## Eggsactly Equivalent

3-5
1

Students use twelve eggs to identify equivalent fractions. Construction paper cutouts are used as a physical model to represent various fractions of the set of eggs, for example, 1/12, 1/6, and 1/3. Students investigate relationships among fractions that are equivalent.

Use the Eggsactly Eggs Overhead to review fractions as part of a set of 12.

For example, ask students how to show 1/2 of a dozen eggs. Accept equivalent fractions [6/12, 3/6, etc.] and all the arrangements of six eggs in a carton that holds a dozen eggs.

Give students paper cutouts that cover various parts of the egg carton, e.g., 1/12, 1/6, 1/4, and 1/3 (see the illustration below). Students need enough cutouts of each fraction to represent the whole. For example, students will need two 1/2s, six 1/6s, four 1/4s, and so forth.

Have students investigate each cutout and identify the fraction that is represented by each. Guide students to label each cutout with the appropriate reduced fraction. For example:

Prompt students to begin looking for fractions that cover the same area, i.e., equivalent fractions. For example, ask students how many 1/12 pieces are needed to cover 1/6 [2.]. Have students record 1/6 = 2/12 on notebook paper. Ask students how many 1/6 pieces are needed to cover 1/3 [2.]. Have students record 1/3 = 2/6 on notebook paper.

Have students work in pairs to continue identifying as many equivalent fractions as possible. Groups should record all equivalent fractions on notebook paper. When finished, have groups take turns reporting the equivalent fractions to the whole class. If any groups did not find the equivalent fraction being shared, they should add the new set to their list. Ensure that all of the following are identified:

• 1/6 = 2/12
• 1/4 = 3/12
• 1/3 = 2/6
• 1/3 = 4/12
• 1/2 = 2/4
• 1/2 = 3/6
• 1/2 = 6/12

Have students explore relationships between the equivalent fractions. For example, students might notice that dividing the numerator and denominator by the same number results in an equivalent fraction. Along the same lines, multiplying the numerator and denominator by the same number also results in an equivalent fraction.

It is important to note one common misconception at this stage. Students assume that multiplying a fraction by 2, for example, will generate an equivalent fraction. That is not the case. Multiplying a fraction by 2/2, for example, will generate an equivalent fraction, because 2/2 is the same as one whole. Be sure to note any students who confuse these concepts so you can address their misconceptions.

• An egg carton designed to hold 12 plastic eggs (or 12 markers) for each pair of students
• Construction paper cut to fit various fractions of an egg carton (1/12, 1/6, 1/4, 1/3, etc.)

Assessment Option
At this stage of the unit, it is important to know whether students can:

• Demonstrate understanding that a fraction can be represented as part of a set.
• Identify fractions when the whole (set) and part of the set are given.
• Identify fraction relationships associated with the set including equivalent fractions (e.g. 1/2 of the set of 12 eggs is the same as 6/12 of the set).

Extensions

1. Continue the activity by using the egg carton that holds 18 eggs. Have students model and record all equivalent fractions. Have students investigate how their fractions would change if the egg carton holds 6 eggs or 24 eggs.
2. Move on to the next lesson, Another Look at the Set Model using Attribute Pieces.

Questions for Students

1. What do you notice about the relationship between 1/2 of a dozen and 6/12 of a dozen?

[Students should be able to tell from their recordings that 1/2 and 6/12 are equivalent fractions.]

2. What do you notice about the relationship between 1/3 of a dozen and 4/12 of a dozen?

[Students should be able to tell from their recordings that 1/3 and 4/12 are equivalent fractions.]

3. What do you notice about the relationship between 1/4 of a dozen and 3/12 of a dozen?

[Students should be able to tell from their recordings that 1/4 and 3/12 are equivalent fractions.]

4. What do you notice about the relationship between 1/6 of a dozen and 2/12 of a dozen?

[Students should be able to tell from their recordings that 1/6 and 2/12 are equivalent fractions.]

5. What other equivalent fractions did you identify using 12 eggs?

[Student responses may vary.]

6. What relationships do you see in the equivalent fractions identified in this lesson? Do you notice any patterns when you multiply a fraction's numerator and denominator by the same number?

[An equivalent fraction results.]

Teacher Reflection

• Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding?
• Which students can identify fractions when the whole (set) and a part of the set is given? What activities are appropriate for students who have not yet developed this understanding?
• Which students/groups can articulate relationships between the fractions? What activities are appropriate for students who have not yet developed this understanding?
• Which students/groups can identify equivalent fractions? 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?

### Eggsactly with a Dozen Eggs

3-5
Students begin to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of 12.

### Eggsactly with Eighteen Eggs

3-5
Students continue to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of eighteen.

### Another Look at the Set Model using Attribute Pieces

3-5
The previous lessons focused on the set model where all objects in the set are the same size and shape. Students also need work with sets in which the objects “look” different. In the real world, we are often faced with fraction situations where the objects in the set are not identical. For this lesson, students use fractions to describe a set of attribute pieces. Students develop skill in problem solving and reasoning as they think about their set and how to create new sets given specific fractional characteristics.

### Class Attributes

3-5
During this lesson, students create their own classroom survey or use previously generated questions to study the class and describe the set [class] in fractional parts. This lesson requires that students identify fractions in real-world contexts from a set of items that are not identical. This lesson is integrated with other areas of the math curriculum, including data analysis and statistics.

### Another Look at Fractions of a Set

3-5
This lesson gives students another opportunity to explore fractions using the set model. This lesson is integrated with other areas of the math curriculum including data analysis and statistics.

### Learning Objectives

Students will:

• Demonstrate understanding that a fraction can be represented as part of a set, given a set of identical items (eggs).
• Identify fractions when the whole (set) and part of the set is given.
• Identify equivalent fractions.
• Identify relationships inherent in equivalent fractions (e.g., 1/2 can be multiplied by 2/2 to get the equivalent fraction 2/4, or 2/4 can be divided by 2/2 to get the equivalent fraction 1/2.).

### NCTM Standards and Expectations

• Use models, benchmarks, and equivalent forms to judge the size of fractions.
• Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.
• Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.

### Common Core State Standards – Mathematics

Grade 3, Num & Ops Fractions

• 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.Content.3.G.A.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 as 1/4 of the area of the shape.

Grade 4, Num & Ops Fractions

• CCSS.Math.Content.4.NF.A.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.

Grade 4, Num & Ops Fractions

• CCSS.Math.Content.4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Grade 5, Num & Ops Fractions

• CCSS.Math.Content.5.NF.B.3
Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.