Key Concept: Division with one or more fractions is a way to find the value of one part of something so it can be shared equally.

Topic Overview | Standards Alignment | #### Common Core

5.NF.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?

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.#### Georgia

IEP Goals

This topic contains lessons to reinforce the conceptual understanding that division is a procedure that can be used to solve equal shares problems with fractions similar to equal shares problems with whole numbers. Students begin with a review of dividing with whole numbers as a vehicle for learning about paraphrasing word problems. They manipulate models to find a quotient that represents the value or unit of each share.

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

1. Given a division story problem involving a fraction divided by a whole number, the student will complete the paraphrase "___ shared equally among ___" with 90% accuracy and use a model to illustrate and solve the problem for 3 consecutive sessions.

2. Given a division story problem involving a fraction or mixed number divided by a whole number and a paraphrasing strategy, the student will paraphrase the problem, write an equation that illustrates the problem, and find the product for 5 out of 6 examples by the end of the first marking period.

3. Given a division equation involving a mixed number divided by a whole number, the student will identify the dividend and divisor and then represent the problem using concrete manipulatives or computer models solving accurately for 5 out of 6 examples for 5 consecutive sessions.

2. Given a division story problem involving a fraction or mixed number divided by a whole number and a paraphrasing strategy, the student will paraphrase the problem, write an equation that illustrates the problem, and find the product for 5 out of 6 examples by the end of the first marking period.

3. Given a division equation involving a mixed number divided by a whole number, the student will identify the dividend and divisor and then represent the problem using concrete manipulatives or computer models solving accurately for 5 out of 6 examples for 5 consecutive sessions.

Unit Launcher

View William's Road Trip: Discussion Guide and KWL Chart

F.9-1-1 Divide Whole Number by Whole Number: Sharing |

View Guided Lesson Write a paraphrase and learn to use a number line model to find the quotient of an equal shares word problem involving whole numbers. (8-15 min)

F.9-1-2 Divide Proper Fraction by Whole Number: Sharing |

View Guided Lesson Write a paraphrase and use a number line or area model to find the quotient of an equal shares word problem involving a fraction and a whole number. (8-15 min)

F.9-1-3 Complete Mixed by Whole Number Division Equations |

View Guided Lesson Write a paraphrase and use a number line or area model to find the quotient and simplify it. (8-15 min)

F.9-1-4 Divide Across Procedure: Sharing |

View Guided Lesson Paraphrase, work with models and complete equations to solve equal shares word problems. (8-15 min)

Real World Investigation Part 1

View William's Road Trip: William's Data

Key Concept: Division can be used to determine how many fractional parts of a specific size are contained in a whole.

Topic Overview | Standards Alignment | #### Common Core

5.NF.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?

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.#### Georgia

IEP Goals

This topic contains lessons to reinforce the conceptual understanding that division is a procedure that can be used to solve measurement division problems. Students begin with a review of division with whole numbers as a vehicle for learning about paraphrasing word problems. They manipulate models and learn the divide across procedure to find a quotient that represents the number of fractional parts contained in a starting amount.

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division

of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

1. Given a division story problem involving a whole number divided by a fraction, the student will complete the paraphrase, "How many ___ in ___?" or "What part of ___ is in ___?" with 90% accuracy and use a model to illustrate and solve the problem for 3 consecutive sessions.

2. Given a division story problem involving a fraction or mixed number divided by a fraction and a paraphrasing strategy, the student will paraphrase the problem, write an equation that illustrates the problem, and find the product for 5 out of 6 examples by the end of the first marking period.

3. Given a division equation involving a mixed number divided by a fraction, the student will identify the dividend and divisor and then represent the problem using concrete manipulatives or computer models solving accurately for 5 out of 6 examples.

2. Given a division story problem involving a fraction or mixed number divided by a fraction and a paraphrasing strategy, the student will paraphrase the problem, write an equation that illustrates the problem, and find the product for 5 out of 6 examples by the end of the first marking period.

3. Given a division equation involving a mixed number divided by a fraction, the student will identify the dividend and divisor and then represent the problem using concrete manipulatives or computer models solving accurately for 5 out of 6 examples.

F.9-2-1 Divide Whole Number by Whole Number: Measurement |

Lesson Plan

View Guided Lesson Write a paraphrase and learn to use a number line model to find the quotient of a measurement word problem involving whole numbers. (8-15 min)

F.9-2-2 Divide Whole Number by Fraction: Measurement |

Lesson Plan

View Guided Lesson Write a paraphrase and use a number line or area model to find the quotient of a measurement word problem involving a fraction and a whole number. (8-15 min)

F.9-2-3 Complete Mixed Number by Fraction Equations |

Lesson Plan

View Guided Lesson Write a paraphrase and use a number line or area model to find the quotient of a measurement word problem involving a fraction and a mixed number. (8-15 min)

F.9-2-4 Divide Across Procedure with Larger Divisors |

Lesson Plan

View Guided Lesson Write a paraphrase and use a number line or area model to find the quotient of a measurement word problem in which the dividend is smaller than the divisor. (8-15 min)

Real World Investigation Part 2

View William's Road Trip: Create Some Data

Key Concept: There is more than one procedure for solving problems that involve dividing with fractions.

Topic Overview | Standards Alignment | #### Common Core

5.NF.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?

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.#### Georgia

IEP Goals

This topic contains lessons that teach both the divide across and the invert and multiply procedures for dividing with fractions. Initial lessons use models to support students as they learn how to apply the “one” fraction procedure to rename the dividend when it is not divisible by the divisor in order to use the divide across procedure to solve for the quotient. In later lessons students learn how to apply the “one” fraction procedure to rename the divisor into its reciprocal in order to use the invert and multiply procedure. Initial lessons are supported with models that are faded as students gain skills with the procedures.

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

6.NS.1 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?

1. Given a division equation with a dividend that is not divisible by the divisor, the student will use a "one" fraction to rename the dividend and divide across to find the quotient with 90% accuracy by completion of the IEP.

2. Given a division equation with a dividend that is not divisible by the divisor, the students will use a "one" fraction to rename the divisor and use the invert and multiply procedure to find the quotient for 5 out of 6 examples by the end of the first marking period.

2. Given a division equation with a dividend that is not divisible by the divisor, the students will use a "one" fraction to rename the divisor and use the invert and multiply procedure to find the quotient for 5 out of 6 examples by the end of the first marking period.

F.9-3-1 Rename to Divide Across Through Sharing |

Lesson Plan

View Guided Lesson Rename by multiplying the dividend by a one fraction and solve equal shares problems using the divide across procedure with the help of models. (8-15 min)

F.9-3-2 Rename to Divide Across Through Measurement |

Lesson Plan

View Guided Lesson Rename by multiplying the dividend by a one fraction and solve measurement problems using the divide across procedure with the help of models. (8-15 min)

F.9-3-3 Rename to Divide Across: Models to Numbers |

Lesson Plan

View Guided Lesson Complete the equation by renaming the dividend and dividing across to find the quotient with and without models. (8-15 min)

F.9-3-4 Rename to Invert & Multiply: Models to Numbers |

Lesson Plan

View Guided Lesson Rename by dividing the divisor by a one fraction and solve using the invert and multiply procedure with and without models. (8-15 min)

F.9-3-5 Rename to Divide Fractions: Numbers Only |

Lesson Plan

Real World Investigation Part 3

View Your Road Trip: Your Data

Key Concept: The procedures for dividing fractions can be used to solve a variety of problems.

Topic Overview | Standards Alignment | #### Common Core

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
#### Georgia

IEP Goals

This topic contains lessons in which students apply the divide across or the invert and multiply procedure for dividing with fractions. Students rename either the dividend to divide across or the divisor to invert and multiply without the use of models. The final lesson reviews multiplication and division with fractions and mixed numbers without the use of models.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

6.NS.1 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?

1. Given a division equation with a dividend and divisor that are not divisible, the student will use a "one" fraction to rename one of the fractions and use either the divide across or invert and multiply procedure to find the quotient for 9 out of 10 problems by the end of the marking period.

2. Given a set of 10 multiplication and division equations and pre-teaching, students will solve for the product or quotient with no more than one teacher problem for 9 out of the 10 problems during 3 consecutive sessions.

2. Given a set of 10 multiplication and division equations and pre-teaching, students will solve for the product or quotient with no more than one teacher problem for 9 out of the 10 problems during 3 consecutive sessions.

F.9-4-1 Divide Fractions: Mixed Procedures |

Lesson Plan

View Guided Lesson Complete the equation and use either the divide across or invert and multiply procedure to find the quotient without models. (8-15 min)

F.9-4-2 Divide Fractions: Choose the Procedure |

Lesson Plan

View Guided Lesson Select either the divide across or invert and multiply procedure, complete the equation and find the quotient without models. (8-15 min)

This Big Idea develops both the conceptual understanding and procedural skills of dividing with fractions. Students learn that by creating a paraphrase for a problem, they can better understand the meaning. Then they create visual models that support how to procedurally represent and solve the problem. The final topic contains lessons that use numbers only without context.

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

6.NS.1 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?