Designing Districts
20 min
Teacher Prep
Setup
Narrative
In the previous voting activities, representatives (“advisors”) were assigned to groups that couldn’t be changed: schools. Sometimes the groups or districts for representatives can be changed, as in districts for the U.S. House of Representatives, and for state legislatures, wards in cities, and so on.
In this lesson, students use geometric reasoning about areas and connectedness to experiment with drawing districts in a way that predicts the outcome of elections. This is often called “gerrymandering.”
As students try different configurations of districts they reason abstractly and quantitatively about how changing shapes affects outcomes (MP2).
In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).
Launch
Arrange students in groups of 2–4. If needed, remind students of the school-level vote for the new mascot from an earlier activity.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Student Task
After the school mascot voting, 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.
- Suppose there were an election with each of the 50 blocks getting one vote. How many votes would be for banana slugs? For sea lions? Which mascot would win this election and what percentage of the votes would they get?
-
Suppose the blocks are in districts 1–5, as shown here. 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 -
Suppose, instead, that the 5 districts are as shown in this new map. 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 -
Suppose the 5 districts are designed in yet another way, as shown in this 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 -
Write a headline for the local newspaper for each of the ways of splitting the town into districts.
-
Which systems of the three maps of districts do you think are more fair? Are any totally unfair?
Sample Response
-
20 votes for banana slugs, 30 votes for sea lions, so sea lions win with 60% of the vote.
-
The people in districts 1 and 2 prefer banana slugs while the people in districts 3, 4, and 5 prefer sea lions. Sea lions win with 3 of 5 representatives.
district number of blocks choosing banana slugs number of blocks choosing sea lions percentage of blocks choosing banana slugs representative’s vote 1 10 0 100% banana slugs 2 10 0 100% banana slugs 3 0 10 0% sea lions 4 0 10 0% sea lions 5 0 10 0% sea lions -
All 5 districts have 4 blocks that prefer banana slugs and 6 blocks that prefer sea lions. Because more blocks in each district prefer sea lions, all 5 representatives vote for sea lions. Sea lions win.
district number of blocks choosing banana slugs number of blocks choosing sea lions percentage of blocks choosing banana slugs representative’s vote 1 4 6 40% sea lions 2 4 6 40% sea lions 3 4 6 40% sea lions 4 4 6 40% sea lions 5 4 6 40% sea lions -
The people in each district are divided in their support for each mascot. 3 of the districts (1, 2, and 3) have more support for banana slugs while the other 2 districts (4 and 5) have more support for sea lions, so 3 representatives vote for banana slugs and 2 vote for sea lions. Banana slugs win with 3 of 5 representatives.
district number of blocks choosing banana slugs number of blocks choosing sea lions percentage of blocks choosing banana slugs representative’s vote 1 6 4 60% banana slugs 2 6 4 60% banana slugs 3 6 4 60% banana slugs 4 1 9 10% sea lions 5 1 9 10% sea lions -
Sample responses:
First map: 60% of Districts and 60% of People Vote for Sea Lions
Second map: All Districts, but Only 60% of People, Vote for Sea Lions
Third map: Banana Slugs Win with 60% of Districts, but Only 40% of People -
Sample response: The first map seems the fairest since the percentages of the people and the representatives match. The second map has the same winner as the vote of the people but different percentages. The third map seems totally unfair: The percentages are reversed. More than half the people voted for sea lions, but banana slugs won.
Activity Synthesis (Teacher Notes)
- Display this first draft:
“Both maps 1 and 2 are equally fair since 60% of people prefer sea lions. That means they show the same results.”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement. If not brought up by students, highlight the number of districts voting for each option in each map and how that influences the final results of the vote.
- Give students 2–4 minutes to work with a partner to revise the first draft.
- Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
Standards
- 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.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
- 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.
- 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
30 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Designing Districts
20 min
Teacher Prep
Setup
Narrative
In the previous voting activities, representatives (“advisors”) were assigned to groups that couldn’t be changed: schools. Sometimes the groups or districts for representatives can be changed, as in districts for the U.S. House of Representatives, and for state legislatures, wards in cities, and so on.
In this lesson, students use geometric reasoning about areas and connectedness to experiment with drawing districts in a way that predicts the outcome of elections. This is often called “gerrymandering.”
As students try different configurations of districts they reason abstractly and quantitatively about how changing shapes affects outcomes (MP2).
In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).
Launch
Arrange students in groups of 2–4. If needed, remind students of the school-level vote for the new mascot from an earlier activity.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Student Task
After the school mascot voting, 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.
- Suppose there were an election with each of the 50 blocks getting one vote. How many votes would be for banana slugs? For sea lions? Which mascot would win this election and what percentage of the votes would they get?
-
Suppose the blocks are in districts 1–5, as shown here. 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 -
Suppose, instead, that the 5 districts are as shown in this new map. 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 -
Suppose the 5 districts are designed in yet another way, as shown in this 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 -
Write a headline for the local newspaper for each of the ways of splitting the town into districts.
-
Which systems of the three maps of districts do you think are more fair? Are any totally unfair?
Sample Response
-
20 votes for banana slugs, 30 votes for sea lions, so sea lions win with 60% of the vote.
-
The people in districts 1 and 2 prefer banana slugs while the people in districts 3, 4, and 5 prefer sea lions. Sea lions win with 3 of 5 representatives.
district number of blocks choosing banana slugs number of blocks choosing sea lions percentage of blocks choosing banana slugs representative’s vote 1 10 0 100% banana slugs 2 10 0 100% banana slugs 3 0 10 0% sea lions 4 0 10 0% sea lions 5 0 10 0% sea lions -
All 5 districts have 4 blocks that prefer banana slugs and 6 blocks that prefer sea lions. Because more blocks in each district prefer sea lions, all 5 representatives vote for sea lions. Sea lions win.
district number of blocks choosing banana slugs number of blocks choosing sea lions percentage of blocks choosing banana slugs representative’s vote 1 4 6 40% sea lions 2 4 6 40% sea lions 3 4 6 40% sea lions 4 4 6 40% sea lions 5 4 6 40% sea lions -
The people in each district are divided in their support for each mascot. 3 of the districts (1, 2, and 3) have more support for banana slugs while the other 2 districts (4 and 5) have more support for sea lions, so 3 representatives vote for banana slugs and 2 vote for sea lions. Banana slugs win with 3 of 5 representatives.
district number of blocks choosing banana slugs number of blocks choosing sea lions percentage of blocks choosing banana slugs representative’s vote 1 6 4 60% banana slugs 2 6 4 60% banana slugs 3 6 4 60% banana slugs 4 1 9 10% sea lions 5 1 9 10% sea lions -
Sample responses:
First map: 60% of Districts and 60% of People Vote for Sea Lions
Second map: All Districts, but Only 60% of People, Vote for Sea Lions
Third map: Banana Slugs Win with 60% of Districts, but Only 40% of People -
Sample response: The first map seems the fairest since the percentages of the people and the representatives match. The second map has the same winner as the vote of the people but different percentages. The third map seems totally unfair: The percentages are reversed. More than half the people voted for sea lions, but banana slugs won.
Activity Synthesis (Teacher Notes)
- Display this first draft:
“Both maps 1 and 2 are equally fair since 60% of people prefer sea lions. That means they show the same results.”
Ask, “What parts of this response are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement. If not brought up by students, highlight the number of districts voting for each option in each map and how that influences the final results of the vote.
- Give students 2–4 minutes to work with a partner to revise the first draft.
- Select 1–2 individuals or groups to read their revised draft aloud slowly enough to record for all to see. Scribe as each student shares, then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.
Standards
- 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.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
- 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.
- 6.RP.A.3.c·Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.