Lesson 3

In this warm-up, students determine the number of dots in an image without counting and explain how they arrived at that answer. The goal is to prompt them to visualize and articulate different ways to decompose the dots, using what they know about arrays, multiplication, and area to arrive at the total number. To discourage counting and encourage students to focus on the structure of the dots, the image will be flashed and hidden a couple of times, rather than displayed the entire time.

Tell students to keep their materials closed. Explain that you will show an image that contains dots for 3 seconds. Their job is to determine how many dots there are and explain how they saw them.

Display the image for 3 seconds and then hide it. Do this twice. Give students up to 1 minute of quiet think time after each flash. Encourage students who have one way of seeing the dots to think about another way while they wait.

Select a couple of students to share the number of dots they saw. To focus students' reasoning on the structure of the dots, ask how they saw them, instead of how they found them. Record and display student explanations for all to see. Consider re-displaying the image to support students in their explanation. 

To involve more students in the conversation, consider asking:

• Who can restate the way ___ saw the dots in different words?
• Did anyone see the dots the same way but would explain it differently?
• Does anyone want to add an observation to the way ____ saw the dots?
• Who saw the dots differently?
• Do you agree or disagree? Why?

This activity allows students to draw diagrams and write equations to represent simple division situations. Some students may draw concrete diagrams; others may draw abstract ones. Any diagrammatic representation is fine as long as it enables students to make sense of the relationship between the number of groups, the size of a group, and a total amount.

The last question is likely more challenging to represent with a diagram. Because the question asks for the number of jars, and because the amount per jar is a fraction, students will not initially know how many jars to draw (unless they know what \(6\frac34 \div \frac34\) is). Suggest that they start with an estimate, and as they reason about the problem, add jars to (or remove jars from) their diagram as needed. 

As students work, monitor for the range of diagrams that students create. Select a few students whose work represent the range of diagrams to share later.

Students in digital classrooms can use an applet to make sense of the problems, but it is still preferable that they create their own diagrams. 

Select previously identified students to share their diagrams, sequenced from the more concrete (e.g., pictures of jars and cups) to the more abstract (e.g, rectangles, tape diagrams). Display the diagrams and equations for all to see. Ask them how they used the diagrams to answer the questions (if at all).

If tape diagrams such as the ones here are not already shown and explained by a student, display them for all to see. Help students make sense of the diagrams and connecting them to multiplication and division by discussing questions such as these:

• “In each diagram, what does the ‘?’ represent?” (The unknown amount)
• “What does the length of the entire tape represent?” (The total amount, which is sometimes known.)
• “What does each rectangular part represent?” (One jar)
• “What does the number in each rectangle represent?” (The amount in each jar)
• “How do the three parts of each multiplication equation relate to the diagram?” (The first factor refers to the number of rectangles. The second factor refers to the amount in each rectangle. The product is the total amount.) 
• “The last diagram doesn't represent all the jars and shows a question mark in the middle of the tape. Why might that be?” (The diagram shows an unknown number of jar, which was the question to be answered.)

Highlight that the last two situations can be described with division: \(7\frac12 \div 5\) and \(6\frac34 \div \frac34\).

In this activity, students continue to investigate division problems in context, think of them in terms of equal-size groups, and represent them using diagrams and equations. 

As students work, monitor the different diagrams and interpretations for each problem. Select students whose diagrams are especially clear in showing the meanings of division to share later.

Both division problems result in quotients that are not whole numbers. As students work, encourage them to use their multiplication equations to check the answers to the division problems.

Keep students in the same groups. Some students may not be familiar with granola and oats, so show or explain what they are. 

Give students 7–8 minutes of quiet work time, and 2–3 minutes to discuss their responses with a partner. During partner discussion, ask them to compare their equations and diagrams in the first question, and their interpretations of division in the second question.

Some students may round their answers to the nearest whole number rather than including the fractions of a scoop or a batch. Ask students to consider if it is possible to have a part of a scoop, a part of a batch, or a part of a unit of weight. Encourage them to think about how to show a part of a whole unit on a diagram.

The aim of the discussion is to solidify students' understanding of the two interpretations of division (“how many groups?” and “how much in a group?”). Ask students who drew effective diagrams to display and explain them. For each interpretation, write a multiplication equation and discuss what each factor represents in the context (e.g., the number of batches, scoops, or bags, vs. how much is in each batch, scoop, or bag).

If no students drew tape diagrams to represent the situations, show the following. Reiterate that each rectangle represents one group (one scoop or 1 bag), the number inside represents the amount in one group, and the number of rectangles tells us how many groups there are.