Year Five - What does it mean to Divide Fractions?
Home -What does it mean to divide a fraction Map
About this Big Idea
In this Big Idea students explore what it means to divide a fraction using concrete, real world examples. While students often learn the "invert and multiply" strategy they may not understand what it actually means to divide a fraction or are unable to describe a situation where this is required other than in a mathematics lesson.
Understanding Goals:
Students will understand:
- what it means to divide a fraction.
- the effect that division has on the size of a fraction.
Background:
In Stage 3, the study of fractions is limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12 and 100 and calculations involve related denominators only
Core Content from the Syllabus:
Working Mathematically
Fraction, Decimals & Percentages
- Find a simple fraction of a quantity where the result is a whole number
- calculate a simple fraction of a collection/quantity
Language:
- In questions that require calculating a fraction or percentage of a quantity, some students may benefit from first writing an expression using the word 'of', before replacing it with the multiplication sign (×).
Connected to:
Mindset Mathematics Learning Activities
Visualise
We begin to explore fraction division by making cards and cutting paper so that students can develop intuition about what division with fractions means. - See page 201
Questions for reflection:
- What strategies did you use to solve this problem?
- How did the number of cards you could make change when you changed the fraction of paper needed to make a card? Why did it change?
- What is left over? Did anyone have leftover paper? How much? How did you know?
- Why is it that division with fractions can lead to a larger answer than you started with?
Play
Students use Cuisenaire rods to explore the patterns that emerge when dividing a fraction into equal-size pieces. - See page 209
Questions for reflection:
- What interesting solutions did you find?
- How did you figure out what each rod was worth? What strategies did you develop?
- What patterns did you notice when partitioning the rods? What patterns did you notice in your (or our) table?
- Which rods had the most solutions? Which rods had the fewest? Why do you think that is?
- Which fractions were the easiest to start with? Which were hardest? Why?
- What is happening when we take a fraction and divide it by a whole number? Draw a picture or use an example to help you describe what is happening.
As understanding builds, students may use the Cuisenaire Rods App on their iPad as another way to model their thinking.
Investigate
Students divide whole numbers by unit fractions to investigate when the quotients are even and when they are odd. They develop visual proofs for dividing these numbers and conjectures for the patterns they notice. - See page 217
Questions for reflection:
- What patterns did you discover dividing whole numbers by unit fractions? What evidence do you have for these patterns? Why do you think these patterns exist?
- Why are the quotients sometimes even and sometimes odd? How can we predict what the result will be?
- What patterns did you see on your classmates' charts that you think are worth discussing, either because they are interesting or because you're not sure whether you agree? (Take some time to discuss these patterns.)
- What are you wondering now?
- What patterns would you predict if you divided a unit fraction by a whole number? Why?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012
