28 Chapter 1

1.4

FOCUSObjectives1.4.1 Identify two general steps in

problem solving.1.4.2 Describe three steps for

solving numeric problems.1.4.3 Describe two steps for solving

conceptual problems.

Guide for ReadingGuide for Reading

Build VocabularyWord Forms Students may benefit from a reminder that certain key words and phrases in each word problem indicate the unknown quantity and its units. These include how much, deter-mine, what is, find, and how long.

Reading StrategyThink Aloud Some students are con-vinced they cannot do word problems because they are too hard. Ask students to describe the process verbally as they attempt to solve a problem.

INSTRUCT

Have students study the photograph and read the text that opens the section. Ask students how they would approach the situation pictured. Ask, What is the most effective way to solve problems?(Approaches vary depending on the given problem; effective approaches are gener-ally organized and well-planned.)

Skills Used in Solving ProblemsDiscussDiscussExplain that the memorization of facts is a relatively small part of learning chem-istry. A person who succeeds in chemis-try has become a good problem solver. Encourage students to share problem-solving methods and techniques. Dis-cuss the supermarket problems. Stress that for these problems, unlike most problems in a chemistry course, answers can vary and still be correct.

11

L2

L2

22

L2 Section ResourcesPrint Guided Reading and Study Workbook,

Section 1.4 Core Teaching Resources,

Section 1.4 Review Transparencies, T7T9

Technology Interactive Textbook with ChemASAP,

Problem-Solving 1.27, 1.29; Assessment 1.4

28 Chapter 1

1.4 Problem Solving in ChemistryProblem Solving in Chemistry

Shape-sorter toys fascinate young children. Typically, the children try placing a shape in different holes until they nd the right one. They may try to place an incorrect shape in the same hole over and over again. An older child has enough experience to place the correct shape in each hole on the rst try. The trial-and-error approach used by young children is one method of problem solving, but it is usually not the best one. In this section, you will learn effective ways to solve problems in chemistry.

Guide for ReadingGuide for Reading

Key Concepts What is a general approach to

solving a problem? What are the three steps for

solving numeric problems? What are the two steps for

solving conceptual problems?

Reading Strategy Identifying Main Idea/DetailsUnder the heading Solving Numeric Problems, there are three main ideas presented as subheads. As you read, list two details that support each main idea.

Skills Used in Solving ProblemsProblem solving is a skill you use all the time. You are in a supermarket. Doyou buy a name brand or the store brand of peanut butter? Do you buy the1-liter bottle or the 2-liter bottle of a carbonated beverage? Do you choosethe express line if there are ve customers ahead of you or the non-expressline with a single shopper who has lots of items?

When you solve a problem you may have a data table, a graph, oranother type of visual to refer to. The shopper in Figure 1.23 is reading thelabel on a can while trying to decide whether to buy the item. She may needto avoid certain ingredients because of a food allergy. Or she may want toknow the amount of Calories per serving.

The skills you use to solve a word problem in chemistry are not that dif-ferent from those you use while shopping or cooking or planning a party.

Effective problem solving always involves developing a plan andthen implementing that plan.

Figure 1.23 A shopper must make many decisions. Some of those decisions are based on data, like the information on a food label.

chem_TE_ch01_FPL.fm Page 28 Tuesday, August 3, 2004 9:21 AM

Introduction to Chemistry 29

Solving Numeric ProblemsDiscussDiscussPoint out that in the laboratory, as well as during students normal routines, they are often presented with more data than is needed to solve a prob-lem. Explain that they will need to sort essential data from extraneous data. In cases with more than the required data, sorting the data into knowns and unknowns helps students to deter-mine the answer to What data do I need to solve this problem?

L2

Answers to...Figure 1.24 Step 1, analyze

Checkpoint If the calculated answer is much larger or smaller than the estimate, check the calcu-lations.

Facts and FiguresHow Teachers Solve Math ProblemsA Professor of Mathematics at Stanford Uni-versity, G.Polya (18871985), wrote a classic book about problem solving in 1945. The second edition of How to Solve It is still in print. Polya used common sense and humor to present problem-solving tactics.

The general four-step approach that Polya outlined in his introduction (understand, plan, try it, and look back) has much in com-mon with the three-step approach for solv-ing numeric problems (analyze, calculate, and evaluate).

Section 1.4 Problem Solving in Chemistry 29

Solving Numeric ProblemsBecause measurement is such an important part of chemistry, most wordproblems in chemistry require some math. The techniques used in thisbook to solve numeric problems are conveniently organized into a three-step, problem-solving approach. This approach has been shown to be veryhelpful and effective. So we recommend that you follow this approachwhen working on numeric problems in this textbook. The steps forsolving a numeric word problem are analyze, calculate, and evaluate.Figure 1.24 summarizes the three-step process and Sample Problem 1.1shows how the steps work in a problem.

Analyze Analyze To solve a word problem, you must rst determine whereyou are starting from (identify what is known) and where you are going(identify the unknown). What is known may be a measurement. Or it maybe an equation that shows a relationship between measurements. If youexpect the answer (the unknown) to be a number, you need to determinewhat units the answer should have before you do any calculations.

After you identify the known and the unknown, you need to make aplan for getting from the known to the unknown. Planning is at the heart ofsuccessful problem solving. As part of planning, you might draw a diagramthat helps you visualize a relationship between the known and theunknown. You might need to use a table or graph to identify data or toidentify a relationship between a known quantity and the unknown. Youmay need to select an equation that you can use to calculate the unknown.

Calculate Calculate If you make an effective plan, doing the calculations isusually the easiest part of the process. For some problems, you will have toconvert a measurement from one unit to another. Or you may need to rear-range an equation before you can solve for an unknown. However, you willbe taught these math skills as needed. There will also be remindersthroughout the textbook to use the Math Handbook in Appendix C.

Evaluate Evaluate After you calculate an answer, you should evaluate it. Is theanswer reasonable? Does it make sense? If not, reread the word problem.Did you copy the data correctly? Did you choose the right equations?It helps to round off the numbers and make an estimate of the answer. Ifthe answer is much larger or much smaller than your estimate, check yourcalculations.

Check that your answer has the correct unit and the correct number ofsignificant figures. You may need to use scientific notation in your answer.You will study significant figures and scientific notation in Chapter 3.

Checkpoint How can making an estimate help you evaluate an answer?

Figure 1.24 This flowchart summarizes the steps for solving a numeric problem.Predicting In which step do you make a plan for getting from what is known to what is unknown?

Analyze EvaluateCalculate

1 32

chem_TE_ch01_FPL.fm Page 29 Tuesday, August 3, 2004 9:21 AM

30

Chapter 1

Section 1.4 (continued)

Sample Problem 1.1

Answers

26.

24 short blocks

27.

24 minutes

Discuss

Write the sample problem on the board and carry out the three-step problem-solving approach. Stress the discipline of writing down the steps that make the solution possible. You may wish to use chalk or markers of different colors for the knowns and unknowns to make the process clearer. Stress the importance of evaluating whether the result makes sense. Stu-dents should make an initial mental estimate to compare with the final result displayed by a calculator.

FYI

Conversion problems are the most common type of numeric problem in an introductory chemistry course. They will be discussed in detail in Chapter 3. In that chapter, students will learn how to combine the two calculate steps in Sample Problem 1.1 into a single step.

L2

30 Chapter 1

withChemASAP

Problem-Solving 1.27Solve Problem 27 with the help of an interactive guided tutorial.

Practice Problems

Estimating Walking TimeYou are visiting Indianapolis for the rst time. Because it is a nice day,you decide to walk from the Indiana State Capital to the Murat Centrefor an afternoon performance. According to the map in Figure 1.25, theshortest route from the capital to the theater is 8 blocks. How manyminutes will the trip take if you can walk one mile in 20 minutes?Assume that 10 short city blocks equals one mile.

Analyze List the knowns and the unknown.

Knowns distance to be traveled 8 blocks walking speed 1 mile/20 minutes 1 mile 10 blocks

Unknown time of trip ? minutes

This problem is an example of what is typically called a conversionproblem. In a conversion problem, one unit of measure (in this case,blocks) must be expressed in a different unit (in this case, minutes).

Divide the distance to be traveled (in blocks) by the number of blocksin one mile to get the distance of the trip in miles. Then multiply thenumber of miles by the time it takes to walk one mile.

Calculate Solve for the unknown

Evaluate Does the result make sense?The answer seems reasonable, 16 minutes to walk 8 short blocks. Theanswer has the correct unit. The relationships used are correct.

8 block 1 mile10 blocks 0.8 mile

0.8 mile 20 minutes1 mile 16 minutes

26. Using the information in the sample problem, how many short blocks can be walked in 48 minutes?

27. There is an ice cream shop 6 blocks north of your hotel. How many minutes will it take to walk there and back?

SAMPLE PROBLEM 1.1

This view of Indianapolis, Indiana, shows part of the historic central canal in White River State Park.

chem_TE_ch01.fm Page 30 Wednesday, April 20, 2005 6:56 AM

Introduction to Chemistry

31

Solving Conceptual Problems

TEACHER DemoTEACHER Demo

Fit an Ice Cube in a Bottle

Purpose

Students suggest different approaches for solving a problem.

Materials

bowl of ice cubes, empty narrow-neck bottle

Procedure

Place a bowl of ice cubes and an empty soda bottle in front of the class. Explain that the problem is to transfer the ice to the inside of the bot-tle. Have students analyze the problem and suggest different approaches.

Expected Outcome

Students may sug-gest crushing the ice so that pieces are small enough to fit through the mouth of the bottle; or melting the ice, pour-ing the water into the bottle, and plac-ing the bottle in a freezer.

L2

Answers to...

Figure 1.25

Meridian Street, Senate Avenue, Market Street, Jackson Place, and Capitol Avenue

Figure 1.26

Solving a conceptual problem usually does not require any calculations.

Section 1.4 Problem Solving in Chemistry 31

Solving Conceptual ProblemsNot every word problem in chemistry requires calculations. Some prob-lems ask you to apply the concepts you are studying to a new situation.In this text, these nonnumeric problems are labeled conceptual problems.To solve a conceptual problem, you still need to identify what is knownand what is unknown. Most importantly, you still need to make a plan forgetting from the known to the unknown. But if your answer is not a num-ber, you do not need to check the units, make an estimate, or check yourcalculations.

The three-step problem-solving approach is modified for conceptualproblems. The steps for solving a conceptual problem are analyze andsolve. Figure 1.26 summarizes the process, and Conceptual Problem 1.1 onthe next page shows how the steps work in an actual problem.

Figure 1.25 Refer to this map of Indianapolis, Indiana, while you do Sample Problem 1.1. Interpreting Diagrams In the section of downtown bounded by north, east, south, and west streets, the main streets and avenues are named for states. What are the five exceptions to this pattern?

Figure 1.26 This flowchart shows the two steps used for solving a conceptual problem.Comparing and Contrasting With a conceptual problem, why is the second step called Solve rather than Calculate?

North Street

Michigan Street

Vermont Street

New York Street

St. Clair Street

Ohio Street

Washington Street

Maryland Street

Georgia Street

Jackson Pl.Louisiana St.

Market Street

South Street

Merrill Street

CANAL

Capi

tol A

venu

e

Illin

ois

Stre

et

Mer

idia

n St

reet

Penn

sylv

ania

Stre

et

Dela

war

e St

reet

Alab

ama

Stre

et

New

Jer

sey

Stre

et

East

Stre

et

Colle

ge A

venu

e

Virginia Avenue

Massa

chus

etts A

venu

e

ONE WAY

ONE WAY

ONE WAY

ON

E W

AY

ON

E W

AY

ON

E W

AY

ON

E W

AY

ON

E W

AY

ON

E W

AY

ON

E W

AY

ONE WAY

ON

E W

AY

ON

E W

AY

ON

E W

AY

Monument Circle

Sena

te A

venu

e

Wes

t Stre

et

IndianaConvention

Center& RCA Dome

Circle Center

ConsecoFieldhouse

CentralLibrary

AmericanLegion Mall

IndianaWorldWar

Memorial

Univ.Park

VeteransMemorial

Plaza

UnionStation

Indiana State

Capitol

IndianaHistorical

Society

MuratCentre

CityMarket

N

Indiana Avenue

Analyze Solve

1 2

chem_TE_ch01.fm Page 31 Wednesday, April 19, 2006 5:53 PM

32 Chapter 1

Section 1.4 (continued)

Section 1.4 Assessment30. Develop a plan and implement the plan.31. analyze, calculate, and evaluate32. analyze and solve33. a. known: 3600 s = 1 h;

unknown: ? s = 1 dayb. 24 h = 1 day

c. 3600 s/h 24 h/day = 86,400 s/dayd. 86,400 seconds in one day seems reason-able in relationship to 3600 seconds in one hour. The answer has the correct units and the relationship used is correct.

CONCEPTUAL PROBLEM 1.1

Answers28. The order of the morning errands

cannot vary. The order of the after-noon errands can vary, as long as the haircut takes place before 3 pm.

29. Possible answers: Do an errand during lunch hour or extend the hours during which he does errands.

DiscussDiscussDiscuss using classification schemes as tools for organizing information. Ask students to suggest classification schemes they encounter in their daily lives (the ways that goods are orga-nized in stores or books in libraries).

ASSESSEvaluate UnderstandingEvaluate UnderstandingTo determine student understanding of the three-step problem solving approach, ask students to suggest ways to evaluate an answer. (Reread the prob-lem to be sure the answer supplies the requested unknown; round off the num-bers and do a quick estimate.)

ReteachReteachHave students try working in pairs to solve problems. One student thinks aloud while trying to solve a problem. The other keeps a careful record of the process. Then they reverse roles and work on another problem.

In both cases, the solver analyzes the problem, makes a plan, and carries out the plan. Problems with numeric answers require that the answer be evaluated to see if it is reasonable.

with ChemASAP

If your class subscribes to the Interactive Textbook, use it to review key concepts in Section 1.4.

L2

33

L2

L1

32 Chapter 1

30. Key Concept What are the two general steps in successful problem solving?

31. Key Concept List the three steps for solving numeric problems.

32. Key Concept List the two steps for solving conceptual problems.

33. Read the conversion problem and then answer the questions. There are 3600 seconds in an hour. How many seconds are there in one day?

a. Identify the known and the unknown. b. What relationship between the known and

unknown do you need to solve the problem? c. Calculate the answer to the problem. d. Evaluate your answer and explain why your

answer makes sense.withChemASAP

withChemASAP

CONCEPTUAL PROBLEM 1.1

Running ErrandsManny has to run 6 errands between 10 and 5 on Saturday. He must get a haircut, wash his car, buy stamps, rent a video, return a library book, and buy some groceries. Assume that each errand will take 30 minutes and that Manny will do only one errand per hour. Manny will stop for a lunch break between 12 and 1. Use the information in the drawing to gure out a way for Manny to accomplish all 6 tasks.

Analyze Identify the relevant concepts.

Each place that Manny needs to visit is open for a limited number of hours on Saturday. Manny must do his errands between 10 and 12, and between 1 and 5. At a rate of one errand per hour, Manny must do 2 errands before lunch and 4 errands after lunch.

Solve Apply concepts to this situation.

The post ofce and library are open only in the morning. The barbershop and the car wash close earlier than the video store. The super-market is open late. One possible order for the errands is post ofce, library, barbershop, car wash, video store, and supermarket.

Problem-Solving 1.29Solve Problem 29 with the help of an interactive guided tutorial.

1.4 Section Assessment

Compare and Contrast Paragraph Write a para-graph comparing the processes for solving numeric problems and conceptual problems. How are the processes similar? In what way are they different?

Assessment 1.4 Test yourself on the concepts in Section 1.4.

8:00 amto 11:00 am

10:00 amto 3:00 pm

7:00 amto midnight

10:00 amto 4:00 pm

10:00 amto 1:00 pm

10:00 amto 6:00 pm

28. Describe two alternative orders in which Manny could complete his errands.

29. What if Manny had 7 errands instead of 6? What would he need to do to adjust for the extra errand?

Practice Problems

chem_TE_ch01_FPL.fm Page 32 Tuesday, August 3, 2004 9:21 AM

chem_TE_028.pdfchem_TE_029.pdfchem_TE_030.pdfchem_TE_031.pdfchem_TE_032.pdf