Fractions Concepts: Coordinating Fractional Units

Overview

Learning Goal

Develop increased sophistication in coordinating and reasoning with a range of fraction units.

Learning Goal Description

The development of knowledge of fractions as quantities opens the door to engaging in a wide range of tasks that involve coordinating and reasoning with fraction units. The simultaneous coordination of the actions of partitioning and iterating supports reasoning flexibly among unit fractions, proper fractions, the whole, and fractions beyond one. The learning progression includes a range of opportunities to engage in creating, comparing, distributing, and transforming fractional units. This approach to reasoning about fractional quantities creates a strong conceptual foundation for more formal approaches to operations with fractions.

Pre-Requisite Knowledge

Students should have facility with the actions of partitioning and iterating fractions. Students should have knowledge of fractions as measures and the notion of unit fractions as the building blocks for other fractions.  (see Ready Set Math module Fraction Concepts: From Part-whole to Measurement)   

Sub-goal A

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Sub-goal B : Reason with fractional units to compare non-unit fractions up to 1

See Instructional Resources
A conceptual approach to comparing fractions involves reasoning about the number of units and the size of the units that make up the fractions. This approach does not rely on finding and using equivalent fractions. If unit fractions are seen as the building block for other fractions, size comparisons can be based on the size of the measure (i.e., unit fraction) and the number of measures. The instructional resources provide opportunities for students to engage in reasoning to compare fractions, order fractions, and place fractions on a number line.
  1. Ask students to compare two written fractions. Note: At this time, fractions are selected strategically so the relative size can be determined by considering the size of the fractional units and number of measures. (e.g., Which is bigger?... 3/7 or 5/7 [same unit of measure]; 2/5 or 2/7 [same number of measures]; 6/7 and 8/9 [one unit of measure from 1]; 1/9 and 5/6 [close to 0 and close to 1]). Intermittently ask questions such as: How did you know? How did you work it out?

If students are not successful, check first for knowledge relating to fractions as measures (see Ready Set Math module Fraction Concepts: From Part-Whole to Measurement). If a measurement concept is in place but students were not able to use that knowledge to compare the fractions, the resources for this sub-goal can support designing instruction to compare fractions by reasoning about fractional units. 

Sub-goal C

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Sub-goal D

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Sub-goal E

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Math Recovery Orange Book

Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing Fraction Knowledge. London: Sage Publications, Ltd. (Chapters 7 - 11)

The knowledge developed in this module supports engaging in operations with fractions in more formal forms. 

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