The clock strikes 10. Then 11. The rest of the world is falling asleep, while you’re still at your desk, staring blankly at a maths problem. Stuck.
As midnight arrives, you search your mind desperately for a solution. But you’re still out of ideas.
For many problems, the hardest part is knowing where to start.
To make your life easier, we at Ned & Tom are sharing our 3 top tips for starting tough questions.
1. Restate the problem in your own words.
Imagine you want to visit a friend, but you don’t know her address.
It doesn’t make sense to start walking without finding out where she lives. You might walk in the wrong direction.
This is like starting a problem without understanding what the question is asking. Understanding what the questioner wants can be hard, because frankly, mathematicians are awkward.
Rephrasing the question in your own words helps you understand what it wants from you. Sometimes this helps you work out every step to the answer – like knowing the exact route to your friends house.
Other times, you can only work out vaguely what the question wants. This is still better than nothing! This is like knowing the neighbourhood where your friend lives. You can at least walk there. Maybe when you get there you’ll see a familiar house and know where to go.
2. Find a related problem you know how to solve.
Sometimes, a question is so hard that we feel we have no hope of solving it.
How would you find the volume of a four-dimensional hyper-cube?
At first, this question seems impossible.
I can’t even imagine a 4-d cube! How on earth could I find its volume?
The trick is to find an easier, related problem. It helps to start small. Find a problem which you know how to solve, and work your way up from there. In this case, a simpler problem is to find the volume of a 3-d cube.
We can go even simpler. A 3-d volume is analogous to an area in 2-d. An even easier problem is to find the area of a 2-d square.
The area of a square is the product of both its side lengths. Since both sides have the same length (call it a), the area of a square with side-length a is , a2, a squared. Similarly, the volume of a cube with side length a is a3, a cubed.
Working this out gives us courage to tackle our original problem.
By analogy, it is reasonable to state that in 4 dimensions, the volume of a 4-d cube is a4.
The question seemed difficult, but by finding a simpler problem and working our way up, we made it easy.
3. Build it and the ideas will come.
Doing maths is like building with Legos.
The building blocks are the information given in the question. When we understand the problem clearly, we know exactly what we want to make. Solving the problem is like following the instructions on the box.
Sometimes, we find a problem unlike any we’ve done before.
We can’t find a related problem. We try to restate it, but we still don’t understand what it wants from us. In this case, the best thing to do is start using the information given in the question.
Solving a problem we don’t fully understand is like building Legos without instructions.
We can’t initially see the finished Lego on the box. But we can start by using the Legos we have. We can build these up into something useful, like a wall, or a doorway. These parts suggest the idea to build a house. We can combine our useful pieces further, eventually ending up with the finished article.
We can start with the information given in the question.
Try and work something useful out from it. After finding something useful, maybe we can use it to get closer to the answer to our original problem.
If you can’t restate the question, and you can’t find an easier one to solve, the best thing to do is start. Put the information in your question together like Legos, and often you’ll build something useful.
If you or your child wants to learn useful problem solving techniques like these, Ned and Tom offer private 1-to-1 and small group tutoring. We teach your children the maths you wish you’d learnt. Sign-up here.