# Fruity Pairs

Today they find a pot of raisins, a banana and two apples in the kitchen. They choose one item each.

What could Kate and Sam choose?

Can you think of a different combination of two items?

Can you find **all** the combinations of two items?

How do you know you have got them all?

Mum complains that all the food they eat costs her a lot of money!

She works out that a pot of raisins is 11p, a banana is 14p and an apple is 27p.

Can you order the different possible choices from the pair that costs the least to the pair that is most expensive?

How did you decide? We would love to hear your reasoning.

Can you find another way of choosing two items? And another?

How will you keep track of the different ways you have found so far?

How will you know when you have got them all?

Which of the totals do you think will be the highest? How do you know?

Which of the totals do you think will be the lowest? How do you know?

Thank you to everyone who shared their ideas with our team.

As you worked on this problem, you may have found several combinations of fruit for Kate and Sam. Drawing a table or a diagram can be a useful way to keep track of your ideas.

Avanish, from the International School in Bangalore, included both a table and a diagram in this solution:

Can you see how many different combinations Avandish found? Does the table show **all** the combinations? How do you know?

How do you think Avanish used the diagram to work out the answer?

Avanish also suggested that the cheapest items would be the pot of raisins and a banana, and that the most expensive items would be the two apples. Do you agree?

### Why do this problem?

The first part of this problem gives learners the chance to work systematically to find all possible combinations, which in this case is a relatively small number. The second part asks learners to order amounts and this gives them the opportunity to calculate and/or to use reasoning to consider relative totals. It is a perfect opportunity to encourage learners to be flexible in their approach to calculation and to select strategies according to the numbers involved.

Read more about the benefits of having a flexible approach to calculation in our Let's Get Flexible! article.

### Possible approach

You may wish to use real fruit as a prop as you introduce this task. Ask for suggestions of one way that Sam and Kate could choose an item each. Take a couple of different options and write them on the board. Invite the class to work in pairs to find all the different combinations. You may like to give each pair a copy of the first set of cards on this sheet, which represent the four different items. (The pictures are in colour but you may prefer to print in black and white.) Alternatively, they could use coloured counters, for example, to represent the four items.

In a mini plenary, bring together everyone's ideas. You may find that there is some disagreement as to what is meant by 'different'. Is an apple and a banana different from a banana and an apple? In this context, as the fruit is for different children then an apple for Kate and a banana for Sam **is** different from an apple for Sam and a banana for Kate. Draw attention to some
good ways of recording thinking that you have observed, which will help keep track of the possibilities. Together as a whole group, add to the list of different solutions on the board and facilitate a discussion about how we know we have them all. (You will find there are seven in total.)

Next introduce the second part of the problem in which the price of each item is given. Invite learners to comment on what they need to do to find out the cost of, for example, an apple for Kate and a banana for Sam. Write up the expression 27p + 14p on the board. What about a banana for Kate and an apple for Sam? That would be 14p + 27p. Give learners a minute to talk to their partners about what they notice and draw out the fact that these totals are the same. Facilitate a discussion about which other pairs of totals would be the same as each other, so that the class can see there will be four different totals.

At this point, you may like to give out a set of the second set of cards on this sheet to each pair. Challenge them to put the totals in order from the one which costs the least to the one which costs the most. As they work, listen out for those learners who are not necessarily calculating each total but are reasoning about the relative sizes of the four using knowledge of place value.

In the final plenary, you could invite comments about how they decided on the order of the four totals. In the discussion, bring out the idea that just because all the calculations were additions, it doesn't mean that all four have to be solved in the same way necessarily. Invite one or two pairs who didn't calculate all the answers to share their reasoning with the class too.

### Key questions

Can you find another way of choosing two items? And another?

How will you keep track of the different ways you have found so far?

How will you know when you have got them all?

Which of the totals do you think will be the highest? How do you know?

Which of the totals do you think will be the lowest? How do you know?

### Possible support

Having cards printed from this sheet to represent the four different items, or using counters/cubes, will help learners try out their ideas.

### Possible extension

What would happen if one more item of fruit was available to choose? You could introduce, for example, another banana. How many possible ways of choosing are there now? How does this affect the different totals?