Year Four - Exploring Fraction Equivalence
Home - Exploring Fractions Equivalence Map
About this Big Idea
In this Big Idea students explore and build their understanding of equivalent fractions. "Equivalence is a key idea in working with fractions and one that underlies the methods of addition, ordering, and subtraction. Equivalence draws on the idea of a relationship, as it involves thinking about the numerator and denominator of a fraction together." (Boaler, Munson, Williams. p. 143).
Understanding Goals:
Students will understand:
- fraction equivalence and why this matters
- that fractions represent a relationship between numerator and denominator
- ways of representing fractions
Background:
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 is midway between 0 and 1 on the number line); and as operators related to division. Fraction equivalence is an important understanding to develop and students need to also develop effective visual and mental strategies for generating equivalent fractions.
Core Content from the Syllabus:
Working Mathematically
Fractions
- model fractions with denominators of 2, 3, 4, 5 and 8 of whole objects, shapes and collections using concrete materials and diagrams
- record equivalent fractions using diagrams and numerals
- identify and interpret the everyday use of fractions
- compare and order simple fractions with related denominators using strategies such as diagrams, the number line, or equivalent fractions
- recognise that as the number of parts that a whole is divided into becomes larger, the size of each part becomes smaller (Reasoning)
Language:
- whole, part, equivalent, equivalence, equal parts, half, quarter, eighth, third, sixth, fifth, tenth, hundredth, one-sixth, one-tenth, one- hundredth, fraction, numerator, denominator, whole number, number line, length, area
Connected to:
Mindset Mathematics Learning Activities
Visualise
Students begin to build understanding of the need for equal-size pieces— a common denominator— by exploring the colors used in a geometric painting. Students try to give fractional names to the colored regions and grapple with the idea of unequal areas. - See page 145
Questions for reflection:
- What strategies did you and your partner come up with to find the fraction each color represented?
- Did different paintings require different strategies? Or could you use the same strategy on all the paintings? Why?
- Did anyone find a different fraction name for the same region?
- Can more than one answer be correct? Why or why not?
- Which colors were the most challenging to figure out? Why? What did (or could) you do to deal with this challenge?
- How did decomposing the paintings help you find the fraction each color represented?
Art by Elsworth Kelly - Variations of Red, Yellow & Blue & Scramble by Frank Stella
Play
In this activity, students color-code different patterns with an assortment of shapes. The goal is for students to understand that fraction models are not limited to circles and squares. - See page 157
Questions for reflection:
- How did you share the brownies among eight friends? What fraction of the pan will each friend get? What different ways did you find to color-code the pan to show the equal shares?
- How did you share the brownies among six friends? What fraction of the pan will each friend get? What different ways did you find to color-code the pan to show the equal shares?
- How did you share the brownies among twelve friends? What fraction of the pan will each friend get? What different ways did you find to color-code the pan to show the equal shares?
- How do our different brownie pans help us to see equivalent fractions?
Questions to extend:
- Ask students to use grid paper (see appendix) to design their own brownie pan shape.
- How many friends will share?
- How many different ways can you color your brownie pan to show possible solutions?
- What fraction of the pan will each friend get?
- Will there be any leftovers? How will the friends handle the leftovers?
Investigate
Students investigate fraction puzzles trying to create rectangles with different proportions of colors. Their goal is to make the smallest rectangle that can satisfy the fraction parameters of the puzzles. The class works together to look for patterns that will help them find rectangles for any set of fractions— a common denominator. - See Page 167
- What patterns do you notice? (Note patterns on the chart.)
- If we wanted to predict what rectangles could be made from a set of fractions, how could we do it just by looking at the numbers?
- If we wanted to use our rectangles to decide which fraction in the set was biggest or smallest, how could we do it?
- How can you find a common denominator for any set of fractions?
Credit:
Boaler, Munson & Williams (2018) - Mindset Mathematics: Visualizing and investigating big ideas Grade 5
NESA - Mathematics K-10 - 2012
