Picking Representatives

10 min

Teacher Prep
Setup
Students in groups of 2. 5 minutes quiet think time followed by work with partner.

Narrative

This activity introduces the question for the remaining activities on voting: How can we fairly share a small number of representatives between several groups of people?

This activity invites students to consider a simpler situation about how to distribute computers fairly to families with children. In the first question, computers can be shared so that the same number of children share a computer in each family. Later in the activity, fair sharing is not possible, so students need to construct arguments to explain which alternative is the fairest, or the least unfair (MP3). 

This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.

Launch

Arrange students in groups of 2. Give students 5 minutes of quiet think time, and then ask them to compare their work with a partner.

Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to create and discuss different methods of distributing computers. Display words and phrases, such as “divide,” “split,” “fair,” “unfair,” “_____ per _____,” and “whole number.”

Engagement: Provide Access by Recruiting Interest. Invite students to share experiences when they have needed to share something. Highlight the difference between times when the item being shared can be divided among all of the people (like a cake) and when the item cannot be divided (like a stuffed toy).
Supports accessibility for: Conceptual Processing, Memory

Student Task

A program gives computers to families with school-aged children. Each month they have a number of computers and need to decide how many computers each family will get.

  1. One month the program has 8 computers and 16 children from 5 families who request computers.

    1. Let PP be the number of children per computer. What is the value of PP?
    2. Fill in the third column of the table. Decide how many computers to give to each family if we use PP as the basis for distributing the computers.
      family number of children number of computers, using PP
      Baum 4
      Chu 2
      Davila 6
      Eno 2
      Farouz 2

    3. Were all 8 computers given out?

  2. The next month they again have 8 computers but now there are 20 children from 6 families.

    1. Let BB be the number of children per computer. What is the value of BB?
    2. Does it make sense that BB is not a whole number? Why?
    3. Fill in the third column of the table. Decide how many computers to give to each family if we use BB as the basis for giving the computers. 
      family number of
      children      		 number of computers,
      using BB       		number of computers,
      your way      		 children per computer,
      your way
      Gray 3
      Hernandez 1
      Ito 2
      Jones 5
      Krantz 1
      Lo 8

    4. Were all 8 computers given out?

    5. Does it make sense that the number of computers for a family is not a whole number? Explain your reasoning.
    6. Find and describe a way to give computers to the families so that each family gets a whole number of computers. Fill in the fourth column of the table.
    7. Compute the number of children per computer in each family and fill in the last column of the table.

    8. Do you think your way of giving the computers is fair? Explain your reasoning.

Sample Response

    1. P=2P=2 children per computer

    2. family number of children number of computers, using PP
      Baum 4 2
      Chu 2 1
      Davila 6 3
      Eno 2 1
      Farouz 2 1
    3. 2+1+3+1+1=82+1+3+1+1=8, so 8 computers have been given out.

    1. B=2.5B=2.5 children per computer. Written as a fraction, B=52B=\frac52.

    2. Sample responses:
      • B=2.5B=2.5 makes sense. It’s an average, not an actual amount for any children or families.
      • BB not being a whole number does not make sense because it is based on the number of children, which must be a whole number.
    3. Divide the number of children by 2.5 or 52\frac52 children per computer for each family to get the number of computers.
      family number of
      children      		 number of computers,
      using BB       		number of computers,
      your way      		 children per computer,
      your way
      Gray 3 1.2 or 65\frac65 1 3
      Hernandez 1 0.4 or 25\frac25 1 1
      Ito 2 0.8 or 45\frac45 1 2
      Jones 5 2 2 2.5
      Krantz 1 0.4 or 25\frac25 1 1
      Lo 8 3.2 or 165\frac{16}{5} 2 4
    4. The sum of entries in column 3 is 8.
    5. It doesn’t make sense for a family to get a fractional or decimal amount of a computer. Computers work only when they are whole. A fraction of a computer is a broken computer.
    6. Sample response from column 4 of the table: This distribution first gives one computer to each family, which uses 6 computers. Then the two last computers are given to the two largest families.
    7. See table.
    8. Sample response: It’s not completely fair because the Hernandez and Krantz children get the computer all to themselves, while the Lo children need to share with 3 others.
Activity Synthesis (Teacher Notes)

The goal of this discussion is to see how distributions are not always equal. First, direct students’ attention to the reference created using Collect and Display. Ask students to share their response to “Do you think your way of giving the computers is fair?” Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases.

Invite students to share their answers. Here are some questions for discussion:

  • “In the first situation, how many children share each computer?” (2)
  • “In the second situation, how many children share each computer?”
  • “Are there multiple ways to give out the computers? Which is the most fair?”
Standards
Addressing
  • 6.NS.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
  • 6.NS.B.3·Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
  • 6.RP.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
  • 6.RP.A.3·Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

15 min

20 min