NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

In grades 3-5

• All students should develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on a number lines, and as divisions of whole numbers.
• Students should use models, benchmarks, and equivalent forms to judge the size of fractions.

In grades 6-8

• All students should compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line.

Compute fluently and make reasonable estimates.

In grades 3-5

• Develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience.

In grades 6-8

• Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods;
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

NCTM Focal Points

Grade 3

• Understand that the size of a fractional part is relative to the size of the whole, and they use fractions to represent numbers that are equal to, less than, or greater than 1
• Understand and use models, including the number line, to identify equivalent fractions.

Grade 4

• Relate their understanding of fractions to reading and writing decimals that are greater than or less than 1, identifying equivalent decimals, comparing and ordering decimals.

Common Core Standards

Grade 3 - Number and Operations - Fractions:
Develop an understanding of fractions as numbers. 

2. Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.

Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

3. Understand a fraction as a number on the number line; represent fractions on a number line diagram.

Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.

Grade 4 – Number and Operations - Fractions:

Extend understanding of fraction equivalence and ordering.

1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Vocabulary

fraction, whole, part, denominator, numerator, models, set, collection

IEP Goals (sample)

• Given a number line and 3 -5 pictorial representations of fractions with common denominators the student will place each in the correct location on the number line with no more than 2 prompts for 5 out of 7 trials.
• Given a number line and 3-5 numerical representations of improper fractions and teacher modeling, the student will drag each fraction to the correct location and explain his reasoning with 100% accuracy for 3 consecutive sessions.
• Given controlled practice and feedback using benchmarking, the student will correctly place fractions on a numberline with 90% accuracy for 4 out of 5 sessions.