Maths tutor Olaf believes one of the most important skills he teaches his students is to first ask themselves "How could I make this problem simpler?". Here he shows us how to "keep things simple" when working with shapes and patterns.
Maths is the study of patterns. For example, which shaped tiles can completely cover a flat surface? Squares work and so do regular hexagons (think of a honeycomb). But try with regular pentagons and there will always be gaps. What is it that lets some of these shapes fit neatly and some not?
This is the kind of question Maths sets out to answer. Breaking patterns. Once you’ve spotted a pattern, you can start to ask what changes you can make without disrupting it. My grid of squares will still work if I make all the squares twice as big. Or I can take the whole thing and ‘squash’ it, turning the squares into rectangles. But change the number of sides — from square to pentagon — and the pattern is broken. Working out which things can be changed and which can’t for a pattern which we’re studying is very useful, because it lets us know what things we can simplify to make our problem easier.
How can I break down a difficult maths problem?
This is one of the most important skills I teach my students, and it starts with a question: "How could I make this problem simpler?". As an example let's think about another pattern — the relationship between speed, distance and time.
Imagine a rocket travelling at 5,218 miles per hour. How long will it take to travel 897 miles? Unfortunately I've forgotten what sum I need to do! Do I need to multiply the two numbers? Divide? How can I remind myself?
Well, how could we make this question simpler? We could replace the numbers with smaller ones: say it's travelling at 2mph and has to go 6 miles. How long does it take? Now the answer's so easy we hardly have to think — it takes 3 hours. And how did we get that? 6 / 2 of course. So, returning to our first problem, we do the exact same sum with the original numbers: 897 / 5,218, which gives us our answer: 0.17 hours (or about 10 minutes).
The point is that the big numbers made the question harder. But the pattern — the relationship between speed, distance and time — doesn't care about the numbers, just like our pattern of squares doesn't care about their size. So, if we can solve it for easy numbers, we can apply the same method to any numbers we like. This can apply to algebra questions too: if the speed is x+y and the distance is 4x, what sum do we do? Exactly the same as we used for the simple numbers! This skill — spotting which parts of the pattern are crucial and which can safely be simplified to help us work out our solution — can really be key to tackling tricky Maths problems at every level from primary school right up through to university.
Blog post crafted by Olaf,
Olaf is one of our top Maths tutors. He studied Mathematics at Cambridge University, and went on to do a Masters degree at Cambridge.
Since then he has been tutoring in a variety of contexts while pursuing a part-time PhD at Oxford University, where he has
also been involved in teaching undergraduates and setting and
marking exams. His research project uses supercomputers to create simulations of dark matter.
Olaf has four degus (a type of
rodent from South America) and is currently teaching them to play football.
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