Lesson 12

This warm-up reinforces students’ understanding of what each of the four operations (addition, subtraction, multiplication, and division) does when performed on fractions. The same pair of fractions are used in each problem so that students can focus on the meaning of the operation rather than on the values. 

Arrange students in groups of 2. Display problems for all to see. Give students 1 minute of quiet think time.

Tell students not to calculate exact values of the expressions. Ask them to estimate the value of each expression by reasoning about the operation and the fractions, and then put the expressions in order based on their values, from least to greatest. Ask students to give a signal as soon as they have determined an order and can support it with an explanation.

Give students 1 minute to discuss their reasoning with a partner and agree on a correct order.

Some students may think the division expression would have the lowest value because they still assume that division always produces a number that is less than the dividend. This is not true for division by a positive number less than 1, which is the case here. If this misconception arises, consider addressing it during whole-class discussion.

Ask 1–2 groups to share how they ordered their expressions from least to greatest. If everyone agrees on one answer, ask a few students to share their reasoning. Record it for all to see. If there are disagreements, ask students with opposing views to explain their reasoning and discuss it to reach an agreement on a correct order.

This activity offers an additional opportunity for students to make sense of word problems, set up an appropriate representation, use that representation for reasoning, and estimate before solving. Students are presented with four situations that involve only fractions. Two of them require multiplication to solve, and the other two require division. Students decide which operation is needed to answer each question, and before solving, estimate the answer based on the given context.

As students work, monitor how they determine appropriate operations to use. Note any common challenges so they can be discussed later.

Keep students in groups of 2. Explain to students that the situations presented in this activity all involve the same two fractions, but they do not all require the same operation to solve. Encourage them to make sense of each situation carefully before calculating or reasoning about the answer. Provide access to geometry toolkits (especially graph paper and colored pencils).

Give students 8–10 minutes to work on the activity either individually or with their partner, and then some time to discuss or check their responses. If time is limited, consider asking students to answer either the first two or the last two questions. 

Display the solutions for all to see and give students time to check their work. If time permits, discuss students’ reasoning. Ask:

• “How did you estimate the answers?”
• “How did you know what operation you needed to perform to find the answer?”
• “For which problems was it difficult to tell what operation to use?” 
• “Did you draw diagrams or write equations? What diagrams or equations were helpful?”

Some students may notice that the second and third questions involve the phrases “how many times?” and “what fraction of,” which suggest that division might be involved. Ask them to identify the size of 1 group in those cases.

This activity prompts students to make sense of and write equations for a variety of situations involving fractions and all four operations. After writing equations, students are assigned two problems to solve, at least one of which is a division problem. Before calculating, students first estimate their answer. Doing so helps them to attend to the meaning of the operation and to the reasonableness of their calculated answer in the context of the situation.

Give students a minute to skim the two sets of problems. Ask them to be prepared to share at least one thing they notice and one thing they wonder. Then, invite a few students to share their observations and questions.

Keep students in groups of 2. Tell students they will practice writing equations to represent situations in context. Ask one person to write an equation for each question labeled with a letter and the number 1, and the other person to do the same for each question labeled with a letter and the number 2. Give groups 4–5 minutes to write their equations, and another 4–5 minutes to check each other’s equations and discuss any questions or issues.

Afterward, briefly discuss and compare the equations as a class. Point out equations that correctly represent the same problem (and are thus equivalent) but are expressed differently. For example, a student may write a multiplication equation with a missing factor, while another writes a division equation with an unknown quotient.

Next, assign at least 1 division problem and 1 problem involving another operation for each student (or group) to solve. Consider preparing the assignments, or an efficient way to assign the problems, in advance. Give students 4–5 minutes of quiet work time or collaboration time.

Much of the discussion will have occurred in small groups, so a whole-class discussion is not essential unless there are common issues or misconceptions to be addressed. Consider having the solutions accessible for students to check their answers.

This optional activity gives students another opportunity to use what they have learned about all operations to model and solve a problem in a baking context. Students need to make sense of the problem and persevere in solving it (MP1).

Students may approach the problem in different ways (by drawing diagrams, making computations, reasoning verbally, etc.). Students may also choose different operations to obtain the information they need. For instance, instead of dividing by a fraction, they may perform repeated subtraction. Notice the different methods students use and identify strategies or explanations that should be shared with the class.

Consider arranging students in new groups of 2. Give students 7–8 minutes of quiet work time and then 1–2 minutes to discuss their response with their partner. Ask students to be prepared to explain their reasoning.

Consider combining every 2–3 groups of 2 students and having students discuss their responses and methods in larger groups of 4–6.

If time permits, reconvene as a class to highlight a couple of strategies and reflect on the effectiveness and efficiency of students’ strategies. For example, if some students performed repeated addition instead of multiplying (or repeated subtraction instead of dividing), ask if repeated addition (or subtraction) is always as efficient as multiplication (or division), or under what circumstances one method might be preferred over the other.