Addition and Subtraction of Positive Fractions and Mixed Numbers

Strand: Number
Outcome: 5

Step 1: Identify Outcomes to Address

Guiding Questions

What do I want my students to learn?
What can my students currently understand and do?
What do I want my students to understand and be able to do, based on the Big Ideas and specific outcomes in the program of studies?

See Sequence of Outcomes from the Program of Studies

Strand: Number

Grade 6

Grade 7

Grade 8

Specific Outcomes

Demonstrate an understanding of factors and multiples by:

determining multiples and factors of numbers less than 100
identifying prime and composite numbers
solving problems using multiples and factors.

Relate improper fractions to mixed numbers and mixed numbers to improper fractions.

Specific Outcomes
5.	Demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially and symbolically (limited to positive sums and differences).

Specific Outcomes
6.	Demonstrate an understanding of multiplying and dividing positive fractions and mixed numbers, concretely, pictorially and symbolically.

Big Ideas

Operations with fractions build on operations with whole numbers. Van de Walle and Lovin (2006, p. 66) reaffirm this in the following excerpt:

The meanings of each operation on fractions are the same as the meanings for the operations on whole numbers. Operations with fractions should begin by applying these same meanings to fractional parts. For addition and subtraction, it is critical to understand that the numerator tells the number of parts and the denominator the type of part.
Estimation of fraction computations is tied almost entirely to concepts of the operations and of fractions. A computation algorithm is not required for making estimates. Estimation should be an integral part of computation development to keep students’ attention on the meanings of the operations and the expected size of the results.

Reproduced from Van de Walle, John A., LouAnn H. Lovin, Teaching Student-Centered Mathematics, Grades 5–8 (p. 66). Published by Allyn and Bacon, Boston, MA. Copyright © 2006 by Pearson Education. Reprinted by permission of the publisher.

Just as integers are an extension of the whole number system to include the opposite of every number, so also fractions are an extension of the whole number system to represent numbers between the whole numbers. Fractions include the whole numbers so the same properties apply but, in addition, fractions are closed under division.

Students need a solid conceptual foundation in fractions as a necessary prerequisite for fraction computation. They must first understand the meaning of fractions, using different models—region, set and length or measurement. To help students add and subtract fractions correctly and with understanding, teachers must help them develop understanding of the numerator and denominator, equivalence and the relation between mixed numbers and improper fractions. Teachers must also encourage students to use benchmarks and estimations (NCTM 2000, p. 218).

Students develop understanding of operations with fractions by making sense of the ideas internally. To guide this process, it is necessary to encourage flexibility in thinking and provide learning opportunities in connecting:

operations with whole numbers to operations with fractions
subtraction of fractions to addition of fractions
concrete, pictorial and symbolic representations
operations with fractions to real world problems. (Alberta Education 2004, p. 3)