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, make an estimate based on the given context. 

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

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?”
• “For which problems was it helpful to draw a diagram?”

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

In this partner activity, students take turns making sense of and writing equations for a variety of situations involving fractions and all four operations. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

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 the reasonableness of their calculated answer in the context of the situation. The work here offers students opportunities to practice reasoning quantitatively and abstractly (MP2).

Much of the discussion takes place between partners.  Use this time to address common issues or misconceptions. Consider having the solutions accessible for students to check their answers.

Math Community

Conclude the discussion by inviting 2–3 students to share a norm they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm ‘’ in action today, and it really helped me/my group because .”

This activity gives students another opportunity to use what they have learned about operations with fractions to model and solve a problem in a recipe context. The question asks for the whole-number of batches that can be made with available ingredients. Because two known quantities don’t translate to the same number of batches, students need to consider how to find the number of batches that can be made and which ingredient limits what is possible. The reasoning here prompts students to make sense of the problem and persevere in solving it (MP1).

Students may approach the problem in different ways (such as by drawing diagrams, making computations, or reasoning verbally). Students may also choose different operations to obtain the information they need. For instance, instead of dividing the available amount of an ingredient by the amount in a batch, they may perform repeated subtraction. Monitor for the different methods that students use, and select strategies or explanations that should be shared with the class.

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

If time permits, reconvene for a whole-class discussion. Highlight a couple of strategies, and invite students to reflect on the effectiveness and efficiency of the strategies. For example, if some students performed repeated addition instead of multiplying (or repeated subtraction instead of dividing), ask students to discuss the efficiency of each operation and consider when one method might be preferred over the other.