Mastering maths

Trains arrive at Euston every 40 minutes from Manchester, every hour from Edinburgh, and every 45 minutes from Liverpool. If all three trains arrive together at Euston at 9am at what time will they next all arrive together at the station?

Three pears cost the same as one orange. Two oranges cost as much as one apple. If a pear, an orange and an apple cost 80p in total, how much does each piece of fruit cost individually?

If a bottle and cork together cost £1.10, and the bottle costs £1.00 more than the cork, how much does the cork cost?

A castle is surrounded by a square moat of a uniform width of 10m. How can those laying siege to the castle breach the moat with just two planks of wood each measuring 9.5m length. No nails, screws, rope or any method of attaching the two planks together can be used. Nor can the planks be weighted at either end to secure them. They can only be placed in such a way as to allow access across the moat. How can the moat be breached?

Pythagoras, the well-known Greek fire fighter arrived at the scene of the fire and quickly realised he needed to extend his ladder to the fifth floor, forty feet from street level, to rescue Mrs Trellis and her cat. Pythagoras knows his ladder will extend fully to fifty feet long.
To prevent Mrs Trellis descending a very steep ladder, what is the furthest away from the building that he can position the end of the ladder? Hint - the ground is flat and meets the base of the building at a 90 degree angle.

Profile

Initially working within banking in London for HSBC, I moved to E&Y and trained as a Chartered Accountant. After fourteen years in audit and accountancy I changed career to school finance whilst I contemplated the prospect of training to teach. At the same time I set up a company to teach swimming – my main sporting obsession - and something I had become involved with as a coach through my children competing for a local club. The business has flourished and produced many successful swimmers who have themselves progressed to competing for the same club, where I still coach.

Several years ago I took the plunge and trained as a Primary mathematics teacher, securing a teaching post at a local school which I relinquished earlier this year to focus on my company and other interests, including mathematics tuition.

Tutor approach

My approach to tutoring mathematics is to create a calm, relaxed teaching environment to optimize pupil learning. Having a consistent and familiar structure to the lesson lets the pupil know what to expect and helps reduce the apprehension many feel towards this subject. Keeping the lessons fun and relevant stimulates student engagement and allows the creation of bespoke lessons through teacher assessment.

A warm up exercise, followed by a review of work previously set to identify areas of weakness or where further support is required, leads on to 20 minutes of teaching a new topic, question practice, a mini-test, feedback and then preparing the next homework task.

Regular, concise tests of exam standard questions targeted at specific topics gives a valuable insight into a pupils grasp of the subject, and as the marks improve is a source of growing confidence, particularly when reviewed over time.

There are key areas that all pupils need to consolidate in order to progress their mathematical understanding. Place value, multiplication, number bonds and the four operations all must be securely embedded. The rapid recall of these facts significantly enhances mental processing of mathematical problems.

Mathematics in the National Curriculum is frequently taught through discrete topics (PV, 4 Op’s, Fractions, Measurement, Shape, Position and Direction) which are repeated year on year, layering up increasing complexity. This requires both a solid foundation of understanding, and the cross referencing of these topics e.g. the importance of multiplication when dealing with fractions. Additional topics are then introduced, Statistics, Estimation, Ratio, Proportion, Algebra and Probability towards the end of KS2.

Many pupils require the extra support that one to one tuition provides. For some it gives them help with their work ethic and self-discipline. For others a different approach may provide that “light-bulb” moment that triggers the connection and understanding to fall into place. For a few they may want to consolidate their existing knowledge. Tutoring can boost confidence and give a lift to underperforming pupils helping create a momentum of its own.

The Tower of Hanoi
The objective is to transfer the set of disks to another column. The rules are simple:-
1) Only one disk may be moved at a time.
2) Only the top most disk of each stack can be moved each time.
3) No disk may be placed on a smaller disk.
What are the minimum number of moves required to transfer the entire column of disks?
The legend is that the temple in which this puzzle was discovered had a column of 64 disks, and an ancient prophecy stated that if the monks in the temple moved one disk per second then before the puzzle could be completed the world would have ended.
Rounded to the nearest billion, how many years would this take? (hint – approx. 42 times the current age of the earth)

Three tins of biscuits are labelled “Digestives”, “Rich Tea”, and “Mixed Biscuits”.
Whilst his mother is out Johnny peels all three labels off and then reattaches them so that no tins contents now matches its label.
When his mother arrives home she can see that the labels have been hastily and incorrectly swapped.
She takes one biscuit out of one tin and eats it. She then correctly reattaches the labels without opening any further tins. Which tin did she take the biscuit from, and how does she know how to correctly reapply the labels?

A shepherd needs to cross a river and take with him a wolf, a goat, and a cabbage.

There is a small boat that can fit himself plus either the wolf, the goat, or the cabbage.

If the wolf and the goat are alone on one shore, the wolf will eat the goat. If the goat and the cabbage are alone on the shore, the goat will eat the cabbage.

How can the shepherd take the wolf, the goat, and the cabbage across the river?