Equivalent Fractions Lesson 2

  • Representing Fractions with Rods

    Lesson 2 of 4
    3rd grade

    60 minutes


    Use Cuisenaire rods to explore fractional relationships. This lays the foundation for work with challenging fraction concepts like equivalence.


    • One set of Cuisenaire rods per student
      Relationship Rods Printout
    • Lesson 2 Activity Sheet
    • Lesson 2 Activity Sheet Answer Key
    • Assessment Downloadable
    • Assessment Downloadable Answer Key
    • Introduce

      To begin the lesson, give students one set of Cuisenaire rods (either homemade or commercial) and a few minutes to explore the materials.

      Ask them "What do you notice?" Examples of things students might mention include:

      • Rods of the same color are the same size.
      • I see ten different sizes.
      • I notice that some rods can be put together to make other rods.
      • Some students will notice that the various colors can be stacked one on top of the other to create a staircase, as in the picture below. This configuration makes comparing the various fractions much easier and is illustrative of the linear model of fractions.


      Distribute the Lesson 2 Fraction Activity Sheet (download from Materials section above). Work together as a class to model the situation where the red is equal to 1. Ask students to grab a red rod and ask:

      • “Let’s say this represents 1 (red rod).  What do the white rods represent?”
      • “How do you know? How did you decide?”

      Direct students to model this relationship with the rods. (See diagram)


      • "What fraction of the red is the white?" [Write the fraction ½.]

      Using the lime green as one whole, work together with the students to find how many white rods it takes to equal one lime green rod.

      cuisinaire rods


      • What fraction of the lime green rod does the white rod represent? 
      • How do you write that number?  [⅓.]  What does the numerator (1) represent?  What does the denominator (3) represent? [The 1 in the numerator represents one of three equal sized pieces. The 3 in the denominator tells us that the whole is divided into 3 equal pieces.]

      Using the purple as one whole, ask:

      • “How many white rods does it take to equal one purple rod?” [4]
      • “What other color rod can be used to make a purple rod?” [2 reds will make a purple.]

      Teacher notes - This activity is used to help students recognize the structure of fractions and build equivalence (SMP 7). For example: When the purple rod represents one whole unit, it takes 4 white rods to equal the purple. Therefore, each white rod equals 1/4 of the purple. It takes 2 red rods to equal the purple, so each red is 1/2 of the purple. (Extension: two whites = one red or 2/4 = 1/2.)

      At this point determine which students need more guided practice and which are ready to work independently. Circulate as they work and monitor for understanding.

      Individual/small group work:

      Dark green = 1 whole (See teacher notes below.)

      Teacher notes - This activity is used to help students recognize the structure of fractions and build equivalence (SMP 7). For example: When the dark green rod represents one whole unit, it takes 6 white rods to equal the dark green. Therefore, each white rod equals 1/6 of the dark green whole. It takes 3 red rods to equal the dark green, so each red is 1/3 of the dark green whole. (Extension: Two whites = one red or 2/6 = 1/3.)

      This is a great opportunity to formatively assess students by observing and listening to students work. Use students' work to facilitate discussion in the class. Have student pair-share their thinking by presenting on one of the questions on the activity sheet.


      To summarize/synthesize what they have hopefully noticed about the structure of the fractions (SMP 7), discuss the following questions:


      "How do the unit fractions we created using the rods help us understand numbers less than one whole? Will these numbers help us solve problems in our everyday lives?" [Students might talk about sharing pizza among 4 friends. Each friend would get ¼ of the pizza.]

      "Let's think about the rods in a different way.

      • “If the white is ½, what is the whole?” [Red.]
      • “If the white is ⅓, what is the whole?” [Lime green.]
      • “If the white is ¼, what is the whole?” [Purple.]
      • “What does the 1 in the numerator stand for in ¼?” [One of the four white rods that make up the whole purple rod.]
      • “What does the 4 in the denominator stand for in this fraction?” [The whole is divided into 4 equal pieces.]


      Exit Ticket (SMP 3):

      Sarah thinks that two white rods are  ⅓  of a dark green. Judd thinks that two whites are 2/6 of the dark green. Who do you agree with and why? 

      Teacher Reflection

      • How can you determine which students understand that a fraction can be represented as part of a linear region? What activities are appropriate for students who have not yet developed this understanding?
      • How can you determine which students can identify fraction relationships using different “wholes” as a reference? What activities are appropriate for students who have not yet developed this understanding?
      • What parts of the lesson went smoothly? What parts should be modified for the future?

      Leave your thoughts in the comments below.

      Related Material

      Equivalent Fractions




      Other Lessons in This Activity

      Lesson 1 of 4

      Create and use fraction strips to discover name fractions and compare two unit fractions.

      Lesson 3 of 4

      Students generate and explore equivalent fractions using Cuisenaire rods.

      Lesson 4 of 4

      Reason and develop an understanding of how to place equivalent fractions on a number line.

  • Comments


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      Fabulous, works every time!

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  • Essential Question(s)

    • How can we use Cuisenaire rods to show fair shares of a whole? Note: 3rd graders are limited to fractions with denominators 2, 3, 4, 6, and 8.


    CCSS, Content Standards to specific grade/standard

    • CCSS.Math.Content.3.NF.A.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.
    • CCSS.Math.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

    CCSS, Standards for Mathematical Practices

    • SMP 3 Construct a viable argument and critique the reasoning of others
    • SMP 7 Look for and make use of structure.

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Implement tasks that promote reasoning and problem solving.
    • Facilitate meaningful mathematical discourse.
    • Support productive struggle in learning mathematics.