Lesson 12: Apportionment & Voting Methods PDF

APPORTIONMENT MAT101: FINALS - Lesson 1  At the end of this lesson, the learners should be able to: 1. Define apportionment and its significance in mathematical contexts. 2. Explain the basic methods of apportionment, including Hamilton’s Method, Jefferson’s Method, and Webster’s Method. 3....

APPORTIONMENT MAT101: FINALS - Lesson 1  At the end of this lesson, the learners should be able to: 1. Define apportionment and its significance in mathematical contexts. 2. Explain the basic methods of apportionment, including Hamilton’s Method, Jefferson’s Method, and Webster’s Method. 3. Apply apportionment methods to solve real-world allocation problems. 4. Evaluate the fairness and limitations of apportionment methods. INTRODUCTION Apportionment is the mathematical practice of distributing a set number of resources (such as seats, goods, or funds) among different groups in proportion to their respective quotas. It is essential in ensuring that resources are allocated fairly, based on criteria such as population, contribution, or other measures of need. KEY CONCEPTS Standard Divisor: The ratio of the total population (or other relevant measure) to the number of resources available. 𝑇𝑜𝑡𝑎𝑙 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 = 𝑇𝑜𝑡𝑎𝑙 𝑅𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠 Quota: Each group’s share of resources, calculated by dividing the group’s population by the standard divisor. 𝐺𝑟𝑜𝑢𝑝 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑄𝑢𝑜𝑡𝑎 = 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 KEY CONCEPTS Lower Quota: The largest whole number less than or equal to the quota. Upper Quota: The smallest whole number greater than or equal to the quota.  BASIC APPORTIONMENT METHODS A. Hamilton’s Method This method ensures fairness by allocating resources based on quotas and then distributing any remaining resources to the groups with the largest fractional parts of their quotas. Steps: 1. Calculate the standard divisor. 2. Compute the quotas for each group. 3. Assign each group its lower quota. 4. Distribute remaining resources (if any) to the groups with the largest fractional parts until all resources are allocated.  BASIC APPORTIONMENT METHODS A. Hamilton’s Method Example: Suppose we need to apportion 10 seats among three regions with populations of 250, 300, and 450. Total population = 1,000 Lower quotas: A = 2, B = 3, C = 4 1,000 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 = 10 = 100 (total = 9 seats). Quotas: Assign the remaining seat to C, 250 which has the largest fractional part o Region A: = 2.5 100 (0.5). 300 o Region B: =3 Final allocation: A = 2, B = 3, C = 5. 100 450 o Region C: = 4.5 100  BASIC APPORTIONMENT METHODS B. Jefferson’s Method Jefferson’s Method modifies the divisor downward until the sum of the rounded-down quotas equals the total number of resources. Steps: 1. Start with the standard divisor. 2. Compute quotas and round each down to the nearest whole number. 3. If the total is less than the available resources, adjust the divisor downward and recalculate.  BASIC APPORTIONMENT METHODS B. Jefferson’s Method Example: Apportion 10 seats to regions with populations of 200, 300, and 500. Total population = 1,000 1,000 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑖𝑣𝑖𝑠𝑜𝑟 = = 100 10 Quotas: 200 o Region A: =2 100 300 o Region B: 100 = 3 500 o Region C: 100 = 5 Total: 2 + 3 + 5 = 10 (no adjustment needed).  BASIC APPORTIONMENT METHODS B. Jefferson’s Method Problem: A university needs to allocate 20 student council seats to its three colleges based on their enrollment numbers. The enrollments are as follows: College of Arts: 1,200 students College of Science: 1,800 students College of Business: 3,000 students Distribute the 20 seats using Jefferson’s Method.  BASIC APPORTIONMENT METHODS B. Jefferson’s Method Solution: 1. Step 1: Calculate the Standard Divisor 3. Step 3: Assign Seats 𝑇𝑜𝑡𝑎𝑙 𝐸𝑛𝑟𝑜𝑙𝑙𝑚𝑒𝑛𝑡 1,200 + 1,800 + 3,000 6,000 o No adjustment is needed as the 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 = = = = 300 𝑇𝑜𝑡𝑎𝑙 𝑆𝑒𝑎𝑡𝑠 20 20 total matches the available seats. 2. Step 2: Calculate Quotas o Final allocation: 1,200 College of Arts: =4 o 300 ▪ College of Arts: 4 seats 1,800 o College of Science: 300 =6 ▪ College of Science: 6 seats o College of Business: 3,000 = 10 ▪ College of Business: 10 seats 300 o Total = 4 + 6 + 10 = 20 (already matches total seats).  BASIC APPORTIONMENT METHODS C. Webster’s Method Webster’s Method uses standard rounding to allocate resources. If rounding leads to over-allocation, adjust the divisor upward; if under-allocation, adjust it downward. Steps: 1. Compute quotas. 2. Round quotas to the nearest whole number. 3. Adjust the divisor if necessary and recalculate.  BASIC APPORTIONMENT METHODS C. Webster’s Method Example: Similar to Jefferson’s Method, but using standard rounding. Problem: A company is distributing 50 new project assignments among three departments based on their project proposals. The numbers of proposals submitted are as follows: Department A: 120 proposals Department B: 180 proposals Department C: 300 proposals Distribute the 50 assignments using Webster’s Method.  BASIC APPORTIONMENT METHODS C. Webster’s Method 3. Step 3: Round Quotas Solution: o Using standard rounding: 1. Step 1: Calculate the Standard Divisor ▪ Department A: 10 (no change) 𝑇𝑜𝑡𝑎𝑙 𝑃𝑟𝑜𝑝𝑜𝑠𝑎𝑙𝑠 120 + 180 + 300 600 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 = = = = 12 ▪ Department B: 15 (no change) 𝑇𝑜𝑡𝑎𝑙 𝐴𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡𝑠 50 50 ▪ Department C: 25 (no change) 2. Step 2: Calculate Quotas 4. Step 4: Assign Assignments 120 o Department A: = 10 o Total matches the available assignments; no further 12 adjustment is needed. 180 o Department B: = 15 12 o Final allocation: 300 o Department C: = 25 ▪ Department A: 10 assignments 12 ▪ Department B: 15 assignments o Total = 10 + 15 + 25 = 50 (matches total assignments). ▪ Department C: 25 assignments Other Example: Cabuyao Elementary School plans to assign 40 teachers to teach each of the grades one to six. The number of students in each grade is given in the table below. Use Hamilton’s method to assign each grade level the appropriate number of teachers. Grade 1 2 3 4 5 6 Total Enrollment 120 90 135 78 200 77 700 Hamilton’s method Grade 1 2 3 4 5 6 Total SD Enrollment 120 90 135 78 200 77 700 Quota Lower Quota Final Apportionment  RESOURCES 1. Interactive Apportionment Applet - Explore methods step by step with an interactive tool. https://apportionmentcalculator.com 2. NCTM Resources on Fair Allocation - In-depth materials and case studies. https://nctm.org/apportionment 3. Video Tutorial: Apportionment Methods Simplified - Detailed explanations with examples. https://youtube.com/apportionment THANK YOU! Voting Methods MAT101: FINALS - Lesson 2  At the end of this lesson, the learners should be able to: 1. Define voting and its significance in decision-making processes. 2. Explain different voting methods and their applications. 3. Analyze the advantages and limitations of each voting method. 4. Solve problems involving various voting scenarios using appropriate methods.  INTRODUCTION Voting is a method of decision-making in which individuals express their preferences or choices among alternatives. - It is fundamental in democratic systems, organizational decision-making, and group settings where collective choices are required.  KEY CONCEPTS Voter Preferences: Each voter ranks or chooses among alternatives based on personal preferences. Outcome: The alternative chosen based on the voting process. Fairness Criteria: A set of principles to evaluate voting methods, such as: o Majority Rule: The option preferred by more than half of the voters should win. o Condorcet Criterion: If an option is preferred in head-to-head comparisons against all others, it should win. o Independence of Irrelevant Alternatives: Adding or removing an option should not affect the outcome.  VOTING METHODS A. Plurality Method The candidate or option with the most votes wins. This method is simple and widely used but can result in a winner who does not have majority support. Example: Three candidates (A, B, C) and their votes: A: 40 votes B: 35 votes C: 25 votes Winner: Candidate A (most votes).  VOTING METHODS B. Borda Count Method Voters rank all options. Points are assigned based on ranks, and the option with the highest total score wins. Example: Three candidates (A, B, C), and three voters rank them: Voter 1: A > B > C Points: A: 2 + 3 + 1 = 6 Voter 2: B > A > C B: 3 + 2 + 2 = 7 Voter 3: C > B > A C: 1 + 1 + 3 = 5 Winner: Candidate B.  VOTING METHODS C. Condorcet Method The winner must defeat every other option in one-on-one comparisons. This method identifies a "true majority" winner but may not always produce a result. Example: Pairwise comparisons between candidates A, B, and C: A vs. B: A wins A vs. C: A wins B vs. C: B wins Winner: Candidate A (wins all comparisons).  VOTING METHODS D. Approval Voting Voters can vote for as many candidates as they approve of. The candidate with the most votes wins. Example: Three candidates (A, B, C), and voters cast approval votes: Votes: Voter 1: A, B A: 2 Voter 2: B, C B: 2 Voter 3: A C: 1 Winners: Tie between A and B.  VOTING METHODS E. Instant Runoff Voting (IRV) Voters rank candidates. If no candidate gets a majority, the one with the fewest votes is eliminated, and their votes are redistributed until one candidate has a majority. Example: First round: Eliminate C, redistribute votes: A: 40 votes A: 40 + 15 = 55 B: 35 votes B: 35 + 10 = 45 C: 25 votes Winner: Candidate A. RESOURCES 1. Khan Academy: Voting Systems Explained https://khanacademy.org/voting 2. Interactive Voting Methods Tool https://votingmethods.org/tools 3. Video: Understanding Voting Systems https://youtube.com/votingsystems THANK YOU!