Lesson 12

Fractional Lengths

12.1: Number Talk: Multiplication Strategies (5 minutes)

Warm-up

This number talk encourages students to think carefully about the numbers in a computation problem and rely on what they know about structure, patterns, and properties of operations to mentally solve it. The reasoning helps students develop fluency and will support students in calculating products and quotients in upcoming work.

Launch

Give students 2 minutes of quiet think time and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find the product mentally.

\(19\boldcdot 14\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

When multiplying \(19\boldcdot 14\), students may only multiply the tens digits and multiply the ones digits and add them to get 136. Ask these students to estimate an answer for the problem and consider whether their answer makes sense.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Ask students if or how the factors in the problem impacted their strategy choice. To involve more students in the conversation, consider asking:

  • “Who can restate ___’s reasoning in a different way?”
  • “Did anyone solve the problem the same way but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to _____’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

12.2: Info Gap: How Many Would It Take? (15 minutes)

Activity

In this activity, students use division to solve problems involving lengths. No methods are specified for any of the questions, so students need to choose an appropriate strategy. 

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for 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). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Four info gap cards.

Launch

Arrange students in groups of 2. In each group, distribute the first problem card to one student and a data card to the other student. After debriefing on the first problem, distribute the cards for the second problem, in which students switch roles.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information needed to solve problems involving lengths using division. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation

Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

  1. Silently read your card and think about what information you need to be able to answer the question.

  2. Ask your partner for the specific information that you need.

  3. Explain how you are using the information to solve the problem.

    Continue to ask questions until you have enough information to solve the problem.

  4. Share the problem card and solve the problem independently.

  5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

  1. Silently read your card.

  2. Ask your partner “What specific information do you need?” and wait for them to ask for information.

    If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.

  3. Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.

  4. Read the problem card and solve the problem independently.

  5. Share the data card and discuss your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Lin has a work of art that is \(14\) inches by \(20\) inches. She wants to frame it with large paper clips laid end to end.

  1. If each paper clip is \(1\frac34\) inch long, how many paper clips would she need? Show your reasoning and be sure to think about potential gaps and overlaps. Consider making a sketch that shows how the paper clips could be arranged. 
  2. How many paper clips are needed if the paper clips are spaced \(\frac14\) inch apart? Describe the arrangement of the paper clips at the corners of the frame.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may not know what is meant by the “spine” of a book. Consider holding up a book and pointing out where its spine is.

If students struggle to represent the situations mathematically, suggest that they draw diagrams to represent the situations. They could start with sketches of the objects and then move toward other simpler or more abstract representations as they make better sense of the problems.

Activity Synthesis

Select previously identified students to share their solutions and reasoning for each question. Start with students who used the more involved methods and move toward the more efficient ones. Record the approaches for all to see.  

Highlight the connections between the different methods (e.g., between diagrams and equations, between a multiplication equation one student wrote and a division equation another person wrote for the same situation, etc.).

12.3: How Many Times as Tall or as Far? (15 minutes)

Activity

In this activity, students practice performing division of fractions and using it to solve multiplicative comparison problems. The activity extends the work students have done earlier in the unit. In the Fractions of Ropes activity (in Lesson 7), students use diagrams to reason about how many times as long one rope is compared to another. Having had more experience in interpreting division situations and having learned a division algorithm, students can solve a wider range of problems that involve a greater variety of fractions. Minimal scaffolding is given here, so students need to decide what representations or strategies would be fruitful.

Launch

Arrange students in groups of 4. Give students 1–2 minutes of quiet time to think about and draw a diagram for each question in the first problem. Ask them also to think about and write two equations that can represent the two questions. Afterward, give each group 2 minutes to compare their diagrams and equations.

Invite a couple of students to share their diagrams and equations. Ask the class whether they agree that the diagrams and equations represent the questions. Once students agree that the representations are appropriate, give them 8–10 minutes to complete the activity, either independently or collaboratively with their group. Encourage students to estimate the answer before calculating and to check their quotients using multiplication.

Reading, Conversing, Writing: MLR5 Co-Craft Questions. To help students make sense of the language of mathematical comparisons, start by displaying only the context for the first question (“A second-grade student is 4 feet tall. Her teacher is \(5\frac23\) feet tall.”). Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the activity’s questions. Help students notice similarities and differences in how they phrased comparison questions and how the activity phrases these questions. For example, “Who’s taller?” versus “How many times as tall . . . ." This will build student understanding of the language of mathematical comparisons and help ensure students interpret the task correctly.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

Student Facing

  1. A second-grade student is 4 feet tall. Her teacher is \(5\frac23\) feet tall.

    1. How many times as tall as the student is the teacher?
    2. What fraction of the teacher’s height is the student’s height?
  2. Find each quotient. Show your reasoning and check your answer.

    1. \(9 \div \frac35\)
    2. \(1\frac78 \div \frac 34\)
  3. Write a division equation that can help answer each of these questions. Then find the answer. If you get stuck, consider drawing a diagram.

    1. A runner ran \(1\frac45\) miles on Monday and \(6\frac{3}{10}\) miles on Tuesday. How many times her Monday’s distance was her Tuesday’s distance?
    2. A cyclist planned to ride \(9\frac12\) miles but only managed to travel \(3\frac78\) miles. What fraction of his planned trip did he travel?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students have trouble drawing and using a diagram to compare lengths, ask them to revisit the Fractions of Ropes activity (in Lesson 7) and use the diagrams there as examples. Suggest that they try drawing a diagram on graph paper, as the grid could support them in drawing and making sense of the fractional lengths.

Activity Synthesis

Consider giving students access to the answers so they can check their work. Much of the discussion will have happened in small groups. If time permits, reconvene as a class to discuss the last set of questions and the different ways they were represented and solved. 

12.4: Comparing Paper Rolls (15 minutes)

Optional activity

This optional activity gives students another opportunity to solve a contextual problem using what they know about fractions, relationships between multiplication and division, and diagrams. Students observe a photograph of two paper rolls of differing lengths and estimate the relationship between the lengths. The photograph shows that the longer roll is about \(2\frac12\) or \(\frac52\) times as long as the shorter roll. Students use this observation to find out the length of the shorter roll.

The two paper rolls are from paper towels and toilet paper. If possible, consider providing one of each roll to each group of students so they can physically compare their lengths in addition to observing the picture.

As students work, notice the different starting equations or diagrams they use to begin solving the problems. Ask students using different entry points to share later.

Launch

Keep students in groups of 4. Ask students to keep their materials closed. Display the image of the paper rolls for all to see. Give students 1–2 minutes to observe the picture. Ask them to be prepared to share at least one thing they notice and one thing they wonder. Then, invite students to share their observations and questions. If no students mention the relationship between the lengths of the rolls, ask them questions such as:

  • “What do you notice about the lengths of the paper rolls or the relationships between those lengths?”
  • “What questions can you ask about the lengths of the rolls?”
  • “What information would you need to answer these questions?”

Then, give students 7–8 minutes of quiet time to complete the questions.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to include a table with the following headings: What do you notice?,  What do you wonder?, What information do you need to answer these questions?
Supports accessibility for: Language; Organization

Student Facing

The photo shows a situation that involves fractions.

3 short cardboard tubes lined up horizontally. 1 long cardboard tube below. Long tube starts at the same spot as the first short tube and ends about half way down the third short tube. 
  1. Complete the sentences. Be prepared to explain your reasoning.

    1. The length of the long tube is about ______ times the length of a short tube.

    2. The length of a short tube is about ______ times the length of the long tube.

  2. If the length of the long paper roll is \(11 \frac 14\) inches, what is the length of each short paper roll?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students might estimate the relationships between the lengths of rolls by rounding too much. For example, they might say that the length of the shorter roll is \(\frac{1}{3}\) the length of the longer roll, or that the longer roll is twice as long as the shorter roll. If this happens, ask students to take a closer look and make a more precise estimate. Suggest that they divide the larger roll into smaller segments, each of which matches the length of the shorter rolls.

Activity Synthesis

Invite previously identified students to share their strategies for finding the length of the short roll. Display their diagram and record their reasoning for all to see.

To find the length of the short roll, some students may use \(11 \frac 14 \div \frac 52 = ?\) and others \(\frac 25 \boldcdot \frac{45}{4} = ?\), depending on how they view the relationship between the rolls. Highlight the idea that to find the length of the short roll, one way is to partition the length of the large roll into 5 equal pieces and find that length and multiply it by 2, because the length of the shorter roll is about \(\frac25\) of that of the longer roll. This is an opportunity to reinforce the structure behind the division algorithm.

If appropriate, discuss the merits of writing the numbers as fractions versus as mixed numbers. Mixed numbers such as \(2\frac12\) and \(11\frac14\) are easier to visualize but \(\frac52\) and \(\frac{45}{4}\) are easier to work with. In fact, we have to use the fractions (rather than mixed numbers) to easily multiply. Explain that two forms serve different purposes and that it is helpful to be able to change from one to the other depending on what we aim to do. When writing them as solutions to problems, both forms are mathematically correct.

Writing, Conversing: MLR3 Clarify, Critique, Correct. Use this routine to help students evaluate the reasonableness of their answers. Display the following incorrect response: “The length of each short paper roll is \(28\frac{1}{8}\) inches because I multiplied \(\frac{45}{4}\boldcdot\frac{5}{2}\).” Share with students that \(\frac{45}{4}\boldcdot\frac{5}{2}\) is indeed \(28\frac{1}{8}\). Give students 1 minute of quiet think time to consider this information. Ask students to discuss with their partner why the answer \(28\frac{1}{8}\) doesn’t make sense (the length of the short paper roll should be less than the length of the long paper roll). Next, ask students to identify why the equation \(\frac{45}{4}\boldcdot\frac{5}{2} = ?\) does not correctly represent this situation. Listen for and amplify corrections that include mathematical language and reasoning.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

In this lesson, we used division to solve problems that involve fractional lengths. For example: How many \(\frac58\)-inch paper clips, laid end to end, are in a length of \(12\frac12\) inches? Review how we can interpret such problems.

  • “How is this question like those we have seen? How can division help us answer it?” (It is a ‘how many groups?’ question. We can think of it as ‘how many \(\frac58\)s in \(12\frac12\)?’ and solve it by finding \(12\frac12 \div \frac58\).)
     
  • “Here is another question: ‘What is the length of one stick if 9 sticks, laid end to end, make \(12\frac38\)-inch?’ How does division help us answer it?” (It is a ‘how much in one group?’ question. We can answer it by finding \(12\frac 38 \div 9\).)

We also saw that division can help us compare two lengths and find out how many times one is as long as the other. For example, suppose one hiking trail, Trail A, is \(1\frac18\) miles and another, Trail B, is \(\frac34\) miles.

  • “How do we find out how many times as long as Trail A is Trail B?” (We can interpret the question as “___ times the length of A equals the length of B” or \({?} \boldcdot 1\frac18 = \frac34\), and then find \(\frac34 \div 1\frac18\).)
     
  • “What is another comparison question we could ask?” (How many times as long as Trail B is Trail A?)
     
  • “How do we represent and answer that question?” (We can interpret it as “___ times the length of B is the length of A” or \({?} \boldcdot \frac34 = 1\frac18\), and then find \(1\frac18 \div \frac34\).)

12.5: Cool-down - Building A Fence (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Division can help us solve comparison problems in which we find out how many times as large or as small one number is compared to another. For example, a student is playing two songs for a music recital. The first song is \(1\frac12\) minutes long. The second song is \(3 \frac34\) minutes long.

Double number line. Top line, first song. 1 and the fraction 1 over 2 units long. Labeled 1 and the fraction 1 over 2 minutes. Bottom line, second song. 3 and the fraction 3 over 4 units long. Labeled 3 and the fraction 3 over 4 minutes.  

We can ask two different comparison questions and write different multiplication and division equations to represent each question.

  • How many times as long as the first song is the second song?

\(\displaystyle {?} \boldcdot 1\frac12 = 3\frac 34\)

\(\displaystyle 3 \frac 34 \div 1\frac 12 = {?}\)

  • What fraction of the second song is the first song?

\(\displaystyle {?} \boldcdot 3\frac 34 = 1\frac 12\)

\(\displaystyle 1\frac12 \div 3\frac34 = {?}\)

We can use the algorithm we learned to calculate the quotients.

\(\displaystyle \begin {align} &= \frac {15}{4} \div \frac 32\\[10px] &= \frac {15}{4} \boldcdot \frac 23\\[10px] &=\frac {30}{12}\\[10px]&=\frac {5}{2}\\[10px] \end {align}\)

This means the second song is \(2\frac 12\) times as long as the first song.

\(\displaystyle \begin {align} &=\frac 32 \div \frac {15}{4}\\[10px] &=\frac 32 \boldcdot \frac {4}{15}\\[10px] &=\frac {12}{30}\\[10px] &=\frac {2}{5} \end {align}\)

This means the first song is \(\frac 25\) as long as the second song.