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MATH 2160 3 rd Exam Review

This resource provides a detailed review of critical geometric concepts, measurement techniques, and problem-solving strategies based on Polya’s four steps. It includes understanding and defining problems, devising plans, executing those plans, and reflecting on the process. Topics cover angles, polygons, special types of triangles and quadrilaterals, properties of circles, and various measurement formulas, including perimeter and area calculations. Explore strategies such as drawing diagrams, looking for patterns, and utilizing algebraic models to enhance problem-solving skills in geometry.

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MATH 2160 3 rd Exam Review

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  1. MATH 2160 3rd Exam Review Geometry and Measurement

  2. Problem Solving – Polya’s 4 Steps • Understand the problem • What does this mean? • How do you understand? • Devise a plan • What goes into this step? • Why is it important? • Carry out the plan • What happens here? • What belongs in this step? • Look back • What does this step imply? • How do you show you did this?

  3. Problem Solving • Polya’s 4 Steps • Understand the problem • Devise a plan • Carry out the problem • Look back • Which step is most important? • Why is the order important? • How has learning problem solving skills helped you in this or another course?

  4. Problem Solving Strategies • Make a chart • Make a table • Draw a picture • Draw a diagram • Guess, test, and revise • Form an algebraic model • Look for a pattern • Try a simpler version of the problem • Work backward • Restate the problem differently • Eliminate impossible situations • Use reasoning

  5. Geometry • Angles and congruency • Congruent– same size, same shape • Degree measure – real number between 0 and 360 degrees that defines the amount of rotation or size of an angle • Sum of the interior angles of any polygon: (n – 2)180o where n is the number of sides in the polygon

  6. Geometry • Special angles • right angle – 90 • acute angle – 0< angle < 90 • obtuse angle – 90< angle < 180 • Sum of the angles • Triangle = 180o • Quadrilateral = 360o • Pentagon = 540o • Etc.

  7. Geometry • Circles • circle – special simple closed curve where all points in the curve are equidistant from a given point in the same plane – NOTE: Circles are NOT polygons! • center – middle point of the circle • diameter – is a chord that passes through the center of the circle • radius – line segment connecting the center of the circle to any point on the circle

  8. Geometry • Polygons – made up of line segments • Triangles – 3-sided polygons • Quadrilaterals – 4-sided polygons • n - gons – the whole number n represents the number of sides for the polygon: a triangle is a 3-gon; a square is a 4-gon; and so on • Regular Polygons – polygon where the all the line segments and all of the angles are congruent

  9. Geometry • Triangles • Union of three line segments formed by three distinct non-collinear points • vertices – intersection points of line segments forming the angles of the polygon • sides – the line segments forming the polygon • height – line segment from a vertex of a triangle to a line containing the side of the triangle opposite the vertex

  10. Geometry • Triangles • equilateral – all sides and angles congruent • isosceles – at least one pair of congruent sides and angles • scalene – no congruent sides or angles • right – one right angle • acute – all angles acute • obtuse – one obtuse angle

  11. Geometry • Quadrilaterals • parallelogram – quadrilateral with two pairs of parallel sides • opposite sides are parallel • opposite sides are congruent • rectangle – quadrilateral with four right angles • a parallelogram is a rectangle if and only if • it has at least one right angle • trapezoid – exactly one pair of opposite sides parallel, but not congruent

  12. Geometry • Quadrilaterals • rhombus – quadrilateral with four congruent sides • a parallelogram is a rhombus if and only if • it has four congruent sides • square – quadrilateral with four right angles and four congruent sides • a square is a parallelogram if and only if • it is a rectangle with four congruent sides • it is a rhombus with a right angle

  13. Geometry • Pentominos • * Won't fold into an open box

  14. Geometry • Pentominos • * Won't fold into an open box

  15. Geometry • Pentominos

  16. Geometry • Patterns with points, lines, and regions • Where k is the number of lines or line segments • P = [k (k – 1)] / 2 • Regions = lines + points + 1 • R = k + P + 1 = [k (k + 1)] / 2 + 1

  17. Geometry • Tangrams • Flips, slides, and turns • Communication • Maps • Conservation of Area • Piaget • If use all of the pieces to make a new shape, both shapes have the same area

  18. Geometry • Polyhedron • Vertices • Edges • Faces • Should be able to draw ALL of the following: • Sphere • Prisms – Cube, Rectangular, Triangular • Cylinder • Cone • Pyramids – Triangular, Square

  19. 5 ft 3 ft Measurement • Rectangle • Perimeter • P = 2l + 2w, where l = length and w = width • Example: l = 5 ft and w = 3 ft • P rectangle = 2l + 2w • P = 2(5 ft) + 2(3 ft) • P = 10 ft + 6 ft • P = 16 ft

  20. 5 ft 3 ft Measurement • Rectangle • Area • A = lw where l = length and w = width • Example: l = 5 ft and w = 3 ft • A rectangle = lw • A = (5 ft)(3 ft) • A = 15 ft2

  21. 3 ft Measurement • Square • Perimeter • P = 4s, where s = length of a side • Example: s = 3 ft • P square = 4s • P = 4(3 ft) • P = 12 ft

  22. 3 ft Measurement • Square • Area • A = s2 where s = length of a side • Example: s = 3 ft • A square = s2 • A = (3 ft)2 • A = 9 ft2

  23. 4 m 5 m 3 m Measurement • Triangle • Perimeter • P= a + b + c, where a, b, and c are the lengths of the sides of the triangle • Example: a = 3 m; b = 4 m; c = 5 m • P triangle = a + b + c • P = 3 m + 4 m + 5 m • P = 12 m

  24. 4 m 5 m 3 m Measurement • Triangle • Area • A = ½ bh, where b is the base and h is the height of the triangle • Example: b = 3 m; h = 4 m • A triangle = ½ bh • A = ½ (3 m) (4 m) • A = 6 m2

  25. 3 cm Measurement • Circle • Circumference • C circle = d or C = 2r, where d = diameter and r = radius • Example: r = 3 cm • C circle = 2r • C = 2(3 cm) • C = 6 cm

  26. 3 cm Measurement • Circle • Area • A = r2, where r = radius • Example: r = 3 cm • A circle = r2 • A = (3 cm)2 • A = 9 cm2

  27. 7 cm 5 cm 6 cm Measurement • Rectangular Prism • Surface Area: sum of the areas of all of the faces • Example: There are 4 lateral faces: 2 lateral faces are 6 cm by 7 cm (A1= wh) and 2 lateral faces are 5 cm by 7 cm (A2 = lh). There are 2 bases 6 cm by 5 cm (A3 = lw) • A1 = (6 cm)(7 cm) = 42 cm2 • A2 = (5 cm)(7 cm) = 35 cm2 • A3 = (6 cm)(5 cm) = 30 cm2 • SA rectangular prism = 2wh + 2lh + 2lw • SA = 2(42 cm2) + 2(35 cm2) + 2(30 cm2) • SA = 84 cm2 + 70 cm2 + 60 cm2 • SA = 214 cm2

  28. 7 cm 5 cm 6 cm Measurement • Rectangular Prism • Volume: • V = lwh where l is length; w is width; and h is height • Example: l = 6 cm; w = 5 cm; h = 7 cm • V rectangular prism = Bh = lwh • V = (6 cm)(5 cm)(7 cm) • V = 210 cm3

  29. 5 cm Measurement • Cube • Surface Area: sum of the areas of all 6 congruent faces • Example: There are 6 faces: 5 cm by 5 cm (A = s2) • SA cube = 6A = 6s2 • SA = 6(5 cm)2 • SA = 6(25 cm2) • SA = 150 cm2

  30. 5 cm Measurement • Cube • Volume: • V = s3 where s is the length of a side • Example: s = 5 cm • V cube = Bh = s3 • V = (5 cm)3 • V = 125 cm3

  31. 7 m 5 m 6 m Measurement • Triangular Prism • Surface Area: sum of the areas of all of the faces • Example: There are 3 lateral faces: 6 m by 7 m (A1= bl). There are 2 bases: 6 m for the base and 5 m for the height (2A2 = bh). • A1 = (6 m)(7 m) = 42 m2 • 2A2 = (6 m)(5 m) = 30 m2 • SA triangular prism = bh + 3bl • SA = 30 m2 + 3(42 m2) • SA = 30 m2 + 126 m2 • SA = 156 m2

  32. 7 m 5 m 6 m Measurement • Triangular Prism • Volume: • V = ½ bhl where b is the base; h is height of the triangle; and l is length of the prism • Example: b = 6 m; h = 5 m; l = 7 m • V triangular prism = Bh = ½ bhl • V = ½ (6 m)(5 m)(7 m) • V = 105 m3

  33. 3 ft 12 ft Measurement • Cylinder • Surface Area: area of the circles plus the area of the lateral face • Example: r = 3 ft; h = 12 ft • SA cylinder= 2rh +2r2 • SA = 2 (3 ft)(12 ft) + 2 (3 ft)2 • SA = 72 ft2 + 2 (9 ft2) • SA = 72 ft2 + 18 ft2 • SA = 90 ft2

  34. 3 ft 12 ft Measurement • Cylinder • Volume of a Cylinder: V = r2h where r is the radius of the base (circle) and h is the height. • Example: r = 3 ft and h = 12 ft. • V cylinder = Bh = r2h • V = (3 ft)2 (12 ft) • V = (9 ft2)(12 ft) • V = 108 ft3

  35. 13 ft 12 ft 5 ft Measurement • Cone • Surface Area: area of the circle plus the area of the lateral face • Example: r = 5 ft; t = 13 ft • SA cone= rt +r2 • SA = (5 ft)(13 ft) +  (5 ft)2 • SA = 65 ft2 +  (25 ft2) • SA = 65 ft2 + 25 ft2 • SA = 90 ft2

  36. 13 ft 12 ft 5 ft Measurement • Cone • Volume: V = r2h/3 where r is the radius of the base (circle) and h is the height. • Example: r = 5 ft; h = 12 ft • V cone= r2h/3 • V = [(5 ft)2 12 ft ]/ 3 • V = [(25 ft2)(12 ft)]/3 • V = (25 ft2)(4 ft) • V = 100 ft3

  37. 8 mm Measurement • Sphere • Surface Area: 4r2 where r is the radius • Example: r = 8 mm • SA sphere = 4r2 • SA = 4(8 mm)2 • SA = 4(64 mm2) • SA = 256 mm2

  38. 6 mm Measurement • Sphere • Volume of a Sphere: V = (4/3) r3 where r is the radius • Example: r = 6 mm • V sphere = 4r3/3 • V = [4 x (6 mm)3]/3 • V = [4 x 216 mm3]/3 • V = [864 mm3]/3 • V = 288 mm3

  39. Measurement • Triangular Pyramid • Square Pyramid

  40. Test Taking Tips • Get a good nights rest before the exam • Prepare materials for exam in advance (scratch paper, pencil, and calculator) • Read questions carefully and ask if you have a question DURING the exam • Remember: If you are prepared, you need not fear

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