Grade 5 AGL The Number System

6.NS.1

About the Math
Preferred Resources
Additional Resources
Independent Work
Assessment

Full Standard

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

Learning Targets (I can)

Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths).
Explain division of a fraction by a fraction using representations.
Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.)
Connect a representation of division of a fraction by a fraction to an equation.
Divide a fraction by a fraction.
Solve word problems that divide a fraction by a fraction.

About the Math

Multiplication and division have an inverse relationship. So we say 10 ÷ 2 = 5 is the same as 2 x 5 = 10. Prior to developing the procedure for division of fractions, students need to experience visual representations of the meaning of division of fractions. Eventually, we get to the procedure for division of fractions which is to multiply by the reciprocal of the divisor. In other words, 10 ÷ 2 is the same as 10 x 1/2. They both result in the answer of 5. So just as 10 ÷ 2 means how many 2's are in 10, a division problem like 7/8 ÷ 1/8 is asking how many one-eighths are in 7/8? There are 7 one-eighths in 7/8. When dividing fractions, you need to think about what the problem is asking. When a greater fraction is divided by a smaller fraction, the quotient is greater than one. For example, 2/3 ÷ 1/3 = 2 because there are two one-thirds in two thirds. When a smaller fraction is divided by a greater fraction, the quotient is less than one. For example 1/3 ÷ 2/3 is 1/2 because there is only part of two-thirds in one-third. Students need to experiment with lots of examples to find this generalization. Essential vocabulary for this standard includes: divisor, dividend, inverse operation, quotient, and reciprocal. Visit the online dictionary or visual math dictionary for vocabulary support.

Progression of Standard within Grade 6 (5 AGL)

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Quarter 1	Quarter 2	Quarter 3	Quarter 4
		Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths). Explain division of a fraction by a fraction using representations. Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.) Connect a representation of division of a fraction by a fraction to an equation. Solve word problems that divide a fraction by a fraction.	Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths). Explain division of a fraction by a fraction using representations. Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.) Connect a representation of division of a fraction by a fraction to an equation. Solve word problems that divide a fraction by a fraction.

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map. Information is not the complete standard.

Grade 5

Grade 7

Divide unit fractions by whole numbers and whole numbers by unit fractions (5.NF.7)

Multiply and divide rational numbers (7.NS.2)

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities (7.SP.3)

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Preferred Tasks and Lesson Seeds

These links provide ideas for lessons connected to this standard. They may support small group instruction or whole group collaborative investigations. They are intended to serve as a starting point for planning.

Explain and Represent Division of Fractions:

Exploring with Pattern Blocks I +++
Exploring with Pattern Blocks II +++
How Many 2s +++
Quick Jog +++
Another Quick Jog +++
Comparing Ropes +++
Making Hot Cocoa +++
Scooping Ice Cream +++
Reasoning About Quotients +++
Bags of Almonds +++
Meaning of Division ( Illustrative Mathematics: Lesson 4. 2 )
What Fraction of a Group? (Illustrative Mathematics : Lesson 4.7)
Baking Cookies (Illustrative Math)
How many _______ are in. . . ? (Links to an external site.) (Illustrative Math)
Cup of Rice (Links to an external site.) (Illustrative Math)
Making Hot Cocoa, Variation 1 (Links to an external site.) (Illustrative Math)
Making Hot Cocoa, Variation 2 (Illustrative Math)
Traffic Jam (Links to an external site.) (Illustrative Math)
How Many Containers in One Cup / Cups in One Container? (Links to an external site.) (Illustrative Math)
Dan’s Division Strategy (Links to an external site.) (Illustrative Math)
The Apple (3 Act Task, G.Fletcher)

Divide Fractions by Fractions:

Jars of Jam +++
Interpreting Division Stories (Illustrative Mathematics: Lesson 4.3)
Using Diagrams to Find the # of Groups (Illustrative Mathematics: Lesson 4.6)
Dividing by Unit and Non-Unit Fractions (Illustrative Mathematics: Lesson 4.10)
Using an Algorithm to Divide Fractions (Illustrative Mathematics: Lesson 4.11
Running to School, Variation 2 (Links to an external site.) (Illustrative Math)
Drinking Juice, Variation 2 (Links to an external site.) (Illustrative Math)
Drinking Juice, Variation 3 (Links to an external site.) (Illustrative Math)
Running to School, Variation 3 (Links to an external site.) (Illustrative Math)
Dan’s Division Strategy (Links to an external site.) (Illustrative Math

Lesson Seeds (Prompts for Rigor)

Rigorous mathematics instruction balances the teaching and learning of conceptual understanding, procedural understanding, and the application of mathematics standards. Below are prompts for applying rigor in your classroom. They may support small group instruction or whole group collaborative investigations. They are intended to serve as a starting point for planning. See the Increasing Rigor page for more information.

You divide two fractions, and the numerator of the quotient is a 4. What could the fractions be?
Compare and contrast the following, using visual models/drawings: ½ ÷ ¾ and ¾ ÷ ½.
Larry ordered 5 ½ subs. He wants to share the subs with 4 friends and himself. Larry thinks they will each receive 1 ¼ subs. Is Larry correct? Why or why not?
Can you use the fractions (2/3), (8/9), and (3/4) to create a true equation? (one answer: 8/9 x ¾ = ⅔ )
Alana wants to run 1 ⅞ miles and the track is ¼ mile around. How many laps will she need to run to make her goal?

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Print Resources

These publications have been provided for each school. They are typically stored in team closets or media centers. Check with your team leader if you cannot find them.

Book Title	Grade	Pages
Teaching Student-Centered Mathematics	6-8	How Much For One?, Activity 8.22, Page 137 Expanded Lesson: Division-of-Fractions Stories, Pages 141- 142
Nimble With Numbers	5-6	85-87 (Fraction Arrangements) 90 (Digits to Fractions II)
Nimble with Numbers	6-7	56-58 (Fraction Formulation) 59-60 (Seeking Fractions) 62 (Finding Fractions Practice) 64 (Creating and Computing Fractions)
Hands-On Standards Common Core	6	22 (Lesson 1)

Additional Tasks and Lesson Seeds

These links provide ideas for lessons connected to this standard. They may support small group instruction or whole group collaborative investigations. They are intended to serve as a starting point for planning.

Dividing with Fractions Part 1 (Lesson)
Dividing with Fractions Part 1 (PPT)
Dividing Fractions Tic-Tac-Toe
Dividing with Fractions (HCPSS Lesson)
Dividing with Fractions (UDL Checklist)
Division of Fractions (MSDE Lesson)
Division of Fractions (MSDE Lesson)
Division of Fractions (MSDE Lesson)
Rabbit Costumes (Inside Mathematics Lesson)

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Student-Facing Centers

These links provide online and print resources to engage students during independent or center time. These resources could be used as lesson seeds.

Braining Camp Dividing Fractions (student-facing resource, pictured)
Braining Camp Dividing Fractions - application (student-facing resource)
Math Soccer (student-facing resource)
Math Playground: Dividing Fractions (student-facing resource)
Math Basketball (student-facing resource)
Kahn Academy: Reasoning about Fraction Division (student-facing resource)
Kahn Academy: Dividing Fractions Word Problems (student-facing resource)

Independent Practice

These resource sheets can be used for independent practice or homework. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.

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Learning Target

Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths).
Explain division of a fraction by a fraction using representations.
Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.)
Connect a representation of division of a fraction by a fraction to an equation.
Solve word problems that divide a fraction by a fraction.

Learning targets identify what students should be able to do. The resources below can be used to measure student understanding of the standard. The recording sheet can be used to collect data. The rubric can be applied to any task. This standard can be continue through all 4 quarters. Select the preview icon to preview the task before downloading.

Recording Sheet (docx)

Rubric (docx)

Assessment Tasks

These tasks align with the learning targets for the standard. They can be combined with other tasks for extended assessment opportunities. Teachers/students are not expected to complete all these tasks.

Tasks Adapted from Smarter Balance and/or PARCC Assessment Sample Tasks

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