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Multiple Remainders

Friday 1st November 2019

This is the Transum Newsletter for the month of November. It begins, as usual with the puzzle of the month.

What is the second smallest number such that when it is divided by 5 the remainder is 4, and when divided by 7 the remainder is 6?

While you think about that here are some of the key resources added to the Transum website during this last month.


I seem to be developing a theme of taking old games and giving them the digital, interactive, mathematical treatment. The latest is the ancient game of Wari, otherwise known as Mancala which is the generic name for a family of two-player, turn-based strategy board games played with small stones, beans, pips or seeds and rows of holes, pots or pits.

The Transum version is called Prime Pips in Pots and requires players to take all the pips from a pot and distribute them, one to each pot moving round the board in a clockwise direction. If the last pip increases the number of pips in a pot to be a prime number then that player wins those pips.

The first player to win 25 pips is the winner. Pupils can play against other pupils or against the computer. The computer only considers one move ahead so pupils do stand a good chance of winning a game against the computer.

I no longer sketch sine, cosine or tangent graphs to show the multiple solutions to trigonometric equations now that I have developed the Inverse Trig Calculator. Unlike a GDC or other graph plotting software this visual aid shows all of the solutions in a given range and makes it easier for pupils to derive the general solutions as they can see the periodic nature of the functions. I have added this interactive tool to the Shine+Write collection as it was really designed for teachers to use to help explain this concept to the whole class.

Inverse sine of a half

Spoiler Alert: The Audible Riddles are presented in an audible format because they wouldn’t work if written down. That’s a huge clue and should help you solve them quite easily but your pupils, who do not know this fact, will have to be really alert to figure out the answers for themselves. Have fun playing the short clips to your class and let me know how it goes.

Audible Riddles

I really enjoy trying new ways of presenting mathematical activities to both promote a different way of learning and to encourage pupils to be ‘drawn in’ to the task. This last month I created Mental Strategies, an online exercise aimed at Year 6 pupils developing a repertoire of different techniques to help them manipulate calculations in their heads. For each calculation there is a blue button which, when clicked, gives a suggested strategy for making the calculation easier.

At the other end of the school range of mathematical teaching I'll quickly mention the new t-Test Revision presentation. Only of interest to A Level or IB teachers but I thought I should mention it.

I notice that there are many subscribers have not entered their school IP address into the system. It's a quick and easy way of ensuring that anyone accessing Transum via your school network sees the site ad-free. You will find the link on you My Account page.

Earlier last month I listened to the Freakonomics podcast episode called America’s Math Curriculum Doesn’t Add Up. Most high-school math classes are still preparing students for the Sputnik era. Steve Levitt wants to get rid of the "geometry sandwich" and instead have pupils learn what they really need in the modern era: data fluency. You can hear an excerpt from the episode on this month's Transum podcast.

In the UK the new A Level specifications require students to explore large data sets. Students are required to perform tasks that assume familiarity with the contexts, the main features of the data and the ways in which technology can help explore the data. Students should also  demonstrate the ability to analyse a subset or features of the data using a calculator with standard statistical functions.

So how can Transum provide resources for learning about data fluency? I have access to large data sets generated by people visiting the website but need you to give me some suggestions for the type of activity you could use.

Date for your diary: My Birthday!

Finally the answer to this month's puzzle:

The answer is 69.

I arrived at the solution by creating the sequences of numbers described in the question until I found a term common to both sequences:

Divided by 5 the remainder is 4: 4, 9, 14, 19, 24, 29, 34.
Divided by 7 the remainder is 6: 6, 13, 20, 27, 34.

So 34 is the smallest number. The second smallest will be one more than double this number; 69.

Still looking for a more elegant way of solving this puzzle.

That's all for now,


PS. What did the zero say to the eight?

Nice belt!

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