This Number Talk encourages students to think about the relationship between division and fractions and the order of operations in order to strategically multiply whole numbers by fractions. The strategies elicited here will be helpful later in the lesson when students find the missing value in multiplication equations for a whole number and a fraction.
To use the properties of operations, students need to look for and make use of structure (MP7). In explaining strategies, students need to be precise in their word choice and use of language (MP6).
Launch
Display one problem.
“Give me a signal when you have an answer and can explain how you got it.”
1 minute: quiet think time
Activity
Record answers and strategy.
Keep problems and work displayed.
Repeat with each problem.
Student Task Statement
Find the value of each expression mentally.
Student Response
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Advancing Student Thinking
Activity Synthesis
“How can rearranging the numbers and operations help find the value of the last expression?” (I know that is 2 so finding this first and then multiplying by 10 is easier than trying to work with the fraction .)
Activity 1
20 mins
Info Gap: Area
Standards Alignment
Building On
Addressing
5.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
This Info Gap activity gives students an opportunity to determine and request the information needed to solve multi-step problems involving multiplication of a whole number by a fraction. In each case, there are multiple ways to solve the problem but the information card does not have the one piece of information—the missing side length—that the student with the problem card could use to find the area. The cards are designed this way to encourage the students to discuss and communicate further.
This activity uses MLR4 Information Gap.
The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then ask for the information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). Moreover, they are prompted by their partner to explain why they need each piece of information so they need to articulate their strategy and reasoning. The Info Gap structure also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Launch
Groups of 2
MLR4 Information Gap
Recall, if necessary, the steps of the Info Gap routine.
“I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.”
Distribute the first set of cards.
Remind students that after the person with the problem card asks for a piece of information, the person with the data card should respond with: “Why do you need to know _____ [that piece of information]?”
Activity
4–6 minutes: partner work time
After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.
Student Task Statement
Your teacher will give you either a Problem Card or a Data Card. Do not show or read your card to your partner.
Information Gap routine. Step 1, both students read the problem card. Cycle. Problem card student says, can you tell me, fill in the blank? Data card student says, why do you need to know, fill in the blank? Problem card student says, I need to know, fill in the blank, because dot, dot, dot. Data card student listens to partner's reason and answers with information from the data cards. Repeat cycle until the problem card student can state, I have enough information to solve this problem. Both solve the problem independently and continue to ask questions if more information is needed. Finally, share data card, then compare strategies and solutions.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards. Repeat the activity, trading roles with your partner.
Activity Synthesis
Select 2–3 students to share the correct answers and discuss the process of solving the problems.
“What questions did you ask to get the information you needed to solve the problems?” (I was missing just one piece of information—the width of the paper towel or the length of the corn field. I asked for this information but my partner did not have it. I was stuck and then my partner said they had information comparing the length and width. I was able to use that to find the width or length and then the area.)
Activity 2
Optional
15 mins
Fill in the Blank
Standards Alignment
Building On
Addressing
5.NF.B
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
The purpose of this activity is for students to find the missing value in equations involving a whole number and a fraction. They find missing factors or products to make equations true. Many of the problems encourage students to think in steps, interpreting a fraction in terms of division by the denominator and multiplication by the numerator.
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most important to solve the problems. Display the sentence frame: “The next time I solve multiplication problems of a whole number by a fraction, I will pay attention to . . . .“ Supports accessibility for: Conceptual Processing, Memory
Launch
Groups of 2
Activity
5–8 minutes: independent work time
1–2 minutes: partner discussion
Monitor for:
Partners who use different strategies.
Students who are challenged but persevere to arrive at a viable solution.
Student Task Statement
Fill in the blanks to make each equation true.
Activity Synthesis
Ask previously selected students to share their responses.
“What was different about you and your partner's strategies?”
“Which problem did you find most challenging and why?”
“Which problem made the most sense to you and why?”
Lesson Synthesis
“Today we multiplied whole numbers by fractions greater than 1 in different forms.”
“We have learned a lot about multiplying a whole number by a fraction or mixed number. What have you learned? What do you still wonder?”
Consider giving students time to respond in their journals before sharing.
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