What fraction of this shape is shaded? Which is bigger, 2 5 or 1 3 ? What is 4 7 of 49 plus 5 8 of 64 plus 6 9 of 81?

Fun with fractions (and decimals and percentages) Dan Kirby Lead Practitioner in Mathematics - SHS

13:00 – 14:10 Building your subject knowledge using pictures and manipulatives 14:10 – 14:30 Break 14:30 – 15:00 “Moving up and down” for students (Concrete, Pictoral, Abstract) 15:00 – 15:30 Making links with decimals and percentages

Sort the questions on fractions into two piles. Questions you feel confident with. Questions you don’t feel confident with. Be honest! This will shape what we look at during this session 

What are fractions all about?

Give me an example of a picture to show… 1 3 and another…

Does this picture show 1 3 ? Why/ why not?

Does this picture show 1 3 ? Why/ why not?

Give me an example of a picture to show… 2 5 and another…

2 5 Make a shape to show me the fraction: Convince me that your shape is right. Where does the 2 appear? Where does the 5 appear?

What fraction is shaded?

Key facts. The denominator tells us how many equal sized pieces an amount or object is shared into. The numerator tells us how many of these pieces we want.

Finding a half and a quarter

6 6

3 3 3 3

Key facts. When we find half of a number, we are dividing it into two equal pieces. Finding 1 quarter is the same as halving and halving again.

Equivalent fractions 1 3 = ? 6 1 3 = ? 9

Equivalent fractions How many fractions can you write down that are equivalent to 1 2 ? How many fractions can you write down that are equivalent to 3 4 ? How many fractions can you write down that are equivalent to 3 5 ?

Finding equivalent fractions

There are 4 different ways to express the blue rectangle as a fraction of the large rectangle. Can you find them all?

Simplifying fractions (cancelling down)

Key facts. Equivalent fractions can have different numerators and denominators, but they represent the same amount. To generate equivalent fractions, we multiply the numerator and denominator by the same number. To simplify fractions, first we find the largest number that goes into the numerator and denominator. We then divide both numerator and denominator by these numbers.

Comparing fractions

Which is bigger?

4 6 or 5 6

2 3 = ? 12 3 4 = ? 12 8 12 or 9 12 ?

Key facts. To compare fractions (say which one is bigger or smaller) they must have the same denominator. Use equivalent fractions to rewrite the fractions with the same denominator and then compare them.

Arithmetic with fractions

Adding and subtracting fractions… 1 5 1 5 + 2 5 = 3 5 2 5 3 5

Adding and subtracting fractions… 1 5 1 5 + 2 5 = 3 5 2 5 3 5

Adding and subtracting fractions… 2 9 2 9 + 5 9 = 3 5 5 9 7 9

Adding and subtracting fractions… 2 9 2 9 + 5 9 = 7 9 5 9 7 9

Adding and subtracting fractions… 6 7 6 7 − 2 7 = 4 7 2 7 4 7

Adding and subtracting fractions… 6 7 6 7 − 2 7 = 4 7 2 7 4 7

Adding and subtracting fractions… 6 7 6 7 − 2 7 = 4 7 2 7 4 7

What if they don’t have the same denominator?

1 4 + 3 8 = 2 8 + 3 8 = 5 8

7 9 − 2 3 = 2 8 + 3 8 = 5 8

7 9 − 2 3 = 7 9 − 6 9 = 1 9

7 9 − 2 3 = 7 9 − 6 9 = 1 9

Key facts. When fractions have the same denominator, we can add the numerators. If the fractions have different denominators, we first make the denominators the same (using equivalent fractions)

Converting improper fractions to mixed numbers 3 1 2 5 3 4 1 5 9 17 2 5 Improper fractions Mixed numbers

3 1 3 10 3

4 1 2 1 7 8

5 2 9 4

Mark this piece of work. For every question that’s wrong, how would you aid a student to correct it? How would you use a diagram in your explanation. Talk it through with your partner.

Key facts. We use improper fractions or mixed numbers to represent numbers that are bigger than 1 using fractions. Mixed number: 4 2 9 Improper fractions: 8 5 (numerator is larger than the denominator) Pictures are a great way of switching between the two and explaining the quicker algorithm for how to do it.

Multiplying fractions… 2 3 × 4 5 = 2×4 3×5 = 8 15

Multiplying fractions… 2 3 x 4 5 2 3 × 4 5 = 2×4 3×5 = 8 15

Multiplying fractions… 2 3 of 4 5 2 3 × 4 5 = 2×4 3×5 = 8 15 4 5

Multiplying fractions… 2 3 of 4 5 2 3 × 4 5 = 2×4 3×5 = 8 15 4 5 2 3

Multiplying fractions… 2 3 of 4 5 2 3 × 4 5 = 2×4 3×5 = 8 15 4 5 2 3

2 3 of 4 5 2 3 × 4 5 = 2×4 3×5 = 8 15 = 8 15 Multiplying fractions…

Use the squared paper to draw pictures to show these calculations: 1 2 × 1 2 3 4 × 2 5 Could you draw a diagram to work out the answer to: 1 2 × 2 3 × 3 4

How might we use diagrams to help students understanding of division? 1 ÷ 1 3 = ? 𝟏 𝟑 How many thirds fit into 1? 3.

Key facts. To multiply two fractions, multiply the numerators together and the denominators together. To divide fractions, flip the 2nd fraction and multiply by it. Pictures can be useful for justifying the method to students.

3 5 of 15 Share the blocks out into 5 equal piles I want 3 of these piles.

2 7 of 14 Share the blocks out into 7 equal piles I want 2 of these piles.

Use this to decide… would you rather have? a) 2 3 of 24 bottles of wine b) 5 6 of 24 bottles of wine c) 5 8 of 24 bottles of wine b) 3 4 of 24 bottles of wine

Stuck on blocks? Make it difficult to use them! What is 2 3 of 501?

Time for a break? What fraction of the area of the square is in each of the triangles in this picture? Finished? How is this shape made using the first shape? What shape has been created in the middle? What fraction of the total area of the large square does this shape take up?

Fractions of amounts. A case study in how to scaffold and fade away.

Scaffolding, also called scaffold or staging, is a temporary structure used to support a work crew and materials to aid in the construction, maintenance and repair of buildings, bridges and all other man made structures. Scaffolds are widely used on site to get access to heights and areas that would be otherwise hard to get to.

Scaffolding, also called scaffold or staging, is a temporary structure used to support a work crew and materials to aid in the construction, maintenance and repair of buildings, bridges and all other man made structures. Scaffolds are widely used on site to get access to heights and areas that would be otherwise hard to get to.

Scaffolding, also called scaffold or staging, is a temporary structure used to support students in mathematics to aid in the construction, maintenance and repair of their mathematical understanding. Scaffolds are widely used in mathematics teaching to get access to heights and areas that would be otherwise hard to get to.

“Unsafe scaffolding has the potential to result in death or serious injury.”

Rank these different ways of teaching fractions of amounts from the most scaffolded to the least scaffolded.

Counting out into piles. Counting up how many blocks Using blocks. Counting out into piles. Counting up how many blocks Doing the calculation without need of manipulatives or picture Drawing a bar to represent what needs to be done “How many boxes do we need to draw How do you know?” “What goes in each box? How do you know” (relate back to cubes and language of sharing) “How many of these do we want? How do you know?” “How many piles are we sharing out into? How do you know?” “How many of these do we want? How do you know?” Where do we start? What question am I going to ask you next? Show me how to do this one. Where do we start? What question am I going to ask you next? Show me how to do this one. Where do we start? What question am I going to ask you next? Show me how to do this one. “We’re sharing out 12 into 3 groups. What calculation do I do? I want 2 of those groups. What calculation do I do?

The art of fading. Direct instruction. Model it for them. Do this. Do this. Do this. Questions and prompts “How many groups?” “How you know?” “How many of these do you want?” Move towards questions like “What am I going to ask you next?” “Talk me through this one from start to finish.” “Convince me you can do it.” Blocks – Pictures – Without (?) They won’t be able to use blocks in an exam. Make sure they don’t become dependent on them.

Multiple representations Converting between fractions, decimals and percentages

Take it in turns to match pairs of cards and place them on the table. As you do this, you must explain to your partner how you know the cards make a pair. When you have given your explanation, your partner should either challenge what you’ve said or say why they agree.

To share with teaching staff from your schools? 101 Red Hot Maths Starters pg 37 Primary UKMT Team Maths challenge Thinkers – ATM publication Fraction fascination – Nrich Standards unit materials – Ordering fractions and decimals

If we were to run another session… What would go on your wishlist? What would you like to spend more time looking at? Is there another topic (within fractions, decimals and percentages) that you would like some input on?

Fun with fractions (and decimals and percentages) Dan Kirby daniel.kirby@sherburnhigh.co.uk