Lesson 5

In this warm-up, students continue to think of division in terms of equal-sized groups, using fraction strips as an additional tool for reasoning.

Notice how students transition from concrete questions (the first three) to symbolic ones (the last three). Framing division expressions as “how many of this fraction in that number?” may not yet be intuitive to students. They will further explore that connection in this lesson. For now, support them using whole-number examples (e.g., ask: “how do you interpret \(6 \div 2\)?”).

The divisors used here involve both unit fractions and non-unit fractions. The last question shows a fractional divisor that is not on the fraction strips. This encourages students to transfer the reasoning used with fraction strips to a new problem, or to use an additional strategy (e.g., by first writing an equivalent fraction).

As students work, notice those who are able to modify their reasoning effectively, even if the approach may not be efficient (e.g., adding a row of \(\frac {1}{10}\)s to the fraction strips). Ask them to share later.

Since the fraction strips do not show tenths, students might think that it is impossible to answer the last question. Ask them if they can think of another fraction that is equivalent to \(\frac{2}{10}\).

For each of the first five questions, select a student to share their response and ask the class to indicate whether they agree or disagree.

Focus the discussion on two things: how students interpreted expressions such as \(1 \div \frac26\), and on how they reasoned about \(4 \div \frac {2}{10}\). Select a few students to share their reasoning.

For the last question, highlight strategies that are effective and efficient, such as using a unit fraction that is equivalent to \(\frac {2}{10}\), finding out how many groups of \(\frac15\) are in 1 and then multiplying it by 4, etc.

This activity serves two purposes: to explicitly bridge “how many of this in that?” questions and division expressions, and to explore division situations in which the quotients are not whole numbers. (Students explored similar questions previously, but the quotients were whole numbers.)

Once again students move from reasoning concretely and visually to reasoning symbolically. They start by thinking about “how many rhombuses are in a trapezoid?” and then express that question as multiplication
(\(? \boldcdot \frac23 = 1\) or \(\frac 23 \boldcdot \,? = 1\)) and division (\(1 \div \frac23\)). Students think about how to deal with a remainder in such problems.

As students discuss in groups, listen for their explanations for the question “How many rhombuses are in a trapezoid?” Select a couple of students to share later—one person to elaborate on Diego's argument, and another to support Jada's argument.

Some students may not notice that in this task, the trapezoid—not the hexagon—represents 1 whole. Encourage them to revisit the task statement to check.

Focus the whole-class discussion on the last two questions, especially on how the visual representation helps us reason about Jada and Diego's points of view, and on the connections between the verbal and numerical representations of the situation.

Select two previously identified students to explain why Diego or Jada are correct. Display a visual representation of “how many rhombus are in a trapezoid?” for all to see (as shown here), or use the applet at https://ggbm.at/VmEqZvke for illustration.

To highlight number of groups and size of one group in the problem, discuss questions such as:

• “This is a ‘how many groups of this in that?’ question. What makes 1 group, in this case?” (One rhombus.)
• “How do we know whether to compare the remainder to the rhombus or the trapezoid?” (Since a rhombus makes 1 group, we need to compare the remainder to the rhombus.)

If students struggle to compare the remainder to a rhombus, ask: “How many triangles are in a trapezoid?” and point out that the answer is “3 triangles.” Here, the answer to “how many rhombuses are in a trapezoid?” would be “(some number of) rhombuses.”

The fact that there are two 1 wholes to keep track of may be a source of confusion (the trapezoid represents the quantity 1 and the rhombus represents 1 group). Students will have opportunities to make clearer distinctions between the two 1 wholes in upcoming activities.

This activity gives students additional practice in using diagrams and equations to represent division situations involving whole numbers and fractions.

For each problem, many kinds of visual representations are possible, but creating a meaningful representation may be challenging nonetheless. Urge students to use the contexts to generate ideas for useful diagrams, and to start with a draft and modify it as needed. Students may also use the fraction strips in the warm-up as a starting point for drawing diagrams.

As students work and discuss, monitor for effective diagrams or those that can be generalized to different situations (e.g., rectangles, tape diagrams, and number lines). Assign one problem for each group to record on a visual display. 

Arrange students in group of 3–4. Give students 8–10 minutes of quiet work time and a few minutes to share their responses with their group. Tell each group they will be asked to present their solution to one problem. Provide access to geometry toolkits and tools for creating a visual display.

During group discussion, ask students to exchange feedback on each other’s diagrams and to notice any that might be particularly effective, efficient, or easy to understand. Then, they should record the diagram and equations for their assigned problem on a visual display and be prepared to explain them.

Students may mistake the divisor and the dividend in the problems. Ask students to discuss (in their groups) the number or quantity being divided, and the reasonableness of the different ways of setting up each problem given the context. Representing the situations with objects may also help.

Invite each group to present the solution on their visual display. Ask the rest of the class to think about two things: whether the equations make sense, and how the presented diagram shows the number of groups, the size of each group, and a total amount. Doing so will help students see the structure of the problems in the equations and diagrams.

Make sure students understand how each situation can be expressed by multiplication and division equations. Students should recognize that a question such as “how many batches are in 4 cups if each batch requires \(\frac23\) cups?” can be written as both \(? \boldcdot \frac 23 = 4\) and \(4 \div \frac 23 = \,?\). With repeated reasoning, they see that a division expression such as \(5 \div \frac 38\) can be interpreted as “how many \(\frac38\)s are in 5?”