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Understanding Algebra Word Problems How Do I Teach? Project a positive attitude Explain types Emphasize systematic approach • Identify type • Identify key unknown • Express all quantities in terms of key unknown • Write equation • Check answers in problem (not equation) Word Problem Types Number Problems Distance Problems Mixture Problems Work Problems Number Problems Types Age Problems Integer Problems Consecutive Integer Problems • Always use 2n for consecutive integer problems Solving Number Problems All unknowns expressed in terms of one number Identify the number Write algebraic expressions in terms of single number Integer Problem The sum of three numbers is 40. The first number is 4 more than the product of 6 and the second number. The third number is 9 less than twice the second number. Find the three numbers. Let n represent the second number • First number is 6n + 4 • Third number is 2n – 9 (6n + 4) + n + (2n – 9) = 40 Benefits of Number Problems Identifying root Writing algebraic expressions Writing equations Distance Problems Distance = (rate)(time) Types Traveling in the Same Direction • Overtaking • Separating Traveling in Opposite Directions • Traveling Toward Each Other • Traveling Away From Each Other Going and Returning Overtaking in the Same Direction Keys • Slower vehicle travels longer ts > tf • Slower vehicle has head start ts = tf Slower/Earlier Vehicle = rsts Faster/Later Vehicle = rftf Simplest Solution: rftf = rsts • When travel times can be determined Alternate Solution: rftf – rstf = Constant • When given head start distance Overtaking Problem A fishing boat leaves Tampa Bay at 4:00 a.m. and travels at 12 knots. At 5:00 a.m. a second boat leaves the same dock for the same destination and travels at 14 knots. How long will it take the second boat to catch the first? Let t be travel time of first boat Then t – 1 is the travel time of second boat 14(t – 1) = 12t Separating in Same Direction Key: Travel time same for both Faster Vehicle = rft Slower Vehicle = rst Solution: rft - rst = Constant • Constant is desired distance between Separating Problem At the auto race one car travels 190 mph while another travels 195 mph. How long will it take the faster car to gain two laps on the slower car if the speedway track is 2.5 miles long? Let t be the racing time 195t – 190t = (2)(2.5) Traveling in Opposite Directions Common variants • Traveling toward each other • Traveling away from each other Key: Net effect is sum of individual distances • Distance covered by vehicle1 = r1t1 • Distance covered by vehicle2 = r2t2 t1 may or may not equal t2 Solution: r1t1 + r2t2 = Constant Coming Together Problem A freight train leaves Centralia for Chicago at the same time a passenger train leaves Chicago for Centralia. The freight train moves at a speed of 45 mph, and the passenger train travels at a speed of 64 mph. If Chicago and Centralia are 218 miles apart, how long will it take the two trains to meet? Let t be time to meet 45t + 64t = 218 Going and Returning Key: Distance between points always same • Speeds usually vary • Times usually vary Common variants • To and from a location • Up and down a waterway Distance going = rgtg Distance returning = rrtr Solution: rgtg = rrtr Going and Returning Problem Brandon and Shanda walk to Grandma’s house at a rate of 4 mph. They ride their bicycles back home at a rate of 8 mph over the same route that they walked. It takes one hour longer to walk than to ride. How long did it take them to walk to Grandma’s? Let h be time to walk Then h – 1 is time to ride 4h = 8(h – 1) Mixture Problems Types Differently Valued Items • Money (coins, bills) • Items priced by pound (candy, nuts, etc.) • Tickets Different Interest Rates Different Liquid Solutions Differently Valued Items Key: Total value equals sum of value of different items • All values must be expressed in same units Total ValueItem 1=(QuantityItem 1)(ValueItem 1) Total ValueItem 2=(QuantityItem 2)(ValueItem 2) Total ValueItem 3=(QuantityItem 3)(ValueItem 3) Solution: (Q1)(V1) + (Q2)(V2) + (Q3)(V3) + … = QtVt Coin Problem A coin bank contains Let n be number of four more quarters nickels than nickels, twice as many dimes as n + 4 = quarters nickels, and five more 2n = dimes than three times as many pennies as 3n + 5 = pennies nickels. If the bank contains $22.25, how many of each coin are in it? (3n + 5)(0.01) + n(0.05) + 2n(0.10) + (n + 4)(0.25) = 22.25 Different Interest Rates Key: Total interest = sum of interest from individual investments • Rate and time must be expressed consistently InterestInvestment 1=(Principal1)(Rate1)(Time1) InterestInvestment 2=(Principal2)(Rate2)(Time2) InterestInvestment 3=(Principal3)(Rate3)(Time3) (p1)(r1)(t1) + (p2)(r2)(t2) + (p3)(r3)(t3) + . . . = Interest Income Interest Problem An investor has $500 more invested at 7% than he does at 5%. If his annual interest is $515, how much does he have invested at each rate? Let p be amount at 5% Then p + 500 is amount at 7% p(0.05)(1) + (p + 500)(0.07)(1) = 515 Different Solutions Key: Solute in parts equals solute in whole Common variants • Adding pure dilutant • Adding pure solute SoluteSolution 1 = QuantitySolution 1ConcentrationSolution 1 SoluteSolution 2 = QuantitySolution 2ConcentrationSolution 2 SoluteSolution 3 = QuantitySolution 3ConcentrationSolution 3 Equation: q1c1 + q2c2 + q3c3 + . . . = qmixturecmixture Solutions Problem How many gallons of cream that is 30% butterfat must be mixed with milk that is 3% butterfat to make 45 gallons that are 12% butterfat? Let c be gallons of cream Then 45 – c is gallons of milk 0.30c + 0.03(45 – c) = 0.12(45) Work Problems work done = (rate of work)(time spent) Types Working Together • Sum of contributions = Complete Task Working Against Each Other • Difference between contributions = Complete Task Work Problem Ron, Mike, and Tim are going to paint a house together. Ron can paint one side of the house in 4 hours. To paint an equal area, Mike takes only 3 hours and Tim 2 hours. If the men work together, how long will it take them to paint one side of the house? Let t be time needed to paint the side. (1/4)t + (1/3)t + (1/2)t = 1 How Do I Test? Select typical problems (10 - 15 per test) • Avoid confusing problems • Avoid “trick” problems Use as bonus questions if used at all Focus on developing equations • Draw diagram (if helpful) • Identify unknown(s) • Write equation Avoid lots of algebra (or arithmetic)

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posted: | 5/8/2010 |

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