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**Contents:**

Johnnie's Math Games. Division Steps Rap. Division Rap. Add and Subtract Negative Integers. Factor Trail game. Factors and Multiples Cartoon Video. Explanation of multiplication clusters parent support. Making Change. Making Arrays Activity. Arrays with Brainpopjr. Rays Arithmetic Book. Singapore Math examples. Place value video on Study Jams. How to read large numbers. What Adds Up to 10? Pictures of Place Value Game. Play Addition Pairs" and speed up your adding - make pairs that add to 20, then and more.

Calculator Game - great for number sense. How Much is a Million? Millions Activities - lots of them! Place Value Puzzler. Line Jumper - learn to use a number line. Let's Do Math. Place Value Mystery Game. Play the Order of Operations Game. Play the Place Value Game. Quiz It - take an interactive quiz and guess the number - click "What am I? Numberland - learn some neat facts about the numbers from Math Libs - click "Lessons" then "Numbers". Print a Yahtzee Scorecard. Play "Donut Doubler" - see how big numbers get when you double them.

Play "Math Millionaire" - test your general math skills. Play "Spin to Win" - build the largest number and win. Play the Place Value Game - try to build a large number. Play "Mend the Number Square" - uses the square. Place Value Calculator. Place Value Darts. Play Fishy Numbers.

Build Numbers with Base 10 Blocks. Mental Math Lesson. Play "Operation Domino". Number Lines - can be used with decimals also. Number Bonds number sense what add to 10? Super Sequencer - great for learning times tables or counting-by Ambleside. Scribble Table - for times table work Ambleside. Complete guide to online Math Resources thanks to Lauren Jackson for sharing. Skills by Ohio Indicators. Coin Combinations Game. Graphing Questions. Pop-up Addition Game. Story Problems Practice. Everyday Math Games. What you know about MATH? Number Sense. Probability and Statistics.

Thompson's Fourth Grade Classroom. Math Sites Of The Week. Create a Bar Graph This wonderful resource is from the U. After learning the basic principles behind bar graphs, students will create data use examples gathered from your classroom, such as the number of people wearing blue pants today, red shirts, etc. ReviseWise - Interpreting Data This web site uses colorful animation and great explanations to teach about interpreting different types of graphs.

Amblegraph This online tools lets you create simple bar charts in your web browser. Children can use it for very quick surveys or a teacher can use it to model the construction of a bar graph with a class or group. Most elements are editable. Logic Zoo In this online game the student needs to place fanciful animals into the proper pens at a zoo based on several attributes. The pens are modeled after Venn diagrams. Venn Diagram Shape Sorter This online activity has the student sort objects into a Venn diagram based on size, color, and shape. The can either choose the rules and then do the sorting to match, or they can do the sorting to try to discover the rules.

Pie Chart Explore percentages and fractions using pie charts. Spinners Work with spinners to learn about numbers and probabilities. Mean - Mode - Median Great website to show the student the difference. Just click on start. ReviseWise - Mode, median, mean This web site uses colorful animation and great explanations to teach about mean, median, and mode.

Comparing Data - Excel Activity This is an Excel spreadsheet activity in which the student enters data and then the spreadsheet generates several measures of that data such as mean, median, mode, quartiles, and range. The student can then investigate the measures to compare the different sets of data. Mean, Median, and Mode Game Great game to teach mean, median, and mode. Finding median Interactive game for finding the median of a series of numbers.

Immediate feedback for student. Box Model - Virtual Manipulative This online activity models a probability experiment of pulling numbers from a box. First, the user decides which numbers, and how many of each, are in a box. Then the computer begins randomly drawing numbers and keeping totals. Superimposed can be a graph of the expected theoretical probabilities.

This is excellent to show real v. Disguise Combos game In this online interactive game the student determines by experiment how many possible disguise combinations there are, given several pieces to work with. After each round, the mathematical formula for combinations is used for the problem. Teachers Tool box Math Dictionary animation for kids This is an outstanding website allowing you to explore a comprehensive view of math with many examples Talking Calculator Students will have fun entering number for all areas.

Created by Martha Portner Franctions and groups This Smart Board lesson will assist students in learning more about fractions. Created by Martha Portner Fractions Same parts different wholes This Smart Board lesson will assist students in learning more about fractions. Discover the thirteen ways half of an eight-piece square can be arranged without rotating or flipping the pattern Fishy Fractions - Improper Fractions In this game you help Ulani the pelican catch fish by matching improper fractions to their equivalent mixed number form, and vice versa. Please download the link above to allow you to utilize this lesson.

Smart Board Lesson - Great Resource 2-Dimensional Animation In this animation, students are shown examples of regular and irregular polygons including triangles, quadrilaterals, pentagons, hexagons and octagons Platonic Solids - Virtual Manipulative This online activity allows you to experiment with five platonic solids, counting their faces, edges, and vertices Smart Board Lesson This lesson will allow the student to learn the names of objects. Super Maths World This is a highly visual game.

Students will be totally engaged. Great Resource Polygon Playground At this interactive web site you'll find hundreds of polygons to drag anywhere you want. Explore symmetry, tessellations, and more! Shapes This website will allow your students to learn all about shapes and have fun at the same time. Chinese Tangrams Create your own tangrams using drag and drop shapes - just like Grandfather Tang's story. You may view a gallery of samples or just create on your own! Chinese Tangrams II Recreate the tangram using shapes.

Make a square, swan, cat, dinosaur, duck, phoenix, or rabbit Geometric Shapes - Spatial Learning Drag the puzzle pieces to create a square. It's not as easy as you think! Dots and Squares Click on the grid to draw one side of a square. Play against the computer. Sorting Triangles This interactive website begins with an animated review of scalene, isosceles, and equilateral triangles.

Then students get to sort the triangles using Venn diagrams based upon single and multiple characteristics. Discover Ohio Interactive map You may log on to this site and observe many different levels. Map of North East Ohio You will find a map with grid lines of this area. See if you can determine which school or object you are looking at. Simple Coordinates Game Students investigate the first quadrant of the Cartesian coordinate system through identifying the coordinates of points, or requesting that a particular point be plotted.

Requires Java Simple Maze Game Students investigate the first quadrant of the Cartesian coordinate system by directing a robot through a mine field laid out on the plane. Eleven sites of interests are shown on a 20 by 20 grid map of a city. The students are to determine the coordinates of the spots using Excel's rows and columns. Good activity for students to work on and printout. Coordinates game This interactive online game displays a 14 by 14 first quadrant grid and asks the student to find specified points when given the coordinates Coordinates II game In this online game, the student is shown a 5 by 5 first quadrant grid.

Then a point is shown on the grid and the student has to identify the coordinates. A new point is given after each is done. What's the Point? In this online activity students identify points on a grid. There are three difficulty levels being identifying points in the first quadrant, identifying points on the entire grid, and clicking on the grid to locate coordinates. Dam Jammer Game - Flips and Turns In this online game you help Bucky the Beaver to plug holes in the dam by flipping and rotating basic shapes, and then placing them on the congruent holes in the dam.

Great Resource Translation Transformation - Virtual Manipulative This online activity allows you to place pattern blocks on a grid and apply a rotation to them. You can do this on your own, or go through three pre-made activities. Reflection Transformation - Virtual Manipulative This online activity allows you to place pattern blocks on a grid and apply a reflection to them. Rotation Transformation - Virtual Manipulative This online activity allows you to place pattern blocks on a grid and apply a rotation to them. TransmoGrapher Students explore the world of translations, reflections, and rotations in the Cartesian coordinate system by transforming squares, triangles and parallelograms.

Parameters: Shape, x or y translation, x or y reflection, angle of rotation. Pattern Video You will view an outstanding video on patterns. It not only explains but will allow the student to answer questions. Patterns This lesson allows students to examine both repeating and growing patterns. Out of racers who started the marathon, completed the race, 14 gave up, and 4 were disqualified. What percentage did not complete the marathon? What percent tip is that? Employees in paid 4.

A project on Kickstarter. How much did they raise? What was their original goal? The population of a town increased from 3, in to 4, in Find the absolute and relative percent increase. Find the absolute and relative percent decrease. Why or why not? They currently have about 80 customers a day. How many customers will they have if their campaign is successful?

If they increase to customers a day, were they successflil? The Walden University had 47, students in , while Kaplan University had 77, students. Complete the following statements: a. In the Olympics, Usain Bolt ran the m dash in 9. Jim Hines won the Olympic gold with a time of 9. What percent of the original price do you end up pajdng?

In each case, what is the net percent gain or loss? Are these two claims equivalent, in conflict, or not comparable because they're talking about different things? Are the values compared in this statement comparable or not comparable? In support of the two wars, more than 6, American soldiers have lost their lives.

Find the equivalent percentage point drop. What percent relative change is this? What percent juice is the final mix? Find a unit rate: Joel ran meters in 4 minutes, 45 seconds. A crepe recipe calls for 2 eggs, 1 cup of flour, and 1 cup of milk. How much flour would you need if you use 5 eggs? How much will it cost to buy 40ft of crown molding? Four 3-megawatt wind turbines can supply enough electricity to power homes. How many turbines would be required to power 55, homes? A highway had a landslide, where 3, cubic yards of material fell on the road, requiring dump truck loads to clear.

On another highway, a slide left 40, cubic yards on the road. How many dump truck loads would be needed to clear this slide? Convert 8 feet to inches. Convert 6 kilograms to grams. How much will 3 kilometers of wire cost? Sugar contains 15 calories per teaspoon. How many calories are in 1 cup of sugar? A car is driving at kilometers per hour.

How far does it travel in 2 seconds?

Math Quiz #7: Maps, Schedules, Graphs, Charts, Data, Probability, and Statistics eBook: Mark J. Curry: rapyzure.tk: Kindle Store. The NOOK Book (eBook) of the Math Quiz #7: Maps, Schedules, Graphs, Charts, Data, Probability, and Statistics by Mark Curry at Barnes & Noble. FREE.

A chain weighs 10 pounds per foot. How many ounces will 4 inches weigh? The table below gives data on three movies. Gross earnings is the amount of money the movie brings in. Compare the net earnings money made after expenses for the three movies. For the movies in the previous problem, which provided the best return on investment?

The population of the U. The population of India is about 1,,,, covering a land area of 1,, square miles. Compare the population densities of the two countries. The population of China is about 1, million, while the population of Sweden is about 9. Compare the GDP per capita of the two countries.

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In June , Twitter was reporting million tweets per day. Each tweet can consist of up to characters letter, numbers, etc. Create a comparison to help understand the amount of tweets in a year by imagining each character was a drop of water and comparing to filling something up. The photo sharing site Flickr had 2. Create a comparison to understand this number by assuming each picture is about 2 megabytes in size, and comparing to the data stored on other media like DVDs, iPods, or flash drives.

Your chocolate milk mix says to use 4 scoops of mix for 2 cups of milk. After pouring in the milk, you start adding the mix, but get distracted and accidentally put in 5 scoops of mix. How can you adjust the mix if: a. There is still room in the cup? The cup is already full? A recipe for sabayon calls for 2 egg yolks, 3 tablespoons of sugar, and 'A cup of white wine.

After cracking the eggs, you start measuring the sugar, but accidentally put in 4 tablespoons of sugar. How can you compensate? The Deepwater Horizon oil spill resulted in 4. Each barrel of oil can be processed into about 19 gallons of gasoline. How many cars could this have fueled for a year? Assume an average car gets 20 miles to the gallon, and drives about 12, miles in a year.

Each yields about 2 tablespoons of juice. Which cylinder would hold more? How much more? Which of these glasses contains more liquid? In the next 4 questions, estimate the values by making reasonable approximations for unknown values, or by doing some research to find reasonable values.

Estimate how many gallons of water you drink in a year. Estimate how many times you blink in a day. How much does the water in a 6-person hot tub weigh? How many gallons of paint would be needed to paint a two-story house 40 ft long and 30 ft wide? During the landing of the Mars Science Laboratory Curiosity, it was reported that the signal from the rover would take 14 minutes to reach earth. Radio signals travel at the speed of light, about , miles per second. How far was Mars from Earth when Curiosity landed?

It is estimated that a driver takes, on average, 1. How far will a car traveling at 60 miles per hour travel in feet before the driver reacts to an obstacle? The flash of lightning travels at the speed of light, which is about , miles per second. The sound of lightning thunder travels at the speed of sound, which is about miles per hour. If you see a flash of lightning, then hear the thunder 4 seconds later, how far away is the lightning? Now let's generalize that result. Suppose it takes n seconds to hear the thunder after a flash of lightning.

How far away is the lightning, in terms of nl Sound travels about miles per hour. If you stand in a parking lot near a building and sound a horn, you will hear an echo. Suppose it takes about Vi a second to hear the echo. Suppose it takes n seconds to hear the echo. How far away is the building, in terms of nl It takes an air pump 5 minutes to fill a twin sized air mattress 39 by 8. How long will it take to fill a queen sized mattress 60 by 8. It takes your garden hose 20 seconds to fill your 2-gallon watering can.

How long will it take to fill a. An infiatable pool measuring 3 feet wide, 8 feet long, and 1 foot deep. A circular infiatable pool 13 feet in diameter and 3 feet deep. You want to put a 2" thick layer of topsoil for a new 20'x30' garden. The dirt store sells by the cubic yards. How many cubic yards will you need to order?

How much would it cost to fill a swimming pool 4 feet deep, 8 feet wide, and 12 feet long with Jell-0? Estimate the cost of having a 18 by 22 ft brick patio installed. Which is the better deal? The grocery store has bulk pecans on sale, which is great since you're planning on making 10 pecan pies for a wedding. Your recipe calls for PA cups pecans per pie. However, in the bulk section there's only a scale available, not a measuring cup. You run over to the baking aisle and find a bag of pecans, and look at the nutrition label to gather some info. How many pounds of pecans should you buy?

Soda is often sold in 20 ounce bottles. The nutrition label for one of these bottles is shown to the right. A packet of sugar the kind they have at restaurants for your coffee or tea typically contain 4 grams of sugar in the U. Drinking a 20 oz soda is equivalent to eating how many packets of sugar? The details may be imprecise; answer the question the best you can with the provided information.

Be sure to justify your decision. You're planning on making 6 meatloafs for a party. You go to the store to buy breadcrumbs, and see they are sold by the canister. How many canisters do you need to buy? Your fi-iend wants to cover their car in bottle caps. You need to buy some chicken for dinner tonight. Is it worth the extra drive? Is upgrading worth it?

Janine is considering buying a water filter and a reusable water bottle rather than buying bottled water. Will doing so save her money? Marcus is considering going car-free to save money and be more environmentally fi-iendly. Is this financially a good decision? For the next set of questions, research or make educated estimates for any unknown quantities needed to answer the question. Compare the costs and time involved with driving, flying, and taking a train.

Assume that if you fly or take the train you'll need to rent a car while you're there. Which option is best? You want to paint the walls of a 6ft by 9ft storage room that has one door and one window. You want to put on two coats of paint. A restaurant in New York tiled their floor with pennies.

Just for the materials, is this more expensive than using a more traditional material like ceramic tiles? If each penny has to be laid by hand, estimate how long it would take to lay the pennies for a 12ft by 10ft room. Considering material and labor costs, are pennies a cost-effective replacement for ceramic tiles? You are considering taking up part of your back yard and turning it into a vegetable garden, to grow broccoli, tomatoes, and zucchini.

Will doing so save you money, or cost you more than buying vegetables fi-om the store? Barry is trying to decide whether to keep his Honda Civic with , miles, or trade it in for a used Honda Civic. Consider gas, maintenance, and insurance costs in helping him make a decision.

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Some people claim it costs more to eat vegetarian, while some claim it costs less. Examine your own grocery habits, and compare your current costs to the costs of switching your diet irom omnivore to vegetarian or vice versa as appropriate. Which diet is more cost effective based on your eating habits? It calls for lYi cups of breadcrumbs. How many meatloafs does the recipe make? It makes 1 meatloaf. How many servings does that recipe make? It says it serves 8. How big is the canister? It is cylindrical, 3. How many breadcrumbs come in 1 canister? How much does a cup of breadcrumbs weight?

I'm not sure, but maybe something from the nutritional label will help How much does a canister cost? A Honda Accord. Everything but the windows and the underside. How big is a bottle cap? Caps are 1 inch in diameter. Info for chicken problem How much chicken will you be buying? Four pounds How far are the two stores? My neighborhood store is 2. The store across town is 8. What kind of mileage does your car get? It averages about 24 miles per gallon in the city. How many gallons does your car hold? About 14 gallons How much is gas?

Problem Solving Info for furnace problem How efficient is the current furnace? How efficient is the new flirnace? What is your gas bill? How much gas do you use? Here is the history for 2 years: Gas Usage Therms Therms 3 il-. How long do you plan to live in the house?

Probably at least 15 years. She normally drinks 3 bottles a day, each How much does a bottle of water cost? She buys packs of How much does a reusable water bottle cost? How long does a reusable water bottle last? Basically forever or until you lose it. How much does a water filter cost? How much water will they filter? The filter lasts for gallons.

Info for car-free problem Where does Marcus currently drive? How will he get to these locations without a car? Sometimes he'll be able to get a friend to pick him up. A few locations he is able to walk to. A couple locations are hard to get to by bus, but there is a ZipCar short term car rental location within a few blocks. How much does gas cost? How much does he pay for insurance and maintenance? How much is he paying for the car?

What mileage does his car get? About 26 miles per gallon on average. How much does a bus ride cost? Problem Solving 29 How much does a ZipCar rental cost? Gas, insurance, and miles are included in the cost. In the United States, federal income taxes help fund the military, the environmental protection agency, and thousands of other programs. Property taxes help fiand schools. Gasoline taxes help pay for road improvements. While very few people enjoy paying taxes, they are necessary to pay for the services we all depend upon. Taxes can be computed in a variety of ways, but are typically computed as a percentage of a sale, of one's income, or of one's assets.

Example 1 The sales tax rate in a city is 9. The sales tax will be 9. When taxes are not given as a fixed percentage rate, sometimes it is necessary to calculate the effective rate. Effective rate The effective tax rate is the equivalent percent rate of the tax paid out of the dollar amount the tax is based on.

What is the effective tax rate? Taxes are often referred to as progressive, regressive, or flat. Tax categories A flat tax, or proportional tax, charges a constant percentage rate. A progressive tax increases the percent rate as the base amount increases. A regressive tax decreases the percent rate as the base amount increases. Problem Solving 3 1 Example 3 The United States federal income tax on earned wages is an example of a progressive tax.

People with a higher wage income pay a higher percent tax on their income.

Notice that the effective rate has increased with income, showing this is a progressive tax. However, in terms of income, a gasoline tax is often considered a regressive tax. Try it Now 1 A sales tax is a fixed percentage tax on a person's purchases. Is this a flat, progressive, or regressive tax? While sales tax is a flat percentage rate, it is often considered a regressive tax for the same reasons as the gasoline tax. Some, for example, have claimed that a flat tax would be fairer. Others call for revisions to how different types of income are taxed, since currently investment income is taxed at a different rate than wage income.

The following two projects will allow you to explore some of these ideas and draw your own conclusions. Imagine the country is made up of households. We are going to determine new income tax rates. The first proposal we'll consider is a flat tax - one where every income group is taxed at the same percentage tax rate.

So, everyone in group A will pay no taxes. For simplicity, we're going to assume that a household is taxed at the same rate on all their income. There is no one right answer here - just make sure you bring in enough money! The cost of basic expenses does increase with income, since housing and car costs are higher, however usually not proportionally.

For each income group, estimate their essential expenses, and calculate their discretionary income. Then compute the effective tax rate for each plan relative to discretionary income rather than income. Which plan seems the least fair to you? Project 2: Calculating Taxes. Visit www. Qualified dividends are earnings on certain investments such as stocks. Based on these three scenarios, what are your impressions of how the income tax system treats these different forms of income wage, dividends, and business income?

Scenario 4: To get a more realistic sense for calculating taxes, you'll need to consider itemized deductions. Calculate the income taxes for someone with the income and expenses listed below. This happens when a group of friends decides which movie to watch, when a company decides which product design to manufacture, and when a democratic country elects its leaders. While the basic idea of voting is fairly universal, the method by which those votes are used to determine a winner can vary.

Amongst a group of friends, you may decide upon a movie by voting for all the movies you're willing to watch, with the winner being the one with the greatest approval.

A company might eliminate unpopular designs then revote on the remaining. A country might look for the candidate with the most votes. In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible. Preference Schedules To begin, we're going to want more information than a traditional ballot normally provides. A traditional ballot usually asks you to pick your favorite from a list of choices. This ballot fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessflil.

Preference ballot A preference ballot is a ballot in which the voter ranks the choices in order of preference. Plurality Method In this method, the choice with the most first-preference votes is declared the winner. Ties are possible, and would have to be settled through some sort of run-off vote.

This method is sometimes mistakenly called the majority method, or "majority rules", but it is not necessary for a choice to have gained a majority of votes to win. Which candidate wins under the plurality method? The election from Example 2 may seem totally clean, but there is a problem lurking that arises whenever there are three or more choices. Looking back at our preference table, how would our members vote if they only had two choices?

That hardly seems fair. Marquis de Condorcet, a French philosopher, mathematician, and political scientist wrote about how this could happen in , and for him we name our first fairness criterion. Fairness Criteria The fairness criteria are statements that seem like they should be true in a fair election. Condorcet Criterion If there is a choice that is preferred in every one-to-one comparison with the other choices, that choice should be the winner. We call this winner the Condorcet Winner, or Condorcet Candidate. Example 3 In the election from Example 2, what choice is the Condorcet Winner?

We see above that Hawaii is preferred over Anaheim. Comparing Hawaii to Orlando, we can see 6 out of 10 would prefer Hawaii to Orlando. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. Analyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons: EUe vs Don: prefer EUe; prefer Don: Don is preferred EUe vs Key: prefer EUe; prefer Key: Key is preferred Don vs Key: prefer Don; prefer Key: Don is preferred So even though Don had the smallest number of first-place votes in the election, he is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.

Try it Now 2 Consider the election fi-om Try it Now 1. Is there a Condorcet winner in this election? Lisincere voting is when a person casts a ballot counter to their actual preference for strategic purposes. In the case above, the democratic leadership might realize that Don and Key will split the vote, and encourage voters to vote for Key by officially endorsing him.

Not wanting to see their party lose the election, as happened in the scenario above, Don's supporters might insincerely vote for Key, effectively voting against EUe. Instant Runoff Voting Instant Runoff Voting IRV , also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting. The choice with the least first-place votes is then eliminated from the election, and any votes for that candidate are redistributed to the voters' next choice. This is similar to the idea of holding runoff elections, but since every voter's order of preference is recorded on the ballot, the runoff can be computed without requiring a second costly election.

This voting method is used in several political elections around the world, including election of members of the Australian House of Representatives, and was used for county positions in Pierce County, Washington until it was eliminated by voters in Example 5 Consider the preference schedule below, in which a company's advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity. A majority would be 1 1 votes.

No one yet has a majority, so we proceed to elimination rounds. There is still no choice with a majority, so we eliminate again. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone's options to fill the gaps. Still no majority, so we eliminate again. Round 3: We make our third elimination.

C has the fewest votes. Find the winner using IRV. The people who voted for Don have their votes transferred to their second choice. We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner.

On the other hand, the temptation has been removed for Don's supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded. Example 7 Consider the voting system below. Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 for Brown.

Wanting to "jump on the bandwagon", 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter. Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with 5 1 votes to Adams' 49 votes!

Even though the only vote changes made favored Adams, the change ended up costing Adams the election. This doesn't seem right, and introduces our second fairness criterion: Monotonicity Criterion If voters change their votes to increase the preference for a candidate, it should not harm that candidate's chances of winning. This criterion is violated by this election. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; just that IRV has the potential to violate the criterion in certain elections.

Borda Count In this method, points are assigned to candidates based on their ranking; 1 point for last choice, 2 points for second-to-last choice, and so on. The point values for all ballots are totaled, and the candidate with the largest point total is the winner. Example 8 A group of mathematicians are getting together for a conference. The members are coming from four cities: Seattle, Tacoma, Puyallup, and Olympia. Their approximate locations on a map are shown to the right. Try it Now 4 Consider again the election from Try it Now 1.

Find the winner using Borda Count. Since we have some incomplete preference ballots, for simplicity, give every unranked candidate 1 point, the points they would normally get for last place. You might have already noticed one potential flaw of the Borda Count fi-om the previous example. In that example, Seattle had a majority of first-choice votes, yet lost the election! This seems odd, and prompts our next fairness criterion: Majority Criterion If a choice has a majority of first-place votes, that choice should be the winner.

Borda count is sometimes described as a consensus-based voting system, since it can sometimes choose a more broadly acceptable option over the one with majority support. In the example above, Tacoma is probably the best compromise location. This is a different approach than plurality and instant runoff voting that focus on first-choice votes; Borda Count considers every voter's entire ranking to determine the outcome.

Because of this consensus behavior, Borda Count, or some variation of it, is commonly used in awarding sports awards. The Copeland Method specifically attempts to satisfy the Condorcet Criterion by looking at pairwise one- to-one comparisons. Copeland's Metliod In this method, each pair of candidates is compared, using all preferences to determine which of the two is more preferred. The more preferred candidate is awarded 1 point. If there is a tie, each candidate is awarded Vi point.

After all pairwise comparisons are made, the candidate with the most points, and hence the most pairwise wins, is declared the winner. Variations of Copeland's Method are used in many professional organizations, including election of the Board of Trustees for the Wikimedia Foundation that runs Wikipedia. Voting Theory 45 Example 9 Consider our vacation group example from the beginning of the chapter. Determine the winner using Copeland's Method. You may recall we did this earlier when determining the Condorcet Winner.

For example, comparing Hawaii vs Orlando, we see that 6 voters, those shaded below in the first table below, would prefer Hawaii to Orlando. Note that Hawaii doesn't have to be the voter's first choice - we're imagining that Anaheim wasn't an option. If it helps, you can imagine removing Anaheim, as in the second table below. Comparing Anaheim to Orlando, the 1 voter in the first column clearly prefers Anaheim, as do the 3 voters in the second column.

The 3 voters in the third column clearly prefer Orlando. The 3 voters in the last column prefer Hawaii as their first choice, but if they had to choose between Anaheim and Orlando, they'd choose Anaheim, their second choice overall. So, comparing Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point.

All together, Hawaii vs Orlando: 6 votes to 4 votes: Hawaii gets 1 point Anaheim vs Orlando: 7 votes to 3 votes: Anaheim gets 1 point Hawaii vs Anaheim: 6 votes to 4 votes: Hawaii gets 1 point Hawaii is the winner under Copeland's Method, having earned the most points.

Notice this process is consistent with our determination of a Condorcet Winner.

Determine the winner using Copeland's method. Notice that in this case, D is not a Condorcet Winner. While Copeland's method will also select a Condorcet Candidate as the winner, the method still works in cases where there is no Condorcet Winner. Try it Now 5 Consider again the election from Try it Now 1. Find the winner using Copeland's method. Since we have some incomplete preference ballots, we'll have to adjust. For example, when comparing M to B, we'll ignore the 20 votes in the third column which do not rank either candidate.

It also satisfies the Majority Criterion and the Monotonicity Criterion. So is this the perfect method? Well, in a word, no. However, the committee then discovers that Dimitry was not eligible for the scholarship he failed his last math class. Even though this seems like it shouldn't affect the outcome, the committee decides to recount the vote, removing Dimitry from consideration.

This leads us to another fairness criterion. Equivalently, if choice A is preferred over choice B, introducing or removing a choice C should not cause B to be preferred over A. The waitress tells him he has two choices: apple pie and blueberry pie. Sidney orders the apple pie. After a few minutes the waitress returns and says that they also have cherry pie at which point Morgenbesser says "In that case I'll have the blueberry pie.

For this reason, Copeland's method is usually the first part of a more advanced method that uses more sophisticated methods for breaking ties and determining the winner when there is not a Condorcet Candidate. So Where's the Fair Method? At this point, you're probably asking why we keep looking at method after method just to point out that they are not fiilly fair.

We must be holding out on the perfect method, right? Unfortunately, no. A mathematical economist, Kenneth Arrow, was able to prove in that there is no voting method that will satisfy all the fairness criteria we have discussed. Arrow's Impossibility Theorem Arrow's Impossibility Theorem states, roughly, that it is not possible for a voting method to satisfy every fairness criteria that we've discussed. This scenario is dubbed Condorcet's Voting Paradox, and demonstrates how voting preferences are not transitive just because A is preferred over B, and B over C, does not mean A is preferred over C.

In this election, there is no fair resolution. Usually the decision of which method to use is based on what seems most fair for the situation in which it is being applied. Approval Voting Up until now, we've been considering voting methods that require ranking of candidates on a preference ballot. There is another method of voting that can be more appropriate in some decision making scenarios. With Approval Voting, the ballot asks you to mark all choices that you find acceptable.

The results are tallied, and the option with the most approval is the winner. Example 12 A group of friends is trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an "X" which movies they are willing to watch. In this vote. The Matrix would be the winner. Try it Now 6 Our mathematicians deciding on a conference location from earlier decide to use Approval voting.

Their votes are tallied below. Find the winner using Approval voting. Approval voting can very easily violate the Majority Criterion. Now suppose that this election was held using Approval Voting, and every voter marked approval of their top two candidates. A would receive approval from 80 voters B would receive approval from voters C would receive approval from 20 voters B would be the winner. Some argue that Approval Voting tends to vote the least disliked choice, rather than the most liked candidate.

Additionally, Approval Voting is susceptible to strategic insincere voting, in which a voter does not vote their true preference to try to increase the chances of their choice winning. For example, in the movie example above, suppose Bob and Alice would much rather watch Scream. They remove The Matrix from their approval list, resulting in a different result. Scream received 6 approvals, and The Matrix received 5 approvals. By voting insincerely.

Bob and Alice were able to sway the result in favor of their preference. Voting in America In American politics, there is a lot more to selecting our representatives than simply casting and counting ballots. The process of selecting the president is even more complicated, so we'll save that for the next chapter. Instead, let's look at the process by which state congressional representatives and local politicians get elected.

For most offices, a sequence of two public votes is held: a primary election and the general election. For non-partisan offices like sheriff and judge, in which political party affiliation is not declared, the primary election is usually used to narrow the field of candidates. Voting Theory 5 1 Typically, the two candidates receiving the most votes in the primary will then move forward to the general election. While somewhat similar to instant runoff voting, this is actually an example of sequential voting - a process in which voters cast totally new ballots after each round of eliminations.

Sequential voting has become quite common in television, where it is used in reality competition shows like American Idol. Congressional, county, and city representatives are partisan offices, in which candidates usually declare themselves a member of a political party, like the Democrats, Republicans, the Green Party, or one of the many other smaller parties. As with non-partisan offices, a primary election is usually held to narrow down the field prior to the general election. Prior to the primary election, the candidate would have met with the political party leaders and gotten their approval to run under that party's affiliation.

In other states, caucuses are used, which are basically meetings of the political parties, only open to party members. Closed primaries are often disliked by independent voters, who like the fiexibility to change which party they are voting in. Open primaries do have the disadvantage that they allow raiding, in which a voter will vote in their non-preferred party's primary with the intent of selecting a weaker opponent for their preferred party's candidate.

Washington State currently uses a different method, called a top 2 primary, in which voters select from the candidates from all political parties on the primary, and the top two candidates, regardless of party affiliation, move on to the general election. While this method is liked by independent voters, it gives the political parties incentive to select a top candidate internally before the primary, so that two candidates will not split the party's vote.

Regardless of the primary type, the general election is the main election, open to all voters. Except in the case of the top 2 primary, the top candidate from each major political party would be included in the general election. While rules vary state-to-state, for an independent or minor party candidate to get listed on the ballot, they typically have to gather a certain number of signatures to petition for inclusion. Using IRV: G has the fewest first-choice votes, so is eliminated first. M wins under Copeland's method. To decide on a new website design, the designer asks people to rank three designs that have been created labeled A, B, and C.

The individual ballots are shown below. Create a preference table. To decide on a movie to watch, a group of friends all vote for one of the choices labeled A, B, and C. The planning committee for a renewable energy trade show is trying to decide what city to hold their next show in. The votes are shown below. How many voters voted in this election? How many votes are needed for a majority?

A plurality? Find the winner under the plurality method. Find the winner under the Borda Count Method. Find the winner under the Instant Runoff Voting method. Find the winner under Copeland's method. A non-profit agency is electing a new chair of the board. The student government is holding elections for president. There are four candidates labeled A, B, C, and D for convenience. The homeowners association is deciding a new set of neighborhood standards for architecture, yard maintenance, etc.

Fo ur options have been proposed. Consider an election with votes. If there are 4 candidates, what is the smallest number of votes that a plurality candidate could have? If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have? If there are 7 candidates, what is the smallest number of votes that a plurality candidate could have? Voting Theory 55 9. Does this voting system havin g a Condorcet Candidate?

If so, find it. Does this voting system having a Condorcet Candidate? The marketing committee at a company decides to vote on a new company logo. They decide to use approval voting. Their results are tallied below. Each column shows the number of voters with the particular approval vote.

Which logo wins under approval voting? The downtown business association is electing a new chairperson, and decides to use approval voting. The tally is below, where each column shows the number of voters with the particular approval vote. Which candidate wins under approval voting? An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third.

If for some reason the election had to be held again and C decided to drop out of the election, which caused B to become the winner, which is the primary fairness criterion violated in this election? If for some reason the election had to be held again and many people who had voted for C switched their preferences to favor A, which caused B to become the winner, which is the primary fairness criterion violated in this election?

If in a head-to-head comparison a majority of people prefer B to A or C, which is the primary fairness criterion violated in this election? If B had received a majority of first place votes, which is the primary fairness criterion violated in this election? Exploration In the election shown below under the Plurality method, explain why voters in the third column might be inclined to vote insincerely.

How could it affect the outcome of the election? In the election shown below under the Borda Count method, explain why voters in the second column might be inclined to vote insincerely. Compare and contrast the motives of the insincere voters in the two questions above. Consider a two party election with preferences shown below. Suppose a third candidate, C, entered the race, and a segment of voters sincerely voted for that third candidate, producing the preference schedule fi-om 17 above. Explain how other voters might perceive cand idate C.

Number of voters 96 1st choice A B 2nd choice B A In question 18, we showed that the outcome of Borda Count can be manipulated if a group of individuals change their vote. Show that it is possible for a single voter to change the outcome under Borda Count if there are four candidates. Show that it is not possible for a single voter to change the outcome under Borda Count if there are three candidates. Voting Theory 57 Show that when there is a Condorcet winner in an election, it is impossible for a single voter to manipulate the vote to help a different candidate become a Condorcet winner.

The Pareto criterion is another fairness criterion that states: If every voter prefers choice A to choice B, then B should not be the winner. Explain why plurality, instant runoff, Borda count, and Copeland's method all satisfy the Pareto condition. Sequential Pairwise voting is a method not commonly used for political elections, but sometimes used for shopping and games of pool. In this method, the choices are assigned an order of comparison, called an agenda.

The first two choices are compared. The winner is then compared to the next choice on the agenda, and this continues until all choices have been compared against the winner of the previous comparison. Using the preference schedule below, apply Sequential Pairwise voting to determine the winner, using th e agenda: A, B, C, D. Show that Sequential Pairwise voting can violate the Pareto criterion. Show that Sequential Pairwise voting can violate the Majority criterion.

The Coombs method is a variation of instant runoff voting. In Coombs method, the choice with the most last place votes is eliminated. Apply Coombs method to the preference schedules from questions 5 and 6. Copeland's Method is designed to identify a Condorcet Candidate if there is one, and is considered a Condorcet Method.

There are many Condorcet Methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist.

GED Math Lessons Part 10 - Data, Statistics, Probability, Permutations & Combinations

Copeland's method does not have a tie-breaking procedure built-in. Research the Schulze method, another Condorcet method that is used by the Wikimedia foundation that runs Wikipedia, and give some examples of how it works. The plurality method is used in most U. Some people feel that Ross Perot in and Ralph Nader in changed what the outcome of the election would have been if they had not run. Research the outcomes of these elections and explain how each candidate could have affected the outcome of the elections for the election, you may wish to focus on the count in Florida.

Describe how an alternative voting method could have avoided this issue. Instant Runoff Voting and Approval voting have supporters advocating that they be adopted in the United States and elsewhere to decide elections. Research comparisons between the two methods describing the advantages and disadvantages of each in practice. Summarize the comparisons, and form your own opinion about whether either method should be adopted.

In a primary system, a first vote is held with multiple candidates. In some states, each poUtical party has its own primary. In Washington State, there is a "top two" primary, where all candidates are on the ballot and the top two candidates advance to the general election, regardless of party. Compare and contrast the top two primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc. In some many states, where voters must declare a party to vote in the primary election, and they are only able to choose between candidates for their declared party.

The top candidate from each party then advances to the general election. Compare and contrast this primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc. Sometimes in a voting scenario it is desirable to rank the candidates, either to establish preference order between a set of choices, or because the election requires multiple winners.

For example, a hiring committee may have 30 candidates apply, and need to select 6 to interview, so the voting by the committee would need to produce the top 6 candidates. Weighted Voting 59 Weighted Voting In a corporate shareholders meeting, each shareholders' vote counts proportional to the amount of shares they own. An individual with one share gets the equivalent of one vote, while someone with shares gets the equivalent of votes. This is called weighted voting, where each vote has some weight attached to it. Weighted voting is sometimes used to vote on candidates, but more commonly to decide "yes" or "no" on a proposal, sometimes called a motion.

Weighted voting is applicable in corporate settings, as well as decision making in parliamentary governments and voting in the United Nations Security Council. In weighted voting, we are most often interested in the power each voter has in influencing the outcome. Beginnings We'll begin with some basic vocabulary for weighted voting systems. Vocabulary for Weighted Voting Each individual or entity casting a vote is called a player in the election. They're often notated as Pi, Pi, Ps, Each player is given a weight, which usually represents how many votes they get.

The quota is the minimum weight needed for the votes or weight needed for the proposal to be approved. A weighted voting system will often be represented in a shorthand form: [q: wi, W2, ws, Example 1 In a small company, there are 4 shareholders. They are trying to decide whether to open a new location. This could be represented by the weighted voting system: [ 30,25,25,20] Here we have treated the percentage ownership as votes, so Mr.

Limits on the Quota The quota must be more than Vi the total number of votes. The quota can't be larger than the total number of votes. If the quota was set at only 3, then player 1 could vote yes, players 2 and 3 could vote no, and both would reach quota, which doesn't lead to a decision being made. In order for only one decision to reach quota at a time, the quota must be at least half the total number of votes.