# Number 1 (N1)

# Fair Sharing

**Which group of teaching assistants get the most to eat?**

In **Fair Sharing (N1)**, students grapple with problems designed to challenge their ideas about **what constitutes the whole**. The initial context of subway sandwiches is deliberately chosen for its bar like potential. When drawing the sandwich, learners inevitably begin to “**mathematise**” the situation. Drawing sandwiches with squared off corners establishes an **emergent **** bar model** of the context, which can gradually be used for other contexts.

Students return again and again to other rectangular food contexts—sectioning up juice bottles, cutting up cheese, and salami sausage. These provide further bar-like representations and help develop associated fraction meanings. The contexts build connections between sharing and fractions, so that learners find ways to think **a fraction of an amount** in sustained relation with** the whole**. In the final section of **Fair Sharing (N1)**, the context of surveys—represented on a **segmented bar**—subtly leads learners to develop their own processes for **adding fractions**. This will be a revelation to students and their teachers alike.

# Overview of Fair Sharing

#### Section A: Sharing the Whole

**Lessons 1 and 2: Subway Sandwiches**

Students examine a context involving groups of people and sandwiches. They are asked to develop strategies for sharing the sandwiches equally. The lessons create a need for students to compare and combine (add together) different pieces of sandwich. In order to decide whether different “sharings” of sandwiches are equivalent, students are forced to name parts of a whole in relation to the whole. These lessons encourage students to draw pictures and, in particular, introduce the use of the bar model for thinking about fractions.

**Lesson 3: Fruit Winders**

Students cut and fold strips of paper into equal parts to model how Fruit Winders can be shared equally. In doing so, they begin to compare simple fractions using a physical bar model and explore the relationship between fractions with unlike denominators.

**Lesson 4: Fruit Juice**

Students compare partially filled fruit juice bottles, transitioning from fractions as portions (as in a fourth of a sandwich or fruit winder) to fractions indicating position (the endpoint of a height/amount on a bottle), moving towards fractions (as points) on a number line. Students informally compare non-unit fractions.

#### Section B: Fractions of an amount

**Lesson 5: The Fun Run**

Thinking through the placement and use of the water-stand on race courses, students continue to develop their understanding of fractions and the importance of the whole. Students are encouraged to use bar models to calculate fractions of an amount within the context of a fun run.

**Lesson 6: Cheese and Sausages**

The relationship between fractions and division is further explored within the context of cutting foods. The context of dividing sausages and cheese gives students the opportunity to visualize concrete examples of part-whole relationships. Students work out a fraction of an amount in both contextual and non-contextual problems.

#### Section C: Seeing fractions differently

**Lesson 7: School Fun Day**

Through the context of a school survey, this lesson develops connections between segmented bars, pie charts and fractions. As a result, students further develop ways of using a bar model to operate with fractions.

**Lesson 8: Games**

In the context of a new survey, to compare differently sized data sets, students make connections between a segmented bar representation of data and fraction descriptors. Students develop the idea of using common multiples for comparing differently sized data sets.

**Lesson 9: Adding Fractions**

Students add fractions using the segmented bar and move towards an understanding of common denominators. Students compare the size of fractions using a range of strategies.

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