40 minutes or so to find a solution that they are confident with and can articulate. The task always leads to children producing various written representations and engaging in heated discussions as they explore their ideas and justify their solutions. Here are two of the explanations given, one presented as a diagram (sorry, poor quality photo!) and the other a written explanation:

If anyone is in need of an extra layer of challenge, the following extension is also provided:

It’s one of the jewels of the First Class Maths resource, released by Alan Peat ltd! http://www.thecepress.com/product/first-class-maths/

## My favourite properties of number question

‘Do all odd square numbers greater than 1 have 3 factors?’

I love this question. Let me explain why, and the train of thought that it can generate.

Children usually start by generating the odd square numbers, identifying that square numbers have an odd number of factors (1, the number and the square root). This helps to underline the uniqueness of square numbers. Nothing new here, really.

So children tend to identify the first two or three odd squares greater than one (9, 25 and 49) and realise that these numbers only have 3 factors. This, I tend to find, is enough evidence to convince most children that the answer to the question must be ‘yes’.

However, this is of course a false presumption, and by making it children realise a crucial mathematical principle: that finding examples to support a theory is not the same as finding a proof. There must be reasoning as well as examples to generate a proof!

So I then ask the children to consider 81. And they soon realise that it is also divisible by 3. What’s happened to the pattern? And why?

Children then investigate further examples, noting that 5 of the first 6 odd square numbers greater than 1 (9, 25, 49, 121, 169) have only 3 factors. But then 225, the square of 15, has 9 factors! There must be some logic here, and of course there is. I ask the children to consider the square roots of each number, as underlined below:

After much discussion and deliberation, and a healthy dollop of struggle, someone makes the breakthrough: the numbers with prime square roots have 3 factors; other numbers can be further divided by the factors of the square.

But does this pattern continue for all odd squares? And why was 1 excluded from the list?

It’s an amazing question for exploring the very nature of the properties of number, and particularly primes and squares. And, in my opinion, for experiencing the joy and beauty of mathematics!

## The Mathematics Apprenticeship Outline

I was sent a private message on twitter from a teacher asking for some more advice on which task to use to start with for the children in his class. As a consequence I decided to write this blog, outlining in brief the content of each task.

The Mathematics Apprenticeship tasks are all non-standard activities, requiring the children to organise and carry out extended tasks. They also require children to consider the ‘real-world’ effectiveness of their solutions. It is difficult to narrow down the specific objectives that are covered in each activity; however, I hope this blog helps to outline the basic context and mathematical content of each task.

The tasks generally speaking get progressively more difficult. I try to match the task chosen to the areas of the curriculum that my class have recently covered. A general outline for each task is included below:

Designing a menu for a stall at a music festival, ordering the necessary ingredients and pitching the proposal to the festival organiser.
Ratio calculations; calculating percentage discounts; scaling calculations.

Classroom Design
Designing a new group room for a primary school, producing a to-scale diagram and matching costings document. The pitch is done to a school-leader or governor.
Finding areas of rectangular and non-rectangular surfaces; a range of multi-step calculations involving money and measures.

Carrying out a research project on behalf of the BBC, presenting findings and recommendations to a BBC commissioner.
Collecting, interpreting and presenting an extended set of data.

Fruit Stall
Designing a stock ordering system for a fruit stall, ensuring that the correct number of baskets of fruit are ordered and that stock remains fresh, whilst minimising costs.
A context for extended division and multiplication; considering a range of factors to minimise costs.

To select a range of activities for a new Adventure Park within a set budget, considering the costs to build and run each activity. Teams must present their plan, which will include details such as toilet blocks and car parking spaces, for their new park to the CEO of Venturemax.
A context for extended calculations involving large monetary amounts; calculating percentages; scaling multiplication.

Action Research
To carry out a research project into one of two possible essay titles, exploring one aspect of mathematics learning in detail. Groups must write an extended report to present their findings.
Opportunity for reflection into the learning process in mathematics, spanning different curriculum areas.

MI5 Security
To use a range of variables to produce two security codes for the Head of Secret Service Headquarters; to produce a to-scale diagram of the office, with security features included. The security proposal is presented to an MI5 official.
Using letters to replace unknown numbers; experimenting with formulas to produce different products; using measures including angle.

Garden Design
To produce a garden design to fulfil a given brief. Groups must produce a to-scale diagram and a detailed costings document, balancing the considerations of quality and cost. The proposals are to be presented to a member of the Marks family.
Extended area and perimeter calculations; calculations involving money and measures.

From the given information, each group needs to decide on the most profitable gadgets for Teknofad to invest in. Children have to consider the cost of both the initial investment in each product, and the potential profits based on different sales figures, when presenting their work to the manager of Teknofad.
Calculating profits using fixed and variable costs and revenue forecasts; considering different probabilities; presenting data in the most appropriate form.

Football Data
Given a range of OPTA Sports information, groups have to analyse the key factors in determining the success of a Premiership football team. They then have to present their findings and recommendations to a member of the League Manager’s Association.
Analysing and presenting data, including the use of scatter graphs.

Educational Resources
The children have to write an educational resource for children at a specific stage of learning, in the process reflecting on what constitutes effective learning in mathematics. They will present their work to you, the teacher.
Opportunity for reflection into the learning process in mathematics, spanning different curriculum areas.

## The rounding question

Yesterday I presented some of the children in my class with this question:

The news reported today that 2000 people have contracted a rare tropical illness. This figure is known to have been rounded to the nearest 100 people. What is the largest possible number of people that could have the condition?

It was fascinating to see the thought progression that so many of the children went through. It looked something like this:

Preliminary idea (in some cases): 1900 or 2100
Idea 1: 1999
Realisation that you can go above 2000
Idea 2: 2040
Realisation that units column can increase
Idea 3: 2044
Realisation that the units column does not determine whether rounding up or down
Idea 4: 2049
Occasional error: 2049.999999
Awareness, in this context, that the answer must be whole number.

It was amazing to see how uniform this pattern of thinking was with different groups of children, particularly as children came to the counter-intuitive realisation that you have to find a number that will round down In order to find the maximum number of people.

There’s no place quite like the classroom!

## Books as context for maths

I recently tweeted a request for people to let me know about books that they have used as a context for maths with nursery, reception and KS1 children. The response was fantastic, thanks to everyone who replied. Here is the final list:

None the Number, a counting adventure
The Hungry Caterpillar (the most mentioned book)
A Place for Zero
How Big is a Million
Guess How Much I Love You (I read this book to my 2 y-o daughter!)
Ten on the Sled
One is a snail, ten is a crab
Benny’s Pennys
Ten Black Dots
Jim and the Beanstalk
George’s Marvellous Medicine
Dear Zoo
Kipper’s Birthday
Percy the Park Keeper (series)
Farmer Duck
Goldilocks
Jasper’s Beanstalk
We’re going on a bear hunt
Six Dinner Did
The Great Pet Sale
Titch
You’ll Soon Grow Into Them

Again, a huge thanks to everyone! I will try to tweet about how these books get used in my school. It shows the power of twitter as an ideas-sharing forum; it’s great to be linked up with so many passionate, knowledgeable people!

## Establishing a Mathematical Culture

How can we learn well in maths? Why do some people find mathematics difficult? What does it mean think and act like a mathematician? These are fundamental questions; the way children answer these questions will to a large extent determine their success as mathematicians. I believe, therefore, that we need to have a stronger dialogue with children about the learning process, specific to maths, that will encourage positive learning dispositions. This is what I call ‘Creating a Mathematical Culture’.

Becoming a successful mathematician is far from being a purely mechanical process. Many people (children and adults) are unable to fulfil their potential specifically because of their inability to deal with their emotional response to the challenge/threat posed by mathematics. The importance of mindset to learning outcomes in maths is supported by neuroscience, the analysis of PISA tests (see below) and, I dare say, the personal experience of many of our teachers.

In my classroom, I aim to establish the following five principles at the start of the year. These are the tenants that I believe are important to building positive attitudes and habits in maths lessons:

Firstly, the children must be convinced that by working hard they can develop their mathematical ability. Any notion that mathematical intelligence is genetically determined, or unchangeable, must be addressed.

In my maths training, I go into the detail of how I introduce these five principles to the children: what exactly each statement means, evidence to show the importance of each statement and how they can work in this way in daily maths lessons. These principles will be referred to, exemplified and celebrated constantly throughout the year, and no doubt amended also. They give the children the framework for thinking about the learning process in maths (metacognition), and critically help pupils to embrace challenge and learn from mistakes.

I also see my ‘Mathematical Culture’ as being like a promise that I am making to my class: that I am promising them a rich mathematical diet, set in a climate of support and trust. Not only is it a guide for the children, but it is a vision for me as a teacher: setting out explicitly what I value, and a standard that I will aspire to fulfil.

## TMA: the ‘how to’ guide

I hope that this blog will help you to know how to deliver the Maths Apprenticeship (TMA) tasks. First of all, there are two different USB sticks. The teachers’ USB, as shown in the picture below, gives information about how to run TMA, and has tips and hints for each task.

The children’s USB sticks are the actual resource, as pictured below. It is suggested that the children work in groups of 3, with one USB stick per group.

You will select a task for the children to complete. They will read the first few pages of each task, which explains what the children have to do. It will also give any information that the children will need to complete their work. For example, below is a picture of two pages from the Fruit Task: a letter from the stall holder and an order form from the supplier.

I strongly recommend that the children print the information that they need for carrying out their calculations (in this case the Harrison’s order forms): this will mean that they can complete the workings for the tasks without being tied to the computer. Then there are pages on each task for the children to complete. For example, in the fruit task they will need to fill in the stallholder’s diary to tell him when to order fruit, and write him a letter. The image below shows two of the pages that the children complete:

Once the children have finished the work for the task, it’s time for them to meet (and impress) the customer! For the fruit task, someone will play the role of the stallholder and the children will explain their proposal to him/her. Then finally, when the work is completed, the teacher will take each group’s USB sticks and mark their work using the teacher evaluation page, as shown below.

So the children work collaboratively on the tasks for an extended period of time, delegating the jobs between team members for each task. Then they pitch their work to the customer, and finally they receive feedback from the teacher.

As they say, a picture is worth 1000 words. So this is what it looks like in action:

For more information and to order TMA, click on The Maths Apprenticeship.