MATH 2160 3 rd Exam Review

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