6 Dot Ns Dot 1 About The Math Learning Targets And Increasing Rigor
Grade 4 G/T The Number System
6.NS.1
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 Links to an external site. or visual math dictionary Links to an external site. 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 |
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Progression of this Standard Across Grades
This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.
Grade 5 | Grade 7 |
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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 the Meaning of the Quotient when Dividing Fractions by Fractions:
- Exploring with Pattern Blocks I Links to an external site. +++
- Exploring with Pattern Blocks II Links to an external site. +++
- How Many 2s Links to an external site. +++
- Quick Jog Links to an external site. +++
- Another Quick Jog Links to an external site. +++
- Comparing Ropes Links to an external site. +++
- Making Hot Cocoa Links to an external site. +++
- Scooping Ice Cream Links to an external site. +++
- Reasoning About Quotients Links to an external site. +++
- Bags of Almonds Links to an external site. +++
- How Many Containers in One Cup / Cups in One Container? (Links to an external site.) Links to an external site. (Illustrative Math)
- Dan’s Division Strategy (Links to an external site.) Links to an external site.(Illustrative Math)
- The Apple Links to an external site. (3 Act Task, G.Fletcher)
Estimate Quotients when Dividing Fractions by Fractions:
- Reasoning About Quotients Links to an external site. +++
- Bags of Almonds Links to an external site. +++
- Baking Cookies Links to an external site. (Illustrative Math)
Divide Fractions by Fractions:
- Jars of Jam Links to an external site. +++
- Using an Algorithm to Divide Fractions Links to an external site.(Open-Up Resources -Illustrative Mathematics: Lesson 4.11)
- Making Hot Cocoa, Variation 1 (Links to an external site.) Links to an external site.(Illustrative Math)
- Running to School, Variation 2 (Links to an external site.) Links to an external site. (Illustrative Math)
- Making Hot Cocoa, Variation 2 (Links to an external site.) Links to an external site.(Illustrative Math)
- Drinking Juice, Variation 2 Links to an external site. (Links to an external site.) (Illustrative Math)
- Drinking Juice, Variation 3 (Links to an external site.) Links to an external site.(Illustrative Math)
- Traffic Jam (Links to an external site.) Links to an external site.(Illustrative Math)
- Running to School, Variation 3 (Links to an external site.) Links to an external site. (Illustrative Math)
- Cup of Rice (Links to an external site.) 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 Thumbnail | Book Title | Grade | Pages |
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Teaching Student-Centered Mathematics | 6-8 | How Much For One?, Activity 8.22, Page 137 Expanded Lesson: Division-of-Fractions Stories, Pages 141- 142 |
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Nimble With Numbers
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5-6 | 85-87 (Fraction Arrangements) 90 (Digits to Fractions II) |
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Nimble with Numbers |
6-7 | 56-58 (Fraction Formulation) 59-60 (Seeking Fractions) 62 (Finding Fractions Practice) 64 (Creating and Computing Fractions) |
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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 Fractions Tic-Tac-Toe Download Dividing Fractions Tic-Tac-Toe
- Dividing with Fractions Download Dividing with Fractions (HCPSS Lesson)
- Dividing with Fractions Download Dividing with Fractions (UDL Checklist)
- Division of Fractions Links to an external site. (MSDE Lesson)
- Division of Fractions Links to an external site. (MSDE Lesson)
- Division of Fractions Links to an external site. (MSDE Lesson)
- Rabbit Costumes Links to an external site. (Inside Mathematics Lesson)
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 Links to an external site. (student-facing resource, pictured)
- Braining Camp Dividing Fractions - application Links to an external site. (student-facing resource)
- Math Soccer Links to an external site. (student-facing resource)
- Math Playground: Dividing Fractions Links to an external site. (student-facing resource)
- Math Basketball Links to an external site. (student-facing resource)
- Kahn Academy: Reasoning about Fraction Division Links to an external site. (student-facing resource)
- Kahn Academy: Dividing Fractions Word Problems Links to an external site. (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.
- Dividing Fractions Links to an external site.
- Dividing Fractions on a Number Line Links to an external site.
- Dividing Fractions Links to an external site.
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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
Download Recording Sheet (docx)
Rubric
Download 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.
- Evaluate Representations of Division with Fractions Situation Links to an external site.
- Evaluate and Create Representations of Division with Fractions Situation Links to an external site.
- Evaluate and Create Representations of Division with Fractions Situation_2 Links to an external site.
- Represent and Compare Fraction Division vs. Multiplication Links to an external site.
- Represent, Solve, and Compare when Dividing Fractions Links to an external site.
- Divide Fractions to Solve Word Problems Links to an external site.
- Divide Fractions and Mixed Numbers to Solve Word Problems Links to an external site.
- Estimate Quotients of Division with Fractions Expressions Links to an external site.
- Estimate Quotients of Division with Fractions Expressions_2 Links to an external site.
- Create Model and Story Problem to Match Division with Fractions Expression Links to an external site.
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