The purpose of this What Do You Know about _____? is for students to share what they know about a sum of fractions. The fractions are selected because they represent whole numbers and the whole-number values are visible. Students will work with expressions such as these throughout this lesson.
Launch
Display the number.
“What do you know about ?”
1 minute: quiet think time
Activity
Record responses.
Student Task Statement
What do you know about ?
Student Response
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Advancing Student Thinking
Activity Synthesis
“What are some expressions that have the same value as ?” (, 11, )
Activity 1
15 mins
Select Expressions
Standards Alignment
Building On
5.NF.B.3
Interpret a fraction as division of the numerator by the denominator . 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 as the result of dividing by , noting that multiplied by equals , and that when wholes are shared equally among people each person has a share of size . If people want to share a -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?
The purpose of this activity is for students to relate their understanding of fractions as representing division to think about decomposing a quotient into partial quotients in a way that simplifies the calculation. To find the value of , students may use:
Their understanding of division.
Multiplication and find how many groups of 6 are in 78.
The fraction expressions from the first part of the problem.
Launch
Groups of 2
Activity
5–8 minutes: partner work time
Monitor for students who use:
Multiplication to find the value of .
The expression to find the value of .
The expression to find the value of .
Activity Synthesis
Invite students to share the expressions that match .
Display:
“How do we know this is true?” (A fraction shows that we are dividing the numerator by the denominator.)
Display:
“How can you use this equation to find the value of ?” (I know is 10 and is 3, so is 13.)
“In the next activity, we will use expressions with fractions to find values of other quotients.”
Activity 2
20 mins
Choose One Expression
Standards Alignment
Building On
5.NF.B.3
Interpret a fraction as division of the numerator by the denominator . 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 as the result of dividing by , noting that multiplied by equals , and that when wholes are shared equally among people each person has a share of size . If people want to share a -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?
The purpose of this activity is for students to find the whole-number values of quotients, using sums of fractions, and to think about which sums were most helpful. They may notice that it is helpful to decompose the dividend into a multiple of the divisor, and that multiples of 10 are particularly helpful. This is closely related to how students found quotients, using partial products, which requires strategically choosing the number of groups of the divisor to subtract.
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.
Representation: Internalize Comprehension. Invite students to identify which details were most useful to solve the problem. Display the sentence frame: “The next time the dividend is not divisible by the divisor, I will look for multiples of 10, or multiples of the divisor, to help me divide more efficiently.“ Supports accessibility for: Conceptual Processing, Memory, Organization
Launch
Groups of 2
Display the expressions:
“Which expression would you use to find the value of ?” (The first expression, because the fractions have nice whole-number values.)
1–2 minutes: partner discussion
“You are going to choose expressions such as this, which are helpful for finding quotients.”
Activity
5–8 minutes: independent work time
1–2 minutes: partner discussion
Monitor for students who explain that:
Numerators that are multiples of the divisor are helpful to divide.
Numerators that are not multiples of the divisor require working with fractions.
Student Task Statement
Use each expression to find the value of . Show your thinking. Organize your work so it can be followed by others.
Choose one of these expressions to find the value of . Show your thinking. Organize your work so it can be followed by others.
Which expressions are most helpful? Which expressions are least helpful? Explain or show your reasoning.
Student Response
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Advancing Student Thinking
Activity Synthesis
MLR1 Stronger and Clearer Each Time
“Share with your partner your response as to why some expressions were helpful and others were not. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
3–5 minutes: structured partner discussion.
Repeat with 2–3 different partners.
(Optional) If needed, display question starters and prompts for feedback.
“Can you give an example to help show . . . ?”
“Can you use the word _____ in your explanation?”
“The part I understood best was . . . .”
“Revise your initial draft, based on the feedback you got from your partners.”
2–3 minutes: independent work time
Lesson Synthesis
Display the expression:
“How do we know this expression has the same value as ?” (, and they are 18ths.)
“How can we use this expression to find the value of ?” ( and there are three of them, so the value of is 30.)
Display the expression:
“How can we use this expression to help us find the value of ?” (, so , and , and .)
Display the expression:
“How do we know this expression has the same value as ?” (, and they‘re 18ths)
“Why is this expression not as helpful as the others?” (The values of those fractions are not whole numbers so we have to calculate, using fractions.)
Standards Alignment
Building On
5.NF.B.3
Interpret a fraction as division of the numerator by the denominator . 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 as the result of dividing by , noting that multiplied by equals , and that when wholes are shared equally among people each person has a share of size . If people want to share a -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?
Select all expressions that have the same value as . Explain or show your reasoning.
What is the value of ? Show your thinking. Organize your work so it can be followed by others.
Student Response
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Advancing Student Thinking
Addressing
Building Toward
Addressing
5.OA.A.2
Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add and , then multiply by ” as . Recognize that is three times as large as , without having to calculate the indicated sum or product.
