Ext 6-8 Sampler

55 Grade8 PartB Lesson28, 8ETeacherGuide 79 Objective56: To find thePythagorean relationship in right triangles. Materials: CentimeterGraphPaper (Master 8), scissors, glueor tape Vocabulary: PythagoreanTheorem, legs, hypotenuse Squaring aNumber In this activity, studentswillmodel anddrawpictures of the areaof squareshaving sides of 1, 2, . . . 10 cm. Display a1-centimeter square anddescribe thenumber of units on each side. This is the smallest square shapewe canmakewith thisgraphpaper.Howmanyunitson the horizontal side? (1) On thevertical side? (1) Howmany small squares in thewhole figure? (1) Writeon theboard: The squareof 1=1 1=1 Ask students to cut out a squarewith2units on each side. This is thenext smallest square shape.Howmany units areon thevertical side? (2) Howmanyunits areon thehorizontal side? (2) Howmany small squares are in thewhole figure? (4) Have students complete the first 3 columns of the following table: Area of Units Sides Squares Square Root 1 1 1 1 1 = 1 2 2 2 4 4 = 2 . . . . . . . . . . . . 10 10 10 100 100 = 10 Be sure students cut out each squareusing the centimeter graphpaper. Theywill use these later in the lesson. SquareRoots What ifwewant toundo the squaringof anumber? If weknowa large squarehas anareaof 9 small squares, howmanyunits areoneach sideof the square? (3) Show the3 cm 3 cm square anddraw apicture to illustrate. Write on the board: 9 3 Repeat the activitywithother squarednumbers.Have students complete the fourth column in the table above. ? ? 9 Discover theRight TrianglePattern Studentswill nowuse trial and error to find3 squares they can fit together on the sides to form a right triangle. Try forminga right triangleby connecting the sidesof any3of your squares.Howmany triangles are right triangles? Ask students todescribe each right triangle they find. (Students should find that 3, 4, and5, and6, 8, and10 form right triangles.) There is a special pattern for the areaof the sidesof every right triangle. Studyyour squares to find thepattern. (The sumof the squares on the2 small sides of a right triangle equals the squareon the large side.) Read the information together at the topof thepage. Ask students to circle the side thatwouldbe the hypotenuse in eachof theproblems 1 to6 (the longest side).Askvolunteers touse thewords “if” and “then” to tell how theywill know if the sides form a right triangle. (Inproblem1, if the sumof the squares of 5 and12 equals the squareof 13, then the triangle is a right triangle.) Have students complete thepageon their own. Part B 79 ©Math TeachersPress, Inc.,Reproduction by anymeans is strictly prohibited. ThePythagoreanTheorem Pythagoras, aGreekmathematician, discovered a special property about right triangles. This property relates to the square that can be drawn oneach side. This right triangle has sides of 3, 4 and5. 3 5 4 The shorter sides, 3and 4, are called the legs of the right triangle. The longest side, 5, is called the hypotenuse. The hypotenuse is the side opposite the right angle. 3 5 4 3 2 = _____ 4 2 =_____ 5 2 = _____ 3 2 + 4 2 = _____ 5 2 = _____ Describe this relationship (known as thePythagoreanTheorem): ___________________________________________________________________ ___________________________________________________________________ Threesidesof a trianglearegiven. Is the trianglea right triangle? 1. 5, 12, 13 _______ 4. 5, 7, 9 _______ 2 . 4, 5, 6 _______ 5. 9, 12, 15 _______ 3. 6, 8, 10 _______ 6. 7, 24, 25 _______ Find the lengthsof the legs and the lengthof thehypotenuseof the right triangle formedby thesquares. 7 . leg=_____ leg=_____ hypotenuse= ______ 8. leg=_____ leg= _____ hypotenuse= ______ ______+ ______= _____________ leg 2 leg 2 hypotenuse 2 ______+ ______=_____________ leg 2 leg 2 hypotenuse 2 9 16 25 25 25 Ina right triangle, the sumof the areasof squareson the legs equals the areaof the square on the hypotenuse. yes no no yes yes yes 6 36 81 64 144 100 225 8 10 9 12 15 8.G.6, 8.G.7 PythagoreanTheorem