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I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask a student to draw any random triangle, find: (1) P(exactly 2 sides congruent) = (2) P(at least 2 angles congruent) = (3) P(2 different triangles with no sides congruent) = Agenda • Go over warm up. • Exploration 8.1--share answers • Review geometry concepts • Discuss attributes: Quadrilateral Hierarchy • Exploration 8.6. • More practice problems. • Assign homework. How did you group the polygons? • For kids… talk about attributes – Shape: # sides, special quadrilaterals – Convex or non-convex – (1 or 2) Pair of parallel sides – (1 or 2) Pair of congruent sides – (1 or 2) Pair of perpendicular sides – Nothing special about it. – Cannot do any proof or justification if kids can’t classify and describe similarities and differences. How do I use a protractor? I forgot! • Line up the center and line. 45˚ 135˚ 90˚ 0˚ 135˚ 180˚ 45˚ 180˚ 0˚ Can you… • Sketch a pair of angles whose intersection is: a. exactly two points? b. exactly three points? c. exactly four points? • If it is not possible to sketch one or more of these figures, explain why. Use Geoboards • On your geoboard, copy the given segment. • Then, create a parallel line and a perpendicular line if possible. Describe how you know your answer is correct. Exploration 8.6 • Do part 1 using the pattern blocks--make sure your justifications make sense. • You may not use a protractor for part 1. • Once your group agrees on the angle measures for each polygon, trace each onto your paper, and measure the angles with a protractor. • List 5 or more reasons for your protractor measures to be slightly “off”. Given m // n. • T or F: 7 and 4 are vertical. 7 6 • T or F: 1 4 3 4 5 2 • T or F: 2 3 1 n • T or F: m 7 + m 6 = m 1 m • T or F: m 7 = m 6 + m 5 • If m 5 = 35˚, find all the angles you can. • If m 5 = 35˚, label each angle as acute, right, obtuse. • Describe at least one reflex angle. More practice problems • Sketch four lines such that three are concurrent with each other and two are parallel to each other. True or False • If 2 distinct lines do not intersect, then they are parallel. • If 2 lines are parallel, then a single plane contains them. • If 2 lines intersect, then a single plane contains them. • If a line is perpendicular to a plane, then it is perpendicular to all lines in that plane. • If 3 lines are concurrent, then they are also coplanar. Pythagorean Theorem • Remember the Pythagorean Theorem? • a2 + b2 = c2 where c is the hypotenuse in a right triangle. • Use your geoboard to make a right triangle whose hypotenuse is the square root of 5. Solution… • If a2 + b2 = c2 is to be used, we want a right triangle whose hypotenuse is square root of 5. 5 • So, a 2 + b2 = 5. • If you do not use a geoboard, there are lots of answers. Van Hiele levels • Formal study of geometry in high school requires that students are familiar and comfortable with many different aspects of elementary and middle school geometry. • Visualization, analysis, informal deduction are all necessary prior to high school geometry. • This means students need to categorize, classify, compare and contrast, and make predictions about figures based upon their attributes. Attributes • Early childhood: – Size: big--little – Thickness: thin--thick – Colors: red-yellow-blue-etc. – Shape: triangle, rectangle, square, circle, etc. – Texture: rough--smooth Why do we need this??? READING!! Talk about polygons What is a polygon? Polygon • A simple, closed, plane figure composed of at least 3 line segments. • Why are each of the figures below not polygons? Convex vs. Non-convex • Both are hexagons. One is convex. One is non-convex. • Look at diagonals: segments connecting non-consecutive vertices. • Boundary, interior, exterior Names of polygons! • Triangle • Quadrilateral • Pentagon • Hexagon • Heptagon (Septagon) • Octagon • Nonagon (Ennagon) • Decagon • 11-gon • Dodecagon Triangle Attributes • Sides: equilateral, isosceles, scalene • Angles: acute, obtuse, right. • Can you draw an acute, scalene triangle? • Can you draw an obtuse, isosceles triangle? • Can you draw an obtuse equilateral triangle? One Attribute of Triangles • The Triangle Angle Sum is 180˚. • This is a theorem because it can be proven. • Exploration 8.10--do Part 1 #1 - 3 and Part 2. Diagonals, and interior angle sum • Triangle • Quadrilateral • Pentagon • Hexagon • Heptagon (Septagon) • Octagon • Nonagon (Ennagon) • Decagon • 11-gon • Dodecagon Congruence vs. Similarity Two figures are congruent if they are exactly the same size and shape. Think: If I can lay one on top of the other, and it fits perfectly, then they are congruent. Question: Are these two figures congruent? Similar: Same shape, but maybe different size. Quadrilateral Hierarchy Quadrilaterals • Look at Exploration 8.13. Do 2a, 3a - f. • Use these categories for 2a: – At least 1 right angle – 4 right angles – 1 pair parallel sides – 2 pair parallel sides – 1 pair congruent sides – 2 pair congruent sides – Non-convex Exploration 8.13 • Let’s do f together: • In the innermost region, all shapes have 4 equal sides. • In the middle region, all shapes have 2 pairs of equal sides. Note that if a figure has 4 equal sides, then it also has 2 pairs of equal sides. But the converse is not true. • In the outermost region, figures have a pair of equal sides. In the universe are the figures with no equal sides. Warm Up • Use your geoboard to make: • 1. A hexagon with exactly 2 right angles • 2. A hexagon with exactly 4 right angles. • 3. A hexagon with exactly 5 right angles. • Can you make different hexagons for each case? Warm-up part 2 • 1. Can you make a non-convex quadrilateral? • 2. Can you make a non-simple closed curve? • 3. Can you make a non-convex pentagon with 3 collinear vertices? Warm-up Part 3 • Given the diagram at A the right, name at least 6 different B F polygons using their vertices. G C D E Agenda • Go over warm up. • Complete discussion of 2-Dimensional Geometry • Polyhedra attributes • Exploration 8.15 and 8.17 • Examining the Regular Polyhedra • 3 Dimensions require 3 views • Assign Homework Quadrilateral Hierarchy • Do the worksheet. Some formulas--know how they work. • Number of degrees in a polygon: Take 1 point and draw all the diagonals. Triangles are formed. Each triangle has 180˚. So, (n - 2)•180˚ is the number of degrees in a polygon. • If the polygon is regular, then each angle is (n - 2) • 180/n. Some formulas--know how they work. • Distance formula: This is related to the Pythagorean Theorem. • If a2 + b2 = c2, then c = a2 + b2 . • Now, if a is the distance from left to right, and b is the distance from top to bottom, then the distance formula makes sense. Some formulas--know how they work. • The distance formula is (x1, y1)• A (x2, y2) • B (x2 - x1)2 + (y2 - y1)2 Some formulas--know how they work. • Midpoint formula: If the midpoint is half way between two points, then we are finding the average of the left and right, and the average of the up and down. • Midpoint: (x2 + x1) , (y2 + y1) 2 2 Some formulas--know how they work. • Slope of a line: change in left and right compared to the change in up and down. • m = (y2 - y1) (x2 - x1) Discuss answers to Explorations 8.11 and 8.13 • 8.11 • 1a - c • 3a: pair 1: same area, not congruent; pair 2: different area, not congruent; • Pair 3: congruent--entire figure is rotated 180˚. More practice problems • Think of an analog clock. • A. How many times a day will the minute hand be directly on top of the hour hand? • B. What times could it be when the two hands make a 90˚ angle? • C. What angle do the hands make at 7:00? 3:30? 2:06?

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