MAE3270 Module 3 - Lecture 3: Fractions and division

Chapter 1: Introduction

Year 6 students learn to solve problems involving addition and subtraction of fractions with the same or related denominators
Year 7 students move into multiplying and dividing fractions using written strategies and digital technologies
When adding and subtracting fractions, ensure the denominators are the same
- Example: 1/4 + 1/4 = 2/4
- Denominator remains the same, numerators are added
When the denominators are different, convert them to the same denominator
- Example: 1/3 + 1/6 = 2/6 = 1/2
- Convert 1/3 to an equivalent fraction with a denominator of 6 by multiplying numerator and denominator by 2
- Add the fractions with the same denominator
- Simplify the result if possible

When multiplying fractions, think about taking a fraction of a fraction
Example: 1/2 * 1/4 = 1/8
Visual representation: Divide a whole into equal sections
- Divide the whole into 4 equal sections
- Take half of one of the sections
- Resulting fraction is 1/8
Multiplying fractions without ones in the numerator
- Example: 2/3 * 4/5
- Visual representation: Divide a whole into equal sections
  - Divide the whole into 5 equal sections
  - Take 4 of the sections
  - Divide each section into 3 equal sections
  - Take 2 of the sections
  - Resulting fraction is 2/3 of 4/5

This represents 2 thirds of 4 fifths.
- There are 15 total equal sections.
- Each section is divided into 3 equal sections.
- There are 8 colored sections out of the 15.
Multiplying fractions
- Multiply the numerators and denominators.
- Example: 1/2 multiplied by 2/5 equals 2/10.
- Simplify the fraction by finding the highest common factor.
- Example: 2/10 simplifies to 1/5.
Division of fractions
- Example: 8/3 divided by 1/3.
- Represented on a number line.
- 8/3 divided by 1/3 equals 8.
- Each jump represents 1/3, so 8 jumps are needed.
- Dividing into thirds means 3 jumps for each whole.

Dividing by 1/3 is the same as multiplying by 3
- 8/3 divided by 1/3 is equal to 8/3 times 3
- 8/3 times 3 = 24/3 = 8
Dividing by 2/3 is the same as multiplying by the reciprocal (3/2)
- 8/3 divided by 2/3 is equal to 8/3 times 3/2
- 8/3 times 3/2 = 24/6 = 4
Explanation of why multiplying by the reciprocal makes sense
- Dividing by a fraction means jumping a certain distance
- When dividing by a fraction, multiply by the reciprocal
- When the numerator is greater than one, each jump is twice as far
- Dividing by a fraction means taking half the number of jumps