Unit 5

Fractions as Numbers on the Number Line

 

Grade 3

Math

Unit Length and Description:

 

22 days

 

In Unit 5 students transition from thinking of fractions as area or parts of a figure to points on a number line. In order to support understanding of this concept, students think of fractions as being constructed out of unit fractions: “1 fourth” is the length of a segment on the number line such that the length of four concatenated fourth segments on the line equals 1 (the whole). Once the unit “1 fourth” has been established, counting them is as easy as counting whole numbers: 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, etc. Students compare fractions, find equivalent fractions in special cases, and solve problems that involve fractions.

 

Standards:

 

Major Cluster: NF: Number and Operations – Fractions

Develop understanding of fractions as numbers. (Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.)

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.

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.

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. (Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.)

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.

c.    Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form of 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

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.

Supporting Cluster: G: Geometry

 

3.G.2

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part is 1/4 of the area of the shape.

Standards for Mathematical Practice: Should be evident in every lesson.

MP.2 Reason abstractly and quantitatively. Students represent fractions concretely, pictorially, and abstractly, as well as move between representations. Students also represent word problems involving fractions pictorially, and then express the answer in the context of the problem.

MP.3 Construct viable arguments and critique the reasoning of others. Students reason about the area of a shaded region to determine what fraction of the whole it represents.

MP.6 Attend to precision. Students specify the whole amount when referring to a unit fraction and explain what is meant by equal parts in their own words.

MP.7 Look for and make use of structure. Students understand and use the unit fraction as the basic building block or structure of all fractions on the number line.

Instructional Outcomes

 

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 b y a parts of size 1/b.

  • I can define a unit fraction.
  • I can recognize a unit fraction as part of a whole.

I can identify and explain the parts of a written fraction. I can compare fractions using equal to, less than, and greater than one.

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.

·         I can define the interval from 0 to 1 on a number line as the whole.

·         I can divide a whole on a number line into equal parts.

·         I can recognize that the equal parts between 0 and 1 stand for a fraction.

·         I can represent each equal part on a number line with a fraction.

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.

·         I can define the interval from 0 to 1 on a number line as the whole.

·         I can divide a whole on a number line into equal parts.

·         I can represent each equal part on a number line with a fraction.

·         I can explain that the endpoint of each equal part represents the total number of equal parts.

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. (Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.)

·         I can describe equivalent fractions.

·         I can recognize simple equivalent fractions.

 

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.

·         I can compare fractions by their size to determine equivalence.

·         I can use number lines, size, visual fraction models, etc. to find equivalent fractions.

c.    Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form of 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.

·         I can recognize whole numbers written in fractional parts on a number line.

·         I can recognize the difference in a whole number and a fraction.

·         I can express whole numbers as fractions.

·         I can explain how a fraction is equivalent to a whole number.

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.

·         I can explain what a numerator means.

·         I can explain what denominator means.

·         I can recognize whether fractions refer to the same whole.

·         I can decide if comparison of fractions can be made (if they refer to the same whole).

·         I can explain why fractions are equivalent.

·         I can compare two fractions with the same numerator by reasoning about their size.

·         I can compare two fractions with the same denominator by reasoning about their size.

·         I can record the results of comparisons using symbols >, =, or <.

 

3.G.2: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part is 1/4 of the area of the shape.

  • I can divide shapes into equal parts.
  • I can describe the area of each part as a fractional part of the whole.
  • I can divide a shape into parts with equal areas and describe the area of each part as a unit fraction of the whole.

 

Enduring Understandings:

 

·         Fractional parts are equal shares of a whole or a whole set.

·         The fraction name (half, third, etc) indicates the number of equal parts in the whole.

·         The more equal sized pieces that form a whole, the smaller the pieces of the whole become.

·         Fractions can be represented on a number line.

·         Fractions can be compared by drawing a model or representation on a number line.

·         When the numerator and denominator are the same number, the fraction equal one whole.

·         Whole numbers can be renamed as fractions.

 

Essential Questions:

 

·         What is a fraction?

·         How do I represent a fraction on a number line?

·         When we compare two fractions, how do we know which has a greater value?

·         How can I represent fractions of different sizes?

·         How can I show that one fraction is greater (or less) than another?

·         How does the numerator impact the denominator on the number line?

·         How are fractions used in problem-solving situations?