Cake
• Twitter has a hashtag, #MTBoS, that’s used by maths teachers to share great ideas and best practices in the classroom.  This past summer, I discovered a creator of math puzzles which were going viral within the maths community.

@mazed70 was on my mind when I was putting together the panel for How Do You Make Maths Fun?  I therefore am delighted to have the opportunity to ask 3 questions on puzzles and problem solving.

Hey @mazed70 !  Can you walk me through the thought process you go through to create elegant maths puzzles like this?

• Note: We are experiencing technical difficulties with Cake’s formatting of @mazed70 ‘s first response. I’ve therefore gone ahead and reposted it here and was able to get things to work.

When our discussion concludes I will hide the unformatted duplicates.

Here’s @mazed70

Thank you for involving me and asking about my twitter posts.

I have been on twitter for a few years and enjoy puzzles and seeing other teachers share ideas for the classroom.
I like to create questions which are linked to a theme or event where possible, so Christmas, Halloween and Bonfire night give a good starting point and provide a hook.

The first puzzle I posted which spread quickly within the maths community was the World Cup question below.

I wanted to post a maths puzzle based around the World Cup and as England were playing that evening I wanted to base it around our star player- Harry Kane.

I started by looking for some numerical facts about his height, weight and his average goal score and then finally decided to base it around his age. The first draft was just a question about Harry being one less than a square number, but as the puzzle evolved I added Jordan Pickford for more interest and reinforcement of square numbers. I think the reason it became popular was firstly the timing, secondly the popularity of football and thirdly the accessibility of the maths.

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This question began life as a fraction question on paper.

"The line AD is such that the ratio of AB to BC is 2:3.

The ratio of BC to CD is 2:3.

If the length of AD is 19cm , what is the length of AC?"

I originally wrote that one length AB was 2/3 of length BC, and BC 2/3 of CD. However I felt that the question didn’t read easily so I decided to use ratios instead and the question changed. I was then happy with the question but still not entirely happy with the presentation so I decided to add the diagram for two reasons :

1) For clarity
2) To make the question more appealing as people are scrolling through their twitter feed.

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The indices problem you quoted evolved from my classroom where I set this question to a very gifted pupil who was working on a completely different topic to the rest of the class. I had already given him some indices work including multiplication and division with algebra but wanted to try something that would make him think a bit differently. His first response was

x = 3 .

I shared it on twitter and enjoyed the responses and reading the different approaches that people take.

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I also like puzzles where there are multiple solutions like the one below.

I know that remembering primes can be difficult, so I thought that this would be a good question to help with remembering them.

"This year Jacks age is under 100 and he is twice as old as Jill. If Jill's age this year is prime ,and next year Jack's age is prime , what ages could Jill be now?"

I wrote out a list of possible primes for Jill first (under 50) and then worked from there.

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The last point I like to consider when setting questions is that as well as being enjoyable and accessible, I try to keep them quite short and often solvable without pen and paper.

• Hey @apm

I was pleased by the number of responses on twitter for my indices question. I said that I liked this question because the first, seemingly obvious answer you want to give is that x=3 . But on closer inspection the correct answer is 8. I thought that this would trick many.

Breaking it down gives:
3^9=3x3x3x3x3x3x3x3x3= 3 x ( 3x3x3x3x3x3x3x3)
which can simplify to 3 x (3^8)= 3^8 + 3^8 + 3^8
therefore x=8 .
………………………………………………………………………………
I enjoyed your question @apm set in November and I shared the way I would solve it then

If I were to use algebra I would write
a+b+c=231
However I know that the difference between a and b and between b and c is the same amount , let's call it x.
Therefore a=27(given in question) b=27+x and c=27+x+x
so a+b+c=27+27+27+3x =231
81+3x=231
3x=150
x=50 and I can solve from here.
………………………………………………………………………..
However, I would not choose this method. I think a far more elegant way is to use bar models. this gives us a visual way of seeing the problem without having to write any equations.

I would draw 3 separate bars for each number and write in the information we know.
(Thank you to @mathsbot for the bar modelling manipulative.)

I know that the total of the 3 bars is 231.
27 x 3=81 so 231 -81=150.
150 divided by 3= 50.
Therefore I know that the blue bars are 50 each and I can solve from here.

• Bar models are a tool that I hadn’t learned growing up. But they can be incredibly helpful in organizing your information in a visual manner.

Organizing a problem’s information in a visual manner helps students to understand what’s missing and the proper calculations to perform.

We want students to understand why they should use a given approach instead of blindly or randomly using algorithms.

@Mathgarden believes that we don’t provide students with enough time to struggle with mathematics.

What general tips do you have for having fun with maths puzzle?

• The use of bar models in the classroom in the UK has grown quite rapidly in recent years. With the emphasis on developing understanding through the CPA approach, pupils start with concrete apparatus, then move to pictorial representations of a problem before moving to the abstract.

I first started using bar models to solve additon,subtraction and fraction problems, now I use them in lots of problems. I have found they have helped understanding a variety of puzzles including percentage, ratio and algebra problems.

Puzzles should be fun and engaging for all ages and as well as yourself, @apm, I enjoy puzzles created by @drewfoster .His puzzles are always creative , fun and often quirky, like this one. I enjoyed this puzzle because although it is not mathematical it is a good logic puzzle .

Drew’s puzzles are mathematical, logical and are presented in a fun way.

With this next problem, I would approach it with bar models . The bars for Burt and Ernie are the same size and the extra piece is 16kg for Burt.

Thank you for asking me to take part apm , and I look forward to sharing and solving more puzzles with you in the future.

Keep puzzling!