Grade 3 Common Core State Standards for Fractions
Develop understanding of fractions as numbers
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Note: This lesson begins with students becoming familiar with models for fractions and describing part-to-whole relationships verbally using whole numbers rather than fraction symbols. The lesson concludes with the introduction of unit fractions, that is, fractions of the form 1/b, where the whole is partitioned into b equal parts.
Note: Fractions a/b are introduced as a parts out of a whole with b equal parts. Fractions are illustrated by showing parts of region models and parts of sets (collections). Further illustrations of fractions for regions and sets with visual models can be seen at Basic Concepts Step 2 TeachingFractionBarVideos
Note: This lesson provides a gradual introduction to the names of fractions. It is suggested that names such as "three-fourths" also be referred to as "3 over 4" in the early stages to be less threatening and to reinforce the part-to-whole language of "3 out of 4." The words "numerator" and "denominator" are also introduced in this lesson, but for simplicity in the early stages are also called the "top number" and the "bottom number."
Note: In this lesson students solve word problems by shading and labeling parts of regions and identifying parts of sets (collections) to illustrate applications of fractions. Students also complete an in-class activity sheet requiring them to sketch and label parts of regions and circle parts of collections. This lesson provides a gradual non-threatening introduction to solving word problems involving fractions. The two worksheets for this lesson have further examples of word problems involving regions and sets.
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. 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.
b. 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.
Note: This lesson addresses the standards in parts a and b above. Students develop a number line for fractions using twelfths bars. Placing the bars on a number line to locate points for fractions provides a connection between the region model and the linear model for fractions, and shows a way to make sense of labeling the points on a number line with fractions. For example, the parts of a bar match the parts of the interval from 0 to 1 and show that the end point of the part based at 0 locates the fraction 1/12 on the number line. Similarly, a bar representing a/b matches the point on the number line for the fraction a/b. In this lesson the number line is also extended to the interval from 1 to 2, providing a natural way to introduce mixed numbers. A similar demonstration can be seen at Basic Concepts Step 3 TeachingFractionBarVideos
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram
Note: The standard from part a above is addressed in this lesson by using visual demonstrations of region models with the same whole and same shaded amount to illustrate equality of fractions. Students form a number line for sixths as an example that equal fractions have the same point on a number line. Part c of this standard is addressed using both the region and number line models. These models show that 12/12 and 6/6 both correspond to the same part of a whole and to the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Note: In this lesson, regions having the same whole and same amount of shading provide a visual model for equality of fractions. By comparing the resulting equalities and looking for patterns, rules for generating equal fractions and writing fractions in simplified form can be formulated. These demonstrations provide a visual illustration of the rule a/b = (n × a)/(n × b). This rule can also be seen at Equality Step 1 TeachingFractionBarVideos
Note: In this lesson conditions are posed for students to sketch and label figures and create word problems. Students also complete an in-class activity by dividing regions into equal parts and creating word problems for conditions given for the regions. The two worksheets for this lesson have further examples of word problems for equality of fractions.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Note: Visual models illustrate inequality of fractions in this lesson. If one bar has a greater amount of shading than another bar, the fraction for the first bar is greater than the fraction for the second bar. Examples of bars having the same number of equal parts and different amounts of shading provide evidence to help students formulate a rule for inequality of fractions having the same denominator. The model is also used to address student misconceptions that larger numbers in the numerator and denominator of a fraction means larger fractions. For example, one such visual shows that 3/12 is less than 1/3.
Note: Various bars are compared to bars that are half shaded. This visual model leads to the conclusion that when a bar is less than half shaded, its numerator is less than half of its denominator; and when a bar is half shaded, its numerator is half its denominator; and when a bar is more than half shaded, its numerator is more than half its denominator. Also see Inequality Step 1 TeachingFractionBarVideos
Note: In this lesson, bars with decreasing shaded amounts lead to two important observations: (1) The fractions for the bars get closer and closer to 0; and (2) The more equal parts something is divided into, the smaller the fraction for one of these parts.
Note: In this lesson, students create word problems and shade diagrams for given information. Further word problems can be seen at Inequality Steps 1 & 2 TeachingFractionBarVideos
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