Key Concept: Multiplication is used to find the product of groups of items and is the same as repeated addition. The groups of items are parts of a whole.

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

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

MGSE4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number e.g., by using a visual such as a number line or area model.

IEP Goals

This topic contains lessons to reinforce the conceptual understanding that multiplication is a procedure that is similar to repeated addition by reinforcing the phrase "groups of". Students begin with a review of multiplying with whole numbers as a vehicle for learning about paraphrasing word problems. They manipulate models to create "groups of" items before moving to multiplication of a fraction by a whole number. All lessons in this unit contain examples in which the numbers have common factors and many require simplification.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

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.5 Interpret multiplication as scaling (resizing), by:b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

MGSE5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.a. Apply and use understanding of multiplication to multiply a fraction or whole number by a fraction. Examples: (a/b) x q as (a/b) x (q/1) and (a/b) x (c/d) = ac/bd

MGSE5.NF.5 Interpret multiplication as scaling (resizing) by:b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1.

MGSE5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.1. Given a multiplication story problem involving a fraction multiplied by a whole number, the student will complete the paraphrase "___ groups of___" with 90% accuracy and use a model to illustrate and solve the problem for 3 consecutive sessions.

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

3. Given a multiplication story problem containing at least one proper fraction, the student will identify the starting value and multiplier 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 multiplication story problem involving a fraction multiplied by a whole number and a paraphrasing strategy, the student will paraphrase the problem, write an equation that illustrates the problem, find the product and simplify the answer for 5 out of 6 examples by the end of the first marking period.

3. Given a multiplication story problem containing at least one proper fraction, the student will identify the starting value and multiplier 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 Springville's Sports Complex: Discussion Guide and KWL Chart

F.8-1-1 Multiply Whole Number by Whole Number: Models |

View Guided Lesson Write a “groups of” paraphrase and use a number line model to find the product of a word problem involving whole numbers. (12-18 min)

F.8-1-2 Multiply Unit Fraction by Whole Number |

View Guided Lesson Write a “groups of” paraphrase and use a number line model to find the product of a word problem involving a fraction and a whole number. (12-18 min)

F.8-1-3 Multiply Unit Fraction by Whole Number: Simplify |

View Guided Lesson Write a “groups of” paraphrase and use either a number line or area model to find the product and simplify the answer. (12-18 min)

F.8-1-4 Multiply Proper Fraction by Whole Number |

View Guided Lesson Paraphrase, work with models and complete equations to solve “groups of” word problems. (12-18 min)

F.8-1-5 Multiply Mixed Number by Whole Number |

View Guided Lesson Paraphrase, work with models and complete equations to solve “groups of” word problems containing whole numbers and mixed numbers. (12-18 min)

Real World Investigation Part 1

View Springville's Sports Complex: Springville's Data

Key Concept: Multiplication with a fraction is a way to find the value of a part of a starting amount.

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

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

#### Georgia

MGSE4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number e.g., by using a visual such as a number line or area model.

IEP Goals

This topic contains lessons to teach the conceptual understanding that multiplication with a fraction is a procedure used to find parts of a whole. Students complete a paraphrase that reinforces the phrase "parts of." Then students manipulate models removing a fractional part of a whole number. The fractional part progresses from a unit fraction, proper fraction, to mixed number. All lessons in this topic contain examples in which the numbers have common factors and many lessons require simplification. The last lesson is a culmination of Topics 1 and 2.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

MGSE5.NF.5 Interpret multiplication as scaling (resizing) by:b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1.

MGSE5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.1. Given a multiplication story problem involving a whole number multiplied by a fraction, the student will complete the paraphrase " ___ parts of___" with 90% accuracy and use a model to illustrate and solve the problem for 3 consecutive sessions.

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

3. Given a multiplication story problem containing at least one proper fraction, the student will identify the starting value and multiplier and then represent the problem using concrete manipulatives or computer models solving accurately for 5 out of 6 examples for 5 consecutive sessions.

4. Given a multiplication story problem, the student will determine if the problem involves finding the result of "groups of" or a "parts of" a given amount and use models to represent and solve the problem using language such as multiplier and starting value and unit with 100% accuracy for 5 consecutive sessions.

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

3. Given a multiplication story problem containing at least one proper fraction, the student will identify the starting value and multiplier and then represent the problem using concrete manipulatives or computer models solving accurately for 5 out of 6 examples for 5 consecutive sessions.

4. Given a multiplication story problem, the student will determine if the problem involves finding the result of "groups of" or a "parts of" a given amount and use models to represent and solve the problem using language such as multiplier and starting value and unit with 100% accuracy for 5 consecutive sessions.

F.8-2-1 Multiply Whole Number by Unit Fraction: Models |

Lesson Plan

View Guided Lesson Write a “parts of” paraphrase and use a number line model to find the product of a word problem. (8-15 min)

F.8-2-2 Multiply Whole Number by Unit Fraction: Simplify |

Lesson Plan

View Guided Lesson Write a “parts of” paraphrase and use a number line or area model to find the product and simplify the answer. (8-15 min)

F.8-2-3 Multiply Whole Number by Proper Fraction |

Lesson Plan

View Guided Lesson Paraphrase, work with models and complete equations to solve “parts of” word problems. (8-15 min)

F.8-2-4 Multiply Mixed Number Types with Common Factors |

Lesson Plan

View Guided Lesson Select and complete the correct paraphrase. Use models and complete multiplication word problems. (8-15 min)

Real World Investigation Part 2

View Springville's Sports Complex: Create Some Data

Key Concept: Multiplication with fractions can result in a product that is less than or greater than the two factors.

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

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

#### Georgia

MGSE4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number e.g., by using a visual such as a number line or area model.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

MGSE5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

IEP Goals

This topic expands on the understanding and skills learned in the previous two topics by introducing students to fractions without common factors and multiplication of a fraction by a fraction. Students work with models to discover that multiplying a given whole number by a fraction can result in a whole number and multiplying a fraction by a fraction results in a product that is less than the two factors. Students bring previous understanding about converting mixed and improper fractions and simplification and apply this knowledge to multiplication procedures.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? 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.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.5 Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

a. Apply and use understanding of multiplication to multiply a fraction or whole number by a fraction. Examples: (a/b) x q as (a/b) x (q/1) and (a/b) x (c/d) = ac/bd

MGSE5.NF.5 Interpret multiplication as scaling (resizing) by:a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. Example: 4 x 10 is twice as large as 2 x 10.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1.

MGSE5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.1. Given a multiplication story problem and concrete or computer models, the student will use the model to solve the problem, represent the problem as an equation, and simplify the product with 90% accuracy by the end of the first marking period.

2. Given a multiplication story problem with 2 fractions with uncommon factors, the student will paraphrase the problem, represent it with models and as an equation, and explain his/her thinking for 9 out of 10 problems by the completion of the IEP.

3. Given a story problem with two mixed numbers, the student will represent the problem as an equation by accurately converting the mixed numbers to improper fractions, simplifying the product without the use of models for 5 out of 6 examples in three consecutive sessions.

2. Given a multiplication story problem with 2 fractions with uncommon factors, the student will paraphrase the problem, represent it with models and as an equation, and explain his/her thinking for 9 out of 10 problems by the completion of the IEP.

3. Given a story problem with two mixed numbers, the student will represent the problem as an equation by accurately converting the mixed numbers to improper fractions, simplifying the product without the use of models for 5 out of 6 examples in three consecutive sessions.

F.8-3-1 Multiply Mixed Number Types with Uncommon Factors |

Lesson Plan

View Guided Lesson Select and complete the paraphrase, work with models and complete equations to solve word problems with uncommon factors. (8-15 min)

F.8-3-2 Multiply Proper Fraction by Proper Fraction |

Lesson Plan

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

F.8-3-3 Multiply Mixed Numbers and Whole Numbers |

Lesson Plan

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

F.8-3-4 Multiply Mixed Number by Mixed Number |

Lesson Plan

View Guided Lesson Paraphrase, work with models and complete equations to solve word problems with two mixed numbers (8-15 min)

Real World Investigation Part 3

View Your Sports Complex: Your Data

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

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

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

MGSE4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number e.g., by using a visual such as a number line or area model.

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

MGSE5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1.

MGSE5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. IEP Goals

Lessons in this topic require students to apply their conceptual understanding of multiplying fractions to solve equations without the use of models. Some lessons provide context, some just paraphrasing, and some with equations only. In one lesson, students are challenged to complete equations with missing unknowns. They bring previous understanding about converting mixed and improper fractions and simplification and apply this knowledge to multiplication procedures.

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

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.5 Interpret multiplication as scaling (resizing), by:b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.a. Apply and use understanding of multiplication to multiply a fraction or whole number by a fraction. Examples: (a/b) x q as (a/b) x (q/1) and (a/b) x (c/d) = ac/bd

MGSE5.NF.5 Interpret multiplication as scaling (resizing) by:1. Given a multiplication story problem without the use of models, the student will verbally paraphrase the problem, represent the problem as an equation and simplify the product with 90% accuracy by the end of the first marking period.

2. Given a multiplication equation with at least one fraction, the student will correctly complete the procedure by multiplying across and simplifying the product and explain his/her thinking for 9 out of 10 problems by the completion of the IEP.

3. Given a multiplication equation with at least one proper fraction or mixed number and when allowed the use of a calculator or multiplication chart, the student will independently rename mixed numbers to solve the equation and simplify the product for 5 out of 6 equations for 5 consecutive sessions.

4. Given a set of five multiplication equations containing fractions and at least one unknown number along with a cognitive strategy, the student will apply the strategy and correctly solve for the answer with 80% accuracy for three consecutive sessions.

2. Given a multiplication equation with at least one fraction, the student will correctly complete the procedure by multiplying across and simplifying the product and explain his/her thinking for 9 out of 10 problems by the completion of the IEP.

3. Given a multiplication equation with at least one proper fraction or mixed number and when allowed the use of a calculator or multiplication chart, the student will independently rename mixed numbers to solve the equation and simplify the product for 5 out of 6 equations for 5 consecutive sessions.

4. Given a set of five multiplication equations containing fractions and at least one unknown number along with a cognitive strategy, the student will apply the strategy and correctly solve for the answer with 80% accuracy for three consecutive sessions.

F.8-4-1 Complete Equations with Proper Fractions |

Lesson Plan

View Guided Lesson Complete equations and solve for the product of two proper fractions and simplify the product without the use of models. (8-15 min)

F.8-4-2 Paraphrase/Multiply Equations: Proper Fractions |

Lesson Plan

View Guided Lesson Paraphrase, write equations and solve word problems for the product of two proper fractions without models. (8-15 min)

F.8-4-3 Complete Equations with Mixed Numbers |

Lesson Plan

F.8-4-4 Paraphrase/Multiply Equations: Mixed Numbers |

Lesson Plan

View Guided Lesson Paraphrase, write equations and solve word problems for the product of mixed numbers and whole numbers without models. (8-15 min)

This Big Idea develops both the conceptual understanding and procedural skills of multiplying 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 both with and without context.

a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. Example: 4 x 10 is twice as large as 2 x 10.