The Core Purpose of Word
Problems
CMC South
30th Annual Conference
3/15/14
Jack Appleman
Irvine Valley College
JAppleman@IVC.edu
CMC3-South Website Distribution Version
The “Common Core” and “Common
Sense”
Students should…
• Make sense of problems
• Persevere in solving them
• Reason abstractly and quantitatively
• Construct and use viable arguments
• Describe with mathematical language
• Use appropriate tools
• Attend to precision
Word Problems effectively promote
these skills when they..
familiar situations
• Help
long-term memory
• Demonstrate real-world value
•
Connect math to
students encode math in
of math
•
Create a
break from
demonstrations, and
the tedium of
worksheets,
repetition.
endlessly boring or confusing
But students don’t like word problems!
• What do you want me to do?
• Which method do you want me to use?
• There are too many numbers and words I
don’t understand !
• I am never going to have a swimming pool for
which I need to know the perimeter ??
They are also not frequently well-used
by teachers
• If I am assessing their ability to complete math
processes, why do I need to add all of the
distractors of a word problem?
• Hard to grade, if I plan to give partial credit.
• Doing word problems is not part of the math
course’s SLO.
• Not applicable to foreign or ESL students
Intro. Statistics Learning Objectives
Upon completion of this course, the student will be able to:
1. Collect, organize, and describe data using graphical and
numerical techniques.
2. Make measurements on a set of data with the aid of a
calculator or computer.
3. Select which probability distribution to use depending on
the problem situation.
4. Formulate hypotheses about the population and test them
by means of the measurements made on the sample.
5. Synthesize all of the above objectives into a statistical
project and explain the procedure by a written and oral
presentation of the results.
6. Evaluate the use of probability distributions mechanisms in
response to particular problem situations.
Learning Objectives: Brief Calculus
1. Graph elementary functions.
2. Compute limits of functions.
3. Differentiate functions.
4. Apply the derivative to find relative and absolute
extrema.
5. Solve real world applications using derivatives.
6. Solve problems involving exponential and
logarithmic functions.
7. Integrate functions, and solve applications using
the integral.
8. Evaluate functions of several variables, and
compute their partial derivatives
“Applied” Math

Many math instructors believe that …
• Applications are only “special” cases of real
math.
• Emphasizing application will undermine the
rigorous assumptions and logic of “real” math.
• Applications will not demonstrate the
“beauty” of mathematical thought or the real
problems that mathematicians are trying to
solve.
A Contrary Argument
• Math for 99% of our students is not like math for
the 1% that will become “mathematicians.”
• For the 99% math is a language of well-defined
terms and symbols, and a grammar(logic) that
permits one to create a narrative about almost
any non-mathematical subject.
• Word problems are our ONLY opportunity to
teach them how to translate “English problem”
narrative to a “math narrative” which assures
them of finding a convincing, fact-based answer.
How should we teach these translation
skills?
Let’s find a word problem template that
• Everyone already knows.
• Is constantly reinforced with conflict, violence
and sex.
• Promotes retention because it is already
linked to well-encoded information and
habits.
• Provides student with a self-validation of math
learning.
Turn to the Master Logicians of the
Modern Era
•
•
•
•
•
•
•
•
•
•
Sherlock Holmes
Castle
Perry Mason
Quincy
Dr. Who
Agents of Marvel
Agents of NCIS
Monk
Major Crimes Detectives
Et. Al.
The Crime Detective’s Approach
Just Five Steps
1. What is the crime and who did it?
2. Collect the facts.
3. Determine which relationships are relevant
[Motive + Means + Opportunity = Probable Suspect]
4. Apply relationship templates to the facts.
5. Build a case against a specific defendant that
will provide guilt beyond a reasonable doubt.
The Math Detective’s Approach
1. What is the question? (English)
2.What are the facts? (English)
3. Determine the relevant quantitative relationships (English)
4. Translate to algebraic definitions and equations (i.e. apply
the relationships to the facts) (Math-lish)
5. Manipulate equations to obtain the answer to the question
(without any doubt!) (Math-lish)
Algebra “Dirt” Example
(Dugopolski 4thEd p.442)
#51. Small Plane. It took a small plane 1 hour
longer to fly 480 miles against the wind than it
took the plane to fly the same distance with
the wind. If the wind speed was 20 mph, then
what is the speed of the plane in calm air?
Algebra “Dirt” Example
(Dugopolski 4thEd p.442)
#51. Small Plane. It took a small plane 1 hour longer to fly 480
miles against the wind than it took the plane to fly the same
distance with the wind. If the wind speed was 20 mph, then
what is the speed of the plane in calm air?
1. What is the question?
 What is the plane’s speed in calm air (air speed) in mi/hr?
2.What are the facts?
 Plane flies a distance of 480 mi (miles)
 Plane takes 1 hr (hour) longer flying against the wind
compared to flying with the wind.
 Wind speed (ground speed) is 20 mi/hr
Algebra “Dirt” Example (continued)
(Dugopolski 4thEd p.442)
3. Determine relevant relationships
 Ground speed is increased when being pushed by wind
 Ground speed is decreased when pushing against the wind
 Distance (mi) is directly proportional to travel time (hr) and speed (mi/hr)
 Things equal to the same thing are equal to each other
4. Translate to algebraic definitions and equations (i.e. apply the relationships
to the facts)
 C = Calm air speed, mi/hr D= distance traveled, mi T = time of travel with
wind, hr.
 (480 mi)/(T hr) = (C + 20)mi/hr
 (480 mi)/(T+1)hr = (C – 20)mi/hr
5. Manipulate equations to obtain the answer to the question (without any
doubt!)
[480/T – 20 ] = [480/(T+1) + 20] [480/(T) – 480/(T+1)] = 40
[ (480(T+1)/(T(T+1) ) – (480T/(T(T+1)) ] = 40
480(T+1) – 480T = 40(T)(T+1)  12=T(T+1)  T=3 hr and C=160 mi/hr
Check: (3hr)(160 mi/hr) = 480 mi; (4hr)(120 mi/hr)=480 mi
Algebra Work Rate Problem
Dugoploski 4thEd p433
#63. Installing a dishwasher. A plumber can
install a dishwasher in 50 min. If the plumber
brings his apprentice to help, the job takes 40
minutes. How long would it take the
apprentice working alone to install the
dishwasher?
Directions: In five minutes begin to write down
the first three steps.
Algebra Work Rate Problem
Dugoploski 4thEd p433
#63. Installing a dishwasher. A plumber can install a dishwasher in 50 min. If the
plumber brings his apprentice to help, the job takes 40 minutes. How long would
it take the apprentice working alone to install the dishwasher?
1. What is the question?
 How long for apprentice to install one dishwasher, min?
2.What are the facts?
 Plumber installs in 50 min.
 Plumber and apprentice install in 40 min.
3. Determine the relevant relationships
 One install takes 50 plumber-minutes;
 One install takes 40 plumber-minutes plus 40 apprentice min.
 Things equal to the same thing are to each other.
Algebra Work Rate Problem
(continued)
4. Translate to algebraic definitions and equations (i.e. apply the
relationships to the facts)
 50 plumber-min = 1 install
 40 plumber-min + 40 apprentice-min = 1 install
5. Manipulate equations to obtain the answer to the question (without
any doubt!)
 50 plumber-min = 40 plumber-min + 40 apprentice-min
 10 plumber-min = 40 apprentice-min
 1 plumber-min = 4 apprentice min
 Therefore: 50 plumber-min = 200 apprentice-min = 1 install
 Check: In 40 min, plumber installs 40/50 of a dishwasher and an
apprentice installs 40/200 of a dishwasher; (40/50 +40/200 =
200/200 = 1)
Extending The Algebra Word Problem
(to calculus, statistics, et. al.)
•
•
•
•
A plumber can install a dishwasher in 50 min. If the plumber brings his apprentice
to help, the job takes 40 minutes. My neighbor is having a dishwasher installed.
The apprentice offers to do it by himself for $60/hr, while the plumber offers to do
it along with his apprentice for $100/hr. How much will my neighbor save?
A sole practitioner plumber decides to hire an apprentice. This will allow him to
do 10 dishwasher installation jobs per day instead of 8. The competitive forces
mean he won’t be able to charge customers more than $100 per install. What is
the most he can afford to pay his apprentice per day?
A sole practitioner plumber decides to hire multiple apprentices. Each one could
reduce his usual 50 minute install time by 10 minutes down to a minimum of 20
minutes. If he charges $100 per install and he pays his apprentices $20/hr, how
many should he hire?
Apprentices can help or hinder a plumber’s installation process. If an apprentice
has a 60% chance of reducing install time by 10 minutes and a 40% chance of
increasing time by 10 minutes, then should a plumber hire an apprentice? What is
the maximum hourly rate he can afford to pay the apprentice?
Algebra of Ratios
(Dugopolski, 4th Ed, p.433)
#61. Capture-recapture. To estimate the number
of trout in Trout Lake, rangers used the capturerecapture method. They caught, tagged, and
released 200 trout. One week later, they caught a
sample of 150 trout and found that 5 of them
were tagged. Assuming that the ratio of tagged
trout to the total number of trout in the lake is
the same as the ratio of tagged trout in the
sample to the number of trout in the sample, find
the number of trout in the lake.
Algebra of Ratios
(Dugopolski, 4th Ed, p.433)
#61. Capture-recapture. To estimate the number of trout in Trout Lake, rangers used the
capture-recapture method. They caught, tagged, and released 200 trout. One week later, they
caught a sample of 150 trout and found that 5 of them were tagged. Assuming that the ratio of
tagged trout to the total number of trout in the lake is the same as the ratio of tagged trout in the
sample to the number of trout in the sample, find the number of trout in the lake.
1. What is the question? (English)
 How many trout are in Trout Lake?
2. What are the facts? (English)
 Large unknown population of trout may be in Lake
 Until two hundred tagged trout were released, no trout in the Lake were tagged.
 One hundred fifty trout were caught of which 5 were tagged
3. Determine the relevant quantitative relationships (English)
 The ratio of all tagged trout to all of the trout in the Lake is number tagged divided by total.
 The ratio of tagged and caught trout to all of the caught trout is the number of tagged caught
to total number caught.
 Ratio of tagged and caught is the same as ratio of all tagged to all trout in the Lake
 Ratios are constants of proportionality.
4. Translate to algebraic definitions and equations (i.e. apply the relationships to the facts) (Mathlish)
 T= total trout in the Lake; (5 caught and tagged)/(150 caught) = (200 tagged in Lake)/(T, total
trout in Lake)
5. Manipulate equations to obtain the answer to the question (without any doubt!) (Math-lish)
(5/150) = (200/T)  5T = (150)(200)  T=(150)(200)/5  T=6000
Check: 5/150 = 1/30; 200/6000 = 1/30
Calculus of Decision-making
Bittinger, 10th Ed, (31, p274)
31. A university is trying to determine what price
to charge for tickets to football games. At an
$18 ticket price, game attendance averages
40K. Every $3 decrease adds 10K to
attendance level. Attendees average $4.50
each on food and drink. What ticket price
should be charged to maximize total revenue?
Calculus of Decision-making
Bittinger, 10th Ed, (31, p274)
31. A university is trying to determine what price to charge for tickets to football games. At an $18
ticket price, game attendance averages 40K. Every $3 decrease adds 10K to attendance level. Attendees
average $4.50 each on food and drink. What ticket price will maximize total revenue?
1. What is the question?
What ticket price will max revenue and what will the max revenue be?
2. What are the facts?
An $18 ticket attracts 40K people.
No matter what the ticket price, people consume $4.50 of food and drink.
No matter what the ticket price, a $3 decrease will increase attendance 10K.
3. Determine the relevant quantitative relationships.
Revenue is the sum of ticket sales and concessions
Attendance is a function of ticket price
Rate of change of attendance as function of ticket price is a constant
A function has a relative extreme where its rate of change changes sign
4. Translate to calculus definitions and equations
R=Revenue; x=ticket price; A=A(x)=attendance at ticket price x; R=Ax+A4.5=A(x+4.5)
dA/dx = -(10K/3)  “derivative is instantaneous rate of change “
5. Manipulate equations to obtain the answer to the question
dA/dx =constant implies A=mX+b where m= -10K/3; since A(18)=40K then
40K= (-10K/3)(18)+b and thus b=100K; R=[[-10K/3]x+100K][x+4.5]= [(-10/3)x^2 + 85x +450 ](K$)
At its relative extreme dR/dx = 0 thus (-20K/3)x+85K = 0 which means x-extremum = 12.75 $
Max total revenue = $992K A($12.75)=(-10K/3)(12/75) + 100K = 57.5 K attendees
Check: at $12 revenue drops to $990K, and at $15 it drops to $975K
Statistics of Knowledge
(Trioloa, 11th Ed; p191, 29)
29. In a test of Micro-Sort genetic selection
technology 13 of 14 babies born using the
method were girls. Does the method yield a
result that is statistically significantly different
from the common result of half of babies
being girls?
Statistics of Knowledge
(Trioloa, 11th Ed; p191, #29)
In a test of Micro-Sort genetic selection technology 13 of 14 babies born using the method were girls. Does the
method yield a result that is statistically significantly different from the common result of half being girls?
1. What is the question?
Are girl babies born “a lot” more often with Micro-Sort than without?
2. List the relevant facts.
Without Micro-Sort the expected number of girls is half the number born.
With Micro-Sort the 13/14 girl/boy ratio could happen, but very seldom.
With Micro-Sort we observe 13 girls in one 14 births trials.
3. Determine the relevant quantitative relationships.
IF [ (a) I believe an event occurs rarely; AND (b) I observe the event ] THEN I should probably change my
belief that the event occurs rarely.  Rare Event Rule
Classical Definition  Probability =
(number of ways A equally likely events can occur)/(number of ways equally likely A and notA can occur)
4. Translate to statistical definitions and equations.
The number of different possible gender sequences in 14 births is 2^14 = 16,384
The number of different gender sequences that generate exactly 13 girls = 14
5. Manipulate the equations to find the answers.
Observing 13 girls in 14 births has probability = (14/16,384) = 0.000854
From the Rare Event Rule it appears that observing 13 girls in 14 births is strong evidence that
Conclusion: Micro-Sort probably increases the probability of girl births over the normal assumption of 0.5
Check: Gather more data to narrow confidence interval and/or improve confidence level.
Objections To The Math Detective’s
Approach
• Solution takes a long time
• Students know answer before they are
finished
• Not enough emphasis on “math procedures”
• No multiple choice format, large effort to
grade
• Students need to know English
Advantages of the Math Detective’s
Approach
• Potential for self-validating learning
experiences.
• Mastery of analytic skills that transcend math
skills and thereby provides incentive to
develop more math skills.
• Approachable and familiar context for
“attacking” quantitative word problems.
• Exercises English narrative presentation of
quantitative analysis of real world problems.
Thank you
Please feel free to offer constructive criticisms
that will move the debate along!
JAppleman@ivc.edu