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In-Class Reference Sheet
Big Idea 8: Multiplication with fractions is similar to multiplication with whole numbers in that you are often
finding the product of groups of items. They are different in that you can multiply with fractions to find parts of an
amount.
Topic 4: Multiplying with Common and Uncommon Factors
Key Concept: The procedure for multiplying fractions can be used to solve a variety of problems.
Prior Knowledge:
1. Can paraphrase a story problem and explain whether the problem is about finding the product of “groups
of” or “parts of” an amount.
2. Fluency with basic multiplication and division facts.
3. Procedure for simplifying fractions.
4. Can rename a mixed number as an improper fraction and an improper fraction as a mixed number.
Lesson 1: Multiply Proper Fractions: Equation Only
Description
Students work with numbers only
practicing the procedure for multiplying
with fractions. Initial problems provide
a paraphrase while problems later in
the lesson contain equations only.
Problems contain proper fractions and
whole numbers.
Vocabulary
Content: paraphrase, product, equation, simplify, improper fraction, mixed number
Process: complete, solve
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© 2011 Conceptua Math LLC
1
In-Class Reference Sheet
Lesson 1: Multiply Proper Fractions: Equation Only
Preparation for
Learning
Open the Multiply Fractions Tool
 Unlink the models from the paraphrase and the equation.
 Complete the paraphrase to read, 5 groups of 1/10.
 Provide students with the following 2 story problems.
1. There were 5 students that participated in a relay race. Each of the
students ran 1/10 of a mile. How many miles did they run altogether?
2. Mary took a 3-mile walk around her neighborhood. She took her brother
with her, but dropped him off at a friend’s house after 1/10 of the walk.
How many miles did Betsy's brother walk with her?

Teacher/student
dialogue
Indicators of
Understanding *
Have students discuss with a partner the context that would best match the
paraphrase. Discuss as a class. As a class, work to complete the equation
and solve the problem.
 Reset and complete a paraphrase to read: 3/4 part of 2 hours. Ask students
to work on dry erase boards or scrap paper to write a context that would fit the
paraphrase. Students compare with a partner.
 Discuss as a class. Complete the equation as a class or have one student
model.
Summarize
 Have students explain how paraphrasing can help to write the equation.
While students are working in the software, be sure to circulate and ask:
 How do you know that your equation matches the meaning of the
paraphrase?
 Explain the steps you used to rename the mixed number as an improper
fraction.
 How do you know the product is in simplest form?
 Explain the multiply-across procedure for multiplying with fractions.
 Places the fraction numbers in the equation that matches the meaning of the
paraphrase.
 Simplifies improper fractions.
 Completes a multiplication equation and solves for the product.
 Recognizes and explains the procedure.
 NOTE: May also recognize the commutative proper of multiplication with
fractions.
* Indicators of Understanding are in addition to the formative assessment at the end of each activity.
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© 2011 Conceptua Math LLC
2
In-Class Reference Sheet
Big Idea 8: Multiplication with fractions is similar to multiplication with whole numbers in that you are often
finding the product of groups of items. They are different in that you can multiply with fractions to find parts of an
amount.
Topic 4: Multiplying with Common and Uncommon Factors
Key Concept: The procedure for multiplying fractions can be used to solve a variety of problems.
Prior Knowledge:
1. Can paraphrase a story problem and explain whether the problem is about finding the product of “groups
of” or “parts of” an amount.
2. Fluency with basic multiplication and division facts
3. Procedure for simplifying fractions
4. Can rename a mixed number as an improper fraction and an improper fraction as a mixed number
Lesson 2: Multiply Proper Fractions: Context and Equation
Description
Students write equations from story
problems. They apply previously
learned skills in converting mixed
numbers to improper fractions and
simplifying.
Vocabulary
Content: paraphrase, product, equation, simplify, improper fraction, mixed number
Process: complete, solve
www.conceptuamath.com
© 2011 Conceptua Math LLC
3
In-Class Reference Sheet
Lesson 2: Multiply Proper Fractions: Context and Equation
Preparation for
Learning
Teacher/student
dialogue
Indicators of
Understanding *
Open the lesson Multiply Proper Fractions: Context and Equation 842 and move to
problem number 1.
 Provide each student with 2 cards, 1 card with the letter G for “groups of” and
one card with the letter P for “part of”.
 Read the problem aloud or use the text-to-speech feature.
 Ask students to raise one of their cards to indicate if the problem is about
finding the product of “groups of” items or “part of” a whole.
 Have one student justify his/her answer and discuss as a class.
 Write the equation as a class. Review how the whole number 4 can be
rewritten as 4/1. Note that students can enter it as a whole number visualizing
the denominator of 1 when completing the procedure.
Repeat the above procedure with problem 2.
 Allow students to justify their answers using the models, manipulatives or
drawings in addition to language.
Summarize
 Have students summarize what it means to multiply by a fraction. How is it
similar to multiplying whole numbers? How is it different? NOTE: The goal is
to have students understand the Big Idea.
While students are working in the software, be sure to circulate and ask:
 What is the multiplier? What is the starting value? How do you know?
 How are you determining the order of the fractions in the equation? Does it
matter?
 Why is the product less than the starting value when multiplying two fractions?
 Can you explain the procedure for multiplying with fractions?
 Can determine if a problem requires finding a product for groups of items or
for part of a whole. Justifies the response.
 Completes a multiplication equation and solves for the product.
 Can simplify a fraction by dividing by a common factor.
 Can explain why multiplication is used to solve the problem.
* Indicators of Understanding are in addition to the formative assessment at the end of each activity.
www.conceptuamath.com
© 2011 Conceptua Math LLC
4
In-Class Reference Sheet
Big Idea 8: Multiplication with fractions is similar to multiplication with whole numbers in that you are often
finding the product of groups of items. They are different in that you can multiply with fractions to find parts of an
amount.
Topic 4: Multiplying with Common and Uncommon Factors
Key Concept: The procedure for multiplying fractions can be used to solve a variety of problems.
Prior Knowledge:
1. Can paraphrase a story problem and explain whether the problem is about finding the product of “groups
of” or “parts of” an amount.
2. Fluency with basic multiplication and division facts.
3. Procedure for simplifying fractions.
4. Can rename a mixed number as an improper fraction and an improper fraction as a mixed number.
Lesson 3: Multiply Mixed Numbers: Equation Only
Description
Students work with numbers only
practicing the procedure for multiplying
with fractions. Initial problems provide
a paraphrase while problems later in
the lesson contain equations only.
Problems contain whole numbers and
mixed numbers.
Vocabulary
Content: paraphrase, product, equation, simplify, improper fraction, mixed number
Process: complete, solve
www.conceptuamath.com
© 2011 Conceptua Math LLC
5
In-Class Reference Sheet
Lesson 3: Multiply Mixed Numbers: Equation Only
Preparation for
Learning
Teacher/student
dialogue
Indicators of
Understanding *
Provide students with fraction circles or fraction strips.
 Have students place one circle in front of them and count out the quantity 11/3
as a class as they fill in each circle with a piece. Draw students’ attention to the
need for an additional fraction circle to represent 4/3, 7/3 and 9/3.
o One-third, two thirds etc.
 Ask a student to write the fraction 11/3 on the board or on individual white
boards.
 Review the term, improper fraction.
 Ask/review the meaning of the numerator 11 in the improper fraction 11/3.
 Review the term, mixed number and have a student write the mixed number 3
2/3 on the board or individual white boards.
 Ask “how do we know that both numbers represent the same value?”
Present students with another improper fraction such as 14/4
 Review the procedure for renaming the improper fraction as a mixed number.
 Ask “how many fourths are in one whole? How many wholes are in 14
fourths?”
o Discuss, then demonstrate/review the procedure of dividing the
numerator by the denominator, in this case 14 ÷ 4.
o Note that 3 2/4 is correct, but is it in simplest form?
Open the Multiply Fractions Tool
 Hide the context, paraphrase and models.
 Put in the equation “4 x 2 2/6 = “
 Have students work in pairs to complete the equation and find the product.
 Have one pair model their process for completing the equation. Discuss as a
class being sure to review the procedure for renaming 2 2/6 as the improper
fraction 14/6 in order to multiply across.
 Have students present their strategy for finding the product of 4 x 14 mentally.
 Review how to rename 56/6 as the mixed number 9 2/6 or 9 1/3.
Summarize
 Write the mixed number 2 1/3 and have students provide the procedure for
renaming it as an improper fraction.
 Write the improper fraction 15/2 and have students provide the procedure for
renaming it as a mixed number.
 Review that when renaming the fraction value does not change. Sometimes
we need to rename in order to use the procedure for multiplying by a fraction.
While students are working in the software, be sure to circulate and ask:
 What is the starting value? Where in the equation does that go?
 What is the multiplier? How are you using the model to show the multiplier?
 Are you using the models or mathematical strategies to simplify the product?
 Why did you decide not to use the model? What mathematical strategies are
you using?
 Reads word problem, selects key information and creates the paraphrase.
 Simplifies improper fractions.
 Completes a multiplication equation and solves for the product.
* Indicators of Understanding are in addition to the formative assessment at the end of each activity.
www.conceptuamath.com
© 2011 Conceptua Math LLC
6
In-Class Reference Sheet
Big Idea 8: Multiplication with fractions is similar to multiplication with whole numbers in that you are often
finding the product of groups of items. They are different in that you can multiply with fractions to find parts of an
amount.
Topic 4: Multiplying with Common and Uncommon Factors
Key Concept: The procedure for multiplying fractions can be used to solve a variety of problems.
Prior Knowledge:
1. Can paraphrase a story problem and explain whether the problem is about finding the product of “groups
of” or “parts of” an amount.
2. Fluency with basic multiplication and division facts
3. Procedure for simplifying fractions
4. Can rename a mixed number as an improper fraction and an improper fraction as a mixed number
Lesson 4: Multiply Mixed Numbers: Context & Equation
Description
Students will multiply mixed
numbers. They will work with models
at the beginning and progress to
solve the problems without models.
Vocabulary
Content: paraphrase, product, equation, simplify, improper fraction, mixed number
Process: complete, solve
www.conceptuamath.com
© 2011 Conceptua Math LLC
7
In-Class Reference Sheet
Lesson 4: Multiply Mixed Numbers: Context & Equation
Preparation for
Learning
Provide students with a 48 cm (18.9 inch) strip of paper that is approximately 2 inches
wide and Cuisenaire rods.
 Fold the strip in half and half again to form 4 sections. Mark each end and
crease with whole numbers.
 Have students determine which rod would allow them to partition the each
“inch” into halves/ (dark green)
 Students use the green bar to mark of halves between the 0 and 3 whole
numbers.
Have students count out the quantity 3 ½.
o One-half, two halves, three halves, four halves, etc.
 Ask a student to write the improper fraction 7/2 on the board or individual white
boards.
 Ask/review the meaning of the numerator 7 in the improper fraction 7/2. Ask
students to use their number lines to determine how we can represent 7/2 as a
mixed number.
 Ask how do we know that both numbers represent the same value?
Have students determine which rod can be used to mark off fourths.
 Have students count out the quantity 2 ¾.
 Ask students to write the improper fraction 11/4 and the mixed number 2 ¾.
 Ask how do we know that both numbers represent the same value?
Repeat with additional rods and fractions until students fluently count fractions and
rename improper fractions as mixed numbers.
Review the procedure for renaming the improper fraction as a mixed number.
 Ask how many eighths are in one whole? How many wholes are in 18 eighths?
o Discuss then demonstrate/review the procedure of dividing the
numerator by the denominator, in this case 18 ÷ 8.
o Note that 2 2/8 is correct, but is it in simplest form?
Open the lesson Multiply Mixed Numbers: Context & Equation 844 and move to
problem number 1
 Work as a class to write the equation for the word problem.
o Review how to write a whole number as a fraction.
o Review how to write the mixed number 2 ½ as an improper fraction.
o Solve and simplify.
Summarize: Sometimes we need to rename in order to use the procedure for
multiplying by a fraction.
Teacher/student
dialogue
While students are working in the software, be sure to circulate and ask:
Indicators of
Understanding *






Why did you represent the mixed number ___ as an improper fraction?
How did you determine that you should divide the number line into___ parts?
Can you explain your answer while using the models?
How do you know you have simplified the product correctly?
Can represent a mixed number as an improper fraction and an improper
fraction as a mixed number.
Explains and demonstrates an understanding of the procedure for multiplying
fractions.
* Indicators of Understanding are in addition to the formative assessment at the end of each activity.
www.conceptuamath.com
© 2011 Conceptua Math LLC
8
In-Class Reference Sheet
Big Idea 8: Multiplication with fractions is similar to multiplication with whole numbers in that you are often
finding the product of groups of items. They are different in that you can multiply with fractions to find parts of an
amount.
Topic 4: Multiplying with Common and Uncommon Factors
Key Concept: The procedure for multiplying fractions can be used to solve a variety of problems.
Prior Knowledge:
1. Can paraphrase a story problem and explain whether the problem is about finding the product of “groups
of” or “parts of” an amount.
2. Fluency with basic multiplication and division facts
3. Procedure for simplifying fractions
4. Can rename a mixed number as an improper fraction and an improper fraction as a mixed number
Lesson 5: Challenge: Multiply Fractions with Unknowns
Description
Students determine the value of
unknowns in an equation.
Vocabulary
Content: product, equation, simplify, unknowns
Process: complete, solve
www.conceptuamath.com
© 2011 Conceptua Math LLC
9
In-Class Reference Sheet
Lesson 5: Challenge: Multiply Fractions with Unknowns
Preparation for
Learning
Open the lesson Challenge: Multiply Fractions with Unknowns 845 and move to
problem 1.
 Share with students that one strategy for finding the unknowns is to work
backward from the simplified product.
 Have students work as a class to convert 4 ½ to the improper fraction 9/2
(product).
 Ask students: “What times 2 equals 2?” Fill in the denominator of the missing
factor.
 Ask students: “What times 1 equals 9?” Fill in the numerator of the missing
factor.
Move to problem 2
 Have students work in pairs to complete the equation.
 Ask one pair to complete the equation and justify their answers before
checking work.
Repeat using the examples from the lesson or create examples using the Multiply
Fractions tool until students demonstrate an ability to work from the simplified product
to find the unknowns.
Summarize: Sometimes we need to rename in order to use the procedure for
multiplying by a fraction. If we convert the simplified product to the improper fraction
we can use what we know about the procedure for multiplying fractions to find the
unknowns.
Teacher/student
dialogue
While students are working in the software, be sure to circulate and ask:



Indicators of
Understanding *



Explain how you determined the mixed number ____ could be written as
____?
Explain your strategy for determining the unknown.
Explain the procedure for multiplying fractions
Can represent a mixed number as an improper fraction and an improper
fraction as a mixed number.
Explains and demonstrates an understanding of the procedure for multiplying
fractions.
Explains a strategy for finding an unknown in an equation involving the
multiplication of fractions.
* Indicators of Understanding are in addition to the formative assessment at the end of each activity.
www.conceptuamath.com
© 2011 Conceptua Math LLC
10