MATH 2160 3 rd Exam Review
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Transcript of MATH 2160 3 rd Exam Review
MATH 2160 3MATH 2160 3rdrd Exam Exam ReviewReview
GeometryGeometry
and and
MeasurementMeasurement
Problem Solving – Problem Solving – Polya’s 4 StepsPolya’s 4 Steps
Understand the problemUnderstand the problem What does this mean?What does this mean? How do you understand?How do you understand?
Devise a planDevise a plan What goes into this step?What goes into this step? Why is it important?Why is it important?
Carry out the planCarry out the plan What happens here?What happens here? What belongs in this step?What belongs in this step?
Look backLook back What does this step imply?What does this step imply? How do you show you did this?How do you show you did this?
Problem SolvingProblem Solving Polya’s 4 StepsPolya’s 4 Steps
Understand the problemUnderstand the problem Devise a planDevise a plan Carry out the problemCarry out the problem Look backLook back
Which step is most important?Which step is most important? Why is the order important?Why is the order important? How has learning problem solving How has learning problem solving
skills helped you in this or another skills helped you in this or another course?course?
Problem Solving Problem Solving Strategies Strategies Make a chartMake a chart Make a tableMake a table Draw a pictureDraw a picture Draw a diagramDraw a diagram Guess, test, and reviseGuess, test, and revise Form an algebraic modelForm an algebraic model Look for a patternLook for a pattern Try a simpler version of the problemTry a simpler version of the problem Work backwardWork backward Restate the problem differentlyRestate the problem differently Eliminate impossible situationsEliminate impossible situations Use reasoningUse reasoning
GeometryGeometry
Angles and congruencyAngles and congruency CongruentCongruent– same size, same shape– same size, same shape Degree measureDegree measure – real number – real number
between 0 and 360 degrees that defines between 0 and 360 degrees that defines the amount of rotation or size of an anglethe amount of rotation or size of an angle
Sum of the interior anglesSum of the interior angles of any of any polygon: polygon: (n – 2)180(n – 2)180oo where n is the where n is the number of sides in the polygonnumber of sides in the polygon
GeometryGeometry
Special anglesSpecial angles right angleright angle – 90 – 90 acute angleacute angle – 0 – 0< angle < 90< angle < 90 obtuse angleobtuse angle – 90 – 90< angle < 180< angle < 180
Sum of the anglesSum of the angles Triangle = 180Triangle = 180oo
Quadrilateral = 360Quadrilateral = 360oo
Pentagon = 540Pentagon = 540oo
Etc. Etc.
GeometryGeometry CirclesCircles
circle circle – special simple closed curve – special simple closed curve where all points in the curve are where all points in the curve are equidistant from a given point in the equidistant from a given point in the same plane – same plane – NOTENOTE: Circles are : Circles are NOTNOT polygons!polygons!
center center – middle point of the circle– middle point of the circle diameter diameter – is a chord that passes – is a chord that passes
through the center of the circlethrough the center of the circle radius radius – line segment connecting the – line segment connecting the
center of the circle to any point on the center of the circle to any point on the circlecircle
GeometryGeometry Polygons – made up of line Polygons – made up of line
segmentssegments Triangles Triangles – 3-sided polygons– 3-sided polygons QuadrilateralsQuadrilaterals – 4-sided polygons – 4-sided polygons n - gons n - gons – the whole number n – the whole number n
represents the number of sides for the represents the number of sides for the polygon: a triangle is a 3-gon; a polygon: a triangle is a 3-gon; a square is a 4-gon; and so onsquare is a 4-gon; and so on
Regular PolygonsRegular Polygons – polygon where – polygon where the all the line segments and all of the the all the line segments and all of the angles are congruentangles are congruent
GeometryGeometry TrianglesTriangles
Union of three line segments formed Union of three line segments formed by three distinct non-collinear pointsby three distinct non-collinear points verticesvertices – intersection points of line – intersection points of line
segments forming the angles of the segments forming the angles of the polygonpolygon
sidessides – the line segments forming the – the line segments forming the polygonpolygon
height height – line segment from a vertex of a – line segment from a vertex of a triangle to a line containing the side of the triangle to a line containing the side of the triangle opposite the vertex triangle opposite the vertex
GeometryGeometry TrianglesTriangles
equilateralequilateral – all sides and angles – all sides and angles congruent congruent
isoscelesisosceles – at least one pair of – at least one pair of congruent sides and anglescongruent sides and angles
scalenescalene – no congruent sides or – no congruent sides or anglesangles
rightright – one right angle – one right angle acuteacute – all angles acute – all angles acute obtuseobtuse – one obtuse angle – one obtuse angle
GeometryGeometry QuadrilateralsQuadrilaterals
parallelogramparallelogram – quadrilateral with two – quadrilateral with two pairs of parallel sidespairs of parallel sides opposite sides are parallelopposite sides are parallel opposite sides are congruentopposite sides are congruent
rectanglerectangle – quadrilateral with four – quadrilateral with four right anglesright angles a a parallelogram is a rectangleparallelogram is a rectangle if and if and
only ifonly if it has at least one right angleit has at least one right angle
trapezoidtrapezoid – exactly one pair of – exactly one pair of opposite sides parallel, but not opposite sides parallel, but not congruentcongruent
GeometryGeometry QuadrilateralsQuadrilaterals
rhombusrhombus – quadrilateral with four – quadrilateral with four congruent sidescongruent sides a a parallelogram is a rhombusparallelogram is a rhombus if and if and
only ifonly if it has four congruent sidesit has four congruent sides
squaresquare – quadrilateral with four right – quadrilateral with four right angles and four congruent sidesangles and four congruent sides a a square is a parallelogramsquare is a parallelogram if and only if if and only if it is a rectangle with four congruent sidesit is a rectangle with four congruent sides it is a rhombus with a right angleit is a rhombus with a right angle
GeometryGeometry PentominosPentominos
* Won't fold into an open box* Won't fold into an open box
f p*
j n
i*
GeometryGeometry
PentominosPentominos * Won't fold into an open box* Won't fold into an open box
t u* v*
GeometryGeometry
PentominosPentominos
w x z
y
GeometryGeometry
Patterns with points, lines, and Patterns with points, lines, and regionsregions Where k is the number of lines or line Where k is the number of lines or line
segmentssegments
P = [k (k – 1)] / 2P = [k (k – 1)] / 2 Regions = lines + points + 1Regions = lines + points + 1 R = k + P + 1 = [k (k + 1)] / 2 + 1R = k + P + 1 = [k (k + 1)] / 2 + 1
1k
1n
xsintPo
GeometryGeometry
TangramsTangrams Flips, slides, and turnsFlips, slides, and turns CommunicationCommunication MapsMaps Conservation of AreaConservation of Area
PiagetPiaget If use all of the pieces to make a new If use all of the pieces to make a new
shape, both shapes have the same shape, both shapes have the same areaarea
GeometryGeometry PolyhedronPolyhedron
VerticesVertices EdgesEdges FacesFaces
Should be able to draw Should be able to draw ALLALL of the of the following:following: SphereSphere Prisms – Cube, Rectangular, Prisms – Cube, Rectangular,
TriangularTriangular CylinderCylinder ConeCone Pyramids – Triangular, SquarePyramids – Triangular, Square
MeasurementMeasurement
RectangleRectangle Perimeter Perimeter P = 2l + 2w, where l = length and P = 2l + 2w, where l = length and
w = widthw = width Example: l = 5 ft and w = 3 ftExample: l = 5 ft and w = 3 ft
PP rectangle rectangle == 2l + 2w2l + 2w
PP == 2(5 ft) + 2(3 ft)2(5 ft) + 2(3 ft) PP == 10 ft + 6 ft10 ft + 6 ft PP == 16 ft16 ft
3 ft
5 ft
MeasurementMeasurement
RectangleRectangle Area Area A = lw where l = length and w = A = lw where l = length and w =
widthwidth Example: l = 5 ft and w = 3 ftExample: l = 5 ft and w = 3 ft
AA rectangle rectangle = = lwlw
AA == (5 ft)(3 ft)(5 ft)(3 ft) A A == 15 ft15 ft22
3 ft
5 ft
MeasurementMeasurement
SquareSquare Perimeter Perimeter P = 4s, where s = length of a sideP = 4s, where s = length of a side Example: s = 3 ftExample: s = 3 ft
PP square square == 4s4s
PP == 4(3 ft)4(3 ft) PP == 12 ft12 ft
3 ft
MeasurementMeasurement
SquareSquare Area Area A = sA = s22 where s = length of a side where s = length of a side Example: s = 3 ftExample: s = 3 ft
AA square square = = ss22
AA == (3 ft)(3 ft)22
A A == 9 ft9 ft22
3 ft
MeasurementMeasurement
TriangleTriangle PerimeterPerimeter PP = a + b + c, where a, b, and c = a + b + c, where a, b, and c
are the lengths of the sides of the are the lengths of the sides of the triangletriangle
Example: a = 3 m; b = 4 m; c = 5 Example: a = 3 m; b = 4 m; c = 5 mm PP triangle triangle == a + b + ca + b + c PP == 3 m + 4 m + 5 m3 m + 4 m + 5 m PP == 12 m12 m
3 m
5 m4 m
MeasurementMeasurement
TriangleTriangle AreaArea A = ½ bh, where b is the base and A = ½ bh, where b is the base and
h is the height of the triangleh is the height of the triangle Example: b = 3 m; h = 4 mExample: b = 3 m; h = 4 m
AA triangle triangle == ½ bh½ bh AA == ½ (3 m) (4 m)½ (3 m) (4 m) AA == 6 m6 m22
3 m
5 m
4 m
MeasurementMeasurement
CircleCircle CircumferenceCircumference
CC circle circle = = d or C = 2d or C = 2r, where d = r, where d =
diameter and r = radiusdiameter and r = radius Example: r = 3 cmExample: r = 3 cm
CC circle circle == 2 2rr
CC == 2 2(3 cm)(3 cm) CC == 6 6 cm cm
3 cm
MeasurementMeasurement
CircleCircle AreaArea A = A = rr22, where r = radius, where r = radius Example: r = 3 cmExample: r = 3 cm
A A circlecircle == rr22
AA == (3 cm)(3 cm)22
AA == 9 9 cm cm22
3 cm
MeasurementMeasurement
Rectangular PrismRectangular Prism Surface AreaSurface Area: sum of the areas of all of the : sum of the areas of all of the
facesfaces ExampleExample: There are 4 lateral faces: 2 lateral : There are 4 lateral faces: 2 lateral
faces are 6 cm by 7 cm (Afaces are 6 cm by 7 cm (A11= wh) and 2 = wh) and 2 lateral faces are 5 cm by 7 cm (Alateral faces are 5 cm by 7 cm (A22 = lh). = lh). There are 2 bases 6 cm by 5 cm (AThere are 2 bases 6 cm by 5 cm (A33 = lw) = lw)
AA11 = (6 cm)(7 cm) = 42 cm = (6 cm)(7 cm) = 42 cm22
AA22 = (5 cm)(7 cm) = 35 cm = (5 cm)(7 cm) = 35 cm22
AA33 = (6 cm)(5 cm) = 30 cm = (6 cm)(5 cm) = 30 cm22
SA SA rectangular prismrectangular prism = 2wh + 2lh + 2lw = 2wh + 2lh + 2lw
SA = 2(42 cmSA = 2(42 cm22) + 2(35 cm) + 2(35 cm22) + 2(30 cm) + 2(30 cm22)) SA = 84 cmSA = 84 cm22 + 70 cm + 70 cm22 + 60 cm + 60 cm22
SA = 214 cmSA = 214 cm22
7 cm
6 cm
5 cm
MeasurementMeasurement
Rectangular Prism Rectangular Prism VolumeVolume: : V = lwh where l is length; w is width; V = lwh where l is length; w is width;
and h is heightand h is height ExampleExample: l = 6 cm; w = 5 cm; h = 7 cm: l = 6 cm; w = 5 cm; h = 7 cm
V V rectangular prismrectangular prism = Bh = lwh = Bh = lwh
VV == (6 cm)(5 cm)(7 cm)(6 cm)(5 cm)(7 cm) VV == 210 cm210 cm33
7 cm
6 cm
5 cm
MeasurementMeasurement
CubeCube Surface AreaSurface Area: sum of the areas of all : sum of the areas of all
6 congruent faces6 congruent faces ExampleExample: There are 6 faces: 5 cm by : There are 6 faces: 5 cm by
5 cm (A = s5 cm (A = s22))
SA SA cubecube = 6A = 6s = 6A = 6s22
SA = 6(5 cm)SA = 6(5 cm)22
SA = 6(25 cmSA = 6(25 cm22)) SA = 150 cmSA = 150 cm22
5 cm
MeasurementMeasurement
Cube Cube VolumeVolume: : V = sV = s33 where s is the length of a side where s is the length of a side ExampleExample: s = 5 cm: s = 5 cm
V V cubecube = Bh = s= Bh = s33
VV == (5 cm)(5 cm)33
VV == 125 cm125 cm33
5 cm
MeasurementMeasurement
Triangular PrismTriangular Prism Surface AreaSurface Area: sum of the areas of all of the : sum of the areas of all of the
facesfaces ExampleExample: There are 3 lateral faces: 6 m by : There are 3 lateral faces: 6 m by
7 m (A7 m (A11= bl). There are 2 bases: 6 m for the = bl). There are 2 bases: 6 m for the
base and 5 m for the height (2Abase and 5 m for the height (2A22 = bh). = bh). AA11 = (6 m)(7 m) = 42 m = (6 m)(7 m) = 42 m22
2A2A22 = (6 m)(5 m) = 30 m = (6 m)(5 m) = 30 m22
SA SA triangular prismtriangular prism = bh + 3bl = bh + 3bl
SA = 30 mSA = 30 m22 + 3(42 m + 3(42 m22)) SA = 30 mSA = 30 m22 + 126 m + 126 m22
SA = 156 mSA = 156 m22
7 m
6 m
5 m
MeasurementMeasurement
Triangular Prism Triangular Prism VolumeVolume: : V = ½ bhl where b is the base; h is V = ½ bhl where b is the base; h is
height of the triangle; and l is length of height of the triangle; and l is length of the prismthe prism
ExampleExample: b = 6 m; h = 5 m; l = 7 m: b = 6 m; h = 5 m; l = 7 m
V V triangular prismtriangular prism = Bh = ½ bhl = Bh = ½ bhl
VV == ½ (6 m)(5 m)(7 m)½ (6 m)(5 m)(7 m) VV == 105 m105 m33
7 m
6 m
5 m
MeasurementMeasurement
CylinderCylinder Surface AreaSurface Area: area of the circles plus : area of the circles plus
the area of the lateral facethe area of the lateral face ExampleExample: r = 3 ft; h = 12 ft: r = 3 ft; h = 12 ft
SA SA cylindercylinder= = 22rh +2rh +2rr22
SA = 2SA = 2 (3 ft)(12 ft) + 2 (3 ft)(12 ft) + 2 (3 ft) (3 ft)22 SA SA == 7272 ft ft22 + 2 + 2 (9 ft (9 ft22)) SASA == 7272 ft ft22 + 18 + 18 ft ft22
SASA = = 9090 ft ft22
3 ft
12 ft
MeasurementMeasurement
CylinderCylinder Volume of a CylinderVolume of a Cylinder: V = : V = rr22h h
where r is the radius of the base where r is the radius of the base (circle) and h is the height.(circle) and h is the height.
ExampleExample: r = 3 ft and h = 12 ft.: r = 3 ft and h = 12 ft. VV cylinder cylinder == Bh = Bh = rr22hh VV == (3 ft)(3 ft)22 (12 ft) (12 ft) VV == (9 ft(9 ft22)(12 ft))(12 ft) VV == 108108 ft ft33
3 ft
12 ft
MeasurementMeasurement
ConeCone Surface AreaSurface Area: area of the circle plus : area of the circle plus
the area of the lateral facethe area of the lateral face ExampleExample: r = 5 ft; t = 13 ft: r = 5 ft; t = 13 ft
SA SA conecone= = rt +rt +rr22
SA = SA = (5 ft)(13 ft) + (5 ft)(13 ft) + (5 ft) (5 ft)22 SA SA == 6565 ft ft22 + + (25 ft (25 ft22)) SASA == 6565 ft ft22 + 25 + 25 ft ft22
SASA = = 9090 ft ft22
5 ft
13 ft
12 ft
MeasurementMeasurement
ConeCone VolumeVolume: V = : V = rr22h/3 where r is the h/3 where r is the
radius of the base (circle) and h is the radius of the base (circle) and h is the height.height.
ExampleExample: r = 5 ft; h = 12 ft: r = 5 ft; h = 12 ft V V conecone= = rr22h/3h/3 V V = = [[(5 ft)(5 ft)22 12 ft ]/ 3 12 ft ]/ 3 V V == [(25[(25 ft ft22)(12 ft)]/3)(12 ft)]/3 VV == (25(25 ft ft22)(4 ft))(4 ft) VV = = 100100 ft ft33
5 ft
13 ft
12 ft
MeasurementMeasurement
SphereSphere Surface AreaSurface Area: 4: 4rr22 where r is the where r is the
radiusradius ExampleExample: r = 8 mm: r = 8 mm SA SA sphere sphere = = 44rr22
SASA = = 44(8 mm)(8 mm)22 SASA = = 44(64 mm(64 mm22)) SA SA = = 256256 mm mm22
8 mm
MeasurementMeasurement
SphereSphere Volume of a SphereVolume of a Sphere: V = (4/3): V = (4/3) r r33
where r is the radiuswhere r is the radius ExampleExample: r = 6 mm: r = 6 mm V V spheresphere == 44rr33/3/3 VV == [4[4 x (6 mm) x (6 mm)33]/3]/3 VV == [4[4 x 216 mm x 216 mm33]/3]/3 VV == [864[864 mm mm33]/3]/3 VV == 288288 mm mm33
6 mm
MeasurementMeasurement
Triangular PyramidTriangular Pyramid
Square PyramidSquare Pyramid
Test Taking TipsTest Taking Tips
Get a good nights rest before the Get a good nights rest before the examexam
Prepare materials for exam in Prepare materials for exam in advance (scratch paper, pencil, and advance (scratch paper, pencil, and calculator)calculator)
Read questions carefully and ask if Read questions carefully and ask if you have a question DURING the you have a question DURING the examexam
Remember: If you are prepared, you Remember: If you are prepared, you need not fearneed not fear