THE 1912 PRESIDENTIAL ELECTION

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1 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY THE 112 PRESIDENTIAL ELECTION S1.1 The 112 presidential election had three strong candidates: Woodrow Wilson, Theodore Roosevelt, and William Howard Taft. Here are the vote totals in that election, which was won by Woodrow Wilson. Woodrow Wilson 6,26,547 Theodore Roosevelt 4,118,571 William Howard Taft 3,486,72 The diagrams in Figure 1 show hypothetical preferences of voters in that election. 1.Represent the vote totals as percentages, and write each below the appropriate diagram. Wilson Roosevelt Taft Roosevelt Taft Roosevelt Taft Wilson Figure 1. Wilson 2. Find the winner of the election by each method. a) Plurality b) Runoff c) A 3, 2, 1 point system 3. Does the 112 presidential election demonstrate flaws in any of the methods? 4.Draw a runoff diagram for this election. Is any candidate capable of beating each of the other two in runoffs? 24

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3 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY THROW THE BUMS OUT S1.2 Since the plurality method can sometimes produce a winner that is ranked last by a majority, some people suggest eliminating any candidates who are ranked last by over half the voters. Figure 1 shows a set of voter preferences to consider. Reese Erskine Robinson 1.Describe how this method would work in this case. Snider Robinson Erskine Robinson Snider Snider Erskine Reese Reese 48% 32% 2% Figure 1. 2.Find the winner of this election by all of the other methods you have studied. 3.Figure 2 shows a different set of voter preferences for the same four candidates. a) Describe how the method that eliminates candidates ranked last by over half the voters would work in this case. Reese Robinson Erskine Snider Erskine Robinson Snider Reese Robinson Erskine Snider Reese Snider Robinson Erskine Reese 46% 22% 1% 13% Figure 2. b) Suppose the 13% of the voters who rank Snider first change their minds. Polls have convinced them that Snider hasn t much of a chance, and they do not want the expense and inconvenience of a second election. They decide to rank Robinson first, Snider second, Reese third, and Erskine last. Does this change the results of the election? 4.Do you think this method is a good alternative to the others you have seen? 251

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5 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY POINT COUNTS WITH REPLAY S1.3 Calculators that are capable of replaying calculations are very helpful when you work with point systems. Figure 1 is a matrix showing the first-, second-, third-, and fourth-place votes for candidates B, C, H, and P in a class of high school students. Use 4 points for first, 3 for second, 2 for third, and 1 for last place. Follow the steps and compare your calculator screen to the one shown. 1st 2nd 3rd 4th 16 B C H P page 1 of 2 Figure 1. If there is a problem, ask someone for help. To find the point total for B, you need to calculate 4 x + 3 x x + 1 x. Type this just as you see it on the calculator screen in Figure 2. Note that the calculator uses * for multiplication rather than x. Figure 2. To get the answer, press ENTER. The calculator has done the four multiplications first, then added the results. In a calculation like 4 * + 3 * 16, the multiplications are always done before the addition unless parentheses are used. In 4 * ( + 3) * 16, the addition is done first (Figure 3). Figure 3. Parentheses are sometimes used to indicate multiplication. Try doing the calculation over like this. Using parentheses takes a little more typing, but some people think this helps keep the points separate from the votes. In this case, the votes are in parentheses (Figure 4). Figure 4. Press ENTER to see the result (Figure 5). Figure

6 SUPPLEMENTAL ACTIVITY Unit 1: PICK A WINNER Mathematics: Modeling Our World S1.3 page 2 of 2 POINT COUNTS WITH REPLAY Now typing those zeros pays off. You are going to use the previous calculation to get the point total for C. Replay the calculation (on many calculators you do that by pressing ENTRY or 2nd ENTER). You ll get the same expression back again (Figure 6). Figure 6. Find the cursor keys on your calculator. The cursor is the flashing object that tells you where you are on the screen, and the cursor keys are used to move the cursor around. Press the cursor key that points left. Hold it down until the cursor stops on the that s in the parentheses after the 4. When the cursor is on the, touch. The should now be a, which is the number of firsts for C (Figure 7). Figure 7. Move the cursor to the right until it is on top of the 1 of the 16. Type a 3 over the 1. The cursor is now on top of the 6. Find the delete key (marked DEL on many calculators) and press it once to get rid of the 6. When you type a one-digit number on top of a two-digit number, you must delete one of the extra digits (Figure 8). Figure 8. Move the cursor to the right until it s on top of the in parentheses after the 2. Type a 1 on top of. Now press the insert key (marked INS on many calculators), then type a 3. You should now have a 13 where the was. When you type a twodigit number on top of a one-digit number, you must insert the extra digit (Figure ). Figure. Move the cursor to the right until you are on top of the in parentheses behind the 1. Type a on top of the. Press ENTER to get the result (Figure 1). Figure 1. Okay, you re on your own. Replay this calculation. Make the changes necessary to get the point total for H. Do the same for P. You should get 61 for both. If you don t, ask for help. 254

7 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY Figure 1 is a matrix showing the first-, second-, third-, and fourth-place votes for candidates B, C, H, and P in a class of high school students. Use 4 points for first, 3 for second, 2 for third, and 1 for last. The rows of this matrix are labeled with places (first, second, third, and fourth) and the columns are labeled with candidates (B, C, H, and P). So, the matrix is a place-by-candidate matrix. If you put the candidates in the rows and the places in the columns, the matrix would be a candidate-by-place matrix. Figure 2 shows a matrix containing the points for first, second, third, and fourth places. It is a point-by-place matrix. Matrices are also referred to by the number of rows and columns. This is also a 1 x 4 matrix. (The number of rows is always stated first.) Matrices are really just tables. They are used pretty much the way you would expect until you try to multiply them. To tell whether it makes sense to multiply two matrices, write their labels side by side, as shown in Figure 3. POINT COUNTS WITH MATRICES Then check to see if the second part of the first label matches the first part of the second label (Figure 4). Point-by-Place Place-by-Candidate Point-by-Place Place-by-Candidate If there is a match, multiplying the matrices makes sense in this situation. (You must be careful that the number of places in each matrix is the same. For example, if the first matrix has only three places and the second has four, then you cannot multiply the matrices. In other words, the number of columns in the first matrix must be the same as the number of rows in the second matrix.) or Point-by-Place Place-by-Candidate or Place-by-Candidate Point-by-Place 1st 2nd 3rd 4th 16 S1.4 page 1 of 2 B C H P Figure 1. 1st 2nd 3rd 4th Points Match No Match Figure 2. Figure 3. Figure 4. If you drop the matching labels, you have the label for the new matrix (Figure 5). Point-by-Place Place-by-Candidate Figure 5. The new matrix will be a point-by-candidate matrix. Since you want to know the point totals for each candidate, the new matrix tells what you want to know. (The same is true of the numbers of rows and columns. The first matrix is 1 x 4, and the second is 4 x 4. The answer is 1 x 4, which you get by dropping the 4 in 1 x 4 and the first 4 in 4 x 4.) 255

8 SUPPLEMENTAL ACTIVITY Unit 1: PICK A WINNER Mathematics: Modeling Our World S1.4 page 2 of 2 POINT COUNTS WITH MATRICES Here is how to multiply the point-by-place and place-by-candidate matrices on a calculator. Enter the point-by-place matrix on the calculator. You need to tell your calculator that the matrix has one row and four columns (Figure 6). Figure 6. Note that you may not be able to see all of the matrix at once. Enter the place-by-candidate matrix. Again, you need to specify that the matrix has four rows and four columns (Figure 7). Figure 7. On the calculator s home screen, multiply the matrices. You need to use the single-letter names that your calculator requires you to use for matrices (Figure 8). Figure 8. Figure. The calculator does the multiplication by multiplying the numbers in the first row of matrix [A] (it has only one row) times the numbers in the first column of matrix [B], then adding up the results. It then does the same thing for the second, third, and fourth columns of matrix [B]. The four totals are the point totals for the four candidates. If you want to compare point totals using two different point systems, just enter each system in a different row of matrix [A]. Suppose you want to compare the totals using the 4, 3, 2, 1 system with the totals using a 5, 3, 2, 1 system. Figure shows the new matrix [A]. Then multiply [A] and [B], just as you did before (Figure 1). The first row has the point totals with a 4, 3, 2, 1 system, and the second row has the point totals with a 5, 3, 2, 1 point system. Figure

9 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY POINT COUNTS WITH SPREADSHEETS S1.5 Figure 1 is a matrix showing the first-, second-, third-, and fourth-place votes for candidates B, C, H, and P in a class of high school students. Use 4 points for first, 3 for second, 2 for third, and 1 for last. 1st 2nd 3rd 4th 16 B C H P Figure 1. Figure 2 is a computer spreadsheet with the same data. Row 2 contains the points to be used for first, second, third, and fourth places. Enter the data on your spreadsheet in the same way, then read on to learn how to do the formulas in row 1. To get B s point total, you must multiply the four points in cell A2 by the zero firstplace votes in cell B5, the three points in cell B2 by the 16 second-place votes in cell B6, the two points in cell C2 by the zero third-place votes in cell B7, and the one point in cell D2 by the nine fourth-place votes in cell B8. Then you must add them together. Here is the formula for cell B1: =B$5*$A$2+B$6*$B$2+B$7*$C$2+B$8*$D$2 On most spreadsheets the formula must start with an equals sign. The dollar signs in the formula are called fixed references. They are there because the spreadsheet has a clever feature that automatically changes cell references when you copy the formula. The dollar signs prevent references from changing. You are about to copy the formula into cells C1, D1, and E1 so that you don t have to type that long formula three more times. You want all the Bs in the formula to change to Cs, but you don t want anything else to change. So there are dollar signs in front of everything but the Bs. Copy the formula in cell B1 into cells C1, D1, and E1. On most computers you do this by clicking on cell B1, then selecting COPY from a menu. You then highlight cells C1, D1, and E1 and select PASTE from a menu. Save your spreadsheet when you are done so that you can use it for additional problems. Figure

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11 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY ENTER ZALINSKI S1.6 An election has three candidates: Alvarez, Boswell, and Shapiro. Polls show that the preferences of the voters fall into three groups (Figure 1). Shapiro Alvarez Boswell Boswell Shapiro Alvarez Alvarez Boswell Shapiro Figure 1. 43% 2% 28% Shortly before the election, a fourth candidate, Zalinski, enters the race. Polls show that Zalinski is a relatively weak candidate who is ranked first by none of the voters. The preferences of voters still fall into three categories. The voters who previously preferred Shapiro rank Zalinski fourth. The voters who previously preferred Alvarez rank Zalinski second. The voters who previously preferred Boswell rank Zalinski third. 1.Does Zalinski have any impact on the election? Answer the question for as many different voting methods as you can. 2.Do your results in Item 1 demonstrate a flaw in any of the election methods? 25

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13 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY IS THERE HOPE IN HYBRIDS? S1.7 Could a combination of two methods be an improvement over both of them? For example, consider a new election method that works like a runoff but uses a point system to determine who gets into the runoff. You find point totals for each candidate, then hold a runoff between the top two. Araya Bolt Campanelli Bolt Bolt Martinez Araya Araya Campanelli Campanelli Martinez Campanelli Martinez Araya Bolt Martinez 34% 2% 32% 14% Figure Figure 1 shows voter preferences in an election. Use this new point/runoff hybrid to find the winner. Use a 4, 3, 2, 1 point system. 2.Does this new hybrid system avoid the flaws you have seen in runoff and point systems? (Use the preferences in Item 1 to help you answer this question or make up preferences of your own.) 261

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15 Mathematics: Modeling Our World Unit 1: PICK A WINNER SUPPLEMENTAL ACTIVITY FINAL PROJECT: HOW DO YOU VOTE? S1.8 1.List some things that you think a good voting method should and shouldn t do. 2.Select a voting method you studied in this unit and describe why you think it does or doesn t do a good job of meeting your conditions. 3.Design your own election process. Why do you think your system is good? What problems do you think it has? 4.Select a topic of interest in your school, community, state, or the country that has more than two choices. Use your method to conduct an election on the topic in your class or school. 263

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