How to multiply by 12, using Trachtenburg’s Method.
This cake contains algebraic sequences, inequalities, and geometric sequences, a sum, a square root and a 3D shape you have never heard of before. If you loved the Da Vinci Code, you will love this…. and if it is almost Christmas, then this is great. You can make it from start to finish on Christmas Eve, if you like. It will keep fine for up to 10 days.
And, it is easy, and vegan.
- Self Raising Flour
- Golden Caster Sugar
- Mixed Spice
- Ground Ginger
- Bicaronate of Soda
- Plain Flour
- Dried Mixed Fruit
- a pack of marzipan
- some plain jam without pips (apricot is best)
- Pack of ready made white icing
- A Mixing Bowl and another bowl
- Wooden Spoon
- Teaspoon measure
- Weighing Scales
- Loaf tin
- Silicone tin liner or greaseproof paper
- rolling pin (or clean glass bottle) and clean flat surface
- the number “200”
An arithmetic sequence is a list of numbers that goes up (or down) the same amount, all the way through.
The first sequence for this recipe starts with the number 200, like this:
- Weigh 200g of dried mixed fruit into a bowl
- Make a nice strong mug of tea with the teabag (or use leftover tea without milk)
- Put the bowl of mixed fruit on the scales and weigh in 180g of the tea.
- Mix together and leave to soak for at least 160 minutes. (you are going to need the number 160 again, so don’t forget it).
- When the fruit has soaked, turn on the oven at 160 degrees (fan), and weigh 160g of plain flour into a sieve that is sitting on top of a mixing bowl.
Geometric Sequence using a teaspoon
A geaometric sequence is made, by repeatedly multiplying by the same number. In this sequence, the multiplier is 0.5 which means that every number is half the size of the previous number.
The cake is very spicy, a little bit gingery, and not very light. So measure into the sieve:
- Two teaspoons of Mixed Spice
- One teaspoon of Ground Ginger
- Half a teaspoon of Bicarbonate of Soda
Tap the sieve carefully until all these dry ingredients have fallen into the mixing bowl.
40+50 = …
… 90g of golden caster sugar, mixed in with the dry ingredients.
Square Root and a Prismoid
You have 90g of sugar… what is the square root of 9?…
THREE teaspoons of rapeseed oil (often labelled “vegetable oil”). Add this to the mixed fruit and tea, stir in, then stirthe wet mixture into the flour and put it all, carefully, into the loaf tin.
A loaf tin is an example of a prismoid. This is such an obscure word that it does not even have its own Wiki page! Poor little prismoid! It just gets a measly mention on the page for prsimatoids. A psimoid is a bit like a cuboid, except that it has sloping sides. A loaf tin has sloping sides, because if they were straight, it would be really difficult to get the loaf out!!!
Upper and Lower Bounds of Cooking Times
Put the tin in the oven and set the timer. Rememeber the sum 40+50=90? You have used the 90, for the sugar. The 40 and 50 are the upper and lower bounds of the cooking time t.
40 <= t <= 50
If you take a look after 40 minutes, it might be cooked. If it sinks as you look at it, it isn;t cooked. If you slide a knife into it, and the knife comes out smeary with mixture, it isn’t cooked. If it isn’t cooked, give it another 10 minutes.
Cool it in the tin for 5 minutes, then turn it out onto a rack to completely cool.
Sprinkle some icing sugar (about a heaped teaspoon) onto the clean surface and roll out enough marzipan to drape right over the cake.
Brush the top and sides of the cake with jam
Drape the marzipan over the jammy cake and press it gently to stick. Trim off any extra.
Roll out the ready made icing and drape over the marzipan.
Voila. Ready to eat. No fuss.
And next year, will you still remember the recipe?
… 200, 180, 160 …. (160 160)
… 2, 1, 1/2
… 40+50=90 … square root of 9 is 3
marzipan and icing and munch.
I am grateful to Mr. Holding who teaches locally for this link. So simple! Lots of GCSE questions presented one at a time. A couple of examples are reproduced below.
The Proof Questions on this site are invaluable – the best collection I have seen.
- There are eight Prime Numbers Under 20
- Many people are unsure whether they have listed them all so it can help to draw the grid out and put them all into it – 4 under 10 then 4 more between 10 and 20
- One is not a prime
- Two IS a prime
For Pupils in Years 7 to 11, I find the “bread and butter” cards on JustMaths really useful – entertaining, and a broad range of revision topics.
Easier (basic skills) stuff is on this page:
and harder stuff (Grade C+, I use these in Years 10 and 11) is:
This excellent puzzle is based on “Rectangles within Rectangles” from the book “Mathematical Snacks” by Jon Millington. Highly recommended as a source of enriching mathematical thinking!
Problem solving is a key part of 1-1 tuition – because the student is not in competition with others, the tutor can provide as much or as little help as is needed and the rewards are potentially very great, in terms of enjoyment, confidence, and increased skill in mathematical approaches.