The purpose of this activity is for students to use an approach that makes sense to them to solve a division problem. Students may apply understanding developed in a previous course about place value and the relationship between multiplication and division, including using partial quotients to divide. They also may apply work from the previous section in which they multiplied, using the standard algorithm.

When students connect the quantities in the story problem to calculations, including the operations of multiplication and division, they reason abstractly and quantitatively (MP2). Students determine the number of groups of 8 people that participated in the record-breaking folk dance. The numbers and context were chosen to encourage students to consider what they know about the meaning of division and to use multiplication to solve the problem.

Monitor for and select students, with the following approaches, to share in the Activity Synthesis:

• Multiply 8 by 100, or by multiples of 100, to get close to 4,704, and then multiply 8 by multiples of 10 and one-digit numbers to find the solution.
• Use a partial-quotients strategy to divide 4,704 by 8.

The approaches are sequenced in order of the ways students have made sense of and solved “how many groups?” situations in previous grades and units. Multiplication is the natural approach many students take when making sense of this situation. When students compare these two approaches side by side in the Activity Synthesis, they make sense of both approaches and deepen their understanding of the relationship between multiplication and division. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For an example of each approach, look at the Student Responses for the last problem.

• Invite previously selected students to share in the given order. Record or display their work for all to see.
• As needed, display the equations or the diagrams from the Student Responses to support students making connections.
• Connect students’ approaches by asking:
  • “¿Cómo están representados los grupos de 8 personas en cada una de estas formas de resolver el problema?” // “How are the groups of 8 people represented in each of these ways of solving the problem?” (In the multiplication equations, we see one factor is 8 and the other factor shows a number of groups of dancers. In the division work, it shows that we’re dividing by 8 to find the number of groups.)
  • “¿Cómo están representadas las 4,704 personas?” // “How are the 4,704 people represented?” (If we add up the partial products in the multiplication work, the sum is 4,704. We can see in the division work that we are dividing 4,704 by 8 and subtracting the partial product each time.)
  • “¿En qué se parecen el trabajo en el que se usa multiplicación y el trabajo en el que se usa división? ¿En qué son diferentes?” // “How is the work that uses multiplication and the work that uses division the same? How are they different?” (They both show the number of groups of 8 people. They show the partial products, but the division shows them being subtracted to see how many people are left after some groups of 8 were made.)
• Connect students’ approaches to the learning goal by asking:
  • “¿Por qué podemos resolver este problema pensando en la multiplicación o pensando en la división?” // “Why are we able to solve this problem by thinking about multiplication or by thinking about division?” (The 4,704 dancers were in groups of the same size and the question was “how many groups?” Multiplication and division are related. You can use both to think about how to find the total number of equal groups.)

The purpose of this activity is for students to solve division problems, using the context from the first activity, in a way that makes sense to them. The sample student solutions for the problems in this activity highlight certain numbers by which to multiply and divide, but students may use multiplication or division in various ways. During the Activity Synthesis, highlight the different ways students used multiplication and division to solve the problems, and focus on the relationship between multiplication and division. When students relate the different numbers of groups of dancers to the different numbers of dancers in each group, they observe structure in the relationship of the quotient to the size of the divisor (MP7).

• Display the expressions: , , and
• “¿En qué se parecen estas expresiones? ¿En qué son diferentes?” // “How are the expressions the same? How are they different?” (All have 4,704 in them, but the divisor is 8, then 4, and then 2.)
• “¿Cómo está representada la situación en cada expresión?” // “How does each expression represent the situation?” (There are always 4,704 dancers, but the size of the groups changes.)
• “¿Cómo se relacionan los valores de las expresiones? ¿Por qué?” // “How are the values of the expressions related? Why?” (When there are 4 dancers in each group there are twice as many dancers as when there are 8 in each group because each group of 8 makes two groups of 4. When there are 2 dancers in each group there are twice as many groups as when there are 4 in each group because each group of 4 makes two groups of 2.)