Mathematics Success – Grade 8 T791

LESSON 30: Pythagorean Theorem Part 1

[OBJECTIVE] The student will explore and apply the concept of the Pythagorean Theorem in mathematical and real world situations.

[PREREQUISITE SKILLS]squaressquare Roots

[MATERIALS]Student pages S390–S404Copies of T805 for each studentCalculatorsScissorsGlue

[ESSENTIAL QUESTIONS]1. Explain the Pythagorean Theorem and how you would apply it to determine the

hypotenuse of a right triangle. 2. What is meant by the converse of the Pythagorean Theorem? 3.Explainhowtofindthelengthofonelegofarighttriangleifthemeasuresofthe

hypotenuse and another leg are given. Justify your answer.

[WORDS FOR WORD WALL] Pythagorean Theorem, Pythagorean triple, square, square root, legs, hypotenuse

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Verbal Description, Concrete Representation, Pictorial Representation, Algebraic Formula, Graphic Organizer

[WARM-UP] (IP, I, WG) S390 (Answers on T803.)HavestudentsturntoS390intheirbookstobegintheWarm-Up.Studentswillfindthesquaresandsquarerootsofnumberstoprepareforfindingthemissinglengthsof triangles using the Pythagorean Theorem. Monitor students to see if any of them need help during the Warm-Up. Have students complete the problems and then review the answers as a class. {Algebraic Formula}

[HOMEWORK]Take time to go over the homework from the previous night.

[LESSON] [2 – 3 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 8T792

LESSON 30: Pythagorean Theorem Part 1

Discovery Activity – Pythagorean Theorem(M, GP, WG, CP) S391, S392, S393 (Answers on T804, T805, T806.)

WG, CP, M, GP: Students will use two different size squares to explore a proof for the Pythagorean Theorem. Each student pair needs a copy of S392. Students should cut out the two squares and label the sides as shown on S391. Assign the roles of Partner A and Partner B. {Concrete Representation, Algebraic Formula, Verbal Description, Pictorial Representation, Graphic Organizer}

MODELING

Discovery Activity - Pythagorean Theorem

Step 1: Model and have students place the small square on top of the larger square as shown in the diagram. Model and have students mark the sections of the sides of the smaller square as c and the sections of the sides of the larger square side as a and b.

a

b

a

b a

b

a

b

c c

c c

• PartnerA,explainwhywecanmarkeachofthesideswiththesameletter. (All sides of a square are congruent.)

• PartnerB,explainhowwemarkedthesidesofthelargersquare.(Thesmaller square divides each of the congruent sides into two sections. Each side has two sections, one labeled a and one labeled b. All the a sections are congruent and all the b sections are congruent.)

SOLVE Problem (GP, WG) S391 (Answers on T804.)

HavestudentsturntoS391intheirbooks.ThefirstproblemisaSOLVEproblem.Youare only going to complete the S step with students at this point. Tell students that theywilllearnhowtousethePythagoreanTheoremtofindthemissingsidelengthofa right triangle. They will use this knowledge to complete this SOLVE problem at the endofthefirstsectionofthelesson.{SOLVE, Verbal Description, Graphic Organizer}

Mathematics Success – Grade 8 T793

LESSON 30: Pythagorean Theorem Part 1

Step 2:HavestudentsturntoS393.PartnerA,explainhowwecanfindtheareaof the large square. (multiply side times side) Record.

• PartnerB,whatisthelengthofeachsideofthelargesquare?(a + b) Record.

• PartnerA,howcanwewriteaformulafortheareaofthelargesquareusing the information given? Justify your thinking. [Area = (a + b)(a + b) which is equal to a2 + 2ab + b2] Record.

Step 3: Let’s look at the area of the small square and the four congruent triangles that are inside the larger square.

• PartnerB,howcanwefindtheareaofthesmallsquare?(multiplyside times side) Record.

• PartnerA, identify the lengthofeachsideof thesmall square. (c) Record.

• PartnerB,explainhowtowritetheformulafortheareaofthesmallsquare using the information given. (c times c which is equal to c2) Record.

• PartnerA,howmanytrianglesareinthelargershape?(4)Record. • PartnerB, explain the formulawe canuse tofind theareaof any

triangle. Defend you thinking. (12bh which is equal to 12ab because the

base of the triangle is represented by the letter a and the height of the triangle is represented by the letter b.) Record.

Step 4: PartnerA,whatisthesumoftheareaofthe4triangles?[4(12ab) which

equals 2ab when we apply the distributive property] Record. • PartnerB,whatistheareaofthesumofthesmallersquareandthe

four triangles? (c2 + 2ab) Record. • PartnerA,whatdoweknowabouttheareaofthelargesquareand

the area of the tilted square plus the four triangles. Explain your answer. (The two areas must be equal because they take up the same amount of space.) Record.

• Modelhowtosetuptheequationthatwillshowthatthetwoareasareequal.

(a + b) (a + b) = c2 + 2ab a2 + 2ab + b2 = c2 + 2ab -2ab -2ab a2 + b2 = c2

• PartnerB,whenwesolvetheequation,wehaveproventhatthesumof the squares of the sides of a right triangle are equal to the square of the hypotenuse.

• ThisiscalledthePythagorean Theorem.

Mathematics Success – Grade 8T794

LESSON 30: Pythagorean Theorem Part 1

Applying the Pythagorean Theorem (M, GP, IP, WG, CP) S394 (Answers on T807.)

WG, CP, M, GP: Students will use representation of triangles and apply the Pythagorean Theorem to determine the measure of the hypotenuse. Calculators may be used to determine the squarerootsasProblems3and4haveanswersthatarenot whole number values. Be sure students know their designation as Partner A and Partner B. {Algebraic Formula, Verbal Description, Pictorial Representation, Graphic Organizer}

MODELING

Applying the Pythagorean Theorem

Step 1: Partner A, identify what information is given with the triangle. (the length of the two legs)

• PartnerB,whatotherinformationdoweknowfromthepictureofthetriangle? Defend your answer. (We know it is a right triangle because of the square in the corner.)

• PartnerA,asthisisarighttriangle,whatcanweusetodeterminethemeasure of side c, which is the hypotenuse? (We can use the formula for the Pythagorean Theorem.)

Step 2: Partner B, what is the formula we can use? (a2 + b2 = c2) Record. • PartnerA,whatisthevaluewecansubstituteinfora? (8) • PartnerB,whatisthevaluewecansubstituteinforb? (15)

Step 3: Partner A, when we substitute in those values what is the equation that we have? (82 + 152 = c2) Record.

• PartnerB,whatisthenextstepintheequation?(Square the value of 8whichequals64andsquarethevalueof15whichisequalto225.)Record.

• PartnerA,whatisthenextstep?(Addthevalues,64and225,ontheleft side of the equation.)

• PartnerB,whatisthesumof64+225?(289)Record.

Mathematics Success – Grade 8 T795

LESSON 30: Pythagorean Theorem Part 1

Step 4: Partner A, what does the equation state now? (289 = c2) Record. • PartnerB,howdowefindthevalueofc?(byfindingthesquare

root of c2) • PartnerA,whenwesolveanequation,whatwillweneed todo to

keep it balanced? (Whatever we do to one side, we must do to the other side.)

• PartnerB,inthisequation,whatwillweneedtodotokeepitbalanced?(Find the square root of 289.)

• PartnerA,whatisthesquarerootof289?(17)Record. • Partner B, what is the value of side c,whichisthehypotenuse?(17cm)

Step 5: Have students look at Problem 3. • PartnerA, identifywhat informationisgivenwiththetriangle.(the

length of two legs) • PartnerB,whatotherinformationdoweknowfromthepictureofthe

triangle? Defend your answer. (We know it is a right triangle because of the square in the corner.)

• PartnerA,asthisisarighttriangle,whatcanweusetodeterminethemeasure of side c, which is the hypotenuse? (We can use the formula for the Pythagorean Theorem.)

Step 6: Partner B, what is the formula? (a2 + b2 = c2) Record. • PartnerA,whatisthevaluewecansubstituteinfora? (9) • PartnerB,whatisthevaluewecansubstituteinforb? (13)

Step 7: Partner A, when we substitute in those values what is the equation that we have? (92 + 132 = c2) Record.

• PartnerB,whatisthenextstepintheequation?(Squarethevalueof 9 which equals 81 and square the value of 13 which is equal to 169.) Record.

• PartnerA,whatisthenextstep?(Addthevalues,81and169,onthe left side of the equation.)

• PartnerB,whatisthesumof81+169?(250)

Mathematics Success – Grade 8T796

LESSON 30: Pythagorean Theorem Part 1

IP CP, WG: StudentswillcompleteProblems2and4topracticeapplicationofthePythagoreanTheoremtofindthevalueof the hypotenuse. After student pairs have completed the problems, review the answers as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation, Algebraic Formula}

SOLVE Problem (WG, GP) S395 (Answers on T808.)

The SOLVE problem is the same one from page S391. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Studentshaveapplied thePythagoreanTheorem inorder tofindthe value of the hypotenuse.) {SOLVE, Verbal Description, Graphic Organizer, Algebraic Formula, Pictorial Representation}

SOLVE Problem (GP, WG) S396 (Answers on T809.)

Have students turn to S396 in their books. This is a SOLVE problem for the next sectionofthelesson.YouareonlygoingtocompletetheSstepwithstudentsatthis point. Tell students that they will learn about the converse of the Pythagorean Theorem. They will use this knowledge to complete this SOLVE problem at the end of this section of the lesson. {SOLVE, Verbal Description, Graphic Organizer}

IP CP, WG: StudentswillcompleteProblems2and4topractice

Step 8: Partner A, what does the equation state now? (250 = c2) Record. • PartnerB,howdowefindthevalueofc?(byfindingthesquareroot

of c2) • PartnerA,whenwesolveanequation,whatwillweneed todo to

keep it balanced? (Whatever we do to one side, we must do to the other side.)

• PartnerB,inthisequation,whatwillweneedtodotokeepitbalanced?(Find the square root of 250.)

• PartnerA,explainhowthisequationisdifferentfromtheequationinProblem 1. (The square root is not a whole number.)

• PartnerB,whatisthesquarerootof250?(15.81whenweroundtotwo place values.)

• PartnerA,whatisthevalueofsidec, which is the hypotenuse? (15.81ft≈c) • PartnerB,whydowenotusetheequalsign?(Becausetheansweris

not exact, we have to round it.)

Mathematics Success – Grade 8 T797

LESSON 30: Pythagorean Theorem Part 1

Exploring Pythagorean Triples(M, GP, WG, CP, IP) S396 (Answers on T809.)

WG, CP, M, GP: Students will be using the Pythagorean Theorem and the concept of Pythagorean Triples to explore and build understanding for the converse of the Pythagorean Theorem. Be sure students know their designated role as Partner A and Partner B. {Algebraic Formula, Verbal Description, Graphic Organizer}

MODELING

Exploring Pythagorean Triples

Step 1: We worked with two squares to explore the Pythagorean Theorem which says that in a right triangle where a and b are legs of the triangle and c is the hypotenuse then: (a2 + b2 = c2) Record.

• Partner A, the Pythagorean Theorem is only true forwhat type oftriangles? (right triangles) Record.

Step 2: Partner B, the numbers used in the triangles are whole numbers that make the (three) sides of a (right) triangle. Record.

• Therearesomegroupsofvaluesthatarecommonlyrecognizedasthe sides of a right triangle. These are called Pythogrean Triples.

• Someofthecommontriplesthatformrighttrianglesare:(3,4,5),(5,12,13),(8,15,17)

Step 3: Each of these triples form a triangle that has values that can be substituted in the Pythagorean Theorem to make a true statement:

a2 + b2 = c2

32+42 = 52

9 + 16 = 25 25 = 25

• Partner A, what is written first in the box? (the formula for thePythagorean Theorem)

• Partner B, when we substitute in the values for the first set ofPythagorean Triples, what is the equation? (32+42 = 52) Record.

• PartnerA,whenwesquareeachvaluewhatisoursolution?(9 + 16 = 25)

• PartnerB, isthisatruestatement?(Yes)Explainwhatthismeans.(Because the two sides of the equation are equal, this is a true statementandthesidesof3,4and5formarighttriangle.)

Mathematics Success – Grade 8T798

LESSON 30: Pythagorean Theorem Part 1

Step 4: Have students work in partners to substitute in the values for the other two sets of Pythagorean Triples to prove that they will form right triangles.

Step 5: We can also multiply the triples by a single whole number value in order to create new right triangles.

• PartnerA,whataretheproductswhenwemultiplythefirsttriplebythe value of 2? (6, 8, 10) Record.

• PartnerB,whataretheproductswhenwemultiplythesecondtriplebythevalueof2?(10,24,26)Record.

• PartnerA,whataretheproductswhenwemultiplythethirdtriplebythevalueof2?(16,30,34)Record.

Step 6: Have student pairs multiply each triple by 3 and then by 5 and record the answers. Review the answers as a whole group.

IP CP, WG: Students will complete the Pythagorean Triples by multiplying the values by 3 and then 5. After student pairs have completed the problems, review the answers as a whole group. {Verbal Description, Graphic Organizer, Algebraic Formula}

Exploring the Converse of the Pythagorean Theorem (M, GP, WG, CP) S397 (Answers on T810.)

WG, CP, M, GP: Students will explore the converse of the Pythagorean Theorem using pictorial representations of triangles. Be sure students know their designated role as Partner A and Partner B. {Algebraic Formula, Verbal Description, Graphic Organizer, Pictorial Representation}

Mathematics Success – Grade 8 T799

LESSON 30: Pythagorean Theorem Part 1

MODELING

Exploring the Converse of the Pythagorean Theorem

Step 1:HavestudentslookatTriangle1onS397.

• PartnerA,whattypeoftriangledoesthisappeartobe?Justifyyouranswer. (right triangle because the angle across from the measure of 10 feet looks like a 90 degree angle)

• PartnerB,isthereawaythatwecanapplywhatweknowaboutthePythagoreanTheoremtoprovethatitisarighttriangle.(Yes,wecansubstitute in the values that are shown on the triangle and simplify the equation to see if it is a true statement.)

• PartnerA,whatvalueswillwesubstitutefora and b? (6 and 8 because those appear to be the legs of the triangle) Record.

• Whatvaluewillwesubstituteforc? (10)

• PartnerB,whatisournextstep?(squarethe6,the8andthe10)

• PartnerA,whatisthisequation?(36+64=100)Record.

• PartnerB,isthisatruestatement?(Yes)Record.

• PartnerA,ifitistruethat36+64=100and100=100thenwhathave we proved? (We proved a2 + b2 = c2 so this means that this is a right triangle.)

Step 2: Have students look at Triangle 2.

• PartnerA,whattypeoftriangledoesthisappeartobe?Justifyyouranswer. (right triangle because the angle across from the measure of 12 feet looks like a 90 degree angle)

• PartnerB,isthereawaythatwecanapplywhatweknowaboutthePythagoreanTheoremtoprovethatitisarighttriangle.(Yes,wecansubstitute in the values that are shown on the triangle and solve the equation to see if it is a true statement.)

• Partner A, what values will we substitute for a and b? (7 and 10because those appear to be the legs of the triangle)

• Whatvaluewillwesubstituteforc? (12)

• PartnerB,whatisournextstep?(squarethe7,the10andthe12)

• PartnerA,whatisthisequation?(49+100=144)Record.

• PartnerB,isthisatruestatement?(No,because49+100isequalto149,not144.)Record

• PartnerA,whatcanweconclude?(Because49+100≠144thisisnota right triangle.) Record.

Mathematics Success – Grade 8T800

LESSON 30: Pythagorean Theorem Part 1

SOLVE Problem (WG, GP) S398 (Answers on T811.)

The SOLVE problem is the same one from page S396. Complete the SOLVE problem with your students. Ask them for possible connections from the SOLVE problem to the lesson. (Students have applied the converse of the Pythagorean Theorem in order to determine if a triangle is a right triangle.) {SOLVE, Verbal Description, Graphic Organizer}

SOLVE Problem (GP, WG) S399 (Answers on T812.)

Havestudents turn toS399 in theirbooks.Thefirstproblem in thissection isaSOLVEproblem.YouareonlygoingtocompletetheSstepwithstudentsat thispoint. Tell students that they will learn how to apply the Pythagorean Theorem in order to determine the unknown measure of a leg when given the hypotenuse and thefirstleg.TheywillusethisknowledgetocompletethisSOLVEproblemattheendof this section of the lesson. {SOLVE, Verbal Description, Graphic Organizer}

Applying the Pythagorean Theorem to Find the Unknown Measure of a Leg (M, GP, WG, CP, IP) S399, S400 (Answers on T812, T813.)

WG, CP, M, GP: Students will be applying the Pythagorean Theorem when given the hypotenuse and the measure of one leg in order to solve for the measure of the second leg. Be sure students know their designated role as Partner A and Partner B. {Algebraic Formula, Pictorial Representation, Verbal Description, Graphic Organizer}

MODELING

Applying the Pythagorean Theorem to Find the Unknown Measure of a Leg

Step 1:WecanalsousethePythagoreanTheoremtofindunknownsidelengthsin right triangles.

• Partner A, we use the same formula, but substitute in the knownvaluesfor(oneleg)andthe(hypotenuse)inordertofindthemeasureof the (unknown leg). Record.

• PartnerB,let’slookatthewordprobleminQuestion1. • Duringatrainingsession,thefiremenleaneda20-footladderupto

a window in a building. The bottom of the ladder is 16 feet from the bottom of the building. How high is the window from the ground to the nearest foot.

• PartnerA,whatstrategycanweusetohelpuswiththisproblem?(Wecan draw a picture so that it is easier to see which sides of the triangle are given in the problem.)

Mathematics Success – Grade 8 T801

LESSON 30: Pythagorean Theorem Part 1

• PartnerB,whatinformationarewegivenintheproblem?(themeasureof one leg and the hypotenuse)

Step 2: Partner A, what is the problem asking us to find? (the height of thewindow from the ground)

Step 3: Partner B, when we draw our picture, what shape does the ladder form with the building? (a triangle)

20 ft

16 ft

a

Step 4: Partner A, what information are we given? (the length of the ladder and the position of the foot of the ladder on the ground)

• PartnerB,what typeof triangle is formedby thebuildingand theladder? (a right triangle)

• PartnerA,sincethisisarighttriangle,whatformulacanweusetofindthemissingsideofthetriangle?(WecanusethePythagoreanTheorem and substitute in the values that we know. a2 + b2 = c2) Record.

• PartnerB,whatisthenextstepinsolvingourequation?(Substitutinginthe values we know from the word problem a2 + 162 = 202) Record.

• PartnerA,whatisournextstep?(squarethe16andthe20.) (a2+256=400)Record. • PartnerB,whatisthenextstep?(subtract256fromeachsideofthe

equation to isolate the variable of a2) Record. • Partner A, after we subtract 256 from both sides, what does our

equation look like? (a2=144) • Partner B, what is our next step? (Find the square root of both

sides.) • PartnerA,whatisthesquarerootofa2? (a) Record. • PartnerB,whatisthesquarerootof144?(12)Record. • Howhighisthewindowfromtheground?(12feet)

Step 5:HavestudentscompleteProblem2tofindthevalueoftheunknownleg.Review the answers as a whole group.