geom. – ch. 2.1 inductive reasoning &...

Geom.–Ch.2.1InductiveReasoning&Counterexamples

Mr.Deyo

HomeWork1‐2‐3:1)StormCheckPastedinNotebook?

NEWUnitTableofContents!2)Section______ 3)Section______RFM/RTProblems_________ NotesCopiedinNotebook?Pasted&SolvedinNotebook?

Learning Target

Studentswillapplyinductivereasoningtoidentifypatterns,makeconjectures,andfindcounterexamplestodisproveconjectures.

Studentswillshowthisbycompletingagraphicorganizerandsolvinginductivereasoningproblemsinapairactivity.

Storm Check (Think, Write, Discuss, Report) Questions on which to ponder and answer: 1.   How are the two pictures similar? 2.   How are they different? 3.   How can these two pictures be related to math?

1)  Conjecture2)  Counterexample

3)  Hypothesis4)  Conclusion

Vocabulary 1.  Teachersaystheword.

•  ALLStudentsrepeattheword.

2.  Teachercountsoutsyllablesass/hesaystheword.

•  ALLstudentsrepeatsyllablecountastheysaytheword.

3.  Studentteamscreateaphysicalrepresentationoftheword.

•  ClasschoosesonephysicalrepresentationforALLstudents.

4.  Studentteamscreateafriendlydefinitionfortheword.

•  Classchoosesonefriendlydefinitionfortheword.

DAY 3 and/or DAY 4 1. Review the word

¨ Friendly Definition ¨ Physical Representation

2. Show how the word works ¨ Synonyms/antonym ¨ Word Problems ¨ Related words/phrases ¨ Example/non-example

Friendly Definition Sketch

Wordwork Sentence

DAY 2 1. Review word

¨ Friendly Definition ¨ Physical Representation

2. Draw a sketch

DAY 5 1. Review the word

¨ Friendly definition ¨ Physical Representation

3. Write a sentence  at least 2 rich words (1 action)  correct spelling  correct punctuation  correct subject/predicate agreement  clear and clean writing

DAY 1 1.   Use Visuals

2.   Introduce the word ¨ Friendly Definition ¨ Physical Representation

3.   Use Cognates

4.   Write friendly definition

5.   Physical Representation

WordList1.  2.  3.  4. 

Notes:

Whenseveralexamplesformapatternandyouassumethepatternwillcontinue,youareapplyinginductivereasoning.

Inductivereasoningistheprocessofreasoningthataruleorstatementistruebecausespecificcasesaretrue.

Youmayuseinductivereasoningtodrawaconclusionfromapattern.

Astatementyoubelievetobetruebasedoninductivereasoningiscalledaconjecture.

A‐BProblemAnotes:

1)Findthenextiteminthepattern. January,March,May,...

2)Findthenextiteminthepattern. 7,14,21,28,…

A‐BProblemACheck:

1)Findthenextiteminthepattern. January,March,May,...

ThenextmonthisJuly.

Alternatingmonthsoftheyearmakeupthepattern.

2)Findthenextiteminthepattern. 7,14,21,28,…

Thenextmultipleis35.

Multiplesof7makeupthepattern.

A‐BProblemBSOLVE!!

3)Findthenextiteminthepattern.

4)  Findthenextiteminthepattern:

0.4,0.04,0.004,…

A‐BProblemBCheck!!

3)Findthenextiteminthepattern. Inthispattern,thefigurerotates90°counter‐clockwiseeachtime.

Thenextfigureis.

4)Findthenextiteminthepattern0.4,0.04,0.004,…Whenreadingthepatternfromlefttoright,thenextiteminthepatternhasonemorezeroafterthedecimalpoint.

Thenextitemwouldhave3zerosafterthedecimalpoint,or0.0004.

StormCheck(Think,Write,Discuss,Report)

Inyourownlife,givemetwoexampleswhereyoulookforpatternsandmakeconjectures?

OneexampleinmylifewhereIusepatternsand

makeconjecturesis______________________.

Anotherexampleis______________________

_______________________________________.

Notes:

InductiveReasoning

1.Lookforapattern.

2.Makeaconjecture.

3.Provetheconjectureorfindacounterexample.

Notes:

Toshowthataconjectureisfalse,youhavetofindonlyoneexampleinwhichtheconjectureisnottrue.Thiscaseiscalledacounterexample.Acounterexamplecanbeadrawing,astatement,oranumber.

Toshowthataconjectureisalwaystrue,youmustproveit.

A‐BProblemAnotes:

Theproductoftwooddnumbersis?.

1)Completetheconjecture(Testexamplesfirst).

4)Makeaconjectureaboutthelengthsofmaleandfemalelionsbasedonthedata.

Average Lion Lengths Length of Female (ft) 4.5 5.1 5.0 4.8 4.9 4.7 Length of Male (ft) 4.7 4.9 5.4 5.2 5.3 4.8

A‐BProblemACheck:

Theproductoftwooddnumbersis?.

1)Completetheconjecture(Testexamplesfirst).

Theproductoftwooddnumbersisodd.

Listsomeexamplesandlookforapattern. 1x1=13x3=95x7=35

4)Makeaconjectureaboutthelengthsofmaleandfemalelionsbasedonthedata.

In5ofthe6pairsofnumbersabovethemaleislonger.

Male lions are longer than female lions.

Average Lion Lengths Length of Female (ft) 4.5 5.1 5.0 4.8 4.9 4.7 Length of Male (ft) 4.7 4.9 5.4 5.2 5.3 4.8

A‐BProblemBSOLVE!!

3)Completetheconjecture(Testexamplesfirst). Thesumoftwopositivenumbersis?.

4)Makeaconjectureaboutthelengthsofmaleandfemalewhalesbasedonthedata.

Average Whale Lengths Length of Female (ft) 49 51 50 48 51 47 Length of Male (ft) 47 45 44 46 48 48

A‐BProblemBCheck!!

3)Completetheconjecture(Testexamplesfirst).

Thesumoftwopositivenumbersis?. Thesumoftwopositivenumbersispositive.

Listsomeexamples&lookforapattern.

1+1=2 3.14+0.01=3.15

3,900+1,000,017=1,003,917

4)Makeaconjectureaboutthelengthsofmaleandfemalewhalesbasedonthedata.

In5ofthe6pairsofnumbersabovethefemaleislonger.

Female whales are longer than male whales.

Average Whale Lengths Length of Female (ft) 49 51 50 48 51 47 Length of Male (ft) 47 45 44 46 48 48

StormCheck(Think,Write,Discuss,Report)

Inyourownwords,howwouldyoudefineaconjectureandacounterexample?

Aconjectureis__________________________.

Acounterexampleis_____________________.

Howdoaconjectureandacounterexamplerelatetoeachother?

Theyrelatetoeachotherbecause____________

________________________________________.

1)  Conjecture2)  Counterexample

3)  Hypothesis4)  Conclusion

Vocabulary Review 1.  Teachersaystheword.

•  ALLStudentsrepeattheword.

2.  Teachercountsoutsyllablesass/hesaystheword.

•  ALLstudentsrepeatsyllablecountastheysaytheword.

3.  Studentteamscreateaphysicalrepresentationoftheword.

•  ClasschoosesonephysicalrepresentationforALLstudents.

4.  Studentteamscreateafriendlydefinitionfortheword.

•  Classchoosesonefriendlydefinitionfortheword.

Learning Target

Studentswillapplyinductivereasoningtoidentifypatterns,makeconjectures,andfindcounterexamplestodisproveconjectures.

Studentswillshowthisbycompletingagraphicorganizerandsolvinginductivereasoningproblemsinapairactivity.

2)Section______ 3)Section______RFM/RTProblems_________ NotesCopiedinNotebook?Pasted&SolvedinNotebook?

HomeWork1‐2‐3:1)StormCheckPastedinNotebook?

NEWUnitTableofContents!

Ticket OUT.

Findthenextitemineachpattern.

1.0.7,0.07,0.007,… 2.

Determineifeachconjectureistrue.Iffalse,giveacounterexample.3.Thequotientoftwonegativenumbersisapositivenumber. 4.Twosupplementaryanglesarenotcongruent.

Ticket OUT.

Findthenextitemineachpattern.

1.0.7,0.07,0.007,… 2.

0.0007

Determineifeachconjectureistrue.Iffalse,giveacounterexample.3.Thequotientoftwonegativenumbersisapositivenumber. 4.Twosupplementaryanglesarenotcongruent.

true

false;90°and90°