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
A Swath of Red Activity Sheet
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
Class Results Overhead
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 cm2 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 cm2 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 cm2? 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
What Happened? Overhead
- 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
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
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.