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Page 1: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

I have 7 triangles, 1 each of:I have 7 triangles, 1 each of:acute scalene, acute isosceles, equilateral, acute scalene, acute isosceles, equilateral,

right scalene, right isosceles, right scalene, right isosceles, obtuse scalene, obtuse isosceles.obtuse scalene, obtuse isosceles.

If I ask a student to draw any random triangle, If I ask a student to draw any random triangle, find:find:

(1)(1) P(exactly 2 sides congruent) =P(exactly 2 sides congruent) =

(2)(2) P(at least 2 angles congruent) =P(at least 2 angles congruent) =

(3)(3) P(2 different triangles with no sides P(2 different triangles with no sides congruent) =congruent) =

Page 2: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 3: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 4: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

How do I use a protractor? I forgot!

• Line up the center and line.

0˚180˚

180˚ 0˚

135˚ 45˚

45˚ 135˚ 90˚

Page 5: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 6: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 7: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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”.

Page 8: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Given m // n.• T or F: 7 and 4

are vertical.• T or F: 1 4• T or F: 2 3• T or F: m 7 + m 6 = m 1• 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.

7 65

43

21

mn

Page 9: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

More practice problems• Sketch four lines such that three are

concurrent with each other and two are parallel to each other.

Page 10: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 11: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 12: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Solution…• If a2 + b2 = c2 is to be used, we want a

right triangle whose hypotenuse is square root of 5.

• So, a2 + b2 = 5.• If you do not use

a geoboard, there are lots of answers.

5

Page 13: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 14: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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!!

Page 15: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Talk about polygonsWhat is a polygon?

Page 16: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Polygon• A simple, closed, plane figure

composed of at least 3 line segments.

• Why are each of the figures below not polygons?

Page 17: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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

Page 18: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Names of polygons!• Triangle• Quadrilateral• Pentagon• Hexagon• Heptagon (Septagon)• Octagon• Nonagon (Ennagon)• Decagon• 11-gon• Dodecagon

Page 19: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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?

Page 20: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 21: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Diagonals, and interior angle sum

• Triangle• Quadrilateral• Pentagon• Hexagon• Heptagon (Septagon)• Octagon• Nonagon (Ennagon)• Decagon• 11-gon• Dodecagon

Page 22: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Congruence vs. SimilarityTwo 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.

Page 23: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Quadrilateral Hierarchy

Page 24: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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

Page 25: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 26: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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?

Page 27: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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?

Page 28: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Warm-up Part 3• Given the diagram at

the right, name at least 6 different polygons using their vertices.

E

G

F

DC

B

A

Page 29: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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

Page 30: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Quadrilateral Hierarchy• Do the worksheet.

Page 31: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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.

Page 32: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Some formulas--know how they work.

• Distance formula: This is related to the Pythagorean Theorem.

• If aa22 + b + b22 = c = c22, then c = a, then c = a22 + b + b22 . .

• Now, if a is the distance from left to right, and Now, if a is the distance from left to right, and b is the distance from top to bottom, then the b is the distance from top to bottom, then the distance formula makes sense.distance formula makes sense.

Page 33: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

Some formulas--know how they work.

• The distance formula is • A

• B

(x1, y1)

(x2, y2)(x2 - x1)2 + (y2 - y1)2

Page 34: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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

Page 35: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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)

Page 36: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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˚.

Page 37: I have 7 triangles, 1 each of: acute scalene, acute isosceles, equilateral, right scalene, right isosceles, obtuse scalene, obtuse isosceles. If I ask.

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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