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Article by Steve Hewson# Getting Started with Solving Rich Tasks

Before we begin, here are two important pieces of general advice:

### DON'T PANIC and IT IS OK TO BE STUCK

Although panic rarely helps you to solve a question, being stuck can, in fact, be a good thing. Why? Because by becoming un-stuck we become better at mathematics. So, as you read on, bear these two pieces of advice in mind. Very quickly you will gain in confidence, skill and understanding.

OK. So, you are getting started with some rich mathematics from the NRICH website. Perhaps you have no idea where to begin, or perhaps you wish to improve your problem solving skills. In fact, you might be wondering what, exactly, is a rich task, and how are the NRICH problems different from, say, standard textbook questions (as, indeed, they are).

To answer this question, let us first look at a piece of school maths: quadratic equations. A standard question might be written as

$$

\mbox{Solve } x^2 -4x +3 = 0

$$

Now, you might look at this question and say to yourself (if you happened to be learning about quadratic equations that week) "Ah, this is a quadratic equation. I know that I am supposed to solve these with a formula, because the teacher has just told us that the solution to the equation

$$

ax^2+bx +c = 0

$$

is given by

$$

x= \frac{-b\pm \sqrt{b^2-4ac}}{2a}

$$

So, using $a=1, b=-4, c=3$ and plugging in the numbers, I get

$$

x = \frac{-(-4)\pm\sqrt{(-4)^2-4\times 1\times 3}}{2\times 1} = 1\mbox{ or } 3

$$

So, the question is finished. There is nothing more to do or think about, so perhaps we might repeat the process with 10 other similar questions.

Is this what mathematics is all about? Is this why people study mathematics to university and beyond?

Not really.

This task is simply the application of a skill, and is entirely routine in nature: so much so, that you could possibly do this question whilst thinking about something else or you could even program a computer to solve it.

Rich tasks are far more interesting. A richer question on quadratic equations might be along the lines

My friend chooses two numbers $a$ and $c$. Can I find a number $b$ such that $ax^2+bx+c=0$ has two real solutions?

To answer this question you need to use the same piece of mathematics (the quadratic equation formula) as before, but it will require you to understand what the formula is trying to say and how it works. Engaging in this task will lead to a far deeper understanding of the mathematics, and longer term assimilation of the process. Furthermore, we could easily extend this task into unfamiliar territory, which will force a really high level of engagement with equations:

My friend then chooses three numbers $a, b$ and $d$. Can I find a number $c$ such that $ax^3+bx^2+cx+d=0$ has a real solution?

Whether you can solve this second part or not, it is certainly more curious. Instead of a full solution, perhaps we might try a few special cases to get a feel for the problem. It starts as an experiment, although understanding, and possibly proof, might follow later.

Rich tasks take you inside the mathematics. They are not a routine application of skills, such as the equation solving above. The task may or may not be particularly difficult, but it will draw you into the mathematics far more than traditional questions. It might be the mathematical process that is particularly interesting, or the result, or it might make links with other areas of mathematics, or it might be something else entirely. One thing is for sure: you will need to think to get the most from rich tasks. This might require more effort than you are usually used to putting into mathematics, but the learning rewards will correspondingly be greater.

How do I get started with the NRICH problems?

The best way forward is to try a few, although without the obvious signposting of 'use this equation' perhaps you may wonder how to begin at first. To help, many of our problems have a 'Solution' and 'Hint' tab at the top of the page. The Hint tab is obvious in function: it will give you some help on how to begin the problem. The Solution tab is more unusual. It does not just give an answer. It gives full solutions to the problems, including all of the working out and thinking that went into the creation of the answer. Moreover, there are often several different types of solution given because there are often many different, equally valid, ways to solve a good problem. Furthermore, these solutions are all sent in by students who are likely to have encountered many of the hurdles that you face. If you are stuck, don't just look at the answer: read through the solutions given. Can you follow them? This is a great way to learn about mathematics.

Soon you will be making progress into problems on your own, and possibly submitting your own thoughts and solutions to the new NRICH problems which appear each month.

In summary, it is always helpful to bear in mind these problem solving tips

1) Don't be afraid to experiment: try a few special case numbers to get a feel for the situation.

2) Don't be afraid to provide a partial solution to a problem. Many rich tasks are 'open': there is sometimes not necessarily a set, final answer.

3) Look at the question carefully: what information does it give us? What mathematics might be helpful in solving it?

4) Don't give in: try different methods: use algebra, draw diagrams, make a model, use numbers or a spreadsheet, or say your thinking out loud. Or even sleep on it and return to the problem at a later time.

5) Think about the answers that you get. What are they telling you? Do they make sense? Are your arguments really correct?

6) And finally: don't forget to do the question your way. There will be no 'set method' for these tasks: if you look at the solutions to some of our problems (see the top of each problem) then you will soon see that there are usually a variety of valid, interesting approaches.

Suggested starter problems

Now that you have an idea about the site, why not try a few of our problems. This list contains some good problems with which to get started:

We do hope that you enjoy getting into these problems. You can find other problems of interest by using the Search facility at the top of the NRICH page or on our list of NRICH starter problems

Or search by topic

Age 11 to 18

Published 2008 Revised 2019

Before we begin, here are two important pieces of general advice:

OK. So, you are getting started with some rich mathematics from the NRICH website. Perhaps you have no idea where to begin, or perhaps you wish to improve your problem solving skills. In fact, you might be wondering what, exactly, is a rich task, and how are the NRICH problems different from, say, standard textbook questions (as, indeed, they are).

To answer this question, let us first look at a piece of school maths: quadratic equations. A standard question might be written as

$$

\mbox{Solve } x^2 -4x +3 = 0

$$

Now, you might look at this question and say to yourself (if you happened to be learning about quadratic equations that week) "Ah, this is a quadratic equation. I know that I am supposed to solve these with a formula, because the teacher has just told us that the solution to the equation

$$

ax^2+bx +c = 0

$$

is given by

$$

x= \frac{-b\pm \sqrt{b^2-4ac}}{2a}

$$

So, using $a=1, b=-4, c=3$ and plugging in the numbers, I get

$$

x = \frac{-(-4)\pm\sqrt{(-4)^2-4\times 1\times 3}}{2\times 1} = 1\mbox{ or } 3

$$

So, the question is finished. There is nothing more to do or think about, so perhaps we might repeat the process with 10 other similar questions.

Is this what mathematics is all about? Is this why people study mathematics to university and beyond?

Not really.

This task is simply the application of a skill, and is entirely routine in nature: so much so, that you could possibly do this question whilst thinking about something else or you could even program a computer to solve it.

Rich tasks are far more interesting. A richer question on quadratic equations might be along the lines

My friend chooses two numbers $a$ and $c$. Can I find a number $b$ such that $ax^2+bx+c=0$ has two real solutions?

To answer this question you need to use the same piece of mathematics (the quadratic equation formula) as before, but it will require you to understand what the formula is trying to say and how it works. Engaging in this task will lead to a far deeper understanding of the mathematics, and longer term assimilation of the process. Furthermore, we could easily extend this task into unfamiliar territory, which will force a really high level of engagement with equations:

My friend then chooses three numbers $a, b$ and $d$. Can I find a number $c$ such that $ax^3+bx^2+cx+d=0$ has a real solution?

Whether you can solve this second part or not, it is certainly more curious. Instead of a full solution, perhaps we might try a few special cases to get a feel for the problem. It starts as an experiment, although understanding, and possibly proof, might follow later.

Rich tasks take you inside the mathematics. They are not a routine application of skills, such as the equation solving above. The task may or may not be particularly difficult, but it will draw you into the mathematics far more than traditional questions. It might be the mathematical process that is particularly interesting, or the result, or it might make links with other areas of mathematics, or it might be something else entirely. One thing is for sure: you will need to think to get the most from rich tasks. This might require more effort than you are usually used to putting into mathematics, but the learning rewards will correspondingly be greater.

How do I get started with the NRICH problems?

The best way forward is to try a few, although without the obvious signposting of 'use this equation' perhaps you may wonder how to begin at first. To help, many of our problems have a 'Solution' and 'Hint' tab at the top of the page. The Hint tab is obvious in function: it will give you some help on how to begin the problem. The Solution tab is more unusual. It does not just give an answer. It gives full solutions to the problems, including all of the working out and thinking that went into the creation of the answer. Moreover, there are often several different types of solution given because there are often many different, equally valid, ways to solve a good problem. Furthermore, these solutions are all sent in by students who are likely to have encountered many of the hurdles that you face. If you are stuck, don't just look at the answer: read through the solutions given. Can you follow them? This is a great way to learn about mathematics.

Soon you will be making progress into problems on your own, and possibly submitting your own thoughts and solutions to the new NRICH problems which appear each month.

In summary, it is always helpful to bear in mind these problem solving tips

1) Don't be afraid to experiment: try a few special case numbers to get a feel for the situation.

2) Don't be afraid to provide a partial solution to a problem. Many rich tasks are 'open': there is sometimes not necessarily a set, final answer.

3) Look at the question carefully: what information does it give us? What mathematics might be helpful in solving it?

4) Don't give in: try different methods: use algebra, draw diagrams, make a model, use numbers or a spreadsheet, or say your thinking out loud. Or even sleep on it and return to the problem at a later time.

5) Think about the answers that you get. What are they telling you? Do they make sense? Are your arguments really correct?

6) And finally: don't forget to do the question your way. There will be no 'set method' for these tasks: if you look at the solutions to some of our problems (see the top of each problem) then you will soon see that there are usually a variety of valid, interesting approaches.

Suggested starter problems

Now that you have an idea about the site, why not try a few of our problems. This list contains some good problems with which to get started:

- Got it
- Cut nets
- Factors and multiples game
- Consecutive sums
- Peaches today, peaches tomorrow
- Isosceles triangles
- Fibonacci factors
- Data matching

We do hope that you enjoy getting into these problems. You can find other problems of interest by using the Search facility at the top of the NRICH page or on our list of NRICH starter problems