# Puzzle it Out – Recreational Maths

As it’s almost the holidays, I thought it might be timely to post some ideas for recreational maths. This may sound a tad geeky, but when we enjoy maths ourselves as teachers, we are much more likely to communicate that enjoyment to the children we teach. So the ideas I’ll be sharing are mainly aimed at adults, but some of them would be accessible to older primary children too.

First of all, a very recent find in the app store. Logic Games has several different types of logic games, all with many different levels. I’ve only tried 2 or 3 of these so far and only the early levels but there’s plenty there to challenge. Unlike some apps, the early levels are already quite challenging (or perhaps it’s just me in need of a holiday). My husband and I are both hooked and I anticipate some healthy competition over the holidays. I suspect these ought not to be beyond brighter children at the older end of primary school but they would need to be prepared to be patient and think pretty hard. A few years ago, I ran an after school maths club which focused on recreational maths, and one of the activities we did was Sudoku puzzles, starting with some fairly easy ones and getting progressively harder. I was amazed when one of the girls, (I’ll call her Sarah) turned out to be easily the best at these. Sarah had struggled with maths throughout school and had a very poor grasp of number bonds and tables and only a shaky idea of basic calculation methods, and yet she could think through a Sudoku puzzle and come up with a solution much more quickly than some of the most able children. It transpired that she had been doing the puzzles with her grandfather every Saturday afternoon for some time. It also occurred to me though, that Sarah was used to finding things difficult and having to really think about them to arrive at a solution, whereas for some of the much brighter children in school, that was quite a novel experience. They were used to grasping things quickly and not having to think too hard. So I think it’s well worth giving our more able mathematicians something that they’ll find quite challenging and have to spend some time on. The ‘Logic Games’ app is free, or for £2.99 you can install a version without adverts – very good value for such a lot of games.

Next, a really good book. I’ve enjoyed the books of Ian Stewart and Marcus du Sautoy, but one of my favourites recently has been by Alex Bellos. ‘Alex’s Adventures in Numberland’ explores our relationship with numbers and mathematics and is very readable. Bellos is a journalist and communicates the ideas involved very effectively without patronising the reader.

Finally, a book aimed at children. Johnny Ball’s ‘Think of a Number’ is a very entertaining book about maths. I’ve used several of the ideas from this in the classroom, including one where he imagines a world without numbers and features pages from a newspaper without any numbers. So one story is headlined ‘Woman has some babies’ and attempts to tell the story of a mother giving birth to sextuplets without actually using any numbers (eg. “it’s quite common for a woman to give birth to a baby and another, but this woman has given birth to a baby and another and another and another and another and another.”)

Why not try one of these over the holidays?

# Higher, Faster, Harder? Supporting More Able Mathematical Learners

**Harder, Faster, Higher? – Supporting More Able Mathematical Learners**

The new curriculum, we are told, is a mastery curriculum. This means there is an expectation that instead of pushing our more able learners on to ever higher level curriculum content, the focus is much more on making sure all our children are secure with the core content for each year group. This leaves us with a challenge for our more able learners, but also with a great opportunity. With less pressure (we assume – the assessment procedures are not yet clear) to push these children through the levels, we have to think of different ways to challenge them – not so much higher and faster as broader and richer content. These are a few ideas about how we can support these children.

**Open ended questions**

Challenging more able children should not be primarily about moving them on to ‘harder’ questions with higher numbers. We want to extend their thinking by asking more open ended questions which challenge them to apply their knowledge in new ways. We also want to develop their reasoning skills by asking them to explain their reasoning. Using Bloom’s question stems can be helpful in planning this. Ask children to explain how to find the answer to a problem and decide which approaches would be best to use. Ask them to explain the rule for a growing pattern or to explain what would happen if … Get them to think of other ways of doing things and compare approaches to decide which is best. Challenge them to explain their reasoning so that a younger child could understand it.

**Mathematically rich activities**

Children need to learn to think mathematically and to apply their skills. The Nrich website provides lots of games, challenges and activities to encourage this. They aim for activities to be ‘low threshold, high ceiling’ ie. accessible to as many children as possible but with enough to challenge more able children. Their curriculum mapping documents are very useful in identifying activities which link to different curriculum areas.

**Investigations**

Investigations can help children to extend their mathematical thinking in a more open-ended way. Typically in an investigation, children are given a starting-point and some ideas of how to get started, but they won’t know what the answer will look like. They need to look for patterns and identify what is happening. The Maths Warriors site has a number of interesting investigations suitable for primary aged children.

**Challenges**

More able children often respond well to challenges. The ‘Mathematical Challenges for more able pupils’ have a number of challenges divided by age group. Another good source of challenges for KS2 are the Challenge cards on the Maths Warriors site.

**Problem Solving**

Whenever a new skills is taught and learned, make sure the children have the opportunity to practise their skills in a real context by applying them to solving problems. It can be particularly meaningful to give the problems a context from another curricular area. As well as regular opportunities to solve problems as part of their maths lessons, children often also respond well to the challenge of a ‘Problem of the Week’ which can be displayed in class for a set time. The Nrich site is a good source of suitable problems, some of which are available as posters. The Numeracy Strategy Logic Problems also have problems at a range of levels.

**Missing Number questions**

For calculation in particular, once a calculation process is learned (eg. column addition), presenting questions with missing digits can extend children’s thinking about the process they have been using.

**Other useful websites**

Mathpickle has some interesting videos and other activities.

Mathsticks has lots of useful resources. There are lots of great activities which are free to download, and some premium resources if you can stretch to a membership.

7puzzle posts a new puzzle every day. The site also categorises the archived puzzles into Easy, Medium and Hard etc.

My Pinterest board has lots of other ideas for investigations, puzzles and challenges.

# There or There Abouts – Using Estimation to Build Number Sense

‘Explore MTBoS’ is a series of challenges put together by a group of experienced maths education bloggers to help those of us with less experience to find our way around the world of maths blogging. I’ve found it a useful way of finding other people who blog about maths teaching and have already encountered lots of new tools to explore and ideas to reflect on. This week’s challenge was to engage with some collaborative sites and although I was already familiar with some of these, many were completely new to me and well worth exploring. I’m sure I’ll be coming back to them.

One that really caught my attention was ‘Estimation 180’. This is a site put together by Andrew Stadel who teaches middle school maths. He has posted hundreds of estimation challenge pictures which could be used as starter activities to lessons. There is a handout that can be used to keep track of estimations over a period of time. Students are encouraged to give an estimate that is too high, one that is too low and then their best estimate. Importantly they are also asked to explain their reasoning, based on contextual clues or pre-existing knowledge. There are lots of ways of using the challenges. Students can submit estimates online and explore the answers that others have submitted and their reasoning. They could fill in the handout each day and keep a record of their estimates. Or the challenges could just be posted up by the teacher at the start of each lesson. The challenges are varied – estimating heights and weights, number of objects, ages etc. and often build from day to day so that the answer to the previous day’s challenge can inform today’s estimate. Key to using this effectively would be giving students the opportunity to explain and share their reasoning. Sharing strategies and approaches could make a valuable contribution to building number sense. I like the fact that many of the challenges involve measures as I often find children find estimating these particularly difficult.

The site is a very useful resource because estimation can be a tricky skill to teach. Give children a typical sheet with pictures of objects and ask them to estimate and then count, and all but the most compliant will probably sneakily count first then make their estimate very close to the actual count (and the reasons why they are so reluctant to risk a wrong answer will probably make a whole new blog some time soon). I’ve found the Primary Strategies Estimation Spreadsheet (shown above) useful as it can be used on an IWB, and the stars can be shown and quickly hidden before the children have a chance to count them. It can be downloaded here. Another interesting looking site is the ‘Guess It’ game on the Problem Site. This gives children a series of estimation challenges by showing dots of different sizes and colours. There is a timer which can be used to adjust the number of seconds the dots are shown for.

I also like the idea of having an Estimation Station in the classroom, a transparent container that is regularly filled with small objects. Children then estimate how many objects are in the container and strategies are taught and compared. Looking at the price of the Amazon one though, I think I could probably come up with a cheaper alternative!

Some of the resources I have mentioned in this blog, can be found on my Number Pinterest board

# Rich List – Using Rich Tasks in Maths Lessons

The new primary maths curriculum has been criticised for its focus on fact fluency and traditional written methods. However, of the three key aims at the beginning of the document, only one focuses on fluency. The other two are that we should ensure that all pupils:

“**reason mathematically** by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.”

and

“ can **solve problems** by applying their mathematic to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.”

It’s very important that we don’t lose sight of these very important aims in the new drive for increased fluency in recall and calculation. Regular use of rich mathematical tasks in our maths classes can really contribute to both of these.

So what is a rich task? In her very helpful Nrich article, Jennifer Piggott describes the characteristics in some detail and some of these are:

- accessibility – they should offer different levels of challenge to learners of different ability, giving opportunities for early success but also scope to extend learning for the more able (low threshold, high ceiling)
- encouraging growing confidence and independence, often by working collaboratively
- potential to link with other areas of maths or to introduce entirely new areas of maths
- encouraging different approaches and creative solutions to problems
- allow learners to pose their own problems and ask questions

Jennifer Piggott also makes the important point that a mathematical task, although it may have the potential to do many of these things, does not become rich unless it is well led by the teacher, asking timely questions and supporting the children just enough to start to construct their own mathematical understanding whilst avoiding ‘spoon-feeding’ them. In practice, this can be difficult to do. In a busy classroom, it can be very tempting to wade in when a child is stuck and show them how to do it, but if we can restrain ourselves and instead offer a hint or a question that might open up a new avenue to explore, the experience will ultimately be much more satisfying and beneficial to our learners.

One good example of the sort of activity that could be used in this way is the ‘Sticky Triangles’ activity from Nrich. Children are presented with a growing pattern of triangles as above. These can be made from lolly sticks or pencils or similar or just sketched. You might like to present just the first two steps to start with and see if the children can suggest how to extend the pattern. Then get them to work on their own or in pairs or groups to explore the patterns. It’s probably best not to ask too many questions to start with. Children often naturally start to notice things about eg. how many triangles are in each row, how many lolly sticks are needed to make each pattern. It can be very interesting to watch the children and see how they approach things. Do they work systematically? Do they record anything? After a while, you might want to suggest some possible avenues for exploration. Can you see any patterns in the way the number of lolly sticks increases with each new row? Can you predict how many triangles will be in the next row? How many triangles would be in the tenth row? How many lolly sticks would be needed by this stage? What about the 100^{th} row? Can you suggest any good ways of recording your findings? Encourage children to explain the patterns they see to each other and to you, and encourage the use of accurate mathematical vocabulary as they do this. The notes on the activity on the Nrich site also offers some other possible ways of extending the task even further.

The Nrich site offers lots of these sorts of activities at all sorts of different levels. As a teacher, I’ve found their curriculum mapping documents for KS1 and KS2 very helpful in identifying activities which might be linked to our other current work. Another source of helpful activities is the BEAM resources which can now be found in the elibrary of the National STEM Centre. You do need to register to access these resources, but registration is free and well worth while as there are a great wealth of resources in the elibrary.

For other suggestions for mathematical investigations, puzzles and challenges, have a look at my Pinterest board.

# Wonder Walls – Maths Working Walls

Have you ever had the experience of looking for a household object, knowing you’ve seen it somewhere but unable to remember where, then finding it in a place that you walk past several times a day? If things are there long enough and we don’t make use of them, they become ‘wallpaper’ and we often stop noticing them altogether.

Unfortunately classroom displays can suffer from the same fate. We can spend hours in the Summer holidays putting up impressive displays, but if we don’t ever refer to them, sooner or later our children will stop noticing they’re there, let alone making use of them. This is where working walls should come into their own. The idea of a working wall is that it should be full of things that will support children’s learning and help them to learn more independently. They should be constantly changing to match our current topic. I appreciate this can be difficult to achieve in the life of a busy teacher and so my top tips for saving time would be:

- Keep things simple – there’s no need for triple mounting and laminating (unless it’s a resource you will use again and again), as long as it’s legible and clear.
- Keep everything – devise a system for filing away your resources so you can dig them out next time you teach this topic. I usually keep things in folders labelled by topic.
- Make use of printable resources – lots are available from sites like Teacher’s Pet and Communication4All.
- Get the children to help – independent or homework tasks could include making posters about your current topic, showing how to use a method or illustrating some new vocabulary.

What should be included on a working wall? This might vary according to the age of your children, but might include:

- Vocabulary related to your current topic (the very useful Cheney Agility Toolkit has this editable word wall which you could use)
- Relevant models and images
- Worked examples of methods – these can be screen shots from your whiteboard or photocopies of children’s work
- Problems and challenges – make these interactive if possible, perhaps by children responding on sticky notes (Nrich have some good posters that could be used for this)
- Number lines or washing lines related to your current learning (eg. lines counting in hundreds or in decimals or in multiples of 2)
- Examples of children’s work (What A Good One Looks Like)
- Real life examples of your current topic (again this is a good task to give for homework – ask children to look for eg. examples of circles, or bar charts or timetables and bring them in)
- Photos of children working on practical tasks
- Practical resources that children can use (eg. mirrors, hundred squares, number lines etc.)
- Success criteria

Whatever you include, make sure you refer to it often and wherever possible refer children to it when they need help.

The Educationworks site has several examples of good working walls for both maths and literacy and there are other examples and ideas on my Working Walls Pinterest board.

# Start Right – Lesson Starters

At the start of the school year, teachers tend to give a lot of thought to how they will make a positive start to the year with their classes: establishing new routines, setting up an attractive and supportive classroom environment, building relationships with children. All of these help to set the tone for the year ahead.

In each new maths lesson, we have the opportunity right at the start of the lesson to set the tone for the whole lesson by the way we start it. There was a very useful and interesting discussion about lesson starters on twitter yesterday evening on the NCETM initiated #mathscpdchat. Lots of different ideas and resources were shared but all agreed that making a positive start to the lesson was crucial. I’m a great believer in having something ready for the children to do as soon as the lesson starts. That way, children get the message that we’re here to work and make the most of every second of lesson time. The actual content of the starter will very much depend on what we are using it for.

**Why have a starter activity?**

In the discussion yesterday, someone made the very good point that we don’t have to have a starter at all. Sometimes it might be appropriate to move straight into the content of the lesson – finishing off work from the last lesson or starting to explore an open ended challenge that will last for the whole lesson time, for instance. However, I’d guess that mostly teachers will want to have a starter for most lessons and so, before we reach for our tried and tested bank of staple activities, we need to ask ourselves just what the starter is for. There are a number of very valid reasons for using a starter activity and we need to choose an activity that is appropriate to our aim. Here are a number of possible reasons for using starters and some suggestions of activities that might be used for each.

**Hooking students in**

Our starter can be a way of engaging children’s interest. Puzzles and challenges can be good for this. The 7puzzle site provides a new number challenge every day and it’s worth looking through the archives as some puzzles are much more challenging for others and could be aimed at different age groups or ability levels. The Chris Moyles quiz show used maths challenges which I’ve found have been popular with KS2 children and there is a bank of them here, although I’d recommend watching them beforehand to check the content is entirely suitable for the age group. Sometimes, just an image on the board which gives something to think about can excite interest. I particularly like the examples of bad maths here. In the twitter discussion yesterday, @PGCEmartin suggested using short video clips or web pages of news and sports items and asking mathematical questions about them.

**Connecting with previous learning**

This might be by recapping a skill learned in the last lesson, giving time to respond to feedback in books or getting children to explain a concept or skill recently taught. Building up a learning wall which is kept current can help with making these connections too.

**A mental ‘warm-up’**

All kinds of activities can be used for this. There are lots of sources of interactive games that can be used on the IWB, including those at mathsframe or Crickweb. Number chains can draw on a range of mental maths skills. As I think I’ve mentioned before, I’m also a big fan of having a structured programme of daily counting.

**Skills practise or skills building**

This very much overlaps with the mental ‘warm up’ idea and similar activities can be used but it’s worth thinking about which particular skills need building up by your group. I’ve used tables practice sheets, for instance with children working on the times table that they’re currently learning, and a similar idea could be used for number bonds and all sorts of calculation skills. I’ve used maths minutes which are available for different primary age groups and found that children respond well to them. Target boards like these can be used in lots of different ways and it’s worth building up a bank of them. Another idea is to have a number of the day which can be done either by using a permanent display or by means of a worksheet. The number can be changed every day and the instructions varied to suit different ability levels.

**Revision**

Particularly as end of key stage tests or end of year assessments draw near, it can be good to choose activities that help children revise previously learned concepts and skills. Sites like the BBC Revisewise or Education City (if your school has a subscription) can be good for this. Games like ‘Who wants to be a mathionaire’ can also be used.

**Introduction to the lesson content**

Sometimes, you might want to use the starter to introduce the main topic in some way. I love this video for introducing division, for instance. A simple idea suggested by @TheMathsMagpie yesterday was to have cards with key questions on one side and the answers on the other, which children could use to test each other.

**Finding starters online**

In the discussion yesterday, we agreed that it’s important to vary the kinds of starters we use. However, it’s always good to have a trusty source of starter activities which we can fall back on. I’ve already mentions 7puzzleblog, Crickweb and mathsframe. Nrich has some suggestions for starters on its excellent site. Other good sites are the Transum Starter of the day site and the Flash Maths site, which is aimed at secondary but has several activities which would be suitable for KS2. Another site mentioned in yesterday’s discussion which I hadn’t come across before was A+click which looks very useful.

As a quick reference, I’ve put together a pinterest board of starter ideas which contains links to most of the sites I’ve mentioned and some others too.