# Go Figure the best problem solving strategy

In a break with tradition I am going to choose a puzzle of the month that I have already used as the monthly puzzle a couple of years ago. The reason is that two weeks ago I heard the most wonderful new solution to the puzzle that I’m sure you will appreciate so let’s start with the puzzle:

Three people enjoyed a meal at a restaurant. The waiter brings the bill for £30 so each person pays £10. Later the chef realises that the bill should have only been £25 so he sends the waiter back to the table with five pound coins. The waiter could not figure out how to divide the £5 so he gave each person a £1 and kept £2 for himself.

So….the three people have paid nine pounds each for the meal:  3 x £9 = £27
The waiter kept two pounds:   £27 + £2 = £29
What happened to the other pound?

The new answer will be at the bottom of this newsletter but before that here are some of the new resources added to the website this last month.

Go Figure is a number placing puzzle in which interconnected addition, subtraction. multiplication and division calculations have to be completed using the digits one to nine.

I got quite excited when I saw pupils using this activity for the first time and heard them talk about their insights. The puzzle can be used to introduce a new problem-solving strategy for this kind of task. Rather than concentrate on which digits could go in the available spaces, make a list of the digits that could not possibly go into the spaces. You really need to try this yourself to see how the properties of the four rules are analysed in the puzzle solving thinking. Make sure you click the ‘Show Tags’ button to assist you find the solution.

Olympic Rings was put together to coincide with the Winter Olympics but the relevance of the puzzle will live on during this inter-Olympic time. There are three levels of difficulty with the lower levels being made easier with some pre-placed digits. This makes the puzzle accessible to younger children but also provides a starting point for an advanced level proof investigation.

Map Scales came about because, after being asked for a good exercise on ratios as used in map scales, I couldn’t find one! There are two levels and the second level introduces the tricky and not necessarily intuitive notion of area scale factors.

Barmy BIDMAS Is a new advanced Lesson Starter. You will need to know about the order of operations and factorial notation to appreciate the subtly of this mathematical wonder. Students could be challenged to make a similar calculation with the surprising value of 6!

Time Sort is the latest additions to the ‘Telling the Time’ collection. There are three levels including digital times, analogue clocks and phrases to represent times. Try using it with pupils working in pairs and listen to the discussion generated.

Sum to One is a set of decimal numbers on virtual cards which can be used for a matching activity. A pairs game, a multiple choice quiz, a tug-of-war game and a snap game. Is that too much choice?

The book I am reading at the moment is Craig Barton’s new book How I Wish I’d Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes. I am a great fan of Craig’s podcasts (I listen to them on my Tuesday morning cycle ride) and this book collects together the insights Craig has collected from all of the educational experts he has interviewed. At the time of writing, 93% of the reviewers on Amazon had awarded the book five out of five stars. I thoroughly recommend this book to you here.

Thanks to those of you who have posted photographs on Twitter of the Transum activities being used in the classroom. It is so good to see that the work that went in to producing the resources was worthwhile. Thanks

If you follow me on Twitter (@Transum) you may have noticed that my list of ‘Hidden Gems on the Transum Website’ has been growing from the 19 included in the last newsletter. I think I will stop when I get to 50.

Now let’s continue the search for the missing pound from the puzzle of the month. One hour later two elderly ladies came into the restaurant and enjoyed a meal together. The waiter brings the bill for £30 so each lady pays £15. The chef again tells the waiter that the bill should have only been £25 so he sends the waiter back to the table with five pound coins. The waiter could not figure out how to divide the £5 so he gave each lady a £1 and kept £3 for himself.

So….the two ladies have paid fourteen pounds each for the meal:  2 x £14 = £28

The waiter kept three pounds:   £28 + £3 = £31

So there is the missing pound! Genius isn’t it? I heard this solution on the Danny Baker radio show and have included the audio excerpt towards the bottom of the June 19th Starter of the Day. It’s worth listening to. Enjoy.

All the best for the month ahead

John

P.S. Always wear glasses to Maths lessons. They help with division!

# 19 Hidden Gems for Deep Maths Learning

This is the Transum Mathematics Newsletter for February 2018. This month’s puzzle comes from The Penguin Book of Puzzles, a collation of great puzzles from old, out of copyright books written by the prolific puzzle setters from way back.

“Divide 45 in four parts, so that the first part with two added, the second with two subtracted, the third divided by two, the fourth multiplied by two, shall be equal to each other.”

That will give you something to think about. It’s not a familiar puzzle format is it? The answer can be found at the end of this newsletter.

The three most noteworthy new learning objects added to the Transum website this last month are as follows:

Frequency Trees: Being able to construct and read these diagrams is a new topic which appears on both the higher and foundation GCSE(9-1) specifications. This interactive exercise requires pupils to fill in the partially completed frequency trees then try some exam-style questions that introduce calculating probability from the numbers in the trees.

Number Grids: I have started collecting a variety of activities that can be enjoyed using this page of customizable number grids. As a Transum subscriber you get access to buttons that can quickly colour in the grids with the most common number patterns for pupils to describe. Please let me know if you have other ideas for number grid learning activities.

Old Equations: Most students can deepen their mastery of linear equation solving with these old, but just as good as new, intelligently varied questions. These linear equations appeared in a book called A Graduated Series of Exercises in Elementary Algebra by Rev George Farncomb Wright published in 1857.

The book I’m reading at the moment is “What does this look like in the classroom?: Bridging the gap between research and practice” by Carl Hendrick and Robin Macpherson. It is a very readable book in which the current educational experts answer the commonly asked questions about learning backed by the latest research. You can dip in and out of the book focusing on the chapters that interest you. The chapter headings are: Assessment, Marking and Feedback, Behaviour, Reading and Literacy, Special Educational Needs, Motivation, Memory and Recall, Classroom Talk and Questioning, Learning Myths, Technology and Independent Learning.

The thought buzzing around my mind at the moment is the notion that “I think the paradox is that the things that make you a good independent learner don’t necessarily look like independent learning.” Or in other words “…independent learning might be a desired outcome, but paradoxically, it may not be the best way to achieve that outcome.”  Get a copy of the book and read about this and other Maths teacher dilemmas.

Over the last couple of weeks I have been posting “Hidden gems on the Transum website” on Twitter. The idea is to make teachers aware of some of the Transum pages I think are really useful but you wouldn’t necessarily think of searching for. They are also like needles in a haystack as there are over 4000 pages indexed by Google on the Transum website. Here are the gems I’ve already tweeted at the time of writing.

More will be tweeted soon as there are many more hidden gems. You can find out about them if you follow me (@Transum) on Twitter.

The answer to the puzzle of the month is 8, 12, 20 and 5

That’s all for this month,

John

P.S. What do you need to calculate the distance around a circle of sheep?

# A Happy MMXVIII From Transum

Happy New Year. I hope that 2018 proves to be a good, positive number for you and that you, and your pupils, achieve all that you want during the next twelve months. If the Roman numeral in the title caught your eye you may like the Roman Numerals Quiz.

This month’s puzzle is taken from the excellent book I have been reading during the holiday called “Can You Solve My Problems” by Alex Bellos. I have just read the problem called “The Shrivelled Spuds” which I present to you here:

A pile of potatoes weighing 100kg is put in the sun. Ninety nine per cent of the weight of the potatoes is made up of water. After a day some of the water evaporates., with the result that 98 per cent of the weight of the potatoes is now made up of water. What’s the new weight of the potatoes?

I thoroughly recommend the book as not only is it an ordered collection of intriguing puzzles but it also has an extensive solution section in which Alex provides insights, history and worked solutions for the puzzles. The chapters are Logic Problems, Geometry Problems, Practical Problems, Problems with Props and Problems for Purists. Here is a link to buy the book from Amazon.

Last month the website was added to and updated as usual but the one new activity I would like draw your attention to is the Area Wall Puzzles. The core concept is a puzzle called Shikaku, an original Nikoli puzzle and though the Transum version refers to area, the activity requires the ability to consider factor pairs of small numbers.

In the process of creating and testing the puzzles I realised how addictive this type of problem solving is. Just like Sudoku solving you will develop strategies as you become more proficient and experience a nice sense of accomplishment when the wall is completely coloured in.

Another new activity is called “Equation of a Line Through Points“. It is a four level exercise requiring users to match the equations of the straight line graphs to the clues about gradients and points. This exercise could be attempted after some of the more basic “y=mx+c” activities have been mastered.

Finally, my answer to this month’s puzzle is 50kg

I found this by first realising that if 99% of the original pile is water then 1% must be other, dry matter.

1% of 100kg is 1kg.

Let the weight of the potatoes after the drying be x.

0.98x + 1 = x

1 = 0.02x

x= 50

That’s all for now

John

P.S. You have to be odd if you want to be Number One.

# Christmaths Activities

As December has begun I hope you don’t consider it too early for me to wish you a Merry Christmas. As usual this newsletter will begin with a puzzle of the month. A slightly more difficult puzzle this month that could keep you thinking right through the holiday.

Young Noel Stocking checked his bank account to see how much money he had to buy Christmas presents. When he recorded the balance he wrote down the number of pence for pounds and the number of pounds for pence. A transposition error. “Wahoo” he exclaimed “I’m rich!”

While in this good mood Noel promised 50p to his younger sister Merry. He adjusted his record accordingly which now turns out to be exactly double the amount in the bank.

How much does Noel have in the bank?

Many of the activities on the website have been updated during November and, in preparation for the festive season, the Christmaths collection has been brightly polished. Here are my favourites:

Christmas Ornaments: A puzzle which can be solved online or by using the printable boards so the task can be done in a very practical way with real Christmas ornaments. Problem solving, trial and improvement and logic all wrapped up in one for Christmas.

Christmas Eve Snow: This is a letters-replace-digits puzzle which comes with clues and, if you scroll down to the bottom of the page, a step-by-step guide for solving this kind of puzzle.

Christmas Tables: This is the special Starter for Christmas Day. When I first saw the Betty Edwards drawing I was amazed that the sense of perspective is so powerful. You will need Flash to view the animation.

Cracker Joke: This is a basic numeracy activity where answers to simple calculations are converted to letters to spell out a maths joke. Each time the page loads the calculations change. You can change the joke too.

The Power of Christmas: This Starter works well with Year 11 and Year 12 pupils as it tests their understanding of indices. Finding one solution is fine but the real challenge is to find all four solutions.

The Twelve Days of Christmas: This is a well-worn and time-honoured problem that’s certainly doesn’t deteriorate with age. I particularly like the solution as sung by Natalie Cole.

Christmas Tree Trim: This is just one of the activities on the Transum website that allows pupils to demonstrate their systematic listing skills. There are eight levels of difficulty and a trophy available for each level.

The links to all of these activities (and more) can be found on the Christmaths page. The last week of term is a great time to do some out-of-the-ordinary mathematics with your pupils and there are plenty of ideas on that page. In my experience some of the end-of-term, fun maths has turned out to be the most memorable and enjoyable learning my pupils did all year!

Have your pupils encountered the binary system? There is a visual aid you can use called Binary Lights to demonstrate how binary works. The reason I mention this is that recently I heard a nice idea about counting on your fingers. It’s common knowledge that you can count to ten with the help of fingers and thumbs but if you use binary you can count up to over 1000! Scroll down the Binary Lights page to see a video demonstration.

The answer to this month’s puzzle is £16.33. I found this answer with the aid of a spreadsheet. Please let me know if you have another way of finding this answer.

That’s all for this month

Enjoy the holiday

John

Q. How is an artificial Christmas tree like the square root of minus nine?

A: Neither has real roots!

# 7 New Maths Resources and Ways to Count Sheep!

Welcome to the Transum Newsletter for November 2017 which has a bit of a sheep theme!

The puzzle for this month is about Percy Cod and Patsy Eal who are both shepherds. They keep sheep in adjacent fields near the source of the river mint. Can you work out how many sheep each person owns from the following clues?

• If Percy sold seven sheep to Patsy, both shepherds would have the same number of sheep.
• If Patsy sold seven sheep to Percy then Percy would have exactly twice as many sheep as Patsy.

Talking of sheep, I would like to recommend the Transum podcast for this month. It has a strong sheep counting theme. Within the podcast there are excerpts from the ‘No Such Thing as a Fish’ podcast and the Chris Evans Radio 2 breakfast show, both discussing the skill of counting sheep.

If you haven’t yet found them there are three sheep related activities on the Transum website: Estimating Sheep is designed to develop your count estimating skills, Herding Sheep introduces Loci  and Coloured Sheep is a quick Maths Lesson Starter involving probability.

If all this sheep counting hasn’t sent you to sleep you might still have the energy to explore the seven new additions to the Transum website last month:

1. Angle Chase – Find all of the angles on the geometrical diagrams. Knowledge of basic angle theorems (in a triangle, on a line and between parallel lines) is required.
2. Snake Sort – Sort the coloured snakes in a logical order. This activity introduces systematic listing.
3. Indices True False – Arrange the given statements involving indices to show whether they are true or false. A knowledge of the laws of indices is required.
4. Yes No Questions – A game to determine the mathematical item by asking questions that can only be answered yes or no. Includes shapes graphs and mathematical words.
5. Combinations and Permutations – Learn how to tell the difference between permutations and combinations and use the formulae to answer questions.
6. Identity, Equation or Formula? – Arrange the given statements in groups to show whether they are identities, equations or formulae.
7. Birthday Card – A mathematical birthday card which prints on A4 card (double sided). Believe it or not this birthday card also has a sheep counting theme!

Thanks to Felton Davis for sharing his “work of a lifetime” on the Pentominoes page. Felton has created an algorithm to create 2339 solutions which are linked to from the bottom of the pentominoes page.

Thanks also to Christopher Allan who has suggested the following extension to the “A Number” Starter: “When written as a word or words, the smallest positive whole number containing the letter ‘a’ is one hundred and one after which they all contain ‘a’ up to X (but what is X ?).”

The answer to this month’s sheep-owning puzzle can be found by setting up two simultaneous equations. Let the number of sheep owned by Percy be x and the number of sheep owned by Patsy be y.

Did you also conclude that Percy has 49 sheep and Patsy has 35 sheep?

That’s all for now,

John

P.S.     f(x) walks into a bar. The barman says, ‘Sorry, we don’t cater for functions!’

# 7, reasons to get an exam question wrong

This is the Transum Mathematics Newsletter for October 2017. As usual I will begin with the puzzle of the month.

In the fantasy world of Maths puzzles there aren’t any stopwatches but there are two egg timers. One of them measures 7 minutes and the other measures 4 minutes. I want to use them to time my daily juggling practice which I want to last for exactly 9 minutes. How can I do it?

The answer, if you don’t manage to figure it out yourself, is at the end of this newsletter.

The most amazing thing I learnt this month was the effect the number seven has on exam performance. I have been teaching since the early eighties and didn’t, until this month, know about this surprising phenomenon.

The news came to me via the wonderful Mr Barton Podcast. Craig was interviewing Trevor Senior who is AQA’s Chief Examiner for the Maths GCSE. When asked about his favourite number seven he related the astonishing fact that if the number seven appears in an exam question the success rate drops by about 10%.

I was fascinated by this and it gave me an idea for a Transum activity that would collect some data. I decided on a ten question exercise intended to test pupils’ written methods. The add, subtract, multiply and divide questions would be randomly generated in pairs. One question of the pair would contain a seven and the other would not. Pupils type in their answers and the system would mark them. The data from those getting at least five questions correct would be recorded in the database and then presented to the pupils at the end of the exercise.

It is intended that pupils do the exercise without knowing the purpose of the data collection. The exercise is a good opportunity to practise written methods so that can be given as the reason for doing it. The title ‘Heptaphobia Research’ is suitably cryptic to disguise the data-collection purpose of the exercise.

At the time of writing only ten people have completed the exercise and the results show that the number seven does not have the effect I was expecting but it’s early days. We need hundreds of people to contribute to the data before we can begin confirming our hypothesis.

With that in mind could I ask that if you have a class that could do with a bit of arithmetic practice, please ask them to do this activity. The more people that take part, the more valuable the data will be. It would be good if the database grows to be the same size as the times tables database which currently has over two hundred thousand sets of data.

An activity that has been on the Transum website for many years now is called Pentransum. It is a collection of multiple choice questions suitable for pupils in Key Stage 3. When a pupils has answered 20 questions correctly they are invited to suggest a question of their own. If I consider it to be suitable it is added to the collection of questions in the database.

When designing a good multiple choice question, as well as coming up with meaningful distractors, you have to choose in which position the correct answer will appear. I found it interesting that about twice as many submissions placed the correct answer in the middle position (C) than the two extreme positions (A) or (E). I guess that’s human nature to subconsciously think that something in the middle is more hidden than something on the outside.

I have used a Casio fx350ms calculator with GCSE students for many years but, to keep up with developments, have now got a Casio fx350es plus. I’m sure many people, like me, don’t have the time to learn all the functions available and sometimes a student will show you something that will stop you in your tracks. Last month I learned that the FACT function will display a number in its prime factors. Very useful for working out if a large number is prime.

One of the new additions to the website last month is a multi-level quiz called Systematic Listing Strategies. Level 1 is a collection of listing puzzles for younger children which encourage them to develop logical methods of listing. The higher levels include standard combination and permutation questions. Let me know what you think.

Here is the answer to this month’s puzzle. Start the seven and four minute egg timers together. As soon as the four-minute timer has finished start it again.

When the seven-minute timer has finished start juggling.

Restart the four-minute timer two more times each time it finishes to obtain the nine minutes of juggling time.

That’s all for this month.

Don’t miss the 31st October Halloween Starter!

John

P.S. Why do mathematicians think that Halloween is the same as Christmas?

Answer: 31 Oct = 25 Dec

[Think number bases!]

# 20 mathematical activities children could be doing during school holidays

Welcome to the Transum mathematics newsletter for August 2017 wherever in the world you are. Some of you are enjoying the long school summer holidays while for others it is business as usual. Before getting on to the main theme of this missive let’s begin with this month’s puzzle.

I have randomly selected two different whole numbers. They aren’t both odd. What is the probability that they are both even?

The answer is at the end of this newsletter but now let’s move on to the main theme of the moment which is about children continuing their learning during the school holidays. Research indicates that by the end of the long summer holiday pupils perform, on average, one month behind where they left off in the spring. The Transum website has plenty of suggestions for activities that will keep children thinking mathematically while having fun and learning new skills. The links for all of these activities are at Transum.org/Holiday

A particularly worthwhile way to spend some of each day during the holidays is to practise the ability to quickly recall multiplication facts. Here are three suggested Transum activities that can be found at the link above.

• Times Tables: A collection of activities to help pupils learn a times table in only 5 days.
• TablesMaster: This extremely popular activity provides feedback to help pupils improve.
• Times Square: Another way to practise mixed tables is by completing this multiplication grid.
1. Newspaper Type Puzzles

Here are some online interactive puzzles of the type often seen in newspapers.

• Suko Sujiko: Interactive number-based logic puzzles (drag and drop).
• Zygo: Randomly-generated, number-based puzzle designed to develop numeracy skills.
• Online Sudoku: An interactive version of the popular number placing puzzle.
1. Exam Preparation

For those who are preparing for forthcoming mathematics examinations.

• Weekly Workout: GCSE-style questions GCSE Mathematics preparation.
• Exam Objectives Checklists: Interactive, printable examination learning objective checklists.
• Formulae to Remember: The traditional pairs or pelmanism game adapted to test recognition for the formulae required to be memorised for GCSE exams.
1. Mathematical Magic

Children can amaze parents and friends with these magic tricks that are based on mathematical concepts.

• Mathemagic: A collection of magical tricks with full explanations.
• Magic Square: Each row, column and diagonal should produce the same sum.
1. Go On A Maths Adventure

Maths adventure games have been around since the first computers appeared in schools.

• Tran Towers: An adventure game requiring pupils to solve puzzles as they move through the old mansion.
• Tran Tunnels: Similar to the above but this time you move through virtual tunnels.
• Pentransum: Mathematical questions with five possible answers. If you get 20 correct you can add your own question to the database.
1. Online Board Games

Whether you prefer the top hat or racing car you don’t need tokens for online board games.

• Mathopoly: A game of buying and selling property with maths questions thrown in for good measure.
• Dump-A-Dice Race: An online board game for two players involving prime and square numbers and making choices.
• Hi-Low Predictions: A version of the Play Your Cards Right TV show. Calculate the probabilities of cards being higher or lower.

Being able to memorise figures, names or concepts is an important skill which these activities develop.

• Kim’s Quiz: The traditional memory game played with mathematical statements.
• Mathematician Pairs: The traditional pairs or Pelmanism game requiring the ability to recognise some of the great mathematicians.
• Mathterpieces: Memorise eight pictures made up of geometrical shapes then sort them into order.
1. Searching For A Strategy

Playing maths games against parents and friends is always more satisfying with a winning strategy.

• Nim is a mathematical game of strategy in which two players take turns removing objects from groups of objects. Whoever takes the last object wins.
• Tantrum: A game, a puzzle and a challenge involving counters being placed at the corners of a square on a grid.
• Fifteen: A strategy game. Play against the computer to select three numbers that add up to 15.
1. Practical Activities

Get the scissors, glue sticks and rulers for some hands-on mathematical tasks.

• Kite Maths: Can you make a kite shape from a single A4 size sheet of paper using only three folds?
• Paper Constructions: Practical mathematical skills are required to work out how to construct these three dimensional items from paper.
• Tangram Table: Use the pieces of the tangram puzzle to make the basic shapes then complete the table showing which shapes are possible.
1. Computer Games

If the child has a computer, iPad or similar, these games have a mathematical link.

• Snooker Angles: An online game for one or two players requiring an ability to estimate angles.
• Digital Darts: An online darts game for one or two players requiring skill, strategy and mental arithmetic.
• Where’s Wallaby?: Find the hidden wallaby using the clues revealed at the chosen coordinates.
1. Mathematical Vocabulary

All about the words used in Maths. Keep a dictionary close at hand.

• Shark’s Dinner: The mathematical version of the classic hangman game. Guess the letters that are contained in the words.
• Mathanagrams: The letters of mathematical words have been mixed up. Can you recognise them?
• Wordles: Remember these mathematical words for their shape and movement.

Enjoying a quiz seems to be a popular activity and there’s nothing better than a Maths quiz.

• Maths General Knowledge Quiz: Questions about general mathematical facts such as ‘What is the name for the longest side of a right angled triangle?’
• Mystery Numbers: If 16 O in a P stands for 16 ounces in a pound, what do you think these mystery numbers are?
• Maths Trivia: Interesting facts about Mathematics for the enthusiast.
1. Another Point Of View

A selection of illusions followed by an exercise on matching the view to the direction.

• First Impressions: This activity will collect data about your first impressions of some optical illusions. You can then analyse the data to come to your own conclusions.
• Optical Illusions: Don’t let your brain be fooled by these geometric optical illusions in this online quiz.
• Plans and Elevations: Interpret plans and elevations of three dimensional shapes.
1. Trains, Cars and Boats

Test your driving skills with these dilemmas and tests of skill.

• Shunting Puzzles: Move the trams to their indicated parking places in the shunting yard as quickly as possible.
• Car Park Puzzle: Can you get your car out of the very crowded car park by moving other cars forwards or backwards?
• Cliff Diving Monkeys: Test your timing skills by clicking on the monkeys so that they jump off the cliff at just the right time to land in the boat.
1. Programs And Codes

Cracking codes and writing computer programs are skills required for these activities.

• Online Logo: An online version of the Logo programming language.
• Code Cracker: Crack the code by finding out which letters replace the encrypted letters in the text given. There are lots of hints provided about code breaking techniques.
• Roman Numerals Jigsaw: An online interactive jigsaw puzzle of a grid of Roman numerals.
1. Amazing Mazes

You can’t get physically lost in these mazes but you may be at a loss mentally.

• Maze: Each visitor to this page has a unique maze constructed for them to find their way through answering mathematical questions on the way.
• Bidmaze: Find your way through the maze encountering mathematical operations in the correct order to achieve the given total.

If you have a steady hand and nerves of steel you may just be able to do these challenges.

• Fizz Buzzer: The digital version of the popular fizz buzz game. Press the buzzers if they are factors of the counter.
• Tables Grab: A one or two player game. The objective is to grab all the multiples of the chosen times table faster than the other player.
• Watsadoo: Rotate the cogs to catch the flying numbers in the correct sections.
1. Arty Maths

Exercise both sides of your brain and show your artistic side with these beautiful activities.

• Mystic Rose: Investigate the properties of the Mystic Rose by using this interactive diagram.
• Tessellations: Which polygons tessellate? Which pentominoes tessellate? Drag the shapes onto the canvas to create tessellating patterns and investigate the laws of tessellations.
• Wordles: Remember these mathematical words for their shape and movement.
1. Investigations

Mathematical investigations are open-ended and could take quite a long time.

• Design a badge: How many different badges can you make using squares put together to make a rectangle. You can use three different colours but the finished design must be symmetrical.
• Dice Investigation: Throw two dice and multiply the scores. Investigate the different products you can obtain. What about adding? What about using three dice?
• Polygon Areas: Investigate polygons with an area of 4 sq. units. Investigate polygons with other areas.
1. Exercises

The final suggestion is for pupils to be quite specific about the work they do so that it closely matches their school curriculum. The Transum website has a Topics page (for teachers) and a Maths Map (for pupils) to help find online exercises on all the different concepts in the school maths curriculum.

The one link for all of the activities mentioned above is in a compact form, perfect for sending out to pupils and parents by email or having on your school website or learning management system.

Transum.org/Holiday

Finally the answer to this month’s puzzle. The incorrect answer is arrived at by thinking that there are two possibilities, the numbers could be both even or the numbers could be different. Two possibilities so the answer is a half.

The correct answer can be found by considering equally likely possibilities. There are in fact three:

• Both numbers are even
• The largest number is odd and the smallest even
• The largest number is even and the smallest odd

Only one of these three possibilities is the one being asked about so the answer is one third.

Enjoy August

John

P.S. An opinion without 3.14159265359 is just an onion

# Six maths learning questions that research can answer

Welcome to the July 2017 Transum Newsletter. Before the news here is this month’s puzzle sometimes known as the Mystic Rose.

Joining points with chords to divide a circle into regions

If there are two distinct points on the circumference of a circle, a chord drawn between these two points will divide the circle into 2 regions.

If there are three points joined by three chords the circle will be divided into 4 regions. Similarly four points joined by six chords produces 8 regions and five points produce 16 regions.

How many regions will be produced by six points joined by fifteen chords?

The answer is not the obvious one! The answer will be at the end of this newsletter after this month’s Transum News.

A brand new online exercise called Train Timetables was written on my laptop as I flew from Malaysia to England then travelled from London to Wolverhampton at the beginning of last month. It is quite overwhelming the number of different styles train, plane and bus timetables take and I’m surprised that people can actually extract relevant information from them. Some of them are particularly hard to read. The online exercise is based on standard train timetables but I have decided to collect photographs of some of the more obscure timetables and add them as an extra level to the exercise. If you have spotted any good specimens please send them to me.

Another brand new exercise called Functions has also been written to cover the GCSE content and provide a strong base for A-level and IB courses. The ordering of the questions was carefully considered to provide progression without forfeiting consolidation. There are six levels and the higher levels include inverse and composite functions.

When I was in London I attended the excellent MathsConf10, a maths conference for the enthusiastic Maths educator. It was a wonderful day and I chose to attend some excellent presentations.

The first was titled ‘Angles in Depth’ and was presented by the prolific blog personality, Jo Morgan. As the presentation progressed I was rapidly making notes on how the Transum angles exercises can be enriched with some to the ingenious examples Jo had found. Though she limited her presentation to adjacent angles on a straight line and the angle sum of a triangle there seemed to be an endless supply of good ideas for activities, puzzles and exercises.

Another presentation I attended was about Filtered Maths Education Research. Cambridge Mathematics produces Espressos for teachers: filtered research reviews to be enjoyed over coffee, discussed at department meetings, or as a basis for digging deeper into CPD issues of interest.

The research answers the following questions:

Colleen Young presented a rich overview of the many excellent free resources for learning A-Level Mathematics. She emphasised the advantages of using the free resources provided to the Boards other than the one you are teaching for to enable your students to appreciate a diverse learning experience.

Liz Henning Investigated making connections from the word problem to bar modelling to abstract approaches with an emphasis on explicit mathematical language and understanding.

Dani Quinn and Rose Dalders shared how they have introduced an alternative approach to marking and feedback that focuses only on quizzes, not books. They have seen improvements in pupils’ outcomes, higher-quality feedback for both pupils and teachers, and – importantly – reduced workload for teachers.

The conference was organized by LaSalle education and you can read more about their forthcoming conferences here: https://completemaths.com/events

The answer to this month’s puzzle actually depends on whether the points (vertices) are evenly spaced around the circumference of the circle or whether they are spaced to produce the maximum number of regions. In the first case the answer for six points is 30 regions.

If however the points are not evenly spaced an additional region exists at the point where the three diameters intersected in the first case. The maximum number of regions is 31 and the formula is:

This formula was brought to my attention by Paul Metcalf, a colleague I had when I first started teaching (at the beginning of the 1980s). It was good to meet up with him on my recent travels and learn how busy he is keeping himself not only running his own hotel but also freely giving his time to support national mathematical organisations.

The wonderful thing about this puzzle is that most of us, given the sequence 2, 4, 8, 16 …, would have been convinced that the answer was 32. Did you think the answer was 32?

That’s all for this month.

John

P.S. A Mathematician can’t remember whether he’s been going out with his girlfriend for one year or two but he knows it’s <3

# Three positive outcomes of failure in Maths

Welcome to the June 2017 Transum Newsletter. This month’s puzzle is about the Numlove family. Can you work out how many children are in the family from the following two clues?

• Each boy has the same number of brothers as sisters.
• Each girl has twice as many brothers as sisters.

While you think about that here are details of some of the more significant new additions to the Transum website last month.

Writing Expressions is designed to provide practice forming simple algebraic expressions for situations described in words. The words come as short audio clips which pupils can play over and over again by clicking a button on the web page. There are three different versions of each question which are independently chosen at random each time the page loads.

Area of a Trapezium is exactly what it says in the title. Level 1 requires finding the areas of the trapezia by using the standard formula. Level 2 requires the application of the trapezium area formula in different ways. There are some nice problem-solving questions here.

Venn Totals completes the Transum collection of Sets activities. It is a multi-level exercise in which you read or enter the total number of elements in regions of two- and three-set Venn diagrams.

Many other activities on the website have been updated during last month with better interfaces or more detailed answers. Talking of answers someone is needed to find the solution to the level 5 Tantrum Puzzle as I am stumped! A screenshot of the solution would be very much appreciated.

The book I am been reading at the moment is “Black Box Thinking: Why Most People Never Learn from Their Mistakes – But Some Do”. The author, Matthew Syed, argues that the most important determinant of success in any field is an acknowledgment of failure and a willingness to engage with it. This theme resonated with me as a teacher of Mathematics and made me think of ways we could better use learners’ failures or mistakes to help them improve.

One example mentioned in the book was about the analysis of a large data set. It was the story of mathematician Abraham Wald who was presented with the following question.

You don’t want your planes to get shot down by enemy fighters, so you armour them. But armour makes the plane heavier, and heavier planes are less manoeuvrable and use more fuel. Armouring the planes too much is a problem; armouring the planes too little is a problem. Somewhere in between there’s an optimum. Wald and his team had to figure out where that optimum is.

The military came to Wald with some data they thought might be useful. When American planes came back from engagements over Europe they were covered in bullet holes. But the damage wasn’t uniformly distributed across the aircraft. There were more bullet holes in the fuselage and not so many in the engines.

Here was an opportunity for efficiency; you can get the same protection with less armour if you concentrate the armour on the places with the greatest need, where the planes are getting hit the most. That would seem to make sense but Wald thought differently.

He reasoned that the armour should go not where the bullet holes are. It goes where the bullet holes aren’t: on the engines.

Wald’s insight was simply to ask: where are the missing holes? The ones that would have been all over the engine casing if the damage had been spread equally all over the plane? Wald was pretty sure he knew. The missing bullet holes were on the missing planes. The reason planes were coming back with fewer hits to the engine is that planes that got hit in the engine weren’t coming back.

Wald’s interpretation of the data with a little out-of-the-box thinking and a lot of common sense provided the solution that the engineers could put into practice.

What a wonderful ‘large data set’ story. Now if only I could get hold of the bullet hole coordinates to create a data analysis activity for the Transum website … !

On the topic of failure, did you know that Steve Ballmer, former chief executive officer of Microsoft and 22nd richest person in the world, was told he was failing at Maths when he was at school? You can hear him talking about it on the podcast version of this newsletter.

The last word on failure is the strategy of trial and improvement. It is valid mathematical technique that might be used in the Where’s Wallaby activity but refined as learners develop and use Iteration for solving equations. Learn from your mistakes!

Now here’s a success story from National Numeracy. They launched a new mobile game called Star Dash Studios, a free game that brings maths to life. The character in the game is a runner on a movie set who has to solve puzzles and carry out tasks for the producer – all of which relate to using numeracy in real life situations.

Finally here is the answer to this month’s puzzle.

Let the number of girls in the family be n.

The number of boys must be n + 1 to satisfy clue number one.

Clue number two produces the following equation n+1 = 2(n – 1)

So n+1 = 2n – 2 or n = 3

Therefore there are 7 children in the Numlove family.

Did you get it?

That’s all for this month

John

P.S. I will do algebra, I’ll do trigonometry and I’ll even do statistics but geometry and graphing is where I draw the line!