Lesson 6Picking Representatives

Let's think about fair representation.

6.1 Computers for Kids

A program gives computers to families with school-aged children. They have a certain number of computers to distribute fairly between several families. How many computers should each family get?

  1. One month the program has 8 computers. The families have these numbers of school-aged children: 4, 2, 6, 2, 2.

    1. How many children are there in all?
    2. Counting all the children in all the families, how many children would use each computer? This is the number of children per computer. Call this number A .
    3. Fill in the third column of the table. Decide how many computers to give to each family if we use A as the basis for distributing the computers.
      family number of children number of computers, using A
      Baum 4
      Chu 2
      Davila 6
      Eno 2
      Farouz 2
    4. Check that 8 computers have been given out in all.
  2. The next month they again have 8 computers. There are different families with these numbers of children: 3, 1, 2, 5, 1, 8.

    1. How many children are there in all?
    2. Counting all the children in all the families, how many children would use each computer? This is the number of children per computer. Call this number B .
    3. Does it make sense that B is not a whole number? Why?
    4. Fill in the third column of the table. Decide how many computers to give to each family if we use B as the basis for distributing the computers. 
      family number of children number of computers, using B number of computers, your way number of children per computer, your way
      Gray 3
      Hernandez 1
      Ito 2
      Jones 5
      Krantz 1
      Lo 8
    5. Check that 8 computers have been given out in all.
    6. Does it make sense that the number of computers for one family is not a whole number? Explain your reasoning?
    7. Find and describe a way to distribute computers to the families so that each family gets a whole number of computers. Fill in the fourth column of the table.
    8. Compute the number of children per computer in each family and fill in the last column of the table.
    9. Do you think your way of distributing the computers is fair? Explain your reasoning.

6.2 School Mascot (Part 1)

A school is deciding on a school mascot. They have narrowed the choices down to the Banana Slugs or the Sea Lions.

The principal decided that each class gets one vote. Each class held an election, and the winning choice was the one vote for the whole class. The table shows how three classes voted.

“Pawsox mascot” by Paul Keleher via Wikimedia Commons. CC BY 2.0.
banana slugs sea lions class vote
class A 9 3 banana slug
class B 14 10
class C 6 30
  1. Which mascot won, according to the principal’s plan? What percentage of the votes did the winner get under this plan?
  2. Which mascot received the most student votes in all? What percentage of the votes did this mascot receive?
  3. The students thought this plan was not very fair. They suggested that bigger classes should have more votes to send to the principal.

    Make up a proposal for the principal where there are as few votes as possible, but the votes proportionally represent the number of students in each class.

  4. Decide how to assign the votes for the results in the class. (Do they all go to the winner? Or should the loser still get some votes?)
  5. In your system, which mascot is the winner?
  6. In your system, how many representative votes are there? How many students does each vote represent?

6.3 Advising the School Board

  1. In a very small school district, there are four schools, D, E, F, and G. The district wants a total of 10 advisors for the students. Each school should have at least one advisor.
    school number of students number of advisors, A students per advisor
    D 48
    E 12
    F 24
    G 36
    1. How many students are in this district in all?
    2. If the advisors could represent students at different schools, how many students per advisor should there be? Call this number A . Show your reasoning.
    3. Using A students per advisor, how many advisors should each school have? Complete the table with this information for schools D, E, F, and G.
  2. Another district has four schools; some are large, others are small. The district wants 10 advisors in all. Each school should have at least one advisor.
    school number of students number of advisors, B students per advisor number of advisors, your way number of students per advisor, your way
    Dr. King School 500
    O’Connor School 200
    Science Magnet School  140
    Trombone Academy 10
    1. How many students are in this district in all?
    2. If the advisors didn’t have to represent students at the same school, how many students per advisor should there be? Call this number B .
    3. Using B students per advisor, how many advisors should each school have? Give your quotients to the tenths place. Fill in the first “number of advisors” column of the table. Does it make sense to have a tenth of an advisor?
    4. Decide on a consistent way to assign advisors to schools so that there are only whole numbers of advisors for each school, and there is a total of 10 advisors among the schools. Fill in the “your way” column of the table.
    5. How many students per advisor are there at each school? Fill in the last row of the table.
    6. Do you think this is a fair way to assign advisors? Explain your reasoning.

6.4 School Mascot (Part 2)

The whole town gets interested in choosing a mascot. The mayor of the town decides to choose representatives to vote.

There are 50 blocks in the town, and the people on each block tend to have the same opinion about which mascot is best. Green blocks like sea lions, and gold blocks like banana slugs. The mayor decides to have 5 representatives, each representing a district of 10 blocks.

Here is a map of the town, with preferences shown.

A figure that represents a map of a town composed of 50 green and gold squares that are arranged in 5 rows with 10 squares in each row. The top 2 rows each contain 10 gold squares and the bottom 3 rows each contain 10 green squares.
  1. Suppose there were an election with each block getting one vote. How many votes would be for banana slugs? For sea lions? What percentage of the vote would be for banana slugs?
  2. Suppose the districts are shown in the next map. What did the people in each district prefer? What did their representative vote? Which mascot would win the election?

    A figure that represents a map of a town composed of 50 green and gold squares that are arranged in 5 rows with 10 squares in each row. The top 2 rows each contain 10 gold squares and are labeled 1 and 2. The bottom 3 rows each contain 10 green squares and the rows are labeled 3, 4, and 5.

    Complete the table with this election’s results.

    district number of
    blocks for
    banana slugs
    number of
    blocks for
    sea lions
    percentage of
    blocks for
    banana slugs
    representative’s
    vote
    1 10 0 banana slugs
    2
    3
    4
    5
  3. Suppose, instead, that the districts are shown in the new map below. What did the people in each district prefer? What did their representative vote? Which mascot would win the election?

    50 gold and green squares are arranged in 5 rows with 10 squares in each row. The top 2 rows are gold and the bottom 3 rows are green. The grid is divided vertically into 5 equal rectangles labeled 1, 2, 3, 4, and 5. Each rectangle contains 4 gold squares at the top and 6 green squares directly underneath.

    Complete the table with this election’s results.

    district number of
    blocks for
    banana slugs
    number of
    blocks for
    sea lions
    percentage of
    blocks for
    banana slugs
    representative’s
    vote
    1
    2
    3
    4
    5
  4. Suppose the districts are designed in yet another way, as shown in the next map. What did the people in each district prefer? What did their representative vote? Which mascot would win the election?

    A 10 by 5 grid of squares with specific area boundaries numbered 1 through 5 indicated. The top two rows are gold and the bottom 3 rows are green. Each numbered area contains a combination of 10 gold and green squares.  Area 1: Starting on the first row, region 1 has the first 4 gold squares. Under the second gold square in the first row is a row of 2 gold squares. Directly under the 2 gold squares in row 2 are 2 green squares. Directly under the 2 green squares are another 2 green squares.  Area 2: Starting on the first row, region 2 has the fifth and sixth gold squares. Under the 2 gold squares and 1 place to the left are 4 gold squares side by side. Under the 4 gold squares are 1 green square, 2 spaces, and another green square. Under that row is an identical row with 1 green square, 2 spaces, and another green square.  Area 3: Starting on the top row, region 3 has the last 4 gold squares in the first row. Under the second gold square in the first row is a row of 2 gold squares. Under the 2 gold squares in row 2 are 2 green squares. Under the 2 green squares are another two green squares. Area 3 is identical to area 1.  Area 4: Starting in row 2, region 4 has starts with a gold square. Directly below in row 3 is 1 green square, then 3 spaces, and 1 green square. Row 4 is identical to row 3. Row 5 has 5 green squares side by side.  Area 5: Starting in row 2, region 5 has the 10th gold square. In row 3 has the 6th green square, then 3 spaces, and 1 green square. Row 4 is identical to row 3. Row 5 has 5 green squares side by side.10 squares over is 1 yellow. In the next row down, 6 squares over is 1 green, 3 spaces, and 1 green square. Under the green square in the previous row is 1 green, 3 spaces, and 1 green square. Under the previous green square on the left, are 5 green squares.

    Complete the table with this election’s results.

    district number of
    blocks for
    banana slugs
    number of
    blocks for
    sea lions
    percentage of
    blocks for
    banana slugs
    representative’s
    vote
    1
    2
    3
    4
    5
  5. Write a headline for the local newspaper for each of the ways of splitting the town into districts.

  6. Which systems on the three maps of districts do you think are more fair? Are any totally unfair?

6.5 Fair and Unfair Districts

  1. Smallville’s map is shown, with opinions shown by block in green and gold. Decompose the map to create three connected, equal-area districts in two ways:

    1. Design three districts where green will win at least two of the three districts. Record results in Table 1.
    A figure that represents a district composed of 30 green and gold squares that are arranged in 3 rows and 10 columns. The squares are arranged in the following order: Row 1: 8 green, 1 gold, 1 green. Row 2: 2 gold, 4 green, 1 gold, 1 green, 1 gold, 1 green. Row 3: 3 gold, 1 green, 1 gold, 2 green, 3 gold.

    Table 1:

    district number of blocks
    for green
    number of blocks
    for gold
    percentage of
    blocks for green
    representative’s
    vote
    1
    2
    3
    1. Design three districts where gold will win at least two of the three districts. Record results in Table 2.
    A figure that represents a district composed of 30 green and gold squares that are arranged in 3 rows and 10 columns. The squares are arranged in the following order: Row 1: 8 green, 1 gold, 1 green. Row 2: 2 gold, 4 green, 1 gold, 1 green, 1 gold, 1 green. Row 3: 3 gold, 1 green, 1 gold, 2 green, 3 gold.

    Table 2:

    district number of blocks
    for green
    number of blocks
    for gold
    percentage of
    blocks for green
    representative’s
    vote
    1
    2
    3
  2. Squaretown’s map is shown, with opinions by block shown in green and gold. Decompose the map to create five connected, equal-area districts in two ways:

    1. Design five districts where green will win at least three of the five districts. Record the results in Table 3.
    A figure that represents a district composed of 100 green and gold squares that are arranged in 10 rows and 10 columns. The squares are arranged in the following order: Row 1: 10 green. Row 2: 2 gold, 5 green, 2 gold, 1 green. Row 3: 3 gold, 1 green, 1 gold, 2 green, 3 gold. Row 4: 5 green, 5 gold. Row 5: 4 green, 2 gold, 1 green, 1 gold, 1 green, 1 gold. Row 6: 3 green, 1 gold, 3 green, 1 gold, 1 green, 1 gold. Row 7: 4 gold, 3 green, 1 gold, 2 green. Row 8: 4 gold, 6 green. Row 9: 4 gold, 6 green. Row 10: 4 gold, 6 green.

    Table 3:

    district number of blocks
    for green
    number of blocks
    for gold
    percentage of
    blocks for green
    representative’s
    vote
    1
    2
    3
    4
    5
    1. Design five districts where gold will win at least three of the five districts. Record the results in Table 4.
    A figure that represents a district composed of 100 green and gold squares that are arranged in 10 rows and 10 columns. The squares are arranged in the following order: Row 1: 10 green. Row 2: 2 gold, 5 green, 2 gold, 1 green. Row 3: 3 gold, 1 green, 1 gold, 2 green, 3 gold. Row 4: 5 green, 5 gold. Row 5: 4 green, 2 gold, 1 green, 1 gold, 1 green, 1 gold. Row 6: 3 green, 1 gold, 3 green, 1 gold, 1 green, 1 gold. Row 7: 4 gold, 3 green, 1 gold, 2 green. Row 8: 4 gold, 6 green. Row 9: 4 gold, 6 green. Row 10: 4 gold, 6 green.

    Table 4:

    district number of blocks
    for green
    number of blocks
    for gold
    percentage of
    blocks for green
    representative’s
    vote
    1
    2
    3
    4
    5
  3. Mountain Valley’s map is shown, with opinions by block shown in green and gold. (This is a town in a narrow valley in the mountains.) Can you decompose the map to create three connected, equal-area districts in the two ways described here? 

    1. Design three districts where green will win at least two of the three districts. Record the results in Table 5.
    A figure that represents a district composed of 18 green and gold squares that are arranged in the following order.  Top row: 2 gold squares and 1 green square each side by side. Second Row: Starting under the green square in row 1, 1 gold square and 3 green squares each side by side. Third Row: Starting under the gold square in row 2, 1 gold square, 2 spaces, 2 green squares side by side, 2 spaces, and 3 gold squares each side by side. Fourth Row: Starting under the first green square in row 3, 3 green squares each side by side, then 1 gold square, and then 1 green square.

    Table 5:

    district number of blocks
    for green
    number of blocks
    for gold
    percentage of
    blocks for green
    representative’s
    vote
    1
    2
    3
    1. Design three districts where gold will win at least two of the three districts. Record the results in Table 6.
    A figure that represents a district composed of 18 green and gold squares that are arranged in the following order.  Top row: 2 gold squares and 1 green square each side by side. Second Row: Starting under the green square in row 1, 1 gold square and 3 green squares each side by side. Third Row: Starting under the gold square in row 2, 1 gold square, 2 spaces, 2 green squares side by side, 2 spaces, and 3 gold squares each side by side. Fourth Row: Starting under the first green square in row 3, 3 green squares each side by side, then 1 gold square, and then 1 green square.

    Table 6:

    district number of blocks
    for green
    number of blocks
    for gold
    percentage of
    blocks for green
    representative’s
    vote
    1
    2
    3