The Core Purpose of Word ProblemsCMC South

30th Annual Conference3/15/14

Jack ApplemanIrvine Valley CollegeJAppleman@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..

• Connect math to familiar situations• Help students encode math in long-term memory• Demonstrate real-world value of math

•Create a break from the tedium of worksheets,

demonstrations, and endlessly boring or confusing repetition.

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 Steps1. 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

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 other4. 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/hrCheck: (3hr)(160 mi/hr) = 480 mi; (4hr)(120 mi/hr)=480 mi

Algebra “Dirt” Example (continued)(Dugopolski 4thEd p.442)

Algebra Work Rate ProblemDugoploski 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 ProblemDugoploski 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 install5. 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 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.

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 tagged3. 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) (Math-lish) 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=6000Check: 5/150 = 1/30; 200/6000 = 1/30

Calculus of Decision-makingBittinger, 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-makingBittinger, 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 $3 decrease will increase attendance 10K.3. Determine the relevant quantitative relationships. Revenue is the sum of ticket sales and concessions Attendance is a function of ticket price Rate of change of attendance as function of ticket price is a constant A function has a relative extreme where its rate of change changes sign4. Translate to calculus definitions and equations R=Revenue; x=ticket price; A=A(x)=attendance at ticket price x; R=Ax+A4.5=A(x+4.5) dA/dx = -(10K/3) “derivative is instantaneous rate of change “5. Manipulate equations to obtain the answer to the question dA/dx =constant implies A=mX+b where m= -10K/3; since A(18)=40K then 40K= (-10K/3)(18)+b and thus b=100K; R=[[-10K/3]x+100K][x+4.5]= [(-10/3)x^2 + 85x +450 ](K$) 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 = 145. 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