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.

The answer can be found at the end of this newsletter.

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:

- 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.
- Snake Sort – Sort the coloured snakes in a logical order. This activity introduces systematic listing.
- 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.
- 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.
- Combinations and Permutations – Learn how to tell the difference between permutations and combinations and use the formulae to answer questions.
- Identity, Equation or Formula? – Arrange the given statements in groups to show whether they are identities, equations or formulae.
- 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!’

]]>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 31^{st} 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!]

]]>How many different ways are there of arranging the digits one to four to make a four digit number? That’s a familiar question in the Maths classroom. This month’s puzzle is to find the sum of all those four digit numbers in a concise, efficient and elegant way. The answer can be found at the end of this newsletter.

For those in the northern hemisphere it’s Back To School time and the Transum website has a list of resources you may find useful at this time of year. Please let me know if you have any suggestions to further develop that area of the website.

The months of July and August have been busy with the Transum laptop being heavily used to create new Maths learning materials for you to use in the classroom. Here are some highlights:

Area Maze is the Transum classroom version of Menseki Meiro, the creation of Naoki Inaba, a prolific inventor of logic puzzles. He came up with the idea after being asked to come up with a puzzle by the head of a school in Japan. Have a look and let me know if I should come up with more levels of difficulty.

Fraction Dissect is an interactive activity. By drawing a straight line between the dots can you split the rectangle to give the target fraction. I was using it this week with pupils of different ages and they all found it a worthwhile learning experience.

The game Skunk is quite new to me but apparently teachers have been playing it in the Maths classroom for years. It gives pupils a feel for probability and generates discussion about the choices made while playing the game. This new Transum version of the game makes life easy for the teacher by providing the dice and results chart.

Numbers in Words is an old Starter but I have updated it as the date, 19^{th} September, I found out is Talk Like A Pirate Day. I couldn’t resist giving it a pirate theme.

The Shine+Write collection has two new resources. Compound Interest calculator and Normal Distribution calculator allow you to make up problems and quickly find the solutions as pupils develop their own calculator skills. While on the subject the Calculator Workout page now has a new ‘skin’ option to accommodate those with Casio fx350es plus, fx83 and fx85GT plus calculators. I am still trying to find data on the types of calculators most popular in schools at the moment. Let me know if you have any information.

There was one National Curriculum statement that didn’t have a related Transum activity. That hole has now been filled by Estimating Powers and Roots which does what the title suggests. It is an animated interface in which pupils have to click on the integer which is closest to the root or power presented.

As part of a lesson introducing the use of the calculator’s degrees, minutes and seconds button to do time calculations I snipped a section of the London Underground map to produce the Walking Times quiz. Half the fun is finding the stations on the map!

The Exam Questions database is now being added to at a rate of one new question each week. Each new question is adapted (inspired by) one of the questions from the recent GCSE papers and a full worked solution is provided. Hopefully this resource will support the cohort that will be taking this exam next year which includes my nephew, Ben. Fingers crossed.

Here is the answer to this month’s puzzle. There are 24 different ways to make a four digit number from the digits one to four. This first few are shown here:

1234

1243

1324

1342

1423

1432

2134

2143

…

It can be seen that each digit appears in each place-value column six times. The sum of the 24 four-digit numbers is therefore:

6 x 1 x (1000 + 100 + 10 + 1) +

6 x 2 x (1000 + 100 + 10 + 1) +

6 x 3 x (1000 + 100 + 10 + 1) +

6 x 4 x (1000 + 100 + 10 + 1) =

6 x 10 x 1111 = **66660**

This method of finding the solution can be extended for situations involving five or more digits.

That’s all for this month,

John

P.S. Did you know that three out of two people have trouble with fractions?

]]>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

**Improve Your Times Tables**

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.

**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.

**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.

**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.**Maths Mind Reader**: Investigate this amazing mind reading animation.**Magic Square**: Each row, column and diagonal should produce the same sum.

**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.

**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.

**Develop Your Memory**

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.

**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.

**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.

**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.

**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.

**How About A Quiz?**

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.

**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.

**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.

**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.

**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.**Navigation NEWS**: Each visitor to this page has a unique maze constructed for them to find their way through answering questions about bearings along the way.

**Steady Hand**

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.

**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.

**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.

**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.

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

]]>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:

- What are the issues in learning and assessing times tables?
- How does assessing confidence affect learning and testing in mathematics?
- Is there any value in applying ‘traditional’ and ‘progressive’ models to mathematics teaching?
- What is ‘number sense’ and how does it affect mathematics learning?
- What are the effects of attainment grouping on mathematics learning?
- How does maths anxiety affect mathematics learning?

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

]]>- 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.

It has been about 6 months since their launch event with Countdown’s Rachel Riley and they are so pleased they have had almost 20,000 downloads to date. If you would like to download it for your mobile device the link is https://www.nationalnumeracy.org.uk/star-dash-studios

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!

]]>Mac has taken seven maths exams this year. His average mark is 78%. What mark must he get on the eighth exam to raise his average to 80%?

While you think about that I’ll alert you to some of the new content added to the Transum website this last month.

Manifest: This is a new game that I have played with some pupils and the excitement it generated was immense. I thoroughly recommend it if you would like to end your lesson with a little bit of strategical thinking thinly disguised as fun. The rules of the game couldn’t be simpler.

Players take turns arranging their cards to make a single digit number, a two digit number, a three digit number and a four digit number. They should do this while the other player is not looking. The player with the largest single digit number wins one point, the largest two digit number wins two points, the largest three digit number wins three points and the largest four digit number wins four points.

The Transum interactive version of the game is designed to build suspense and anticipation at the ‘compare numbers’ stage. Give it a go, you will love it.

Screen Test: A selection of short (one to two minutes) factual videos have been chosen for this test of memory. When the video has finished playing you can reveal the mathematical questions about the video. The first questions test recall while the latter questions require application of the facts. A little bit of variety in a Maths lesson can be provided by this five-minute activity.

Bearings: A five-level, self-marking quiz on three-figure bearings. Level 2 is a measuring exercise with an online protractor provided. If you are planning on using this activity with your pupils be sure to develop bearing estimation skills first by using the Plane Bearings visual aid.

Box Plots: Level 3 of this online exercise is a major new manipulative feature. Pupils can drag the handles on the box-and-whisker diagrams to create the correctly-aligned box plot.

Averages: Not strictly a new activity (3528 people have already earned a Transum Trophy for completing it) but this online exercise has been updated and some of the levels changed.

For most of the online exercises the Check button can now be double clicked to make it float at the bottom of your screen. This makes it much easier to check answers as you are working through an exercise rather than just at the end. Transum excises are designed for frequent checking and the pupil is encouraged to change wrong answers and click the Check button again.

Many schools have direct links to Transum activities in their schemes of work and at least one of the major exam boards links to Transum activities in their publications. There is now a short URL for each activity (scroll down the page to find it) making it easier for teachers to include the link in their scheme of work or learning management system. The short link is also ideal to show or send to pupils.

It is not too long now until some of your students sit the brand new GCSE(9-1) examinations. Many people have talked about how the new numbered grading system will be understood by those outside education who have been used to the lettered grades. When I took my O Level Maths exam in 1974 it was also a set on numbered grades but going in the opposite direction; the top grade was a one!

An amusing reflection on the new 9-1 grades was included in the News Quiz on BBC Radio 2 last week. The excerpt has been included in the podcast version of this newsletter.

Talking of Maths exams, I would like to invite you to send your pupils a pre-exam present. Send them the URL of the Transum Exam Revision page. The page contains tips and links to useful revision resources such as my Exam Tips Presentation (a poem) and the Exam-Style questions.

But if they have done too many past papers already how about inviting them to take the Tran Tunnels adventure which is full of GCSE style activities and accompanied by music (The Goldberg Variations).

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

In order for his average mark on eight exams to be 80% the total of the percentages on all eight exams must be 8 x 80% = 640%

His total percentage on the first seven exams is 7 x 78% = 546.

Therefore the mark he must earn on the eighth exam is 640 – 546 = 94%

That’s all for this month.

John

P.S. Theorem: a cat has nine tails.

Proof: No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails.

]]>

Imagine a delicious coffee and walnut cake (my personal favourite) that has almost completely been eaten except for a lonely slice that needs to be cut in two. Rather than cutting the cake in the normal way, from circle centre to the midpoint of the arc (which can be tricky), you decide to cut it in a direction perpendicular to that. So you are not cutting the cake horizontally but vertically so that the centre of the original cake is on one side of your cut and the arc of the remaining slice is on the other.

So the question is where do you make the cut so as two get two equal volumes of mouth-watering cake? The answer can be found at the end of this newsletter.

The most popular lesson Starters (according to the mean of the scores collected on each page) are, in reverse order:

10. Addle (Mental addition)

9. Flabbergasted (Factors and multiples)

8. One Out of Ten (For April Fool’s Day)

7. Ice Cream (Click the buttons on the machine)

6. Mystic Maths (Scroll down the page for the best mind reader)

5. Broken Calculator (A number of versions available)

4. Sum Square (One of many puzzles)

3. 9:50 Puzzle (Just for fun)

2. Maths Crossword (My sense of humour, sorry)

1. How many? [Triangles, Rectangles or Squares)

The most customisable Starter is Refreshing Revision.

March was a busy month with many hours given to updating the website. In addition to the pages that were improved, the following is a description of the new features:

Iteration is an online exercise requiring learners to find approximate solutions to equations numerically using a looping or repeating process. There are a number of levels including one which focusses on the use of a flowchart to show the steps required.

The Brackets online exercise has been around for a long time but now a tenth level has been added for those wanting to practice multiplying three binomials and simplifying the result.

Both Circle Equations and the equations of the tangents are included in this new online exercise.

These aforementioned topics are new for this year’s Higher GCSE(9-1) so are particularly useful for students who have been using past papers to revise as they won’t have seen those topics represented.

Superior and Nevertheless are variations on the ever-popular Great Expectation activity. Guaranteed to create excitement while thinking mathematically. Try one of them during the last ten minutes of your next Maths lesson.

A completely new Cumulative Frequency exercise is now live including grouped data with exam-style questions.

The suite of Pairs programs are always useful for providing variety in the Maths lesson, or, for those individual tutors out there, great for one-to-one tutorials. Three new topics have been added: Units Pairs (both metric and imperial), Fill Graphs Pairs (Great to use after doing the Desmos Waterline simulation) and, inspired by the differences in American and British English, Math vs Maths Pairs.

As April begins it would be remiss not to remind you that there are 20 Weekly Workouts and 20 Practice Papers available on the Transum website for your students taking the GCSE exams soon to use. The Weekly Workouts focus on the higher grades of the Foundation tier while the Practice Papers are strictly for the Higher Tier candidates. As a subscriber you have access to worked solutions for the Higher Tier questions.

For future reference there are two ‘mirror’ sites that contain all the Transum Starters and activities. They are at www.transum.com and www.transum.info The only difference is that they don’t contain the details of your Transum subscription account so you won’t be able to log in there. If it looks like Transum.org will be offline for a long time then I will transfer the database containing your details to Transum.com so you will eventually be able to log in there too.

The answer to the puzzle is that the position of the cut depends on the angle between the straight horizontal edges of the slice of cake. The maximum ratio is:

**1 : √2 − 1**

You can see the working here.

That’s all for this month,

John

P.S. If you are asked to subtract five squared from the square root of six hundred and twenty five, say nothing and you’ll be correct!

]]>A 10cm long cylindrical hole is drilled straight through the centre of a solid sphere. What is the volume of the remaining part of the sphere? The answer is at the end of this newsletter.

A good mathematics exercise is structured to lead the learner through from simple, confidence-building questions to harder questions that deepen understanding. In addition to this an exercise should provide the repetitive practice needed to fuse the method into long term memory.

To prevent this repetitive practice becoming boring the exercise could be presented in the form of a game so that the same objectives are achieved in a more fun-filled, exiting manner. Here are seven Transum games that work well with learners of a wide range of ages.

- Pairs Games for remembering connections whether it be parts of a circle and their names or common fractions and their decimal equivalents.
- Digital Darts is an online game for one or two players requiring skill, strategy and lots of mental arithmetic.
- Rounding Snap is a fast and furious game. If the last card put down equals the previous card to the nearest whole number then all players race to shout SNAP!
- Connect 4 Factors a game for one or two players. The winner is the first to line up four numbers with a common factor.
- Choose Your Average is a game for two players. You should know how to find the mean, median and range of a set of numbers before starting this game.
- Two Dice Bingo is a whole class game. Use it to provide a purpose for drawing up a possibility space.
- Hi-Low Predictions A version of the Play Your Cards Right TV show. Calculate the probabilities of cards being higher or lower.

And two brand new games this month both provide practice related to number sequences:

Watsadoo requires quick thinking. Players need to quickly identify the falling numbers as odd, even, square, cube, triangular or prime. At the time of writing no one has successfully complete a level greater than two.

The Square Pairs Game is for two players who take turns to select two numbers that add up to a square number. Please try this game and let us know any strategies you invent!

Another new activity is called Mix and Math. Determine the nature of adding, subtracting and multiplying numbers with specific properties.

Upper and Lower Bounds is a traditional exercise with five levels of difficulty. Learners have to determine the upper and lower bounds when rounding or truncating quantities used in calculations.

Finally an new set of Circle Game cards has been added so you can play this whole class game with simple equations.

Now for the answer to the puzzle presented at the beginning of this newsletter.

One approach is to let the sphere have a radius, say R, and then do some calculations using the formulas for the volumes of a sphere, cylinder and a spherical cap. A much quicker method however requires a little creative thinking. As the puzzle has been presented in this newsletter you can assume that there is a solution. As the size of the sphere is not given it is reasonable to assume the solution is independent of any sphere dimensions. In that case you can consider the limiting case, one in which the radius of the sphere is 5cm and the radius of the spherical hole is zero. The remaining volume would then be the total volume of the sphere which equals 524 cubic centimetres to three significant figures.

Don’t forget to look at all of the GCSE(9-1) questions on the Transum website now that we are in countdown mode.

That’s all for this month,

John

P.S. There are 10 kinds of people in this world. Those that understand binary and those that don’t.

]]>This month’s puzzle is about a restrained flea that jumps one foot at a time either north, south, east or west. At how many different places could he end up after 8 jumps?

While you think about that I would like to tell you of five resources on the Transum website that have been updated recently. Although they are perfect for the UK’s GCSE exam preparation they could also be used in different ways for younger learners.

**Weekly Workouts**. These question papers (5 more have just been added) are designed for students on the Mathematics GCSE(9-1) Foundation level courses who are hoping to achieve one of the higher grades available. Each Weekly Workout contains 7 exam-style questions. The first six can be answered online but the seventh requires the student to draw something that needs the teacher to check. Each Workout can also be printed onto one A4 page.**Practice Papers**. These printable papers are designed to challenge students on the Mathematics GCSE(9-1) Higher level courses. Each question is similar to a question on one of the specimen papers produced by the exam boards for the 2017 exams. Full worked solutions are available for each question for Transum subscribers.**Revision Tips**. This is a page of suggestions and links to resources for anyone preparing for a mathematics exam. There are links to self-marking exercises on all the basic school mathematics concepts along with puzzles, games and investigations all designed to support revision.**Syllabus Checklists**. This part of the Transum Mathematics website contains a growing list of objective checklists for various common mathematics exams. Students can go through each objective and classify them as easy, OK or help! They can then print the objectives they have classified as requiring help and fill in the space for notes as their understanding develops.**Exam Presentation**. Save this for a week before the exam. It contains the tips and tricks that students might find useful when doing their last-minute preparation. You, as a subscriber, can download the PowerPoint version of the presentation so that it can be customised to suit your situation.

In addition to the items mentioned above, many other pages on the Transum Mathematics website have been updated or changed. A Starter called Tindice provides a quick, fun (when you know the answer) Starter to a busy Maths lesson but it can also be used to initiate an investigation into the sum and product of odd and even numbers.

I often help older students with their understanding of significance testing in statistics. In particular the chi-squared test is often clouded with strange precedents and terminology. A very short presentation called Significance has been developed to simplify the concept and to get the student to analyse the data provided by the Optical Illusions survey. As a subscriber you can see the results of the significance testing in real time. The students can use their GDCs to find the connections themselves. The presentation focusses on the big picture idea and leaves you as the teacher to fill in any gaps.

The Transum website was particularly busy in the weeks leading up to Christmas. Some of the ChristMaths activities had been updated and clearly people all over the world were enjoying them. If you missed out this year why don’t you send yourself a time-delated email (to arrive on the 1^{st} December) reminding you of the URL. An email to yourself can be flagged as ‘Delay Delivery’ in many email programmes such as Outlook.

The answer to this month’s puzzle can be found by considering the following:

Think of the flea on a coordinate grid starting at the origin. If the flea only jumps in one direction it would end up at either (0,8), (8,0), (0,-8) or (-8,0).

Now consider the possible points in the first quadrant, (x,y) where x is the number of jumps east minus the number of jumps west and y is the number of jumps north minus the number of jumps south. It is probably a good idea to sketch these points on some graph paper and you will see the pattern created by the locations. Multiply the number of points in the first quadrant by four and add the ‘return-to-origin’ possibility to find the total.

The answer is 81 different places.

That’s all for now

John

P.S. If a got 50 pence for every time I failed a maths exam I’d have about £6.30 now.

]]>