In mathematics, diagrams are often a useful way of organising information and help us to see relationships. A diagram can be a rough sketch, a number line, a tree diagram or two-way table, a Venn diagram, or any other drawing which helps us to tackle a problem.

Labels (e.g. letters for vertices of a polygon) are useful in a diagram to help us be able to refer to items of interest.

A diagram can be updated as we find out new information.

**Examples of using a diagram to tackle a problem**

First we will read all three examples and have a quick think about them and then we will look at how a diagram can help us with each one:

**Restaurant Example**

A restaurant offers a “business lunch” where people can choose either fish or chicken or vegetables for their main course, accompanied by a side portion of rice, chips, noodles or salad. How many different combined meals can they choose between?

**Rectangle Area Example**

To the nearest centimetre, the length and width of a rectangle is 10cm and 8cm.

- What are the limits of accuracy for the area of the rectangle?
- What is the difference between the minimum and maximum values for:
- the lengths of the sides?
- the area?

**Prime Numbers Example**

Masha says that if she writes out numbers in rows of six then all of the prime numbers will either be in the column that has 1 at the top, or they will be in the column that has 5 at the top. How can you find out if she is correct?

**Worked Solutions to Examples**

**Restaurant Example**

A restaurant offers a “business lunch” where people can choose either fish or chicken or vegetables for their main course, accompanied by a side portion of rice, chips, noodles or salad. How many different combined meals can they choose between?

**Solution**

One way to tackle this would be to write out a list, being systematic to ensure that all combinations are considered.

Another is to draw out a diagram like the one below. As shown, you actually don’t need to finish the diagram in order to conclude how many combinations there are:

You could also use a 2-way table as shown below:

Rice | Fries | Noodles | Salad | |

Fish | ||||

Chicken | ||||

Vegetable |

**Rectangle Area Example**

To the nearest centimetre, the length and width of a rectangle is 10cm and 8cm.

- What are the limits of accuracy for the area of the rectangle?
- What is the difference between the minimum and maximum values for:
- the lengths of the sides?
- the area?

**Solution**

Drawing a rough sketch of the rectangle labelled with the boundaries of its side lengths can really help us to visualise the situation here:

It can then be helpful to draw sketches of the smallest possible rectangle and the largest possible rectangle:

We can now answer the questions, so (a) the smallest possible area is 7.5 x 9.5 = 71.25cm^{2} and the largest “possible” area is 8.5 x 10.5 = 89.25cm^{2}. So the limits of accuracy are [71.25,89.25) cm^{2}.

For (b), we can see from the sketches that the difference between the minimum and the maximum values is 1cm in the case of both the width and the lenght. For part (ii) we simply subtract the numbers above to give 89.25-71.25 = 18cm^{2}.

**Prime Numbers Example**

Masha says that if she writes out numbers in rows of six then all of the prime numbers will either be in the column that has 1 at the top, or they will be in the column that has 5 at the top. How can you find out if she is correct?

**Solution**

Here, listing out numbers, especially for the first few is going to be helpful. We should list them as specified in the question, and we can highlight the prime numbers:

1 | 2 | 3 | 4 | 5 | 6 |

7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 |

19 | 20 | 21 | 22 | 23 | 24 |

25 | 26 | 27 | 28 | 29 | 30 |

31 | 32 | 33 | 34 | 35 | 36 |

Because we know that no even numbers other than 2 are prime, we know that further prime numbers cannot be in the second, fourth or sixth column. The third column keeps adding 6s, so it is adding multiples of 3 to multiples of 3, so the numbers will always be divisible by 3, so further numbers in this column cannot be prime. So she is correct that the prime numbers must be in the first or the fifth column.

**31 Questions of increasing difficulty**

**Worked Solutions to Questions**