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# Elem Math for Teachers II MTH 202

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This 14 page Class Notes was uploaded by Donny Graham on Saturday September 19, 2015. The Class Notes belongs to MTH 202 at Michigan State University taught by Jia He in Fall. Since its upload, it has received 49 views. For similar materials see /class/207293/mth-202-michigan-state-university in Mathematics (M) at Michigan State University.

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Date Created: 09/19/15

Final Exam Review 81 msualiz on N ets gm 0 Tetrahedron Triangular Based Pyramid 0 Square based Pyramid 82 Angels De nitions ofAngel 1 An angle is an amount of rotation about a xed point 2 Suppose there are two rays In a plane that meet at a common endpoint The two rays and region between them is called the angle Three Facts 1 Vertical Angles are Congruent b 4 Yb 4 241 936 Kw 4a Lc 2 ParaHe Postmate Correspondmg Angres are congruent b Anemate ntenor Angres are congruent Caunxpuudmg Angles Aneume Inlenm Ange 3 The Sum of Angres m aTrrangre 180 degrees an mg m rxn Proofs for Verucar Angres are congruent Ang e1 Angre 3 7180 degrees gt Ang e1180 degrees rAnge 3 Angre 2 Angre 3 7180 degrees gt Angre 2 180 degreesrAnge 3 gt Ang e 1 Angre 2 Tummg Angres Drrecuon Probrerns 33 39rcles and Spheres Derrnmons A crrcre rsme coHchon of 5 me perms m a pranemat are a certam xed drstance away rrorn a certam xed perm m the pane o Tms xed porntrs caHedme center of me crrcre o Tms xed drsrance rs caHed me radms orme crrcre L 5 the orms m d drstance away rrorn a certam xed perm m space 0 Tms xed perm rs caHed the center of the sphere o This distance is called the radius of the sphere 84 Trian les Quadrilaterals and Other Pol ons De nition 0 A polygon is a closed connected shape in a plane consisting of a finite number of line segments that do not cross each other Triangles o A triangle is a closed shape in a plane consisting of three line segments 0 Right Triangle Has1 right angle C is called the hypotenuse C b o Equilateral Triangle Three equal side lengths B o lsosceles Triangle 1 At least two sides of equal lengths um as Lrlml Quadrilaterals o A quadrilateral is a closed shape in a plane consisting of four line segments 0 Straight lines 0 Close 0 No crosses Square quadrilateral with four right angles whose sides all have the same length Rectangle quadrilateral with four right angles 0 The diagonals meet at a pint halfway across each diagonal o Diagonals are only perpendicular when it s a square 0 The diagonals always have the same length Rhombus quadrilateral whose sides all have the same lengths o The diagonals meet at a point that is halfway across each diagonal o Diagonals are perpendicular o Diagonals have different lengths unless they are a square 0 Diagonals cut a rhombus in half Parallelogram quadrilateral for which opposite sides are parallel o Trapezoid quadrilateral with at least one pair of opposite parallel sides Polygons o A polygon is a closed connected shape in a plane consisting ofa finite number of line segments that do not cross each other Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Octagon 8 N sides ngon 000000 85 Constructions With Straightedge and Compass Constructing a line that is perpendicularto a line segment and divides it in half 0 Starting with a line segment AB open a compass to at least half the length ofthe line segment and draw part of a circle centered at A 0 Keep the compass open to the same width and draw part of a circle centered at B Draw a line through the two points where the two circles meet This line is perpendicularto AB and divides AB in half Dividing an Angle in Half 0 Starting with two rays that meet at a point P draw a circle centered at P Let q and R be points were the circle meets the rays 0 Keep the company open to the same width Draw a circle centered at Q and another circle centered at R 0 Draw a line though point P and the other point where the last two circles drawn meet 86 Polydedra and Other Solid Shapes A closed shape in space that is made out of polygons is called a polyhedron o The plural of polyhedron is polyhedra o The polygons that make up the surface of the polyhedron are called the faces of the polyhedron o The place where two faces come together is called an edge of the polyhedron o The corner point where several faces come together is called a vertex or corner of the polyhedron o The plural of vertex is vertices Prisms Cylinders Pyramids and Cones o A right prism is polyhedron that roughly speaking can be thought of as going straight up over a polygon o The two polygons that you started with are called bases ofthe right prism o If faces are now places so as to connect corresponding sides ofthe two polygons then the shape formed this way is a prism o A prism that is not a right prism is an oblique prism A prism with a triangle base can be called a triangular prism A prism with a rectangle base can be a called a rectangular prism o A cylinder is a tubeshaped object kind of like a prism but with a closed curve bases an oval or a circle and one curved face 0 A pyramid is made by a polygon and separate single point that does not line in the plane of the polygon o The original polygon is called the baj of the pyramid o A right pyramid is a pyramid for which the point lies straight up over the center of the base 0 A pyramid that is not a right pyramid can be called an oblique pyramid o Cones can be described by starting with a closed curve in a place and a separate point that does not lie in that plane Platonic Solids 0 Five special polyhedra are called the platonic solids o Tetrahedron is made of 4 equilateral triangles with 3 triangles coming together at each vertex Cube is made of 6 squares with 3 squares coming together at each vertex Octahedron is made of 8 equilateral triangles with 4 triangles coming together at each vertex 0 Dodecahedron is made of 12 regular pentagons with 3 pentagons coming together at each vertex 0 lcosahedron is made of 20 equilateral triangles with 5 triangles coming together at each vertex 0 The platonic solids are the only convex polyhedra having these two properties O O 0 They are made of only one kind of regular polygon o The same number of polygons comes together at each vertex 0 A shape in the plane or in space is convex if any line segment connecting tow points on the shape lies entirely with the shape 91 ReflectionI Translations and Rotations Transformation o A transformation ofa plane isjust what the name implies an action that changes or transforms a plane A reflection or flip ofa plane across a chosen line called the line of re ection is the following kind oftransformation o For any point Q in the plane imagine a line that passes through Q and is perpendicular the line of reflection Now imagine moving the Q to the other point on the line that is the same distance from the line of reflection A translation or slide ofa plane by a given distance in a given direction is the end result of moving each point in the plane the given distance in the given direction 0 A rotation orturn about a point through a given angle is a transformation ofa plane that is the end result of rotating all points in the plane about a fixed point through a xed angle 0 A glide re ection is the end result of combing a reflection and THEN a translation in the direction ofthe line re ection Re ections Translations and Rotations in a Coordinate Plane 0 A coordinate plane is a plane together with two perpendicular number lines in the plane that meet at the location ofO on each number line 0 The two number lines are called the axes ofthe coordinate planes The horizontal axis is called the xaxis The vertical axis is called the yaxis o A point can be designated 45 3 0 We say that 45 and 3 are the coordinates ofthe point 45 is the rst coordinate or the xcoordinate 3 is the second coordinate or the ycoordinate 92 Symmetry Re ection Symmetry o A shape or design in a plane has reflection symmetm or mirror symmetry if there is a line in the plane such that the shape or design as a whole occupies the same place in the plane both before and after reflecting across the line 0 This line is called the line of symmetry Translation Symmetry o A design or pattern in a plane has translation symmetm ifthere is a translation ofthe plane such that the design or pattern as a whole occupies the same place in the plane both before and afterthe translation Rotation Symmetry o A shape or design in a plane has rotation symmetm if there is a rotation of the plane of more than 0 but less than 360 such that the shape or design as a whole occupies the same points in the plan both before and after rotation o A design has nfold rotation symmet provided that a rotation of 360 n degrees takes the design as a whole to the same location GlideReflection Symmetry o A design or pattern in a plane has glidere ection symmetm ifthere are a re ection and a translation such that after applying the reflection followed by the translation the design as whole occupies the same location in the lane 0 First the reflection then the glide 93 Congruence Congruent o lnformally two shapes either in a plane or solid shapes in space that are the same size and shape are called congruent o Formally two shapes or designs in a plane are congruent if there is a rotation reflection translation or a combination of these transformations that takes one shape or design to the other shape or design SideSideSide SSS 0 Given a triangle that has sides length a b and c units it is congruent to all other triangles that have sides of length a b and c o Fortriangles it is the mathematical way to say that triangles are structurally rigid AngleSideAngle ASA o If we specify the length of a side of a triangle and if we specify two angles that add to less than 180 at the two ends ofa line segment than al triangles formed with those specifications are congruent SideAngleSide SAS o If we specify the lengths of two sides ofa triangle and if we specify the angle between these two sides than all triangles formed with those speci cations are congruent 94 Similarity Similar lnformally we say that two objects that have the same shape but not the same size are similar Two objects or shapes are similar if one object represents a scaled version of the other o Formally we say two shapes or objects are similar if every point on one object corresponds to a point on the other object and there is a positive number k such that the distance between any points on the second object is k times as long as the distance between the corresponding points on the first object o Scale Factor This number k is called the scale factor from the first object to the second object Three Methods for Solving Problems about Similar Objects or Shapes 0 Scale Factor Method 0 Since the scale model and the actual sculpture are to be similar there is a scale factor k from the model to the actual sculpture such that every length on the actual sculpture is k times as long as the corresponding length on the scale model 0 Example Scaled Model 10X20 Actual Picture 60X Because 10 times 6 equals 60 Therefore 20 times 6 equals actual height 0 Relative Sizes Method 0 Since the scale model is 10 inches wide and 20 inches tall it is 2010 or 2 times as tall as it is wide The actual sculpture should therefore also be 2 times as tall as it is wide 0 Example Scaled Model 10X20 Actual Picture 60X Because the scaled height is two times the size of the width Therefore the real height is 60X2 o Proportions Method 0 As before since the scale model and the actual sculpture have to be similar there is a scale factor k from the model to the actual sculpture k can be calculated as the width of sculpture width of model and height ofsculptureheight of model 0 Therefore width of sculpturewidth of model height of sculptureheight of model 0 Example 6010 20 60X2010X 120010X 120 X Triangle Similarity Criterion 0 Here is a criterion for similarity oftriangles Two triangles are similar exactly when the two triangles have the same size angles To clarify two triangles are similar exactly when it is possible to match each angle ofthe first triangle with an angle ofthe second triangle in such a way that matched angles have equal measures 0 Parallel Lines created similar triangles 0 Parallel lines on opposite sides of the point where two lines meet create similar triangles Sighting 0 Creating a pair of similar triangles to measure distant objects or to measure their distance 101 Fundamentals of Measurement US water boils at 212 F Metric System Units of Len th UNIT ABBREVIATION SOME RELATIONSHIPS Fundamental Relationship o1mL1g1cm3 102 LengthI AreaI Volume and Dimension Length 0 Describes the size of something that is one dimensional the length ofthat one dimensional object is how many of a chosen unit of length it takes to cover the object without gaps or overlaps OneDimensional o Roughly speaking an object is onedimensional if at each location there is only one independent direction along which to move within the object Perimeter o The perimeter ofa shape is the distance around a shape it is the total length ofthe outer edge around the shape AreaSurface Area 0 An area or a surface area describes the size of an object or a part of an object that is two dimensional the area of that two dimensional object is how many ofa chosen unit of area it takes to cover the object without gaps or overlaps o TwoDimensional o Roughly speaking an object is twodimensional if at each location there are two independent directions along which to move within the object Volume 0 A volume describes the size of an object tat is three dimensional the volume of that three dimensional object is how many ofa chosen unit of volume it takes to ll the object without gaps or overlaps where it is understood that we may use parts of a unit too 0 ThreeDimensional o Roughly speaking an object is threedimensional if at each location there are three independent direction along which to move within the object 103 Calculating Perimeters of Palmons Areas of Rectanqles and Volumes of Boxes Calculating Perimeters of Polygons 0 Add the lengths of all four sides Calculating Areas of Rectangles 0 0 Area squared 0 Length X width Calculating Volumes of Boxes 0 H X W X L 0 Cubic units 0 Height X Width X Length 104 Error and Accuracy in Measurements Significant Digits o In a measurement the digits that we can accurately report are called significant digits Reported and actual distance 0 The reported distance is the number used while rounding o The actual distance is the span that the number could be in 105 Converting from One Unit of Measurement to Another Sometimes there is not a direct comparison 0 So you must ratchet down then over and back up Dimensional Analysis 0 Repeatedly multiplying by the number 1 expressed as a fraction that relates two different units Area and Volume Conversion 0 Take into consideration that area and volume you must also square the numbers not just the unit 0 One square yard is 9 square feet not 3 square feet 0 First determine what onefoot is in terms of meters Then use that information to determine what one square foot is in terms of square meters Approximate Conversions and Checking your Work 0 An inch is about 2 12 centimeters so 2 inches is about 5 centimeters o A meter is a little more than a yard 0 A kilometer is about 610 ofa mile so little over halfa mile 0 A liter is a little more than a quart o A kilogram is a bit more than 2 pounds 111 The Moving and Additivity Principles about Area Moving Principle o If you move a shape rigidly without stretching it then its area does not change Additivity Principle o If you combine shapes without overlapping them then the area of the resulting shape is the sum of the areas of the individual shapes Subdividing into 2 rectangles 112 Usin the Movin and Additivit Princi les to Prove the P ha orean Theorem Pythagorean Theorem 0 A theorem about right triangles Recall that in a right triangle the side opposite the right angle is called the hypotenuse o The Theorem Says In a right triangle the square ofthe length the hypotenuse is equal to the sum of the squares ofthe lengths of the other two sides In other words ifc is the length ofthe hypotenuse ofa right triangle and ifa and b are the lengths of the othertwo sides 0 a2 b2 02 Proof 0 A proof is a thorough precise logical explanation for why a statement is true based on assumptions or facts that we already know or assume to be true 113 Area of Triangles Base 0 The base of a triangle can be any one of its three sides 0 The length of the base can be represented by b Height Perpendicularto the base and Connects the base or an extension ofthe base to the vertex ofthe triangle that is not on the base 0 The length of the height can be represented by h The Formula for the Area of a Triangle o 12b X h square units 0 Why is this valid 0 2 X area of triangle area of rectangle 0 but since the rectangle has an area of b X h o 2 X area of triangle b X h 0 so Area of triangle 12 b X h 114 Areas of Parallelograms Base 0 The base of a parallelogram can be chosen to be any one of its four sides Height 0 Perpendicularto the base and o Connects the base or an extension of the base to a vertex ofthe parallelogram that is not on the base Area of Parallelogram o b X h square units 0 Why is this valid 0 You can show it by subdividing and recombining to create a rectangle o Enclosing the parallelogram in a rectangle abxhaxhbxh remove the two additional triangles will removed the area of a X h left with the remaining parallelogram to make b X h 115 Cavalieri s Principle about Shearing and Area Sheanng o The process of sliding infinitesimally thin strips is called shearing Cavalieri s Principle 0 During shearing each points moves along a line that is parallel to the fixed side During shearing the think strips remain the same width and length The strips just slide over they are not compressed either in width or in length Shearing does the change the height of stack of thin strips In other words if you think of shearing in terms of sliding toothpicks the height of the stack of toothpicks doesn t change during shearing 116 Areas of Circles and the Number Pi Area of Circle 0 TEI392 Circumference o The circumference of a circle is the distance around the circle 0 Recall the radius of a circle is the distance from the center ofthe circle to any point on the circle 0 Recall the diameter of circle is the distance across the circle going through the center its twice the radius 0 For any circle whatsoever whether huge tiny or in between the circumference divided byt eh diameter is always equal to the same number this number is called pi and is written with the Greek letter 11 o Circumference diameter 11 o Circumference 11 X diameter 0 Circumference 11 X 2r 2nr Circle Area Formula 0 nr2 square units 117 Approximating Areas of Irregular Shapes Under and Over Estimates for the area of an irregular shaped object 0 Count the small squares in the smaller shape Figure out how big one square is Multiply times the number of squares Count the small squares in the bigger shape Do the same The area of the irregular shape will be somewhere in between 118 Relating the Perimeter and Area of a Shape The following is true in general 0 Among all shapes of a given fixed perimeter P the circle of circumference P has the largest area and every positive numberthat is less the area of that circle is the area of some shape of perimeter P Among all rectangles of a given fixed perimeter P the square of perimeter P has the largest area and every positive numberthat is less than the area ofthat square is the area of some rectangle of perimeter P 119 Principles For Determining Volumes The moving and Additivity Principles about Volumes o If you move a solid shape rigidly without stretching or shrinking it then its volume does not change

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