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September has arrived, bringing a fresh collection of Transum ideas for the classroom. Before we explore them, here is a taxi-fare puzzle that may prompt rather more debate than its £45 price tag suggests.
It’s a long way to Tipperary and the taxi fare for the return journey, there and back, is £45. Two-thirds of the way there, the taxi picks up Mr Pernickety and his large suitcase; it drops them off one-third of the way back. If the suitcase counts as an extra person, what is the fairest way for Mr Pernickety and me to share the fare?
If you find an answer, I'd love to hear how you solved the puzzle. Fire off an email with your answer to gro.musnarT@rettelsweN
While you think about that, here are some of the key resources added to the Transum website during the last month.
During the summer, a second level was added to the How Many Triangles? exercise. There are five questions and students are told that a trophy is available for getting the first two questions right and to not attempt questions three to five as they are too difficult for humans. I wonder if this reverse psychology will have any effect?
Then, another idea struck me: Level 3! I decided that what students would really like is the opportunity to count the number of triangles in the wireframe drawings of three-dimensional solids. It would be a chance to develop their spatial awareness skills as well as reinforcing the need to be systematic. So the third level of the How Many Triangles? exercise contains an octahedron, an icosahedron, a prism, a cube and a hexagonal-based pyramid.
A new Periodic Graphs exercise has been added to the online resources. Its five levels range from piecewise graphs based on real-life situations to transformed trigonometric functions. Along the way, students identify cycles, calculate periods and amplitudes, and use repeating patterns to extrapolate beyond the part of the graph shown. It provides a gradual introduction to the key features of periodic graphs before moving on to some rather more challenging exam-style questions.
I have updated the Calculator Words activity for anyone who remembers typing a number into a calculator, turning it upside down and being delighted by the word that appeared. Many modern calculator displays have sadly lost that ability, so the activity now includes a specially designed old-style seven-segment calculator that can still perform the trick. Students do each of the given calculations using the on-screen calculator, which they then rotate to find the hidden words.
I began by sending a congratulatory email to anyone who had earned 100 trophies, but soon had to extend this to those who had earned any multiple of 100 (yes, some people really do earn that many!). In these emails, I ask whether they have any suggestions for new activities on Transum. Last month, someone replied with the suggestion, “Could you create a Two-Way Tables exercise?” Well, here it is, taking you from zero to hero, or at least from very easy (Primary) to quite difficult (A Level, IB): Two-Way Tables.
The same prolific trophy earner also suggested that I create an activity in which students could rotate a 3D model in order to help draw its plan and elevation views. There's already a Transum activity called Plans and Elevations, so I cunningly named the new activity Elevations and Plans.
If it is the start of a new school year for you and you have a class you have not taught before, do you go straight into the scheme of work or spend the first lesson laying down the law? Do you have a good idea of what the pupils already know, or is it still one of the great unknowns?
The Back To School page contains some suggestions and some resources to help you create a good impression and establish a good culture for learning from day one.
I was inspired by a newspaper article from the mid-1980s about maths teachers holding placards at a demonstration against the way calculators were being used in the classroom. We have come a long way since then, but many students still need guidance about when a calculator is helpful, when it is unnecessary and how to use one efficiently.
This gave me the idea for a twelve-question survey asking students for their views on calculator use. The questions are multiple choice, but the available answers are also designed to encourage students to reflect on good practice and perhaps reconsider some of their habits. In other words, the survey collects useful data while also encouraging students to think more carefully about their calculator use. Here it is: Calculator Survey.
Last month, I mentioned that Transum occupies a valuable niche by providing resources that can be accessed extremely quickly, without requiring users to sign in. My latest example is how quickly you can throw together a colourful pie chart to display in front of your class. I used the Pie Chart Creator to produce the chart below showing the number of visits to Transum from clicks in Google searches for the month of July 2026.
Visitors to Transum from Google.
Online Logo, one of the most viewed pages on the site, has been updated so that it now awards stars for completed challenges. Earning seven or more stars unlocks a trophy, while earning all ten also triggers a jolly jingle!
Don't miss international Talk Like a Pirate Day on the 19th of September.
I was interested to read in the news that the publication of this summer’s GCSE results has highlighted an interesting contrast in mathematics. More pupils are achieving the highest grades, yet the number of older students repeatedly resitting GCSE Maths continues to grow, with relatively few securing the grade 4 they need. At the same time, Cambridge International is placing greater emphasis on non-calculator mathematics, while new Australian guidance is encouraging teachers to make better use of formative assessment. There's never a slow news day in the world of mathematics education!
Also in the news ... Britain’s first specialist maths secondary school opens this week. 1729 Maths School, based within the Mill Hill campus in north London, is about to welcome its first pupils. It describes itself as the UK’s first specialist mathematics school for children from 11 to 18, rather than the sixth-form maths schools that already exist.
Finally, the answer to last month's puzzle, which was:
Monsieur Thénardier has been doling out the charm, ready with a handshake and an open palm. Meanwhile Madame Thénardier has been cookin’ the books and contemplating interesting facts about the numbers in her life. She has realised that the digits of her age are the digits of her husband’s age reversed. Although he is the older of the two, the difference between their ages is only one-eleventh of their sum. How old are the Thénardiers?
The puzzle was adapted from Amusements in Mathematics Henry Ernest Dudeney
That's all for now,
John
P.S. I have just found out that the company that makes metre rulers won't be making them any longer.
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Mala, New Zealand
Saturday, August 1, 2026
"Let x and y be the 2 digits that make up the ages of the husband and wife.
Husband’s age, H= 10x + y and wife’s age, W = 10y + x, where x is greater than y as the husband is older.
H + W = 11x + 11y and H-W= 9x – 9y. Solving the two simultaneously, we get 5y = 4x. The values of x and y that satisfy this equation are x=5 and y=4, which means that the husband is 54 and the wife is 45 years old. Those are the ages of the Thenardiers. "
Chris, Scotland
Sunday, August 2, 2026
"Monsieur T could be 10a+b (a and b are single-digit positive integers).
Madame T would then be 10b + a.
The sum of their ages:
(10a + b) + (10b + a) = 11a + 11b = 11(a + b)
we're told the difference of their ages is one eleventh of this which is simply (a + b)
The difference of their ages is also given by:
(10a + b) - (10b + a) = 9a - 9b
so
a + b = 9a -9b
10b = 8a
5b = 4a
b = 4/5a
The only single-digit integer which we can work out 4/5 of is 5 so a = 5 and then b = 4.
So Monsieur T is 54 and Madame T is 45.
Check: 54 + 45 = 99, one eleventh of which is 9 and that's the difference since 54 - 45 = 9. "
Leonard, United States
Sunday, August 2, 2026
"Here are my thoughts on the Master of the House puzzle.
Rather than start with the two ages as variables, keep it simple and start with the sums, which are a multiple of 11 (Column 1 below).
The age difference (Col 2) is now found by dividing by 11.
The two ages (Cols 3&4) can now be found quite simply from the sum and difference.
Finally, compare the ages to see if one is the reverse of the other.
Our answer is 45 and 54.
he other.
Our answer is 45 and 54. .
Note: If we ignore the implicit assumptions (names, caricatures, reference to novels from the 1800s, ages expressed in years, etc.), we could include the solutions:
4545 and 5454, as well as 4995 and 5994. "
Rick, United States
Sunday, August 2, 2026
"This is one of the easier puzzles in the recent past.
First, let’s assume that both are less than 100 years old. Then, the two digits of their age must differ by only one and the difference between their ages is nine. If their digits were two or more, then the minimum difference between their ages would be 18 at a minimum. Eleven times 18 would require that one be older than 100.
With the difference being 9, the sum of their ages must be 11 times that number, or 99. The only ages that satisfy this is that Monsieur Thénardier is 54 and Madame Thénardier is 45.
Of course, they could be vampires and then Monsieur Thénardier is 594 and Madame Thénardier is 495. "
Kevin, Australia
Sunday, August 2, 2026
"Ages AB BA
Sum = 10A + B +10B + A = 11(A + B)
1/11 sum = A +B = 10A – 10B + B – A (diff)
So 8A =10B and 4A = 5B so A = 5 B = 4
54 and 45
Sum 99 Difference 9 checks
=================
And the advanced one
A^2-B^2 = A – B
(A – B)(A + B) = (A – B)
A + B = 1
Height = 1m
I may give this to my Y9 group as we have been doing difference of two squares "