# 3.3 Using ratio tables

One way to support learners in developing their own mental strategies for solving proportion problems is through the use of ratio tables.

Ratio tables are a way to **symbolise** the problem and can support learners in finding strategies for solution. They encourage approaches such as halving, doubling, and multiplying by 10.

## Activity 20 Reflecting

Two learners have used ratio tables to work on the problem shown below. They have both taken slightly different approaches.

Have a look at their workings below. Can you identify each of their strategies?

**Problem: **Seedling plants come in boxes of 35 plants. How many plants would be in 16 boxes?

Boxes | 1 | 2 | 4 | 8 | 16 |

Plants | 35 | 70 | 140 | 280 | 560 |

Boxes | 1 | 10 | 2 | 6 | 16 |

Plants | 35 | 350 | 70 | 210 | 560 |

### Discussion

**Sophie’s strategy**

*Repeated doubling to get to 16 boxes.*

This works because 16 is a power of 2. If the question was asking for 15 boxes, she could use the same strategy but then subtract one lot of 35.

**Alejandro’s strategy**

*Multiply by 10.**Separately also multiply original amount by 2, then multiply by 3 to get 6 lots.**Add 10 lots and 6 lots to get 16.*

Alejandro has used a **building-up strategy**. He did not just use a scalar multiplier. Instead, he had to find different parts and sum these. This can be known as the **addition and scaling method**.

In the next example, Alejandro was given the unit amount (that 1 box contained 35 seedlings). This meant that he only needed to use multiplication.

In the second example below, the unit cost is not given, so some division, as well as multiplication, is required.

## Activity 21 Proportion problem

Think about how you might approach this problem before reading the learner’s response below.

**Problem: **Mangoes are 2 for £3. How many could you buy for £7.50?

Mangoes | 2 | 4 | 1 | 5 |

Price | £3 | £6 | £1.50 | £7.50 |

### Discussion

This problem is slightly different from the previous examples because it is not as clear how to get from £3 to £7.50 as it was to get from 1 to 16. Because £7.50 is not a multiple of 3, it cannot be made using either Sophie’s or Alejandro’s strategy.

**Yasmin’s strategy: **

*Double to get £6.**Need £1.50 more so adding another £3 will be too much.**Half the original amount to get £1.50.**Add £1.50 to £6 (1 lot to 4 lots) to get £7.50.*