Understanding Algebra Word Problems

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Understanding Algebra Word Problems
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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)

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Word Problem Types

• Number Problems
• Distance Problems
• Mixture Problems
• Work Problems

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Number Problems
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Types

• Age Problems
• Integer Problems
• Consecutive Integer Problems
• Always use 2n for consecutive integer problems

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Solving Number Problems

• All unknowns expressed in terms of one number
• Identify the number
• Write algebraic expressions in terms of single
  number

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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

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Benefits of Number Problems

• Identifying root
• Writing algebraic expressions
• Writing equations

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Distance Problems

• Distance (rate)(time)

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Types

• Traveling in the Same Direction
• Overtaking
• Separating
• Traveling in Opposite Directions
• Traveling Toward Each Other
• Traveling Away From Each Other
• Going and Returning

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Overtaking in the Same Direction

• Keys
• Slower vehicle travels longer
• ts gt 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

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Overtaking Problem

• A fishing boat leaves Tampa Bay at 400 a.m. and
  travels at 12 knots. At 500 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

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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

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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)

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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

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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

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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

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Going and Returning Problem

• Brandon and Shanda walk to Grandmas 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
  Grandmas?
• Let h be time to walk
• Then h 1 is time to ride
• 4h 8(h 1)

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Mixture Problems
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Types

• Differently Valued Items
• Money (coins, bills)
• Items priced by pound (candy, nuts, etc.)
• Tickets
• Different Interest Rates
• Different Liquid Solutions

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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

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Coin Problem

• A coin bank contains four more quarters than
  nickels, twice as many dimes as nickels, and five
  more than three times as many pennies as nickels.
  If the bank contains 22.25, how many of each
  coin are in it?

• Let n be number of nickels
• n 4 quarters
• 2n dimes
• 3n 5 pennies

(3n 5)(0.01) n(0.05) 2n(0.10) (n
4)(0.25) 22.25
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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
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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

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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

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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)

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Work Problems

• work done (rate of work)(time spent)

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Types

• Working Together
• Sum of contributions Complete Task
• Working Against Each Other
• Difference between contributions Complete Task

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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

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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)