Lesson 10

The purpose of this warm-up is to introduce the context of a world record event about the largest Peruvian folk dance, which will be useful when students solve problems about this event in the lesson. While students may count many things in the image, the number of groups of 8 people is the important discussion point.

• Groups of 2
• “How many do you see? How do you see them?”  
• Display image.
• 1 minute: quiet think time

• Display image. 
• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “How could we figure out how many people are in the picture?” (There are 8 people in each circle so we could count the number of circles we can see and multiply it by 8 and then add in the extra people that are parts of circles on the edges of the image.)
• “This picture shows the largest Peruvian folk dance. Today we are going to solve some problems about this event.”

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 strategies to share in the synthesis:

• Students do not get the correct solution and can explain the mistake they made.
• Students multiply 8 by 100, or multiples of 100 until they close get 4,704 and then multiply 8 by multiples of tens and single digit numbers to find the solution.
• Students use a partial quotients strategy like the one in the student responses.

When students connect the quantities in the story problem to calculations, including the operations of multiplication and division, they reason abstractly and quantitatively (MP2). 

• Groups of 2
• “What do you know about dancing?” (There are lots of different kinds of dances. Sometimes people dance in pairs.)
• 30 seconds: quiet think time
• 1 minute: partner discussion

• 10 minutes: partner work time
• As students work, consider asking:
  • “How do your diagrams or expressions represent the problem?”
  • “Why did you decide to multiply?”
  • “What do the numbers in your calculations mean, in terms of the situation?”

If students do not have an entry point to determine how many groups of 8 people were at the record breaking event, ask, “How could you use multiplication to solve this problem?”

• Ask selected students to share in the given order.
• “Why did you decide to multiply 8 by a multiple of 100?” (There are 8 people in a group and there 4,704 people so I need to multiply 8 by a large number to get the total number of groups.)
• Display student work or use work from student responses:
  \(500 \times 8 = 4,\!000\)
  \(50 \times 8 = 400\)
  \(30 \times 8 = 240\)
  \(8 \times 8 = 64\)
  \(500 + 50 + 30 + 8 = 588\)
• “Where are the groups of 8 people represented in this work? Where are the 4,704 people represented in this work?”(The 8 represents each group of 8 and the other factor tells how many groups of 8 dancers there are. If we add up the partial products, it will equal 4,704.)
• “What division expression can represent this problem?” (\(4,\!704\div8\))
• Ask a student to share who used division or display the following diagram:

• “How does this division work relate to the multiplication work?” (They both show the number of groups of 8 people. Both 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.)

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 to multiply and divide by, but students may use multiplication or division in various ways. During the 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 number of groups of dancers to the number of dancers in each group they observe structure in the relationship of the quotient to the size of the divisor (MP7).

• Groups of 2

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion
• Monitor for students who:
  • use multiplication to solve the problems.
  • use division to solve the problems.
  • use the solution from one problem to solve another problem.

• Display expressions: \(4,\!704 \div 8\), \(4,\!704  \div 4\) and \(4,\!704 \div 2\)
• “How are the expressions the same? How are they different?” (They all have 4,704 in them, but the divisor is 8, then 4, then 2.)
• “How does each expression represent the situation?”(There are always 4704 dancers, but the size of the groups changes.)
• “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.)