# Three positive outcomes of failure in Maths

Welcome to the June 2017 Transum Newsletter. This month’s puzzle is about the Numlove family. Can you work out how many children are in the family from the following two clues?

• 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!

# October 2016 News

This is the Transum Newsletter for October 2016, the 10th month of the year. Have you ever noticed that the month name begins with the suffix ‘Oct-‘ suggesting eight and not ten. There is a reason for that and a quick internet search will reveal it to you.

Let’s begin with the puzzle for this month which is about three hungry children.

There was a short queue in the school canteen. Ayden was directly in front of Betsy who was directly in front of Carl.

Aden’s age is an even number but Carl’s is odd. Is a person with an even age directly in front of a person with an odd age? The answer is at the end of this newsletter.

I am very keen to tell you about some of the new additions to the Transum website that appeared last month. The first is Maths Mind Reader. Absolutely everyone I’ve used it with have been extremely impressed with this clever web page. As a Transum subscriber you will be see the mathematics that makes it work and Secondary pupils should be able to understand and even prove the concept.

A Transum website visitor, Les Page, sent me an addictive little puzzle he has devised called Zygo. He has kindly allowed a Transum interactive version to be created which is now ready to improve the numeracy and problem solving skills of your pupils. Thanks Les.

Pupils quickly learn to recognise and name regular polygons but the new activity called Polygon People may help younger pupils to name irregular polygons too. The activity has three levels and only accepts the correct spellings.

For the older pupils (14+) the Completing the Square and Proof of Circle Theorems activities should support those entered for the higher tier of the GCSE exams (or equivalent).

At times when I have not been creating new content for the website I have had a small amount of time to look at an updated app that I have downloaded to my iPhone. Photomath has been around for a couple of years but I’ve been very impressed with the recent improvements. You point your phone camera at an equation, and it will give you the answer and show you the working. I’m still amazed it can read my handwriting!

Photomath supports arithmetic, integers, fractions, decimal numbers, roots, algebraic expressions, linear equations and inequalities, quadratic equations and inequalities, absolute equations and inequalities, systems of equations, logarithms, trigonometry, exponential and logarithmic functions, derivatives and integrals.

My only reservation against using it with pupils is some of the phrases used to explain the stages of solving an equation. “Move constant to the right side and change its sign. Move variable to the left side and change its sign” is less helpful than the notion of doing the same thing to both sides in my opinion.

The answer to this month’s puzzle is yes. We don’t know Betsy’s age but we do know it is either even or odd. Let’s consider the two possibilities.

If Betsy’s age is odd then Ayden (even) is in front of Betsy (odd) and the answer is yes.

If Betsy’s age is even then Betsy (even) is in front of Carl (odd) and the answer is yes.

So regardless of Betsy’s age, the answer is always yes.

A similar problem was devised by Hector Levesque and it was included in Alex Bellos’ Guardian blog. Unbelievably 72 per cent of the 200,000 people who answered the question got it wrong.

That’s all for this month.

John

P.S. Why do mathematicians think that Halloween and Christmas are the same?

Because 31 OCT = 25 DEC (You need to know about the octal number system to understand this month’s joke 318 = 2510)