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This is the Transum Newsletter for the month of September 2024. It's the month that sees the first day of autumn in the northern hemisphere. It occurs on the 1st of September according to the meteorological calendar but not until the 22nd (the Equinox) according to the astronomical calendar.

It is autumn that has inspired the puzzle of the month:

In preparation for the cold weather, a large group of people dressed warmly in different ways.

- One quarter of the people who wore hats also wore scarves and gloves.
- One fifth of the people who wore scarves also wore hats and gloves.
- One sixth of the people who wore gloves also wore hats and scarves.
- There were twice as many people wearing exactly 2 items as were wearing exactly 3 of the mentioned items of clothing.

What fraction of the group wore all three items: hats, scarves and gloves?

If you get an answer I'd love to hear how you solved the puzzle (or your students solved it). Drop me an email 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.

Heron's Formula is another new exercise in three levels. This is a strange concept as it does not appear in any of the syllabus specifications that I work with. Anyway, it's there if you need a diversion or a broadening activity.

Maths Minds is designed for a bigger picture moment with your Year 13 students. You can tell them "Understanding the contributions of famous mathematicians and the history behind their discoveries is not just about memorising names and dates. It’s about appreciating the journey of human thought and how these brilliant minds have shaped our world. By learning about these mathematicians, you gain insight into the origins of the concepts that you study in your Maths." and then challenge them in pairs to match names with dates and mathematical concepts. There are clues for the dates as I don't expect many people to know that information.

If your new school year begins in the next week, I have some useful resources for you. If your school year is already in full swing, I have some useful resources for you to bookmark!

In most first maths lessons of the year, the teacher will let their students know what equipment they need to bring to each lesson. Here is a Maths equipment list, including pictures, that you can customise and project while the students with good intentions, obediently copy down the list in the front of their exercise books.

Talking of exercise books, depending on the class, you may need to demonstrate how you expect their work to be laid out in their books. This resource is an annotated image of a page of an exercise book.

Does your school or department have a homework policy? What are your expectations? This projectable image provides a prompt for discussion and reminds you to answer all of the age-old questions about the standards you expect.

Finally, what mathematical activity will you choose for that very first lesson? You want to make a good, memorable start and win the enthusiasm of the students. My Back To School page provides links to many ideas that you can use during this crucial lesson.

**Late Review:** During the research for the History of Mathematics page that I'm creating, I stumbled upon the film *The Man Who Knew Infinity* (2016). Ramanujan has now become one of my favourite mathematicians. His story is fascinating, and I have to admit I was quite moved by the end of the film. If you haven’t watched it yet, I highly recommend it. It’s easy to find and inexpensive to stream online.

I’ll be keeping an eye out for the number 1729 the next time I take a taxi (if you know, you know!).

I read an article in a magazine where Dr Jonathan Jang, a NASA scientist, talked about the search for extraterrestrial life. He mentioned broadcasting a message to the Milky Way, and the first thing he would want to say is, "Hey, we understand mathematics, and we count in tens." That's quite an opener, and it made me feel proud to have been involved in Maths education for most of my life.

Trophy News: Of the last 10,000 trophies claimed, 473 were for Equations, 368 were for Coordinates, 326 were for Measuring Angles, 227 were for Coordinate Plotter and 210 were for Substitution. So I assume those five activities are currently the most popular.

The person with the most trophies currently has 847. Now there's a challenge for you!

Thursday, 19th September this year is International Talk Like a Pirate Day. This must be the most ridiculous annual celebration acknowledged on the Transum website, and the suggested pirate activity is suitably silly. Having said that, I believe that if you can get the pupils involved with the big number chanting, it will create a positive memory of mathematics, even if the learning is minimal. So give it an 'Arrr matey!' or a 'Shiver me timbers!' but I think a daily rum ration might be out of the question!

Don't forget you can listen to this month's podcast which is the audio version of this newsletter. The podcast is called Transum Mathematics Puzzles.

Almost forgot: Today (1st September 2024) is also the first day of the last third of the year. 2024 has 366 days, a number divisible by three, so mention it in conversation and people will admire you even more!

Finally the answer to last month's puzzle which was:

I am thinking of three numbers.

If I double them all the mean stays the same.

If I square all the numbers the new median is nine.

The range of the numbers is seven.

The square of the product of the numbers is 144.

What are the three numbers?

The first five correct answers were received from Chris, Wil, Paul, Kevin and Mala.

The three numbers satisfying all four conditions are: {-4, 3, 1} and {-3, -1, 4}.

Enjoy reading a sample of the methods used below. Many, many thanks to those who shared their thinking.

That's all for now,

John

P.S. There are more hydrogen atoms in a single molecule of water than there are stars in our entire Solar System.

What's even more intriguing, to be mathematically precise, there are exactly twice as many! ... think about it :-)

Do you have any comments? It is always useful to receive feedback on this newsletter and the resources on this website so that they can be made even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

Chris,

Thursday, August 1, 2024

"The answer to the weekly puzzle is pi-reminiscent: 3, 1, -4.

The mean clue told me I needed to consider negatives.

The median clue told me that one of the values was 3 or -3.

The range clue meant that I was restrained to quite a small range of 7.

The square of the product meant that the size of the three original numbers' product (ignoring the sign) was 12, which was handy since I knew that I already had a factor whose absolute value was 3.

I then revisited the clues and 3,1,-4 sort of dropped into my lap (yes, that final line of explanation is a bit dissatisfying but I'm just being honest). "

Wil,

Thursday, August 1, 2024

"The numbers are -3, -1 and 4. It took me more than a moment to work it out: since doubling the numbers doesn't double the mean, at first my thought was that the middle number was 0, and they were -n, 0 and n, maybe -3½, 0 and 3½ because the range is 7. But that doesn't work because the product is 0 and we need the product to be 12 or -12. So try -3, 1, 4: everything OK now except that the mean doesn't stay the same when the numbers are doubled; the mean must be 0. So try -3, -1, 4"

Paul,

Thursday, August 1, 2024

"If I square all the numbers the new median is nine - requires a 3 in the answer.

The square of the product of the numbers is 144 - product of the three numbers is 12 - 1 a 3 and a 4

The range of the numbers is seven - lowest possible range of positive numbers is 1 - 8 -> put thinking cap on

Tried using a 0 to help things out but the product requirement ruled that out so only option was to go negative

Tried -3,1,4 didn't meet the mean rule but then -3,-1,4 did. "

Leonard,

Friday, August 2, 2024

"If the mean stays the same after doubling, we can deduce that the three numbers span the origin. That is, at least one is positive and at least one is negative.

The median clue tells us that one of the numbers is either 3 or -3.

The square clue tells us that the product of the three numbers is either 12 or -12.

We can factor twelve into integer triplets as follows: {1, 1, 12}, {1, 2, 6}, {1, 3, 4} and {2, 2, 3}. The first two are eliminated for not containing a 3.

There is no way to get a range of 7 with {2, 2, 3}, so we focus on {1, 3, 4}.

Adding the appropriate signs, we get two solutions: {1, 3, -4} and {-1, -3, 4}. "

Rick,

Sunday, August 4, 2024

"Here is my answer to the August 2024 podcast puzzle:

I am thinking of three numbers. If I double them, the mean stays the same.

Hence, if the numbers are a, b, and c:

(a+b+c)/3=(2a+2b+2c)/3 or

a+b+c=0

This means that some of the numbers (I am assuming integers) are positive and some are negative.

If I square all the numbers, the new median is nine.

Therefore, one of the numbers is three or negative three. Because nine is the median, this number is also not the largest or smallest, in absolute value.

The square of the product of the numbers is 144.

This means that the product of the numbers is 12 or -12. Since one of the numbers is three or negative three, and it is the middle number, the other numbers must be, in absolute value, one and four.

The range of the values is seven.

I am not sure why this clue is needed. There are no other numbers that satisfy the other criteria other than one, three, and four. Now, which ones are negative and which ones are positive? I see two possible solutions:

Negative three, negative one, and four, or

Negative four, one, and three,.

Both seem to satisfy all the clues. "