The Joy of Algebra Word Problems

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The Joy of Algebra Word Problems Powered By Docstoc
					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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