## A Swath of Red

A political map of the United States after the 2000 election is largely red, representing the Republican candidate, George W. Bush. However, the presidential race was nearly tied. Using a grid overlay, students estimate the area of the country that voted for the Republican candidate and the area that voted for the Democratic candidate. Students then compare the areas to the electoral and popular vote election results. Ratios of electoral votes to area are used to make generalizations about the population distribution of the United States.

If students have not seen the Who Won This Election? graphic from Lesson 2, use it as an opener for this lesson. Hide the statements at the bottom, and ask students who won. Many students will say "white" because it covers so much of the area of the map. Reveal the first paragraph. Ask students if they are surprised that the "gray" candidate got more popular votes, meaning that more voters cast their votes for that candidate. Now display the second paragraph. Again ask who won the election. The answer, as stated on the overhead, is the "white" candidate, George W. Bush, who received the most electoral votes. The map displays the winning political party for each state in the 2000 presidential election. Ask students why a larger area represents nearly the same number of voters. Explain that they will be investigating this conundrum in this lesson.

Who Won This Election? Overhead

The election of 2000 was politically charged, so you should be prepared to address the issue. The merits of the Electoral College are called into question during each presidential election, and some people have strong opinions.

On a blank sheet of transparency film, draw a random irregular shape. Explain
that you will be finding the area of "the state of Confusion." Ask
students to brainstorm ways to estimate the area of the state. If no
student suggests it, introduce the idea of estimating area using a
square grid overlay. Lay the transparent grid paper over the state of
Confusion and trace the border. Demonstrate that you can count squares
to find the area. Be sure to point out partial squares, and ask
students how they should be addressed. What should a student do if a
square is ^{1}/_{4} full? A good beginning strategy is
to group half squares together to make 1. As your students work, you
can help them do other groupings, such as a ^{1}/_{4} square paired with a ^{3}/_{4} square
to make a whole square. Students will need to find the total number of
squares to arrive at a final estimate of the area.
Students can work either individually or in groups.

Pass out the A Swath of Red Activity Sheet and assign each group a political party and its corresponding color. Make sure to assign the colors to the groups (or individuals) so that both colors are being counted by the same number of groups. Red/white is likely to be an easier area to estimate because many of the states are contiguous. The blue/gray states have a spottier distribution and will require students to work with several smaller areas and more partial squares. Use this insight to differentiate the task.

As students begin reading over the assignment, point out some areas that may cause confusion. Since students are trying to find the area covered by a single color, there is no need to estimate the individual areas of states that border one another. Students may attempt to do this even if you indicate that state boundaries within a color are not meaningful. Watch for students who find the area as a color block, which simplifies their task. Once the area is determined, students should also count the total electoral votes that are assigned to the states of that color. The data on this activity sheet are from the 2000 election and are therefore based on the 1990 census. This means it does not match the data students collected in the Lesson 1.

Have each group send a member up to record their data on the Class Results overhead as they complete Questions 2 and 3. When all groups have reported their results, ask two students to average the two rows in each color. This is a way of checking for accuracy. As groups enter their data in the table, they can see if their calculation is an outlier. It allows them the opportunity to self-check. It is unlikely that any two groups will get identical answers for the area, but each column should be close to the other columns: this is an opportunity for conversations about precision. Finding the mean of each row can balance out any over-counting and under-counting. Encourage the class to discuss the answers. What strategies did those groups use? Have them demonstrate their technique and allow the rest of the class to ask questions and determine the validity of their approach.

Questions 1 and 2 on the overhead ask students to come up with a unit rate comparing electoral votes to the total area of the states won. Students may try to do this state by state, but encourage them to focus on the entire colored area as a voting block. Give students the opportunity to study the two ratios on the overhead table. Then ask them to discuss the results and their observations. You may choose to use the following questions to lead this discussion:

- What do the ratios mean?
[The number of votes cast per square unit of area.]

- Why are the units cm
^{2}rather than just cm?[This activity was done on a 2-dimensional map.]

- What would the ratios be if you counted the actual square miles in a state, as opposed to cm
^{2}on a scale map?[The ratios would be much smaller, but the actual unit of measurement is not important in this activity. The comparison between the red and blue state ratios is.]

- Which candidate has a higher ratio of electoral votes to cm
^{2}? What does that tell you about the states that voted for that candidate?[Students should find that the blue states have a higher ratio than the red states. The population density in the blue states is greater because there are more people living closer together. Since the number of electoral votes is related to the population of a state, comparing electoral votes to area is like comparing population to area, which is the definition of population density.]

You may also wish to use the What Happened? overhead in your closing discussion. It shows a typical map of the United States and a cartogram below it. The cartogram is drawn so the ratio of electoral votes to area is 1:1. In other words, all states with 3 electoral votes are the same size and smaller than all other states.

- Transparent sheet of centimeter (or similar) grid paper
- Markers
- Calculator
- Who Won this Election? Graphic
- A Swath of Red Activity Sheet
- Class Results Graphic
- What Happened? Graphic (optional)

**Assessments**

- Conduct the same activity using the results of the most recent presidential election. You could also have students compare results from the 2000 election, which were explored in this lesson, to the results from the most recent election.
- As students work on the irregular area problems, evaluate their strategies. Do they find whole rectangles of area first or do they count squares one at a time? How do they count partial squares? Are they using reasonable pairs of partial squares to make a whole? While there are several strategies that may provide reliable results, assess students on their overall strategies.
- Ask students to describe each state or color-block of states in terms of the voters who live there. In the class discussion, listen to hear if students understand that a higher ratio indicates people living in closer proximity. A lower ratio indicates more empty land for each person in that geographic area. Population density, indeed the idea of density in general, is a key idea that connects mathematics with science.

**Extensions**

- Guess My State game: Students choose one state and calculate the ratio of its electoral votes to 1 cm
^{2}on the map. Call on students and let them state their ratio. The first student to guess the state based on the ratio wins that round and gets to pick the next mystery ratio. - Research the results of the 1900 election. This election looks nearly tied based on area, but the electoral votes for the winner (McKinley) are nearly double those cast for the loser. Be sure to discuss the beige states, which had not yet joined the union.

**Questions for Students**

- What strategies are most useful for finding the area?
[Answers will vary. One strategy is to find the area of full blocks of rectangles first, and then pair up half squares and other fractional pairs.]

- Why did each group arrive at a different ratio?
[Due to the variations in counting methods and estimation.]

- Which number in the ratio varied the most?
[The area is most likely to vary. The number of electors is an exact value, as opposed to the areas which were estimated.]

- What does this ratio mean? Explain your answer in the context of the problem.
[This means that for each square unit of land, there are

*x*electors who represent the population of that area. If the ratio is 20 electors per cm^{2}, then for every square centimeter of land represented on the map there are 20 electors who represent the population of that area.] - Why does it look like the “red” candidate won the election by a lot of votes, when in reality he barely won?

[Overall, the states that voted red are less densely populated than the blue states. Land area is not correlated to population or electoral votes. Remind students of the opening overhead, which gave data about population and electoral votes.]

**Teacher Reflection**

- What were some of the difficulties students encountered when finding irregular areas?
- Did students stay on task during the data-gathering and estimation process?
- Did the lesson address the needs of a variety of learners?
- Were students able to understand the meaning of the ratios in this context?
- Do students understand that the colored map is not a representation of the popular vote?

### Why Is California So Important?

### How Could That Happen?

### There is a Difference: Histograms vs. Bar Graphs

### Will the Best Candidate Win?

### Learning Objectives

Students will:

- Investigate the equivalence of two symbolic expressions.
- Transition from numeric to algebraic notation.

### NCTM Standards and Expectations

- Understand and use ratios and proportions to represent quantitative relationship.

- Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

### Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

- CCSS.Math.Content.6.RP.A.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

Grade 6, Geometry

- CCSS.Math.Content.6.G.A.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.