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How Could That Happen?

Number and Operations
Kimberly Morrow-Leong
Gainesville, VA

This problem-solving lesson challenges students to generate election results using number sense and other mathematical skills. Students are also given the opportunity to explore the mathematical questions in a politically challenging context. Calculations can be made using online or desktop tools or using the data gathered on the Lesson 1 activity sheet, Why California?  Additional resources are introduced to extend the primary activity. 

Begin the lesson by displaying the Who Won This Election? Graphic. Ask students to examine the numbers carefully. Students should notice that the "gray" candidate received more of the popular vote, yet the "white" candidate received more of the electoral vote. If students notice that the number of electoral votes is different from those in Why California?, explain that the Why California? data is current and the data in the graphic is from a previous election.

overhead Who Won This Election? Graphic 

Engage students in a class discussion about whether or not they feel the results of the election are "fair." Many students will have their own opinions, but try not to influence students with your own opinion. You may wish to share with them that these are real election results from the 2000 presidential election between Al Gore and George W. Bush. Gray represents Gore and white represents Bush. See the Extensions section below for possible follow-up questions.

You may also choose to bring in other political information and data. For example, since the electoral college is part of our Constitution, any proposal to change the electoral college process would need the support of \frac{3}{4} of the states (not \frac{3}{4} of the population). Many citizens have strong feelings on this topic. Some are highly critical of the Electoral College for the reason just stated, while others believe we should continue with the Electoral College since it benefits the least populous states.

Tell students that their goal is to investigate situations in which the Electoral College system creates unusual election results. This is a good activity for students to work on in groups because discussions arise naturally from the investigation. However, you may wish to collect activity sheets from each student to assess individual understanding. One approach is to organize students into groups of three with the following roles:

  • Researcher: looks up the electoral votes for each state
  • Recorder: records the data on the activity sheet and leads the group in arranging and displaying results
  • Executive: makes the ultimate decisions about what to do and keeps the group on task

Display and discuss the rubric found with the How Could That Happen? Answer Key before work begins. Allow students to ask questions to ensure they understand how they will be assessed on the activity.

pdficon How Could That Happen? Answer Key and Rubric 

2825 win 2825 lose 2825 win lose 

Distribute the How Could That Happen? Activity Sheets. There are three problems included to choose from. The first two problems, How Could I Win? and How Could I Lose?, are nearly equivalent in difficulty. The third problem, How Could I Win and Lose?, is more challenging and open-ended, giving you several options for differentiation among individuals and groups. Alternatively, you could assign How Could I Win? or How Could I Lose? to groups as a beginning activity and use How Could I Win and Lose? as the main challenge of the lesson. Note that this may require a second class period.

pdficon How Could That Happen? Activity Sheets 

Search for the Electoral College through the National Archives And Records Administration to gather the most current electoral vote counts as well as the most up-to-date teacher resources. The activity described in this lesson can be done with an online tool, by organizing data in a spreadsheet, or by using a calculator. Students can use the table created in Why California? as a resource as well. If you choose to go directly to this lesson without completing Lesson 1, you will need to provide students with the table from the Why California? Answer Key.

pdficon Why California? Answer Key  

There are several ways to address the question in How Could I Win and Lose? One possibility is to let students make assumptions to simplify the problem. Since Why California? uses population data and the solution to the problem requires data on registered voters, allow students to assume the same portion of the population in each state is registered to vote, which will allow them to use the available population data. The actual percentages vary between states, and fall within the range of about 50% to 80% of a state's population being registered to vote. The U.S. Census Bureau offers extensive information about the demographics of the electorate, including the number of registered voters in each state.

Leave time toward the end of class to draw students back to a class discussion and review of the rubric requirements. Emphasize that a level 5 answer will include all elements outlined on the activity sheet, and remind students to focus on developing a mathematically sound explanation despite the strong political content of the activity.

Display the shaded maps from How Could I Win? together. Then display the maps from How Could I Lose? together, and finally, display the maps from How Could I Win and Lose? together. Discuss the similarities as each set of maps is put up. Ask students questions to encourage a discussion of these similarities. For example:

  • What state is always shaded for How Could I Win?
  • What other states should be on the list for How Could I Win? Why?
  • Which states are necessary on the list for How Could I Lose?
  • Which of the three tasks requires the most states to be shaded in?
  • What strategies did you use to help you find the solution to How Could I Win and Lose?

If time permits, allow students to volunteer to share some of the other results and discoveries they made during the activity.


Assessment Options 

  1. Use the How Could That Happen? Rubric to evaluate final student work on the problems completed. The individual assignments can be evaluated for a student’s progress toward the process standard of presenting a solution to a problem. As you read through students' solutions, ask yourself:
    • Are students using information they learned as they were working on the problem to answer it?
    • If students tried a strategy and modified it as they worked the problem, is that shared in the write-up?
    • Did students use mathematics appropriately?
  2. Have students find an election outcome in which the candidate wins exactly 270 electoral votes.
  3. Using the rubric as a guide, allow students to evaluate the work of classmates from other groups. You may also have students write journal entries on what they learned while doing the activity and what they learned by reading the results of other groups.


  1. Have students answer the question: If a candidate can’t win California, what is the minimum number of states the candidate needs to win the election?
  2. Explore the mathematics between the Constitutional requirements to alter how a president is elected. Ask questions such as the following: How many states is \frac{3}{4} of the states in the United States? What is the greatest population that can be represented by \frac{3}{4} of the states? What is the least population that can be represented by \frac{3}{4} of the states?
  3. The How Could I Win and Lose? problem is related to the topic of bin-packing. Have students research the topic to explore different ways in which solutions are possible. The problem could be very helpful to students learning about algorithms.
  4. Move on to the last lesson, A Swath of Red.


Questions for Students 

1. What kind of numbers did you look for to find the states necessary to win with the fewest votes? 

[I looked for the highest number of electoral votes.] 

2. What numbers did you look for when you needed to find the states that resulted in a loss even though the candidate had the most votes? 

[I looked for the lowest number of electoral votes.] 

3. Are there times when it doesn't matter which state you add to the list? 

[States with equal numbers of electoral votes can be interchanged in the list.] 

4. Are there any geographic patterns to the states you shaded in? 

[The states in the middle of the country have fewer electors, and the coastal states have more electors. You may choose to discuss the historical reason for this difference.] 

5. How likely do you think it is that a candidate would win 40 states without winning the election? 

[Answers will vary.] 

6. How likely do you think it is that a candidate would only win 11 states and win the election? 

[Answers will vary.]  

Teacher Reflection 

  • Did students have a systematic strategy for finding a solution to the problem? If students did not have a strategy, how did they approach the problem?
  • What problems did students have solving the problems? Were the problems mechanical (e.g., The student did not having a correct table or was not using it)? Did students struggle with a mathematical concept?
  • Did students understand that they should look for the states with the highest or lowest number of electoral votes?
  • Did the results of the rubric assessment show a growth in problem-solving strategies?
  • Were the problems too easy? too difficult?

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.
Data Analysis and Probability

Why Is California So Important?

In this lesson, students learn about the mechanics of the Electoral College and use data on population and electoral votes for each state. Students calculate the percentage of the Electoral College vote allocated to each state, and use mathematics to reflect on the differences. Several questions are provided to strengthen understanding of measures of central tendency and fluency with decimals and percents.
Data Analysis and Probability

There is a Difference: Histograms vs. Bar Graphs

3-5, 6-8
Using data from the Internet, students summarize information about party affiliation and ages at inauguration of Presidents of the United States in frequency tables and graphs.  This leads to a discussion about categorical data (party affiliations) vs. numerical data (inauguration ages) and histograms vs bar graphs.
Data Analysis and Probability

Will the Best Candidate Win?

This lesson plan for grades 9‑12 is adapted from an article in the January 2000 edition of Mathematics Teacher. The following activities allow students to explore alternative voting methods. Students discover what advantages and disadvantages each method offers and also see that each fails, in some way, to satisfy some desirable properties.

Learning Objectives

Students will:
  • Use the data in the table in the Why California? activity sheet to solve a problem.
  • Compare their problem-solving methods with those of other students.
  • Present a solution to a problem in an organized manner, representing it verbally, numerically, graphically, and systematically.

NCTM Standards and Expectations

  • Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.
  • Develop meaning for integers and represent and compare quantities with them.

Common Core State Standards – Mathematics

Grade 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.A.3
    Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Grade 4, Num & Ops Base Ten

  • CCSS.Math.Content.4.NBT.B.4
    Fluently add and subtract multi-digit whole numbers using the standard algorithm.

Grade 6, Stats & Probability

  • CCSS.Math.Content.6.SP.A.3
    Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Grades 9-12, Number & Quantity

  • CCSS.Math.Content.HSN-Q.A
    Reason quantitatively and use units to solve problems.