### Pizza Portions

My friends and I love pizza. Can you help us share these pizzas equally?

### Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

### Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

# Rod Fractions

## Rod Fractions

What fraction is the yellow rod of the orange rod?

Use this picture to help you. Note that it only uses orange and yellow rods.

You might like to use the interactivity further down this page to help you answer the following problems:

Using as many brown and red rods as you like, but no rods of any other colours, work out what fraction the red rod is of the brown one.

Using as many red and orange rods as you like, but no rods of any other colours, work out what fraction the red rod is of the orange one.

Can you find any other pairs of rods so that the length of the shorter rod compared with the longer rod is a fraction with 1 as its numerator?

Given an unlimited supply of any two differently coloured rods, can you find a general rule to work out what fraction the shorter rod is of the longer one?

You may like to explore this by using the pairs of colours suggested below as starting points.

Dark green and blue:

Pink and orange:

Light green and orange:

Why does your rule work?

Now, looking back at the different pairs of rods you have explored, can you find a way to express the longer rod as a fraction of the shorter rod?

### Why do this problem?

Cuisenaire rods are the ideal context in which to give learners opportunities to express one quantity as a fraction of another (in this case lengths). There is no need to know the quantitative value of the rods' lengths, which means learners can focus on the comparisons, and the relationship between the lengths.

### Possible approach

Ideally, learners would have real Cuisenaire rods to use, so that they can solve this problem practically as well as virtually. If they are not already familiar with Cuisenaire rods, it is essential to give them time to 'play' before having a go at this activity.

Introduce the task by showing the second image of the two yellow rods and orange rod. Try not to say anything by way of explanation, simply ask, "What do you see? What do you wonder?".  Give learners a few minutes of thinking time on their own before suggesting that they talk to a partner.  Invite pairs to share their noticings, or wonderings, with the whole group, writing them up on the board without offering comment yourself. Encourage members of the class to respond to anything you have written. If it does not come up naturally during the conversation, ask, "What fraction is the yellow rod of the orange rod?".

At this point, introduce Cuisenaire - either the physical rods, or the interactivity. If you do not have real rods, it would be useful for students to have access to the interactivity in pairs, for example on a tablet or computer. (Throughout this activity, try to refer to the rods using their colours, rather than giving them any numerical values.) Show the image of the red rod and brown rod, and ask, "What fraction is the red rod of the brown rod?".  Explain that learners can use as many brown and red rods as they like to find out, and give them a minute to work in pairs. You could invite a pair up to demonstrate what they did using the interactivity on the interactive whiteboard (IWB).

Set the class off on comparing the red and orange rods in a similar way, and share responses before giving them time to find other pairs of rods so that the length of the shorter rod compared with the longer rod is a fraction with 1 as its numerator. In a mini plenary, invite a pair of students to share their findings. Encourage learners to explain how they know they have found all the possible pairs of rods that satisfy this criteria.

Next, pose the question, "Given an unlimited supply of any two differently coloured rods, can you find a general rule to work out what fraction the shorter rod is of the longer one?". You could leave the images of the dark green/blue, pink/orange and light green/orange pairs on the board for learners, should they need a starting point. Allow some thinking and working time, then you may want to draw attention to some different ways of recording that you have observed.

In the final plenary, you could work on a full generalisation as a whole class. Invite a couple of pairs to use the IWB to share what they have done, or use a visualiser to capture them arranging real rods. You may find that some learners express the relationship between the rods in terms of a ratio, for example "For every three dark green rods, there are two blues", while others may be happy to go straight to a fractional relationship. Articulating why the rule works is not necessarily straightforward, and you may need to help learners make the link between expressing the relationship as a ratio and as a fraction.

### Key questions

Tell me about what you're doing/what you've done.
Have you tried using more than one of the longer rods?

### Possible support

It might be helpful for students to have squared paper for recording purposes.

### Possible extension

The final question in the task where learners are asked to express the longer rod as a fraction of the shorter rod makes a good challenge.