Learning from each other; locally, nationally & internationally Helping teachers to develop as reflective practitioners

Using the digits 3,4,5,& 6 once only find the smallest answer?

Planning to prevent misconceptions Session 3 “Roadmap to Mastery” Programme

Potential misconceptions Fractions are read as pieces rather than equal part / whole relationships. The fraction of the shape that is shaded is 3 / 5

Abigail Bobby Charlie

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What language do we want our pupils to use when exploring fractions? When might they experience fractions in real life? Design fraction stories that illustrate and develop an idea OR what real life contexts can be used to explore the depth of their understanding?

Each piece of the pie is one third. Misconception Fractional pieces have to be congruent (the same shape) to be the same fraction.

Can you draw shapes that can be used to represent quarters that are NOT divided into congruent pieces?

Can we build in variation to the ways we introduce recognising fractions? Can we also use different representations of lines, groups of objects……?

Fractions as numbers

What does the research say? …there is evidence that pupils are commonly introduced to fractions in the context of part-whole relations but are not exposed as widely to fractions in other forms. Although part-whole and quotient situations are important, so too is an understanding that fractions are numbers with magnitude in their own right and that they hold positions on a numberline. ….the numberline is a useful way of teaching ‘fraction density’ i.e. understanding that there are infinite fractions between any two points on a numberline

Misconception The number 1 / 2 lies half-way between any two labelled integers on a number line

Appreciating the size Estimation of how full?

And another, and another… Give an example of a fraction that is less than a half. (Y3) Give an example of a fraction that is more than one half but less than a whole. (Y4) Give an example of a fraction that is more than three quarters. (Y5) Give an example of a fraction that is greater than 1.1 and less than 1.5 (Y6)

Misconception The number 1 / 5 is smaller than 1 / 6

Would you rather have ½ of the bun or ¼ of the cake ?

Which is the bigger whole?

Which is the bigger whole (ii)

Misconception: Fractions have to be smaller than 1 whole. When counting in fractional steps, provide visual representations to represent the count. Then introduce the symbolic representation alongside. What ‘variations’ could you provide?

Video Counting in Fractional Steps

1515

2525

3535

4545

5555

1

1 5 1

2 5 1

Small step or conceptual leap from counting to addition & subtraction?

One whole

Subtraction fallacy You have four £1 coins and a 10 p piece. You owe someone £2.30

11 6 / 17 – 5 15 / and 6 / and 23 / 17

= Equivalent Fractions … == ===

Misconception: Cancelling is division 12 3 x x 5 5 ==

1 / / 2

How many ways can you share 3 pizzas between 4 people? Making links

Fraction notation can be used… to represent a proportion of a whole or of a unit; to represent the point on a line; to represent a proportion of a set; to model a division problem;

Planning activity

Working towards Teaching for Mastery Clear learning objectives for all concepts, facts, procedures – small steps Learning activities include CPA, variation, deeper learning, intelligent practice To develop Fluency, understanding and reasoning

Previously

Strands

Size of a fraction Simple addition Equivalence Complex addition Basic concept Fraction of a quantity Fraction as a number Percentages, decimals, multiplication etc

Consider the links Recognising Fractions & Fractions as Numbers Addition & Subtractions of Fractions Equivalence of Fractions Decimals & Percentages Fractions as a Proportion of a Quantity

What do pupils need to answer? Look at the A3 sheet Addition & Subtraction of Fractions Look at the A4 sheet Exemplar Questions and the Big Ideas. NB the 3 exemplar questions have been transferred for you. Consider the development of the students in KS 1 & 2. From your own experience, add further key questions that pupils will encounter as they progress in this topic area.

What are the concepts? From your own experience, what are the concepts and the key understanding that you need to focus on? Look at the Big Ideas sheet to see if this helps. Write down your ideas in the column; Progression Which of the key questions would suitably assess their understanding? Do we need to ask additional questions? DOSHOW EXPLAIN

Repeat for the other strands In groups, each consider a different aspect of fractions. Consider the development of questions and add further key questions that show progression in this strand of fractions. Consider the concepts that need to be developed in order to make progress in this strand. Consider how you can assess a pupils understanding beyond just being able to “Do” an arithmetic procedure. For later – what potential variation would help pupils? - what potential aspects of “going deeper”?

Common agreement vs Calculation policy Consider the ideas that have been presented to you. You may also wish to look back at the “large sheets” representing the different aspects of fractions. What learning activities and models are you going to use across the whole school to develop the pupils understanding? Focus initially on the skills & understanding at the level of pupils you teach

Lesson planning & collaboration Developing ideas together (pairs?) Reflecting on what you learn from trials Learning from others’ experiences Adapting resources & strategies for your own classroom and pupils.