In this activity, students use division to solve problems involving lengths but do not initially have enough information to do so. To bridge the gap, they need to exchange questions and ideas.

The Information 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 to ask increasingly more precise questions until they get the information they need (MP6).

After students have completed their work, share the correct answers, and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “How are the situations about stickers and binder clips alike?” (They are both about finding how many of a fraction are in a larger number. They both involve lengths. The answers are both whole numbers.)
• “How are the problems different?” (One dividend is not a whole number and the other one is.)
• “Was your strategy for finding different from the one you used to find ? Why or why not?”

Highlight for students that we can reason about equal-size groups in the context of lengths using the same representations and strategies as used in other contexts. In some cases, it might be easier to reason in terms of multiplication, such as by thinking. “What number times is 20, or ?” Other times, it might be quicker to draw a tape diagram or use an algorithm.

In this optional activity, students practice dividing fractions as they solve multiplicative comparison problems. The reasoning here builds on the work in an optional activity earlier in the unit, “Fractions of Ropes.” There, students used diagrams to reason about how many times as long one rope is compared to another. Here, students can decide what representations or strategies would be fruitful. If needed for scaffolding, consider facilitating that activity before this one.

As students create expressions, equations, or diagrams to make sense of quantities and answer questions in context, they practice reasoning abstractly and quantitatively (MP2).

Students may be unsure how to use the algorithm to divide two mixed numbers. Ask them to recall or revisit how they divided a mixed number by a fraction or by a whole number in earlier activities. Remind them that each mixed number could be written as a fraction with only a numerator and denominator.

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

This activity gives students another opportunity to solve a contextual problem using what they know about fractions, the relationship 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  or () 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, monitor for the different starting equations or diagrams that they use to begin solving the last problem. Ask students who use different entry points to share later.

Select 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 and others , 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, find that length, and multiply it by 2, because the length of the shorter roll is about of that of the longer roll. This is an opportunity to reinforce the structure behind the division algorithm.