Fractions Concepts: From Part-Whole to Measurement

Overview

Learning Goal

Develop knowledge relating to two meanings for fractions, beginning with part-whole concepts and advancing to measurement concepts.

Learning Goal Description

Knowing fractions as a part of a whole focuses on the action of equal partitioning. This involves counting parts within a whole. While this is fundamental, it is important to progress to other meanings for fractions. Knowing fractions as measures focuses on the action of iterating to create fractional quantities. This supports the development of a multiplicative relationship between fractional quantities. The learning progression includes a range of opportunities to engage with these ways of knowing fractions through equal sharing, creating fractions, naming fractions, and working with unit fractions. This knowledge creates a strong foundation for coordinating and reasoning with a range of fractional units.

Pre-Requisite Knowledge

Students should have facility building and working with units involving whole numbers. The ways that students work with composite units in additive and multiplicative reasoning can influence and support the development of fraction knowledge. 
(see Ready Set Math modules Multiplication and Division: Counting-based Strategies and Addition and Subtraction: Strategies and Basic Facts to 20)    

Sub-goal A

Purchase the module to unlock this Description.
Purchase the module to unlock this Student Knowledge Check.

Sub-goal B

Purchase the module to unlock this Description.
Purchase the module to unlock this Student Knowledge Check.

Sub-goal C

Purchase the module to unlock this Description.
Purchase the module to unlock this Student Knowledge Check.

Sub-goal D : Create and use unit fractions as measures

See Instructional Resources
Fractions as measures comes from knowing fractions as quantities being produced from iterations (i.e., connected copies) of a unit fraction. A unit fraction becomes known as having a 1 to n multiplicative relationship with the whole. Unit fractions then become fractional units that are the building blocks for other fractions. This can be the basis for determining a unit fraction from a given whole, re-creating the whole from a unit fraction, and creating non-unit fractions from a unit fraction. Fractions as measures also provides the basis for fractions on a number line. The instructional resources provide opportunities for students to engage with fractions as measures.
  1. Ask students to verify the size of a given unit fraction (e.g., [Provide the student with one strip representing the whole and another strip that is 1/4 of the whole.] I think that this strip [point to the small strip] is 1/4.  How could you find out if it really is 1/4? [Note: The verifying should not include partitioning, rather, look for the student to use the unit fraction to measure for the number of iterations to produce the whole.])
  2. Ask students to use a unit fraction to re-create the whole (e.g., [Provide the student with a small paper strip to represent the unit fraction and blank paper] If this is 1/6 of the whole, draw the whole.)
  3. Ask students to produce a non-unit fraction from a given unit fraction (e.g., [Provide the student with a small paper strip to represent the unit fraction and blank paper] This strip is 1/5 of the whole. Draw what 3/5 would look like.)
  4. Ask students to mark the location of fractions on a number line from 0 to 1. (e.g., [Show students a number line with hash marks indicating 0 and 1.] Mark the number line to show 1/3. Label the mark. How did you know where to put the mark? [Continue with other fractions such as 2/5, 3/4, 1/6, 5/6, etc.])

If students are not successful, check first for knowledge relating to fractions removed from the whole in Sub-goal C. If Sub-goal C are met and the students are not successful with these tasks, the resources for this sub-goal can provide support in designing instruction for creating and using unit fractions as measures.  

Math Recovery Red Book

Wright, R. J., Ellemor-Collins, D., & Tabor, P. D. (2012). Developing number knowledge: Assessment, teaching & intervention. London: Sage Publications, Ltd. (Chapters 10 & 11)

Math Recovery Orange Book

Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing Fraction Knowledge. London: Sage Publications, Ltd. (Chapters 4 - 6)

The knowledge developed in this module can be extended to learning that involves working flexibly with a range of fractional units. 
(see Ready Set Math module Fraction Concepts: Coordinating Fractional Units) 
 
Further learning with fractions is supported by a strong foundation with multiplicative structuring and strategies.  
(see Ready Set Math module Multiplication and Division: Strategies and Basic Facts) 

© 2022 US Math Recovery Council®. All rights reserved. Permission is hereby granted for individual use only to support student learning. No other part of this product may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise without the written permission of the US Math Recovery Council®.