Tag Archives: Probability

Before getting into the main theme of this newsletter let’s begin with this month’s puzzle.

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.

7 New Resources for the Maths Classroom

Happy Christmas and welcome to the December 2016 edition of the Transum Mathematics newsletter. We will begin with the puzzle for this month: How many positive two-digit numbers are there whose square and cube both end in the same digit? The answer is at the end of this newsletter.

While you think about that, here are the seven new resources that have appeared on the Transum website since the last newsletter.

1. First Impressions

I was given the idea to create this fun data collecting application by Year 13 students working on projects including the chi-squared test. It was proving difficult and time consuming for them to collect their own data in sufficient quantities in order to meaningfully apply statistical tests. First Impressions asks the pupil for their initial perceptions of optical illusions. When the activity has been completed (it takes less than two minutes) the pupil is presented with the data collected from all of the other people who have also used this app. This data can then be used by the pupil for all sorts of graphs, charts and statistical analysis. Give it a go and share your ideas.

1. Weekly Workout

With questions similar to those on the specimen papers produced by the exam boards for the forthcoming Maths CGSE(9-1), the Weekly Workouts provide half-hour revision papers for Foundation students aspiring to achieve the higher grades. The first six questions can be answered online just like the other Transum online exercises but the seventh question on each paper requires more drawing and is best done on paper with feedback from the teacher. The number of Weekly Workouts for Foundation level pupils is growing week by week. You have probably already seen the twenty Practice Papers for Higher students haven’t you?

1. Brainbox

This number arranging puzzle was devised by Les Page and adapted as a Transum Mathematics interactive numeracy puzzle. There are twelve levels (and a few hidden bonus levels) arranged in increasing order of difficulty and there are efficient solving strategies that you will probably soon discover for yourself. Perfect for Year 5 pupils up to pensioners.

This simulation describes the motion of a ball falling through a Quincunx (Galton Board) made out of pegs. In the intro tab, a ball has an equal probability of going to the left or right of the peg. The pupil can choose to send 1, 10 or all the balls though the board (up to a maximum of hundred) and watch how the balls fall into the different containers at the bottom of the board. A nice introduction to the normal distribution.

A self-marking exercise on the sine rule, cosine rule and the sine formula for finding the area of a triangle. The questions are carefully arranged in increasing order of difficulty preparing pupils for the linked exam-style questions.

1. Triangle Solver

This new, powerful resource is a large triangle to project on to your whiteboard. Drag the vertices to make the triangle roughly the shape you want then type in three measurements, a mixture of sides and angles, then within the blink of an eye the other measurements magically appear. The triangle is solved!

This Solver is not only intended to be used with standard trigonometry or Pythagoras questions but also as a resource for students learning the basic construction skills with a rulers and pair of compasses. It also works well for a class practicing drawing angles using a protractor.

The teacher could manipulate the triangle to show a base of say 13cm. Either side of this base angles of 50° and 70° are shown. The class is then challenged to make an accurate drawing of the triangle and their accuracy can be measured against the actual values the Triangle Solver produces when everyone has finished their drawings.

Similarly a triangle with only the three sides given can be projected for a class practicing ruler and compass constructions. This time it is fun to compare the measured angles of the finished triangle with the ones the Triangle Solver calculates.

1. ChristMaths Activities

Not strictly a new resource but certainly an updated one. Don’t be tempted to stray from Mathematics when planning those festive, end-of-term lessons when there are so many Yuletide treats in this collection.

The answer to this month’s puzzle is:

• The nine two digit numbers that end in a zero;
• The nine two digit numbers that end in a one;
• The nine two digit numbers that end in a five;
• The nine two digit numbers that end in a six;

These added together give a total of 36.

Enjoy the Christmas holiday and enjoy the ChristMaths activities,

John

P.S. Calendars, their days are numbered.

November 2016 News

Welcome to yet another newsletter from Transum Mathematics. As has become traditional I will start off with the monthly puzzle.

Trains from Punspace station go either north or south. Those going north leave hourly, those going south leave hourly. If I arrive at the station at a random time the probability that the next train to leave will be going north is five times the probability that the next train to leave will be going south. How could that be?

While you are thinking about that here is some news about the latest additions to Transum Mathematics.

Kite Maths is a very visual, practical colourful activity leading to the discovery of important geometrical theorems. A new page of interactive animations created in Geogebra has now been added. These dynamic images are great visual aids for classroom use.

Some excellent interactive activities have been created by an organisation called PhET (Physics Educational Technology) and the mathematical simulations are being added to the Transum website. Founded in 2002 by Nobel Laureate Carl Wieman, the PhET Interactive Simulations project at the University of Colorado Boulder create these appealing Maths and Science simulations. They are based on extensive education research and engage students through an intuitive, game-like environment where students learn through exploration and discovery.

During October the following PhET activities have been added: Area Builder, Grid Arithmetic, Fraction Matcher, and Function Builder. You can find them by searching for activity title using the Transum search box (in the footer of every page) or by looking them up on the relevant topic page.

The activity called Clouds was updated last month. It has now been split into five levels and the higher levels include decimals and fractions. The idea of this activity is that clouds have magically floated across some calculations obscuring one of the numbers. Pupils need to find a strategy for working out what is behind each of the clouds. Teachers will see the link with algebra, rearranging formulae or solving equations.

The Mixed Numbers exercise has also been update. Level 5 now includes a variety of questions with words and diagrams! You as the teacher can decide whether pupils should be using this exercise to practice their pen and paper techniques or use it as a calculator exercise, making sense of the calculator’s strange fraction notation.

Coming very soon (hopefully by the end of this week) are some GCSE(9-1) practice papers for pupils on a Foundation level course. 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.

The answer to this month’s puzzle is best understood by considering what the timetable for departures might look like. If the northbound trains leave at 10 minutes past the hour and the southbound trains leave at 20 minutes past the hour then there’s only a ten minute window for you to arrive at the station for the next train to be going south. There is however a fifty minute window for arriving to find the next train is northbound. Hence the probability that the next train to leave will be going north is five times the probability that the next train to leave will be going south.

Enjoy November

John

P.S. I don’t understand how to double 2n. It sounds 4n to me.

April 2016 News

Easter has come early this year which means that many schools are currently still closed for the Easter holidays. That’s a pity! It means that you don’t get the opportunity to fool your pupils with the 1st April Starter. Next year maybe?… No, April 1st falls on a Saturday next year. Perhaps you could use the ‘One Out Of Ten’ joke on another day of the year.

The puzzle for this month is about three cars arriving at a three way junction at high speed. The junction has a triangular (ish) traffic island at the centre and each car has a 50% chance of turning left and a 50% chance of turning right when they arrive at the island. What is the probability of no collisions taking place?

March was another busy month for adding content to the website. The most significant addition is Refreshing Revision, the ultimate customisable Starter. It is called Refreshing Revision because every time you refresh the page you get different numbers and diagrams in the questions. Scroll down the page to see and select the concepts you want to be included in the Starter. It should be useable with pupils in Year 5 (with the times tables questions) all the way up to Year 11 (revising for exams).

I will continue to add more concepts to this during this month but if you have any suggestions please let me know.

The leader boards for TablesMaster and Fast Factors have been adapted so that you can filter out all but pupils from your own school. Instructions explaining how to do that are available on the Times Table Filter page. Many thanks to Matt Curtis from Edgewood School for suggesting this idea.

The Times Tables page contains links to many activities pupils can do to improve their recall of multiplication facts. A new activity was created last month called Times Square. It has nothing to do with that famous location in New York but lots to do with providing yet another way for pupils to practise their tables. The completed tables square comes complete with buttons to show some of the geometric patterns created by sets of numbers in the grid.

A number of videos found on YouTube have been added to the curated list but the one that stands out is the one about the mathematical puzzles found in an episode of the Simpsons called Mathlete’s Feat.

I was surprised recently that one of my pupils, who has strong abilities in most areas of mathematics, didn’t know the order of the months of the year. A drag and drop activity was created which he will use as part of his regular recap activities and also help him to remember the number of days in each of the Months of the Year.

Whenever I am teaching probability I would try hard to include a little bit of fun with the Snail Race. It can be adapted to a wide range of abilities and lead to some interesting questions. Last month Ben from New Zealand asked the ultimate question about snail number seven’s chance of winning the race. We have still not been able to come up with the answer but you can follow the discussion on the Snail Race Teacher’s Version page. If you are an expert in Negative Binomial Distributions we need to hear from you!

Previous newsletters can be found with podcasts (the audio versions of the newsletters) online and for the latest news of Transum updates follow @Transum on Twitter. Thanks to all you who have left comments on the Transum web pages and sent feedback about how your pupils have enjoyed using the resources.

Finally the answer to the Tri-junction puzzle. As a subscriber you can see a tree diagram which can be used for solving this puzzle on the Tree Diagrams Challenge page. The answer can be found by considering the probability of all three cars turning left or all three cars turning right. The answer is 0.53 + 0.53 = 0.25 or 25%.

Enjoy the month of April.

John

ps. Parallel lines have so much in common it’s a pity they’ll never meet.

September 2015 News

It seems such a long time since the last newsletter but here it is again to welcome the month of September 2015. I have had a wonderful summer (northern hemisphere) and managed to visit lots of places and catch up with many old friends. I always had my laptop with me so found the time to write more puzzles, quizzes and activities to add to the ever-growing Transum collection.

Before going any further let’s go straight to the puzzle for this week. This is to give you something to think about as you commute, exercise or relax today so don’t look at the answer at the bottom of this newsletter if you want to enjoy the challenge.

Three very logical mathematicians walk into a bar (change this to a sweet shop if you are telling it to your pupils). The barman asks if they all wanted a drink. The first mathematician said “I don’t know”, the second mathematician said “I don’t know” but the third mathematician said “Yes we do”. Can you explain how the third mathematician could be so certain?

While you think about that here’s a mention of some of the new activities created since the last newsletter.

The Fractal Mosaic is an interactive, animated version if the Snowflake Squares activity. A fractal-like pattern can be created by following a very simple set of instructions that are repeated a number of times. The end result makes excellent display material. Talking of which, a new section called Maths On Display has been created giving you some ideas of how your pupils can engage in mathematical activities that produce very pleasing results.

Follow the rule repeatedly to create this beautiful pattern

Following the success of the Bridge Crossing starter a challenge called ‘Without Lifting the Pencil’ provides pupils with six designs that they have to decide whether they can trace according to the rules. Instead of actually using a pencil and paper the Transum interactive version allows them to click dots (the nodes) to simulate the tracing and makes it easier for them to try many different routes quickly and not worry about making mistakes.

A Frequently Asked Questions section has been added for subscribers but, thus far, there are not many questions and answers posted. The plan is to add to this slowly but surely over the coming months. If you have any questions about using Transum Mathematics please let me know as your question might be suitable for this FAQ page.

The One Torch Tunnel challenge has been added as an interactive page which I hope will be as popular as River Crossings is. It is quite easy to get all four people through the tunnel but not so easy to do it in the optimal time.

A Fibonacci Quest section has been added. The plan is to, over a period of time, add fascinating Fibonacci facts to these pages but each new page will have an interactive aspect to it so that pupils learn by doing!

The Probability Washing Line is certainly not a new idea. Pupils are invited to hang out the shirts, each containing a probability word, on a line stretching from impossible to certain. The software will check their attempts according to some very loose rules.

There was nothing on the website to practise Telling the Time … but now there is! Lots of analogue clocks with some mixed questions thrown into level four. As with all other activities on the site let me know if you think this section should be expanded.

The answer to the puzzle asked at the beginning of this newsletter can be explained as follows.

If the first mathematician did not want a drink he would be certain that not all three wanted a drink as there was at least one of them (himself) that did not. He would have therefore replied “No” instead of “I don’t know”. The third mathematician was therefore sure that the first mathematician did want a drink due to his answer.

The same argument can be also applied to the second mathematician who must also have wanted a drink.

The third mathematician was quite certain he wanted a drink himself bringing the total to three.

That’s all for now. Enjoy September and if it’s the beginning of your school year enjoy all that newness and new-beginning enthusiasm that’s around at the moment.

John

ps How many times can you subtract 7 from 83, and what is left afterwards?

Answer: I can subtract it as many times as I want, and it leaves 76 every time!

June 2015 News

This is the June 2015 Transum Newsletter bringing you the latest news from Transum Mathematics.

The puzzle for this month is about a game in which a coin is tossed repeatedly. If it comes up heads Pascal gets a point but if it comes up tails Fermat wins a point. The first person to win three points is the winner and receives the prize of £12.

Unfortunately the game had to end abruptly after three tosses of the coin. Pascal had two points and Fermat had one point. They decided to share the £12 in a ratio that matched the probability of them winning the game if it had continued. How should they divide the £12?

That should give you something to mull over as you go about your activities today. It is worth noting however that if you think the answer is £8 and £4 you may have some more thinking to do. The answer is at the end of the newsletter but don’t look if you want to figure it out for yourself.

Hundreds of virtual Transum Trophies are earned by pupils around the world each day for completing a Transum online activity. The trophies are collected in their own virtual trophy cabinet online which the pupils can proudly show to their teacher, parents or friends.

As a Transum subscriber you can award Teacher Trophies to your pupils for all sorts of achievements. You could issue a trophy for a good effort, taking part in a group project or any reason you see fit. Each pupil can receive a maximum of one Transum Teacher Trophy each day.

The button linking to the trophy awarding page is on your ‘My Account’ page.

As usual there have been a number of additions to Transum Mathematics last month and here is a roundup of those worth mentioning.

For those of you involved with teaching Formal Written Methods there are some examples of standard ways to set out the calculations along with buttons linking to online exercises providing some practice.

Five related activities are provided to support your teaching of correlation. The starting point would be Cartoon Scatter Graph  which is a very visual way to convey the concept of the scatter graph. Pupils have to place cartoon characters on a set of axes to show their heights and ages.

To introduce the concept of correlation there is nothing more memorable than being a part of the graph yourself so Human Scatter Graphs is a highly recommended next step. This resource provides pairs of variables, each relating to the pupils, and asks them to find their position in the room relative to a set of very large axes.

Next is a sorting activity called Correlation which consolidates the understanding of different types of correlation.

The basics are covered by Plotting Scatter Graphs which ensures that the pupils practise the manual point plotting skills before they learn to leave the hard work to the calculator when answering Exam-Style Questions.

Now special awareness skills are a funny old phenomenon. Some children find activities so easy while others struggle, and it is not always the best mathematicians that do well. To provide practice there are a number a different ways you can approach this. Try Net or Not and Plans and Elevations. There is no substitute for making models and setting up situations in the classroom to really get to the bottom of this topic though.

The most popular online exercises, if the number of trophies earned is the defining criteria, are the basic arithmetic quizzes. A new basic exercise called Powten has now been added which has a number of levels, containing twelve questions each, requiring an understating of the easy methods of multiplying and dividing by powers of ten.

Thanks to everyone who have ever given a Starter a score. If you click on the stars on a Starter page you can see a bar chart showing the distribution of the votes. Currently the most popular starter of the year is How Many Triangles and the least popular is Chin Ups. I wonder why!

Finally the answer to the puzzle posed at the beginning of this newsletter. Consider the following equally likely scenarios for the outcome of the next two tosses:

• Tails tails.

Three of the four possibilities would make Pascal the winner so the prize should be shared in the ratio 3:1 meaning that Pascal should receive £9 and Fermat should receive £3.

Not many people get the right answer first time! This puzzle is just one of the Advanced Starters designed for older students.

That’s all

John

Ps. What did the mathematical acorn say when it grew up?

A. Gee I’m a tree (Geometry).

April 2015 News

Welcome to the April 2015 edition of the Transum Mathematics Newsletter. Did you start the month off with the April Fool’s Starter? Did your pupils fall for it?

Your puzzle for this month is about a game called Best Dice in which two people roll a dice and whoever gets the higher number wins. A prize is awarded to the person winning most times after 100 games. The catch is the dice don’t have the numbers one to six on their faces.

There are four different dice and you are allowed to choose which dice you will play with.

Best Dice

• The red dice has threes on all of its faces.
• The blue dice has four fours and two zeros.
• The yellow dice has three fives and three ones.
• The green dice has four twos and two sevens.

Which dice would you choose to give you the best chance of winning the prize? The answer can be found at the end of this newsletter.

There have been many pages added and updated during this last month. A new puzzle called Numskull  is designed to provide a relaxing logic challenge where the mathematics involved is suitable for upper Primary pupils. There are five levels differing by the number of clues available.

For older students a Number Systems Venn Diagrams activity provides a quick but effective revision task. The objective is to drag the numbers in to the correct layer of the concentric circles. The software checks the correctness of the placings.

Also for older students is a rapidly growing database of Exam-Type Questions and their worked solutions. There are currently 90 questions and answers in the database but more are being added regularly. They are similar to questions that have appeared on IB Standard, Maths Studies and GCSE examinations but have all had the wording and numbers changed to make them different to past-paper questions you may find elsewhere. The solutions can be revealed line by line making a great teaching tool for the classroom.

Though not specifically mathematical a Scheduling puzzle has been added to provide a little more variety to the Transum Puzzles page. It’s not too difficult and the software shows you which criteria you have and have not fulfilled when you choose to check your solution. I’d love to know if you decide to use it with your pupils.

I’m not sure how we managed so long with out a traditional fractions, decimals and percentages  conversion activity on the Transum website. Now there’s a Starter, an interactive pupil activity and a revision presentation on this important topic.

The answer to the puzzle posed at the beginning of this newsletter is a bit like rock, paper, scissors. Whichever dice you choose, your opponent could always pick one of the remaining dice which has a better chance of beating you in the long term. Construct the possibility spaces for the possible dice pairings to see for yourself.

Blue beats red, red beats green, green beats yellow and yellow beats blue! You can see why in the answers section of the Best Dice Starter page.

Have a happy Easter, Songkran or whatever you may be celebrating in April.

John

ps . What do you call a saucepan of simmering soup on top of a mountain?

… A high-pot-in-use!

March 2015 News

Welcome to the March 2015 edition of the Transum Mathematics Newsletter. This month the focus is on probability and the puzzle for this month is one that all Maths teachers will probably (?) know the answer to. What is the minimum number of pupils there needs to be in a class such that the probability of two or more of them having the same birthday is greater than a half? If you don’t know have a guess (make an estimate).

The answer is at the end of this newsletter but for now let’s look at what was new or updated on the Transum website last month.

The copyright holders of the puzzles Suko and Sujiko granted Transum permission to create versions of the puzzles online. The Transum version allows the user to have up to 5 clues to help find the solution. This allows the puzzles to be used with classes of different abilities as the teacher can decide just how difficult each puzzle should be.

Beat the Clock is great fun no matter how good your numeracy skills are. Race to answer the mental arithmetic questions as a curtain slowly lowers concealing the questions on the screen. There are nine levels of difficulty but you can make the challenge easier by choosing to only answer the questions in the left column.

Assuming your school does not have a policy of using traditional playing cards in the classroom you’ll find they are a versatile resource for mathematical activities. Visit the new Playing Card Maths section of the website to see all the ideas that have been gathered together so far. Please get in tough if you have any other suggestions. If you are a one-to-one Maths Tutor you may want to always have a pack of cards in your bag as a plan ‘B’ just in case the other activities you have planned for the tutorial go horribly wrong.

If you are interested in times table skills I’m sure you are aware of one of the most popular Transum activities called TablesMaster. Now when you claim your trophy for completing an exercise you have the option to view your very own ‘reverse bar chart’, comparing your trophy winning times for each of the times tables. It has proved to be a great motivator for pupils working on their times tables.

Other less newsworthy activities have been added to the site and many updated but you’ll probably come across those when you browse the topic index.

Probability is a unique topic in the school mathematics syllabus. As well as learning the techniques, formulas and procedures pupils should develop a ‘feel’ for what probability means. Most adults have a limited understanding of the concept and would think that if they had tossed a coin nine times and it had landed heads each time that the probability of the tenth toss landing tails is more likely as it hasn’t happened for so long! So does the coin have a memory? Does it know that if it has landed heads many times that it should be time to give tails a chance? No of course not. The probability is still 50% (unless the coin is biased in some way).

There are some nice examples of probability in the real world I have just been listening to on the excellent podcast called ‘No Such Thing As A Fish’. The QI Elves tell of how Spotify changed its random play list because people didn’t think it really was random enough! You can hear that excerpt as part of the Transum Podcast for this month.

The surprising answer to this month’s question is that there need only be 23 pupils for the probability of two or more of them to have the same birthday to be greater than a half! The reason is well documented on the web and in particular in an article called ‘The Birthday Problem’ on Wikipedia. If you’d like to try to prove it yourself you may want to consider the question of the birthdays not being on the same day and subtract your result from one. Good Luck.

John

ps 3.14% of Sailors are PI rates!