This activity helps to transition students from thinking about “How many groups?” to “What fraction of a group?” Students make sense of division situations in which the number of groups could be greater than 1 or less than 1.

Given the amount of milk required for 1 batch of soup (or the size of 1 group in this situation), students find out how many batches or what fraction of a batch can be made with different amounts of milk. They write multiplication and division equations to represent the questions. To support their reasoning, students may also choose to use the partially started tape diagrams they previewed in the Warm-up.

As students work, monitor for various ways of reasoning and of marking up the diagrams. Select students who reason effectively and in different ways, and ask them to share later.

If students are unsure how to write equations for each situation, urge them to start with a multiplication equation. Remind them that a question such as “How many groups of 5 are in 20?” can be thought of as “What number times 5 is 20?” or . Ask them what the equal-size groups are in this situation. Once students see the questions as “How many groups of 9 are in some number?” and write multiplication equations, ask them: “How would you write a division equation that represents the same question?”

Ask previously identified students to share their responses (including diagrams) and to explain their reasoning. Display the diagrams for all to see.

To help students see the structure in the diagrams, ask: “How are the diagrams for Monday and Tuesday like the ones for Thursday and Friday? How are they different?” If not mentioned by students, point out that:

• The size of 1 group (or the amount of milk in 1 batch) is the same in every diagram, but the amount we are comparing to 1 batch varies. On Monday and Tuesday, that amount is each greater than 1 batch (9 cups). On Thursday and Friday, it is less than 1 batch (9 cups).
• Comparison to the size of 1 group is also reflected in the language used. The questions start with “How many batches?” for the first two days and “What fraction of a batch?” for the other two.

To help students notice the structure in the equations, display the four multiplication and division equations (showing the quotients).

Ask students: “What do the multiplication equations have in common? What about the division equations?” Highlight that:

• The four questions can also be phrased as “How many groups of 9 are in each amount of milk?” Represented as multiplication, they all have the structure of “What number times 9 equals a given amount of milk?” or .
• Whether the questions are asking “How many batches (of 9 cups)” or “What fraction of a batch (of 9 cups),” they can be expressed with a division by 9 or .

This activity offers students another opportunity to make sense of division situations in which the quotients can be either greater than 1 or less than 1. Students examine different lengths of ropes and compare their lengths relative to one another in multiplicative terms. For example, Ropes B and C are 5 and times as long as Rope A, respectively. Rope D, which is shorter than Rope A, is  times as long as Rope A.

To make such comparisons, students need to recognize that equal-size groups are also at play in this context, and that the size of 1 group is the length of one rope to which another rope is compared. As students work, monitor for students who make these connections.

In the digital version of the activity, students use an applet to compare the lengths of rope to one another. The applet allows students to move the ropes around and mark off increments. This activity works best when each student has access to the applet because students will benefit from being able to move the shorter ropes around as they compare them to the longer ones. If students don't have individual access, displaying the applet for all to see would be helpful during the synthesis.

Display the completed table for all to see. Give students a minute to check their answers and ask questions. Consider displaying the diagram and marking off the lengths of Rope A on each of the other ropes.

Next, discuss the connections between “How many times as long?” questions and “How many groups?” questions. Ask students:

• “How are these two types of questions alike?” (They both involve finding how many groups of one amount are in another amount. The number of groups may or may not be a whole number.)
• “How are they different?” (The equal-size groups here are lengths. The questions involve the phrase “times as long as.”)

Then discuss the values that complete the comparison statements. Ask students:

• “How are the values in the last two statements different from the first two?” (In the former two, each value is greater than 1. In the latter two, each value is less than 1.)
• “In this situation, what do the values that are less than 1 mean about Rope D?” (Rope D is shorter than the other ropes, so its length is a fraction of that of the other ropes.)

If time permits, discuss how the multiplication and division equations that students wrote represent comparisons between Rope D and Ropes A and C.

In this activity, students apply their insights from earlier activities to make sense of division questions without a context. While the questions begin with the phrase “what fraction of”— suggesting that the situations deal with an amount that is less than the size of 1 group, students still need to determine which of the two given values in each question represents the size of 1 group.

As students make sense of quantities described in words, represent them with diagrams and equations, and interpret numerical answers in terms of the questions, they reason abstractly and quantitatively (MP2).

In earlier tape diagrams, students saw that each square represented 1 unit. That scale worked well then and would also work fine for representing the first question here. If students default to using the same scale for the last question, however, the parts showing would be very small, making the diagram difficult to use. Urge students to think strategically when deciding the overall length or the scale to use so that the diagram can be helpful. Ask questions such as:

• “What is the smallest value you need to represent? What is the largest value?”
• “What should each grid square represent to best show those values? How many squares should represent 1 whole?”

Invite students to share their responses. Display their equations and diagrams for all to see. Discuss how the two given values and the answer to the question can be seen in each representation. Ask questions such as:

• “What do the quotients and mean in terms of the questions?” (There is of 9 in 3, or 3 is of 9.
  There is of 5 in , or is of 5.)
• “How can we check if the answers are correct?” (We can use multiplication: and )