Year Three - Understanding ¹⁄₂

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About this Big Idea

In this Big Idea students develop a deep understanding of what fraction is and why a fraction is best seen as a relationship between the numerator and denominator that creates one number.

"I have found in my teaching and work with students that the most important idea for students when learning fractions is the idea of a relationship. I teach students that what is special about a fraction is that the numerator relates to the denominator and that we do not know anything about the fraction without knowing what that relationship is." (Boaler, Munson & Williams. 2018 p149)

Understanding Goals:

Students will understand:

• that a fraction is a relationship between the numerator and denominator.
• that a fraction is one number, not two.
• visual representations of fractions.

Background:

In Stage 2 Fractions and Decimals 1, fractions with denominators of 2, 3, 4, 5 and 8 are studied. Denominators of 6, 10 and 100 are introduced in Stage 2 Fractions and Decimals 2. Fractions are used in different ways: to describe equal parts of a whole; to describe equal parts of a collection of objects; to denote numbers (eg ¹⁄₂ is midway between 0 and 1 on the number line); and as operators related to division (eg dividing a number in half).

Core Content from the Syllabus:

Working Mathematically

Fractions & Decimals

• recognise that as the number of parts that a whole is divided into becomes larger, the size of each part becomes smaller (Reasoning)
• recognise that fractions are used to describe one or more parts of a whole where the parts are equal,
• interpret the denominator as the number of equal parts a whole has been divided into
• interpret the numerator as the number of equal fractional parts

Language:

• whole, part, equal parts, half, quarter, eighth, third, fifth, one-third, one-fifth, fraction, denominator, numerator, fractional part, number line

Connected to:

• Being Flexible with Number
• Seeing Fractions: The parts and the wholes
• Tiling to understand area
• Thinking in Equal Groups

Mindset Mathematics Learning Activities

Visualise

Students build a deep and flexible understanding of (Communicating, Problem Solving) ¹⁄₂ and whole by trying to spy ¹⁄₂ in a complex geometric image. - See page 151

Questions for reflection:

• How did you know you were seeing ¹⁄₂?
• What strategies did you use for constructing or testing whether a portion was ¹⁄₂?
• How can you see when something is ¹⁄₂ of a whole?

Play

Students play with finding half of numbers using the Number Visuals Sheet and looking for patterns connecting ¹⁄₂ to even and odd, doubling, and equal groups. - See page 160

Questions for reflection:

• Which of these numbers can show ¹⁄₂ (without cutting dots in half)?
• What is ¹⁄₂ of each of these numbers?
• How do we know when we have found them all?
• What patterns do you see?
• How can we predict what numbers you can find ¹⁄₂ of? What makes you say that?

Investigate

Students connect previous work with half of shapes and half of numbers to find half of the area of different sizes of squares. Students generate multiple ways to partition squares in half, justify those as half, and look for patterns. - See page 171

Questions for reflection:

• How can we cut each square in ¹⁄₂?
• How do you know that you have cut the square in half? What are your strategies for justifying that you have ¹⁄_2?
• What strategies do we share? What ways of cutting the boards in half were common?
• What were the most creative ways you saw? What did you see that surprised you?
• What is ¹⁄₂ of each square? Why does that make sense?
• What patterns did you notice in our solutions?
• What are you wondering now?
• What does ¹⁄₂ mean? Tell or show as many different ways of thinking about ¹⁄₂ as you can.

Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012

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