PRINT VERSION - Modeling and Explaining Fractions

This toolkit will support students who struggle with understanding what unit and non-unit fractions are and what the numbers in a fraction represent. Students will engage with visual representations such as paper folding and tape diagrams and make connections to the fraction form.

Goals and Standards

Colorado Academic Standards:

• Describe 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: 3.NF.A.1)

Mathematical Practice Standards:

• Make sense of problems and persevere in solving them (MP1)
• Model with mathematics (MP4)
• Attend to precision (MP6)

Learning Goal:

• Students will understand and precisely model and describe the meanings of the numbers in a fraction.

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Progression of Skills (outline)

Before the lesson:

• Students should have received some instruction on what fractions represent. This toolkit supports students who struggle to understand what the numbers mean and/or how to model both unit and non-unit fractions. If students can complete the Equal Parts Task with little or no guidance, they are ready for this intervention.

During this toolkit lesson:

• Students will learn to name fractional parts, partition a whole into unit fractions and iterate unit fractions to create non-unit fractions. Precise language for naming fractions will be explicitly taught throughout the intervention.

After the lesson:

• Students will be ready to learn how to place fractions on number lines, compare unit fractions and model improper fractions.

Session 1: Partitioning and Unit Naming (30 mins)

The purpose of this session is to use a real-world context and partition a model to show equal-sized parts. The models will serve as a visual for understanding the names of fractional units (for example: fourths, thirds).

Sharing Task #1: Folding paper strips to represent a context

1. Read from note catcher Sharing Task #1: 
   • Suppose there was one candy bar to be shared among four children so each child gets the same amount. Show how much each child would get by folding your paper strip.
2. Allow students to share their thoughts. If students have different representations, discuss similarities and differences.
   • Possible models:
     • Bring student attention to the vertically-partitioned model (model on the right). Encourage them on future problems to split vertically only. This will support transitioning paper folding models to tape diagrams.
3. Ask: Would it be reasonable to think of each of these parts as being worth one whole?
   • Tell students we need a new way to name these parts because each child will get less than one candy bar but more than zero candy bars.
   • This is one example of a fractional unit. There are 4 parts that make one candy bar so each of those parts is called a fourth. In this case, our fractional unit is fourths.
4. Fill in the blanks on our note catcher:
   1. There are __4___ parts that make 1 candy bar, so each of those parts is called a ___fourths______.

Sharing Task #2: Folding paper strips with more difficult units

1. Read from note catcher Sharing Task #2:
   • Suppose there was one candy bar to be shared among three children so each child gets the same amount. Show how much each child would get by folding your paper strip.
   • Instructor’s Note: Allow students some time to try to create thirds through folding and have extra strips on hand for when mistakes are made. Remind students that the children are sharing equal-sized parts.
2. Allow students to share their thinking.
   • Ask students: How is this problem the same as before? How is it different?
   • Tell students this is another example of a fractional unit. There are three parts now.
   • Each child will get less than one candy bar, but more than zero candy bars. There are 3 parts that make one candy bar so each of those parts is called a third. Our fractional unit is thirds.
3. Fill in the blanks on our note catcher:
   • There are ___3___ parts that make 1 candy bar, so each of those parts is called a third.
4. Bring students’ attention to what they notice about the fractional unit and the size of the part.
   • Allow students to discuss why it is reasonable for fourths to be smaller than thirds. Prompt as needed until they conclude that splitting the whole into more parts results in a smaller size.

Sharing Task #3: Transitioning from paper strips to tape diagrams

Instructor’s Note: Students may still want to use paper strips to scaffold their partitioning. Feel free to offer this option, then transfer the model to the tape diagrams.

1. Read from note catcher Sharing Task #3:
   • James and his sister want to do some arts and crafts. He has one brick of clay to split equally between himself and his sister. Use the tape diagram below to show what the brick will look like and name the fractional unit.
   • There are __2___ parts that make 1 brick of clay, so each of those parts is called a _____half____.
2. Allow students time to discuss their ideas for partitioning the tape diagram. Complete the task together.
3. Ask students: What connections can you make between the tape diagram and the paper strip? When creating your tape diagram, how do you know your model is showing halves?
4. Session #1 Practice:
   • As a group, complete the first tape diagram together, ensuring students understand the difference between creating the fractional unit and how many lines are needed on the model.
   • Allow students to complete the rest of the practice with limited support.

Complete the Progress Monitoring chart for each student for Session #1.

Session 2: Writing and Naming Fractions

The purpose of this session allows students to make connections between the visual models in Session 1 to the numerical/symbolic form of a unit fraction. Students will learn precise language for what the numbers in a fraction mean, which is an important component of understanding fractions.

Naming a Fraction Task #1: Understanding what the numbers mean

1. Read from note catcher Naming a Fraction Task #1:
   • José has a long piece of ribbon for an art project. He wants to split the ribbon equally to make six (6) projects. What fraction of the ribbon will each art project need?
   • Allow students to use either a paper strip or their note catcher tape diagram to demonstrate their thinking and fill in the blanks:
   • There are ___6__ parts that make the whole ribbon, so each of those parts is called a __sixth_______.
2. Bring student attention back to the question in the original problem, “What fraction of the ribbon will each project need?” Let students know this is NOT asking for the fractional unit. While we may know the ribbon is partitioned into the fractional unit of sixths, the question is asking us to write our answer as a fraction.
3. A fraction is a special type of number that describes an amount between the values of whole numbers.
   • Show students the fraction 1/6 and read it aloud for students as “one sixth”.
4. Continue referring to the numerical form 1/6. Explain that this is how we write a fraction and each part of the fraction has a different meaning:
   • The value on the top tells us how many parts are being counted
   • The value on the bottom tells us how many parts to make 1 whole
5. Model for students using the context of the ribbon to explain how 1/6 answers the question from the problem:
   • “We are counting one part of the ribbon to be used for a project. There are six parts needed to make 1 whole ribbon.”
   • Speaking aloud, have students practice explaining the meaning of the numbers in the fraction 1/6 to connect to the context of the ribbon.

Session #2: Practice

1. Look at the Session #2 Practice on the note catcher. Support students with not only writing the correct fraction, but also continuously referencing what the numbers mean:
   • The value on the top tells us how many parts are being counted
   • The value on the bottom tells us how many parts to make 1 whole
   • Support students as needed with completing the note catcher. The meaning of the numbers should be revisited often.
2. Discussion: Have students begin to reason about the size of different fractional units.
   • Encourage students to use the meaning of the denominator (a term they will learn in the next session) to justify the size of the part. Students should understand that if there are more parts to make 1 whole, those parts will be smaller in size.
   • Complete the Progress Monitoring chart for each student for Session #2.

Session 3: Iterating Fractional Parts to Create Non-unit Fractions

The purpose of this session is to help students understand fraction composition through iterating parts. Allowing students to build one whole will support their learning of non-unit fractions in this session.

Composing Task #1: Iterate a unit fraction to build 1 whole

1. Give each student a 1-inch cubic block or Unifix cube and have them trace the block on their note catcher. Let students know this block is going to represent one fifth. Label 1/5 underneath the right side of the block.
2. As a group, allow students to discuss the following questions:
   • “What do we already know about one fifth?”
   • Instructor’s note: Review what the numbers of a fraction mean as needed:
   • The value on the top tells us how many parts are being counted
   • The value on the bottom tells us how many parts to make 1 whole
   • How many one fifths would we need to create one whole? There are five fifths to make 1 whole.
   • How could we use this 1/5 block to build one whole? We could make copies or repeat the 1 fifth block end to end until we have 5 fifths.
3. Support students as they trace/draw to build one whole on their note catcher.

Composing Task #2: Iterate to build non-unit fractions

1. Revisit the model in Task #1. Tell students each one of these parts is 1/5 which is an accurate way to talk about each individual piece of our model.
   • Bring their focus back to the next part of the note catcher:
   • How might we name the fraction if we were counting two of these parts on our model? 2/5
   • How might we shade our model to show we are counting two of these parts? We can shade two parts.
2. Ensure students understand the structure of the fraction: 2/5 is formed by using two parts that are each one-fifth in size, or 2/5 is the same as two one-fifth units.
   • 2/5 is formed by counting __2__ parts that are each one _one-fifth_ in size.
3. Address misconceptions and ensure students understand how we can build 2/5:
   • If we wanted to show 2/5, could I simply draw two squares? Why or why not?
   • Use this as an opportunity to check in if students understand the relationship between the denominator and one whole.
4. We now know that we can count more than one fractional part of a whole. If we are only counting one part, we call this a unit fraction. If we are counting more than one part, we call this a non-unit fraction.
5. Beneath the tape diagram created in Iterating Task #1, label the partitioned units as 0/5, 1/5, 2/5 and so on, focusing on the concept that we are counting more one fifth units to build the non-unit fractions.
6. Session #3 Practice A:
   • Students should move to their note catcher, practicing partitioning and iterating to create non-unit fractions.

Composing Task #3: Applying fraction vocabulary

1. We now know what the numbers in a fraction mean and how to draw a model of fractions. Both of these are very important skills when it comes to understanding fractions.
   • With this understanding, we will now use the precise mathematical vocabulary for referring to the top and bottom numbers of a fraction:
   • Numerator- how many parts are being counted
   • Denominator- how many parts to make 1 whole
   • Bring student attention to their note catcher and apply this language to the given fraction of 2/3.
2. Session #3 Practice B:
   • Students should move to their note catcher, naming fractions and identifying numerators and denominators.

Complete the Progress Monitoring chart for each student for Session #3.

• Halves, thirds, and sixths
• Naming the Whole for a Fraction
• Understanding Fractions in a Recipe Task
• Students will view a recipe for banana bread from My Favorite Banana Bread Recipe - Sally's Baking Addiction. Ask students to create tape diagrams to demonstrate each ingredient and its quantity needed.

• Teaching Fractions for Understanding: Addressing Interrelated Concepts- Getenet & Callingham
• Van de Walle, J. A., Karp, K. S., Bay-Williams, J. M., & Wray, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Pearson.
• Eureka Math Grade 3 Module 5
• Creating, Naming, and Justifying - FRACTIONS
• Watanabe, T. (2002). Representations in Teaching and Learning Fractions. Teaching Children Mathematics, vol. 8: issue 8, pg. 457-463.
• Understanding Fractions Begins with Literacy-Bingham & Rodriguez
• Monson, D. et al. (2020). Using Models to Build Fraction Understanding. Mathematics Teacher: Learning and Teaching PK-12, vol. 113: issue 2, pg. 117-123.
• Empson, S. B. et al. (2020). Unit Fractions as Superheroes for Instruction. Mathematics Teacher: Learning and Teaching PK-12, vol. 113: issue 4, pg. 278-286.