Chapter-3-PROBLEM-SOLVING.pdf hhhhhhhhhh

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2. Generally, it is a situation you want
tochange!
What is a problem?

Aproblemisasituationthat
conformsthelearner,thatrequires
resolution,andforwhichthepathofthe
answerisnotimmediatelyknown.
Thereisanobstaclethatpreventsone
fromsettingaclearpathtotheanswer.

3. Problem Solving has been defined
as higher-order cognitive process
that requires the modulation and
control of more routine or
fundamental skills"(Goldstein
& Levin,1987).
What is a ProblemSolving


4. 
A. UnderstandingReasoning
Mathematicalreasoningreferstothe
abilityofapersontoanalyzeproblem
situationsandconstructlogical
argumentstojustifyhisprocessor
hypothesis,tocreatebothconceptual
foundationsandconnections,inorder
forhimtobeabletoprocessavailable
information.

5. People who can reasonand think analytically
tend
To note patterns, structure, or regularitiesin
both real-world situations and symbolic
objects;
Toaskifthosepatternsareaccidentalorif
they occur for areason
To conjecture andprove
NCTM pointed outthat….


6. 
Students are expectedto:
1.Define astatement
Reasoning
2.Identify the hypothesis and conclusion
in astatement
3.Write conditionalstatements
4.Write the Converse, Inverse,
Contrapositive of a given conditional
statement.

7. 
What kind of thinking isused
when solvingproblems?
Inductive ordeductive?
B. Inductive and DeductiveReasoning

8. Thetypeofreasoningthatformsaconclusion
basedontheexaminationofspecificexamplesiscalled
inductivereasoning.
InductiveReasoning

Conclusion
Specific
Examples
Theconclusionformedbyusinginductive
reasoning isoftencalled aconjecture,sinceitmayor
may not becorrect.

9. 
Example1:
Ababycries,thencries,thencriestogetamilk.
We conclude that if a baby cries, he/she gets amilk.
6,9,12,
Example2:
Here is a sequence of numbers:3,
____
What is the 5
th
number?
Examples
We can easily conclude that the next number is15.

10. 
Example3:
7
th
Youareaskedtofindthe6
th
and
term in thesequence:
1,3,6,10,15,______ ,_____
Thefirsttwonumbersdifferby2.The2
nd
and3rdnumbersdifferby3.Thenextdifference
is4,then5.So,thenextdifferencewillbe6and
Thusthe6
th
termis15+6=21whilethe7
th
is
21 + 7 =28.

11. Takenote!

Inductivereasoningisnotusedjustto
predictthenextnumberinalist.
Weuseinductivereasoningtomakea
conjectureaboutanarithmetic
procedure.
Make a conjectureabouttheexample2
and 3 in the previousslide…

12. Exercise
Use Inductive Reasoning
to Make aConjecture
A. Consider the followingprocedure:
1.Pick anumber.
2.Multiply the number by8,
3.Add 6 to theproduct
4.Divide the sum by 2,and
5.Subtract3.
Completetheaboveprocedureforseveraldifferentnumbers.
Useinductivereasoningtomakeaconjectureabouttherelationship
betweenthesizeoftheresultingnumberandthesizeoftheoriginal
number.

13. Conjecture
Solution :
Let n represents the number
Multiply the number by 8 8n
Add 6 to the product 8n + 6
Divide the sum by 2
8??????+6
2
= 4n +3
Subtract 3 4n + 3 –3 = 4n
we start with n and end with 4n . This means
that the number is 4 times the original
number.

14. Exercise
Use Inductive Reasoning
to Make aConjecture
B. Considerthefollowingproce
dure:
1.Pick anumber.
2.Multiply the number by9,
3.Add 15 to theproduct,
4.Divide the sum by 3, and
5.Subtract5.
Completetheaboveprocedureforseveraldifferentnumbers.
Useinductivereasoningtomakeaconjectureabouttherelationship
betweenthesizeoftheresultingnumberandthesizeoftheoriginal
number.

15. 
C. Consider the followingprocedure:
1.List 1 as the first oddnumber
2.Addthe nextodd number to1.
3.Add the next odd number to thesum.
4.Repeat adding the next odd number to the previoussum.
Construct a table to summarize the result. Use inductive
reasoning to make a conjecture about the sumobtained.
Exercise
Use InductiveReasoning
to Make aConjecture

16. 
D. Observe the two sets of polygonsbelow:
Whatisthenameofapolygonthatcanbeused to
describe the polygons in column2?
Useinductivereasoningtomakeaconjectureaboutthe
polygons in column2.
Exercise
Use InductiveReasoning
to Make aConjecture

17. 
Exercise
Use InductiveReasoning
to Make aConjecture
Scientists often use inductive reasoning. Forinstance, Galileo
Galilei(1564–1642)usedinductivereasoningtodiscoverthatthetime
requiredforapendulumtocompleteoneswing,calledtheperiodof
thependulum,dependsonthelengthofthependulum.Galileodid
nothaveaclock,sohemeasuredtheperiodsofpendulumsin
“heartbeats.”Thefollowingtableshowssomeresultsobtainedfor
pendulumsofvariouslengths.Forthesakeofconvenience,alength
of10incheshasbeendesignatedas1unit.
Use the data in the table and inductive
reasoning to answer each of the following
questions.
a.If a pendulum has a length of 49 units,
what is itsperiod?
b.If the length of a pendulum is
quadrupled, what happens to its
period?

18. 
Takenote:
Conclusions based on
inductive reasoning may be
incorrect.
Asanillustration,consider
thecirclesshown.Foreach
circle,allpossibleline
segmentshavebeendrawnto
connecteachdotonthecircle
withalltheotherdotsonthe
circle.Foreachcircle,count
thenumberofregionsformed
bythelinesegmentsthat
connectthedotsonthecircle.

19. Astatementisatruestatement
providedthatitistrueinallcases.
Ifyoucanfindonecaseforwhicha
statementisnottrue,calleda
counterexample, then the
statement is a falsestatement
Counterexamples


20. Verify that each of the following
statements is a false statement by finding
a counterexample.
For all numbersx:
a.??????>�
b. ??????
�
>??????
c. ??????
−�
<??????
Exercise1

MMW by Joseph G. Taban ,UNP

21. Verify that each of the following statements
is a false statement by finding a
counterexample.
For all numbersx:
Exercise2


22. Anothertypeofreasoningiscalled
deductivereasoning.
Deductivereasoningisdistinguished
frominductivereasoninginthatitis
theprocessofreachingaconclusion
byapplyinggeneralprinciplesand
procedures.
DEDUCTIVE REASONING:


23. 
Mathematicsisessentiallydeductive
reasoning
Deductive reasoning is alwaysvalid
Deductivereasoningmakesuseof
undefinedterms,formallydefined
terms,axioms,theorems,andrulesof
inference.

24. 
Example1:
Ifanumberisdivisibleby2,thenitmustbeeven.
12 is divisible by2.
Therefore,12isanevennumber.
Example2:
All math teachers know how to playsudoku.
Resty is a mathteacher.
Therefore, Resty knows how to playsudoku.
Examples of DeductiveReasoning

25. 
Example3:
IfastudentisaDOSTscholar,hereceivesa
monthlyallowance.
Ifastudentreceivesamonthlyallowance,his
parents will behappy.
Therefore,ifastudentisaDOSTscholar,his
parents will behappy.
Example4:
If ∠A and ∠B are supplementaryangles.
If m∠A = 100º, then m∠B =80º
Examples of DeductiveReasoning

26. Theessenceofdeductivereasoningis
drawingaconclusionfromagiven
statement.
Thedeductivereasoningworksbest
whenthestatementsusedinthe
argumentaretrueandthestatements
in the argument clearly follow from one
another.
Takenote:


27. 
LogicPuzzlescanbesolvedbydeductivereasoningandachart
thatenablesustodisplaythegiveninformationinavisual
manner.
Example1:
Each of four neighbors, Sean, Maria, Sarah, and Brian, has a
differentoccupation(editor,banker,chef,ordentist).
Fromthefollowingclues,determinetheoccupationofeach
neighbor.
1.Mariagetshomefromworkafterthebankerbutbeforethedentist.
2.Sarah, who is the last to get home from work, is not theeditor.
3.The dentist and Sarah leave for work at the sametime.
4.The banker lives next door toBrian.
LogicPuzzles

28. SOLUTION
CLUES:
1.Maria gets home from work after the banker but before thedentist.
2.Sarah, who is the last to get home from work, is not theeditor.
3.The dentist and Sarah leave for work at the sametime.
4.ThebankerlivesnextdoortoBria

n.
From clue 1: Maria is not the
banker or thedentist.
From clue 2, Sarah is not the
editor.
We know from clue 1 that the
banker is not the last to get home,
and we know from clue 2 that
Sarah is the last to get home;
therefore, Sarah is not thebanker.
From clue 3, Sarah is not the
dentist.
As a result, Sarah is theChef.
Maria is theEditor.
From clue 4, Brian is not thebanker.
Brian is theDentist.
Sean is theBanker.
EditorBankerChefDentist
Sean
X XX
Maria
XXX
Sarah
X X X
Brian
X XX

29. Brianna,Ryan,Tyler,andAshleywererecentlyelectedasthe
newclassofficers(president,vicepresident,secretary,
treasurer)ofthesophomoreclassatSummitCollege.
Fromthefollowingclues,determinewhichposition
eachholds:
1.Ashleyisyoungerthanthepresidentbutolderthanthe
treasurer.
2.Briannaandthesecretaryareboththesameage,and
theyaretheyoungestmembersofthegroup.
3.Tylerandthesecretaryarenext-doorneighbors.
EXERCISE


30. GroupActivity:

Distribute the Activity Sheet:
Watch the movie after 20minutes.
Canyousolve_Einstein’sRiddle-
Dan Van derVieren.mp4

31. 
1.INTUITION
Intuitionistheabilitytoacquire
knowledgewithoutproof,evidence,or
consciousreasoning,orwithoutunderstanding
howtheknowledgewasacquired.
“Intuitionisasenseofknowinghowto
actspontaneously,withoutneedingtoknow
why”–SylviaClare
C.INTUITION,PROOF,ANDCERTAINTY

32. MinaandSaraharegettingreadyfor
school.Minasaid,“Ihaveaverystrong
feelingthatitwillrainthisafternoon.Let
useachbringajacket”
Example ofIntuition


33. 
Byintuition,weknowtruthsimply
bytheprocessofintrospectionand
immediateawareness.

34. A proofis a sequence of statements that
form anargument.
There aretwocommonmethods of
proof:
1.DirectProof
2.IndirectProof
2.Proof


35. In a directproof
You assume the hypothesisp
Give a direct series (sequence) of
implications using definitions,
axioms, theorems and rules of
inference
Show that the conclusion qholds.
DirectProof


36. 
Show that the square of an even number isan
even number.
Rephrase: If n is even, then n
2
iseven.
Assume n iseven
–Thus,n=2k,forsomek(definitionofeven
numbers) –n
2
= (2k)
2
= 4k
2
=2(2??????
2
)
–As n
2
is 2 times an integer, n
2
is thuseven.
Direct proofexample

37. The best way to improve proof skills is
PRACTICE.
Let them prove inAlgebra
Ex. Prove that “If 8x –5 = 19, then x=3.”
The sum of two odd integers iseven.
Forstudents


38. 
When weuse
IndirectProof
an indirect
proof toprove
a theory, we
follow three
steps.
Anindirectproofisalsocalledaproofby
contradiction,becauseweareliterallylooking
foracontradictiontoatheorybeingfalseinorder
toprovethatthetheoryistrue.

39. If ??????
&#3627409360;
is an odd integer then n is an oddinteger.
Proof:
Assumetheconclusiontobefalse.nisaneven
integer
-n=2kforsomeintegerk(definitionofeven
numbers)
-n
2
= (2??????)
2
= 4??????
2
= 2(2??????
2
)
-Since n
2
is 2 times an integer, it iseven.
Indirect proofexample


40. 
Therearethreelinesofinquirytoaddress
theproblemofcertaintyinmathematics.
1.Lookatthehistoricaldevelopmentof
mathematics
2.Sketchtheindividualcognitive
developmentinmathematics
3.Examinethefoundationsofcertainty
formathematicsandinvestigateits
strengthsanddeficiencies
3.Certainty

41. 
Ancientmathematicianswhowere
interested in problem-solving are Euclid,
Rene Descartes, and Gottfried Wilhelm
Leibnitz.
One of the foremost recent mathematicians
to make a study of problem solving was
George Polya (1887–1985). He was born in
Hungary and moved to the United States
in1940.
D. PROBLEM -SOLVINGSTRATEGIES

42. 
POLYA’S STEPSIN
PROBLEMSOLVING
Understandthe
Problem
Devise aPlan
Carry out thePlan
LookBack

43. 
Do you understand all the words used in stating the
problem?
What are you asked to find orshow?
Can you restate the problem in your ownwords?
Can you think of a picture or diagram that might help
you understand theproblem?
Is there enough information to enable you to find a
solution?
Devisea
Plan
Carryout
thePlan
Look Back

44. 
Findtheconnectionbetweenthedataand
theunknown.Youmaybeobligedtoconsider
auxiliaryproblemsifanimmediateconnection
cannotbefound.Youshouldobtaineventuallya
planofthesolution.
Polyamentionsthattherearemany
reasonablewaystosolveproblems.Theskillat
choosinganappropriatestrategyisbestlearnedby
solvingmanyproblems.Youwillfindchoosinga
strategyincreasinglyeasy.
Understand the
Problem
Carry out
the Plan
LookBack

45. 
Apartiallistofstrategiesisincluded:
j
Understand the
Problem
Carry out
the Plan
LookBack
Make a list of the known
information.
Make a list of information
that isneeded.
Draw adiagram.
Make an organized list that
shows all thepossibilities.
Make a table or achart.
Workbackwards.
Try to solve a similar but
simplerproblem.
Look for apattern.
Write an equation. If
necessary,define what each
variablerepresents.
Perform anexperiment.
Guess at a solution and then
check yourresult.

46. 46

47. 
■Workcarefully.
■Keep an accurate and neat record of all
yourattempts.
■Realize that some of your initial plans will
not work and that you may have to devise
another plan or modify your existingplan.
Understand the
Problem
Devisea
Plan
Look
Back

48. 
Once you have found a solution,check the solution.
■Ensurethatthesolutionisconsistentwiththe
facts of theproblem.
■Interpret the solution in the contextof the
problem.
■Askyourselfwhethertherearegeneralizationsof
the solution that could apply to otherproblems.
Understand the
Problem
Devisea
Plan
Carryout
thePlan

49. Discuss the 5 examples and give
comments/suggestions on how to improve the
strategies
Apply Polya’s four steps in problemsolving
Activity sheets -STRATEGY in PROBLEMSOLVING
.pdf
Group Activity


50. 
DEMONSTRATION

51. 
Predict the next term in asequence
nth-term Formula for aSequence
Word Problems which involves
numericalpattern
E. MathematicalProblems
InvolvingPatterns

52. An ordered list of numbers suchas
5, 14, 27, 44, 65,...
iscalledasequence.Thenumbersinasequencethatare
separatedbycommasarethetermsofthesequence.Inthe
abovesequence,5isthefirstterm,14isthesecondterm,27
isthethirdterm,44isthefourthterm,and65isthefifth
term.Thethreedots“...”indicatethatthesequence
continuesbeyond65,whichisthelastwrittenterm.Itis
customarytousethesubscriptnotationa
ntodesignatethe
nthtermofasequence.Thatis,
TERMS OF ASEQUENCE


53. Give problems involving sequence of
numbers and worded problemsinvolving
numericalpatterns
Ex. 1. Find the 10
th
term in thesequence
3,7,11,15,…
2.Marksavesmoneyfromhisallowance.
Eachdayhesaves12pesosmorethanthe
previousday.Ifhestartedsaving8pesosinthe
firstday,howmuchwillhesetasideinthe5
th
day?
Exercise:


54. 
Sudoku
MagicSquares
A magic square of order n is an arrangement of
numbers in a square such that the sum of the n numbers in
eachrow,column,anddiagonalisthesamenumber..
KenKenPuzzles
KenKenisanarithmetic-basedlogicpuzzlethatwas
inventedbytheJapanesemathematicsteacherTetsuya
Miyamotoin2004.Thenoun“ken”has“knowledge”and
“awareness”assynonyms.Hence,KenKentranslatesas
knowledgesquared,orawarenesssquared.
KenKenpuzzlesaresimilartoSudokupuzzles,but
theyalsorequireyoutoperformarithmetictosolvethe
puzzle.
F. Recreational Problemsusing
Mathematics

55. KenKenPuzzle

SOLUTION

56. Distribute ActivitySheets:
KENKENPUZZLE.docx
Solve a KenKenPuzzle


57. ActivitySheet -TOWER ofHANOI.docx
Activity:Exploration


58. 
QUIZ

59. 
Problem Set #1

60. Mathematical Excursions (Ch. 1) by R. Aufmann , et
al.
Mathematical Excursions Ch. 2) by R. Aufmann etal.
References
