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4.2 HTML v1

Building Blocks of a Standard

  • 4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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.
  • 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.

Building Blocks of a Standard Notes
Understand that fractions are equal if they are the same size in an area model or if fractions label the same point on a number line. Last year’s standard: 3.NF.3a

(Potential challenge) Students may incorrectly identify the unit (whole) on a number line as the entire portion of the number line in the diagram. For example, when representing ¾, students may incorrectly identify the number 3 on a number line labeled from 0 to 4. Encourage them to first use strips of paper, which can be subdivided in a similar way to the number line, to represent fractions.

Sample instructional strategy) Ask students to represent various fractions, including several that are equivalent, using area models or the number line. Compare their sizes or positions and discuss why some fractions are the same size or position on the number line (they are equivalent).

Recognize and generate simple equivalent fractions and explain why they are equivalent. For example 1/2 = 2/4 because they are the same size when represented using an area model. Last year’s standard: 3.NF.3b

Potential challenge) Students may use different sized wholes when drawing area models. Different wholes may lead to incorrect representations/comparisons of the sizes of different fractions.

Write whole numbers as fractions and recognize fractions that are equal to whole numbers. Last year’s standard: 3.NF.3c.

Note that expressing whole numbers as fractions is a special case of recognizing equivalent fractions. For example, students may notice that the point labeled 2 on the number line may also be labeled by 2/1, 4/2, and so on.

Sample instructional strategy) Ask students to express 2 as a fraction, and compare the different ways they write it (2/1, 4/2, 8/4, and so on). Then repeat the process for 1, which is a more important value for students to understand fluently.

Display a fraction as a shaded portion on an area model, or as a point on the number line, and partition each unit fraction part into smaller equivalent parts. Recognize that this does not change the shaded portion of the area model, or the position on the number line, and thus does not change the value of the fractions. (Potential challenge) Students may struggle to understand that they may further partition a representation of a fraction as long as they partition all of the parts in the representation, and as long as all of the smaller parts are equal.
Understand how multiplying the numerator and denominator of a fraction by the same number corresponds to the fraction representation on the number line or area model (partitioning each unit fraction piece into smaller pieces). Note that multiplying the numerator and denominator by the same number, n, can be represented on an area model by partitioning the whole into n times as many pieces, leading to n times as many unit fraction pieces as in the original representation of the fraction.

Sample instructional strategy) Ask students to represent a fraction using an area model or the number line. Then encourage them to use that same diagram to represent an equivalent fraction that could be created by multiplying the numerator and denominator of the original fraction by the same number. Discuss student strategies and support students to notice how the multiplier is used in each strategy and each representation.

Find equivalent fractions by multiplying the numerator and denominator of a fraction by the same number. Note that when students understand that multiplying the numerator and denominator of a fraction by the same number generates an equivalent fraction, they are laying a conceptual foundation for fraction multiplication, which they will encounter next year. In particular, generating equivalent fractions in this way is closely related to the mnemonic “multiplying by 1.” Some students (who have a firm understanding of the visual representations of equivalent fractions) may be ready to begin thinking symbolically like this.

Formative Assessment Lesson Plan:
 Equivalent Fractions Lesson, Grade 4

Common Core Standard 4.NF.1
 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x 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.
Display a fraction as a shaded portion on an area model, or as a point on the number line, and partition each unit fraction part into smaller equivalent parts. Recognize that this does not change the shaded portion of the area model, or the position on the number line, and thus does not change the value of the fraction.

Learning Goal(s) Success Criteria
  • Understand the relationship between the number of wholes and the number of parts in equivalent fractions.
  • Connect the visual/geometric logic of equivalent fractions to the symbolic process of multiplying the numerator and denominator by the same number.
  • Explain why two or more fractions are equivalent by referencing the number of parts in the fractions (numerator) and the overall number of wholes (denominators).
  • Generate equivalent fractions by multiplying the numerator and denominator of a fraction by the same number.
Misconceptions students are likely to have as they work toward the Goals:
  • Use different sized wholes when drawing models for comparing the values of fractions.
  • Write fractions as the total number of parts divided by the number of parts being examined.
  • Compare fractions by focusing on the numerator and denominator within a fraction, instead of focusing on the numerators of the fractions and the denominators of the fractions.
Strategies to share Learning Goals and Success Criteria with students
  • Learning Goals and Success Criteria will be shared after the whole group discussion of strategies to create the fraction strips.
  • Students will do self-assessment during the whole group discussion of the shaded fraction strips
  • Self-assessment against Success Criteria occurs at the end of the lesson.
How will I gather evidence of student learning – Classroom strategies to elicit evidence
Collecting Evidence:
 Start of Lesson

  • Students individually create three (identical) fraction strips for thirds by folding and then drawing along the folds
  • Partners use one of the thirds fraction strips to create a fraction strip for sixths and use another thirds fraction strip to create a fraction strip for ninths.
  • Whole group discussion of strategies for creating the fraction strips for sixths and ninths.
Collecting Evidence:
 Middle of Lesson
  • Students individually shade part of each fraction strip to show two thirds.
  • Whole group discussion of strategies, during which the three equivalent fractions are determined (2/3, 4/6, and 6/9).
  • Cooperative groups draw pictures or write expressions to show why their strategy works.
  • Cooperative groups draw pictures or write expressions to show why their strategy works.
Collecting Evidence:
 End of Lesson
  • Gallery walk for students to view the strategies.
  • Whole group discussion to reconcile the strategies.
  • Students individually use the strategy to find two fractions that are equivalent to three fifths.
  • Reflection and self-assessment of Success Criteria.
Key discussion questions I will pose during instruction
Discussion Questions:
 Start of Lesson

  • How did you fold the strip to create thirds?
  • How do you know each section of this fraction strip is one sixth (or one ninth)?
  • How did you fold the thirds fraction strip to represent sixths (or ninths)?
  • Is there another way you could have folded the fraction strip?
Discussion Questions:
 Middle of Lesson
  • Please show me on your fraction strip how you know the shaded portion is two thirds.
  • What does it mean when two fractions make the same amount?
  • How does the number of parts on the fraction strip change when you partition each third into smaller equal parts?
  • How does the number of shaded parts on the fraction strip change when you partition each third into smaller equal parts?
Discussion Questions:
 End of Lesson
  • Please show me how each strategy relates to the shaded fraction strips.
  • How are the numerators related in equivalent fractions?
  • How are the denominators related in equivalent fractions?
When will I provide descriptive feedback to students?
  • During the creation of the fraction strips.
  • As students shade their fraction strips to show two thirds.
  • During cooperative group discussions
  • Students will learn to use the following terms with increasing precision: Equivalent, numerator, denominator, value, multiple, partition.
Strategies for student self- and peer assessment
Self- and Peer Assessment:
 Start of Lesson

Students will receive feedback from their partners as they compare their fraction strips that show thirds and investigate how to create fraction strips for sixths and ninths.

Self- and Peer Assessment:
 Middle of Lesson

Students will compare their shaded fraction strips as they check their work for the group discussion.

Self- and Peer Assessment:
 End of Lesson

Students will check their calculations for fractions that are equivalent to three fifths as part of their self-reflection.

DMU Timestamp: November 28, 2015 16:47





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