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2:1:33:Polygons, Quadrilaterals
TITLE OF LESSON
Geometry Unit 1 Lesson 33 – Polygons, Quadrilaterals
Prove it! What’s on the outside? What’s on the inside? Of Geometry
TIME ESTIMATE FOR THIS LESSON
One class period
ALIGNMENT WITH STANDARDS
California – Geometry
Introductory lesson necessary for:
8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral
area, and surface area of common geometric figures.
10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi,
parallelograms, and trapezoids.
11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common
geometric figures and solids.
12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to
classify figures and solve problems.
13.0 Students prove relationships between angles in polygons by using properties of complementary,
supplementary, vertical, and exterior angles.
MATERIALS
None
LESSON OBJECTIVES
To introduce
• The general concept of a
polygon
• To look at quadrilaterals in
particular
• Polygons
• Convex polygons
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Concave polygons
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
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Octagon
Nonagon
Decagon
FOCUS AND MOTIVATE STUDENTS
1) Homework Check – Pass back binders. Pass back graded work and have students place in the appropriate
sections of their binders.
2) Agenda – Have students copy the agenda.
3) Homework Review – (5 minutes) Review homework from Lesson 32. The homework was to solve the
following:
If two parallel lines are cut by a transversal and m∠2 is 3x + 5 and m∠8 is x + 10 what is x?
What is the size of angle 2?
What is the size of angle 8?
If m∠1 is 3x + 50 and m∠7 is x + 90 what is x?
What is the measure of angle 1?
What is the measure of angle 7?
Have one student answer each part. How did you get the answer? Demonstrate on the board. Answer questions
from students who did not solve or came up with a different answer. Collect homework.
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2:1:33:Polygons, Quadrilaterals
ACTIVITIES – INDIVIDUAL AND GROUP
1.
Definitions: Polygons – (20 minutes) Have the students add the following definition to the terms and definitions
section and copy the notes that follow into the notes section. Write the following on the board:
A closed shape or figure with three or more sides in a plane is called a polygon.
Poly – (from Greek) is a prefix meaning more than one. So, we know that a polygon has sides, but it can have
any number of sides, as long as it has more than one. But we already know what we call a shape that has one
side – a line! An example of a polygon that we have seen is a triangle. (Anyone know what tri- means? It’s a
prefix –from Latin and Greek – that means three, so a polygon with three sides is called a triangle—which
actually means it has three angles, but that works too.) What do you think it means that the figure is closed? (All
the line segments are connected.)
Figure 33.1Closed Figure
Figure 33.2 Not Closed
Ask for a volunteer to demonstrate a closed figure on the board or in the classroom. Ask for a volunteer to
demonstrate a figure that is not closed.
What does it mean that it is in a plane? (That means it is two-dimensional.)
Lets look at some definitions:
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A triangle is a three sided polygon
A Quadrilateral is a four sided polygon (quad- means 4)
A Pentagon is a five sided polygon (pent- or penta- means 5)
A Hexagon is a six sided polygon (hex- means 6)
A Heptagon is a seven sided polygon (hept- means 7)
An Octagon is an eight sided polygon (oct- means 8)
A Nonagon is a nine sided polygon (non- means nine)
A Decagon is a ten sided polygon (deca- means 10)
After each one has been defined, ask a student to draw an example on the board.
Starting with the triangle and ending with the decagon (if you have at least 10 students), have the students form
each of the figures by holding pieces of string, or stretching one piece of string, between them to form the sides
of the figure in question. For instance the triangle is formed by three students each being one vertex. As we go
from the triangle to the decagon we need to add one more student for each new side. So for each side increase,
the shape adds one vertex too. This implies that a 10-sided figure has 10 vertices. How many vertices does a
259-sided figure have? (259.)
2.
Definitions: Concave and Convex Polygons – (10 Minutes) Have the students open their binders to the terms
and definitions section and copy the following definitions. Write the following definition on the board:
A convex polygon is a polygon in which the measure of each interior angle is less than 180º.
A concave (or non-convex polygon) is a polygon where at least one of the interior angles is more than 180º.
Ask for a volunteer to draw a five-sided convex polygon (pentagon) on the board. Then, ask for a volunteer to
draw a five-sided concave polygon (pentagon) on the board. Next, have 5 students hold string to form a fivesided convex polygon. Then, have 5 students hold string to form a five-sided concave polygon.
2
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Prove It
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2:1:33:Polygons, Quadrilaterals
Figure 33.3 Convex Pentagon
Figure 33.4 Concave Pentagon
Hint: To remember
which is which, the
concave polygon forms
a cave outside!
Ask student what they see as differences between concave and convex. They may come up with a number of
things. Ask them to draw one of each in their binders, then to draw interior line segments. Do they notice
anything as they do this? Point out, if no one brings it up, that on the inside of a convex polygon you can choose
any two points, draw a line segment between them, and the line segment will always be in the interior of the
polygon. With the concave polygon this is not the case. Can anyone show this? Have a student demonstrate at
the board.
Figure 33.5 Concave Pentagon
Note that if a line is drawn between the two points represented by the small squares, the line will go outside the
concave polygon.
3.
Quadrilateral – (10 minutes) Lets us look at one particular type of polygon—the quadrilateral. The quadrilateral
is a four-sided polygon. Can anyone draw a four-sided figure? Can anyone draw a four-sided figure where all of
the sides are of equal size? What do we call this? Can anyone draw a four-sided figure where the sides are not
equal? Can anyone draw a four sided figure where all of the angles are 90º? Can anyone draw a four-sided
figure where the top and bottom are parallel with each other and the sides are parallel with each other? Is it
possible to draw more than one figure like this and make the figures NOT congruent?
4.
Definitions: Parallelograms – (3 minutes) Write the following definitions on the board and have the students
copy them into the terms and definition section of their binders.
3
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Prove It
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2:1:33:Polygons, Quadrilaterals
A parallelogram is a quadrilateral that has two pairs of parallel sides.
A square is a parallelogram with all the angles equal to 90º and all the sides equal to each other.
A rectangle is a parallelogram with all the angles equal to 90º. (This includes the square but also includes
parallelograms with sides not all equal to each other.)
5.
Homework – (2 minutes) Have students copy their homework. Answer any questions.
HOMEWORK
1) Look for examples in the world of the following. Come up with ten examples total. The more diverse your
examples the better! Finding one of each will be the best. You can draw the example, cut the example out of a
newspaper or magazine, or write a description of the example.
• Polygons
• Quadrilateral
• Octagon
• Convex polygons
• Pentagon
• Nonagon
• Concave polygons
• Hexagon
• Decagon
• Triangle
• Heptagon
2) Create outline for final project: proof scramble. Due Lesson 35.
GROUP ROLES
None
DOCUMENTATION FOR PORTFOLIO
None
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© 2001 ESubjects Inc. All rights reserved.