Transcript Slide 1

Four rules or facts about probability:
1.
2.
3.
4.
The probability of an event that cannot
occur is 0.
The probability of an event that must
occur is 1.
Every probability is a number between 0
and 1 inclusive.
The sum of the probabilities of all
possible outcomes of an experiment is 1.
How will you dress for the weather?
0
.5
Certain
Likely to
occur
50-50 chance
of occurring
Not Likely to
occur
Impossible
What are the chances?
1
Homework
 Textbook
734 #47, 48, 59, 60
Geometric Conclusions
 Determine
if each statement is a
SOMETIMES, ALWAYS, or NEVER
Who Am I?
 My
total angle measure is 360˚.
 All of my sides are different lengths.
 I have no right angles.
Who Am I?
I have no right angles
My total angle measure is not 360˚
I have fewer than 3 congruent sides.
Who Am I?
My total angle measure is 360˚ or less.
 I have at least one right angle.
 I have more than one pair of congruent sides.

Who Am I?

I have at least one pair of parallel sides.
 My total angle measure is 360˚.
 No side is perpendicular to any other side.
Types of curves
simple curves: A curve is simple if it does
not cross itself.
Types of Curves
closed curves: a closed curve is a curve
with no endpoints and which completely
encloses an area
Types of Curves
convex curve: If a plane closed curve be
such that a straight line can cut it in at
most two points, it is called a convex
curve.
Convex
Curves
Not Convex Curves
Triangle Discoveries
Work with a part to see what discoveries can
you make about triangles.
Types of Triangles
Classified by Angles
 Equiangular: all angles congruent
 Acute: all angles acute
 Obtuse: one obtuse angle
 Right: one right angle
Classified by Sides
 Equilateral: all sides congruent
 Isosceles: at least two sides congruent
 Scalene: no sides congruent
Triangles
Scalene
(No sides
equal)
Isosceles
(at least two
sides equal)
Equilateral
(all sides
equal)
What’s possible?
Equilateral
Equiangular
Scalene
NO
Acute
Right
NO
Obtuse
Isosceles
NO
Homework
Textbook pages 444-446
#9-12, #23-26, #49-52
Pythagorean Theorem
c2
a2
b2
a2 + b2 = c2
Pythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Pythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Pythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Pythagorean Theorem
http://regentsprep.org/Regents/Math/fpyth/PracPyth.htm
Testing for acute, obtuse, right
2 + b2 = c2
a
Pythagorean theorem says:
What happens if
a2 + b2 > c2
or
a2 + b2 < c2
Testing for acute, obtuse, right
Right triangle: a2 + b2 = c2
Acute triangle: a2 + b2 > c2
Obtuse triangle: a2 + b2 < c2
Types of Angles
Website
www.mrperezonlinemathtutor.com
 Complementary
 Supplementary
 Adjacent
 Vertical
Transversals
Let’s check the homework!
Textbook pages 444-446
#9-12, #23-26, #49-52
What is the value of x?
2x + 5
3x + 10
Angles in pattern blocks
Diagonals
Joining two nonadjacent vertices of a
polygon
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Trapezoid
Parallelogram
Rhombus
Rectangle
Square
Kite
Are diagonals
perpendicular?
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Are diagonals
perpendicular?
Trapezoid
maybe
Parallelogram
Rhombus
Rectangle
Square
Kite
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Are diagonals
perpendicular?
Trapezoid
maybe
Parallelogram
maybe
Rhombus
Rectangle
Square
Kite
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Are diagonals
perpendicular?
Trapezoid
maybe
Parallelogram
maybe
Rhombus
yes
Rectangle
Square
Kite
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Are diagonals
perpendicular?
Trapezoid
maybe
Parallelogram
maybe
Rhombus
yes
Rectangle
maybe
Square
Kite
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Are diagonals
perpendicular?
Trapezoid
maybe
Parallelogram
maybe
Rhombus
yes
Rectangle
maybe
Square
yes
Kite
For which shapes will the diagonals
always be perpendicular?
Type of
Quadrilateral
Are diagonals
perpendicular?
Trapezoid
maybe
Parallelogram
maybe
Rhombus
yes
Rectangle
maybe
Square
yes
Kite
yes
If m<A = 140°, what is the m<B, m<C and
m<D?
A
B
C
If m<D = 75°, what is the m<B, m<C and
m<A?
B
C
A
Sum of the angles of a polygon
Use a minimum of five polygon pieces to create a
5-sided, 6-sided, 7 sided, 8-sided, 9-sided, 10sided, 11-sided, or 12-sided figure. Trace on
triangle grid paper, cut out, mark and measure the
total angles in the figure.
2
1
3
4
9
http://www.arcytech.org/java/patterns/patterns_j.shtml
8
2
1
5
7
3
6
4
7
5
Sum of the angles of a polygon
Polygon
#
sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
Nth
N
Total degrees
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
Octagon
8
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
1440
Undecagon
11
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
1440
Undecagon
11
1620
Dodecagon
12
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
1440
Undecagon
11
1620
Dodecagon
12
1800
Triskaidecagon
13
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
1440
Undecagon
11
1620
Dodecagon
12
1800
Triskaidecagon
13
1980
nth
n
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
1440
Undecagon
11
1620
Dodecagon
12
1800
Triskaidecagon
13
1980
nth
n
?
What patterns
do you see?
Sum of the angles of a polygon
Polygon
#
sides
Total degrees
Triangle
3
180
Quadrilateral
4
360
Pentagon
5
540
Hexagon
6
720
Heptagon
7
900
Octagon
8
1080
Nonagon
9
1260
Decagon
10
1440
Undecagon
11
1620
Dodecagon
12
1800
Triskaidecagon
13
1980
nth
n
180(n-2)
What patterns
do you see?
Total degree of angles in polygon
Area Ideas

Triangles
 Parallelograms
 Trapezoids
 Irregular figures
Homework
TB445 #13, 17, 18, 33, 34
Area Formulas: Triangle
http://illuminations.nctm.org/LessonDetail.aspx?ID=L577
Area Formulas: Triangle
1. Using a ruler, draw a
diagonal (from one corner
to the opposite corner) on
shapes A, B, and C.
2. Along the top edge of shape
D, mark a point that is not
a vertex. Using a ruler,
draw a line from each
bottom corner to the point
you marked. (Three
triangles should be
formed.)
3. Cut out the shapes. Then,
divide A, B, and C into two
parts by cutting along the
diagonal, and divide D into
three parts by cutting along
the lines you drew.
4. How do the areas of the
resulting shapes compare
to the area of the original
shape?
Area Formulas: Triangle
Area Formulas: Triangle
Area Formulas: Trapezoids
http://illuminations.nctm.org/LessonDetail.aspx?ID=L580
Area Formulas: Trapezoids
Do you have suggestions for finding area?
What other shapes could you use to help you?
Are there any other shapes for which you already know how to find the area?
Area Formulas: Trapezoids
18cm
15 cm
13 cm
11cm
24 cm
Connect Math Shapes Set
http://phcatalog.pearson.com/component.cfm?site_id=6&discipline_id=80
6&subarea_id=1316&program_id=23245&product_id=3502
CMP Cuisenaire® Connected Math
Shapes Set (1 set of 206)
ISBN-10: 157232368X
ISBN-13: 9781572323681
Price: $29.35
Area Formulas: Trapezoids
A = ½h(b1 + b2)
When triangles are removed from each corner and rotated, a rectangle will be formed. It’s important for
kids to see that the midline is equal to the average of the bases. This is the basis for the proof—the
midline is equal to the base of the newly formed rectangle, and the midline can be expressed as
½(b1 + b2), so the proof falls immediately into place. To be sure that students see this relationship, ask,
"How is the midline related to the two bases?" Students might suggest that the length of the midline is
"exactly between" the lengths of the two bases; more precisely, some students may indicate that it is
equal to the average of the two bases, giving the necessary expression.
Remind students that the area of a rectangle is base × height; for the rectangle formed from the
original trapezoid, the base is ½(b1 + b2) and the height is h, so the area of the rectangle (and,
consequently, of the trapezoid) is A = ½h(b1 + b2). This is the traditional formula for finding the area of
the trapezoid.
Area Formulas: Trapezoids
18cm
15 cm
13 cm
11cm
24 cm
Area Formulas: Trapezoids
Websites:
http://argyll.epsb.ca/jreed/math9/strand3/tra
pezoid_area_per.htm
Parallelograms
A=
Length x width
http://illuminations.nctm.org/LessonDetail.aspx?ID=L578
Area of Parallelogram
Can you estimate the area of Tennessee?
Area of irregular figure?
Find the area of the irregular figure.
Area of irregular figure?
Area of irregular figure?
Circles
Area = πr2
Circumference = 2 πr
or
Circumference = π d
Circles
Otis is drawing a circle with a 4 inch radius.
He wants to double the radius. How will
this affect the area of the circle?
Circles
Su is selling 12 inch diameter pumpkin pies
for $6.50. How should she adjust her
price (if she wants to be fair) when she
reduces her pies to a 10 inch diameter?
Circles
Javier’s bicycle tire has a 12 inch radius.
How far will he travel. . .
. . . in one rotation of the tire?
. . . in 10 rotations of the tire?
1
V 
Bh
3
1
V 
lwh
3
Fact:
m<1 = 30˚ and m<7 = 100 ˚
Find:
5
6
m<2
8
m<3
7
m<4
m<5
m<6
2
1
4
3
9
10
11
m<8
12
m<9
m<10
m<11
m<12
Fact:
m<1 = 30˚ and m<7 = 100 ˚
80 ˚6
100 ˚
5
7
100 ˚
30˚
150 ˚
2
1
3
4 30˚
150 ˚
8
80 ˚
50 ˚ 9
130 ˚
10
11 12
50 ˚
130 ˚
m<1 + m<5 + m<12 = _______
m<2 + m<8 + m<11 = _______
The sum of which 3 angles will
equal 180˚?
2
3
4
9
11
10
12
8
7
1
5
6
The sum of which 3 angles will
equal 360˚?
2
3
4
9
11
10
12
8
7
1
5
6
Polyhedron
A polyhedron is simply a three-dimensional
solid which consists of a collection of
polygons, joined at their edges.
A polyhedron is said to be regular if its faces
and vertex figures are regular polygons.
Platonic Solids
What do these polyhedra have in
common?
Name that figure. . .
Triangular Prism
Hexagonal Prism
Rectangular Prism
Heptagonal Prism
What do these polyhedra have in
common?
Name that figure. . .
Triangular Pyramid
Pentagonal Pyramid
Rectangular Pyramid
Hexagonal Pyramid
Prisms vs. Pyramids

Two congruent,
parallel faces are the
bases
 Sides are
parallelograms

Named by its base

One base

Sides are triangles

Named by its base
http://www.math.com/school/subject3/lessons/S3U4L1GL.html
Polyhedra

Faces: Polygonals regions
that make up the surface of a
solid

Edges: The line segments
created by the intersection of
two faces of a solid

Vertices: The points of
intersection of two or more
edges
Counting Parts of Solids, Navigations (Geometry), Grades 3-5
Figure
Rectangular
Prism (Cube)
Pentagonal
Prism
Rectangular
Pyramid
Pentagonal
Pyramid
Number of
Faces
Number of
Vertices
Number of
Edges
Counting Parts of Solids, Navigations (Geometry), Grades 3-5
Figure
Number of
Faces
Rectangular 6
Prism (Cube)
Pentagonal
Prism
Rectangular
Pyramid
Pentagonal
Pyramid
Number of
Vertices
Number of
Edges
8
12
Counting Parts of Solids, Navigations (Geometry), Grades 3-5
Figure
Number of
Vertices
Number of
Edges
Rectangular 6
Prism (Cube)
8
12
Pentagonal
Prism
10
15
Rectangular
Pyramid
Pentagonal
Pyramid
Number of
Faces
7
Counting Parts of Solids, Navigations (Geometry), Grades 3-5
Figure
Number of
Vertices
Number of
Edges
Rectangular 6
Prism (Cube)
8
12
Pentagonal
Prism
7
10
15
Rectangular
Pyramid
5
5
8
Pentagonal
Pyramid
Number of
Faces
Counting Parts of Solids, Navigations (Geometry), Grades 3-5
Figure
Number of
Faces
Number of
Vertices
Number of
Edges
Rectangular 6
Prism (Cube)
8
12
Pentagonal
Prism
7
10
15
Rectangular
Pyramid
5
5
8
Pentagonal
Pyramid
6
6
10
6
12
8
6
12
8
5
9
6
7
15
10
8
18
12
5
8
5
4
6
6
6
10
6
7
12
7
Explain the relationship that
exists among the number of
faces, edges, and vertices of
each solid in the chart.
Faces + vertices = edges + 2
F+v=e+2
F+v=e+2
A polyhedron has 7 faces and 15 edges.
How many vertices does it have?
F+v=e+2
A polyhedron has 10 edges and 6 vertices.
How many faces does it have?
F+v=e+2
A polyhedron has 6 faces and 8 vertices.
How many edges does it have?
Geometry
July 1, 2008
Connect Math Shapes Set
http://phcatalog.pearson.com/component.cfm?site_id=6&discipline_id=80
6&subarea_id=1316&program_id=23245&product_id=3502
CMP Cuisenaire® Connected Math
Shapes Set (1 set of 206)
ISBN-10: 157232368X
ISBN-13: 9781572323681
Price: $29.35
Surface Area
10 inches
3 inches
5 inches
Surface Area
2 inches
6 inches
5 inches
5 inches
4 inches
3 inches
Surface Area
2 in
4’
2 in
4’
2 in
4’
Pentominos
How many ways can you arrange five tiles
with at least one edge touching another
edge?
Use your tiles to determine arrangements
and cut out each from graph paper.
Pentominos

http://www.ericharshbarger.org/pentominoes/
Which nets will form a
box without a lid?
Building a Box
Illuminations:
How many different nets can you draw that
can be folded into a cube?
http://illuminations.nctm.org/activitydetail.aspx?ID=84
It’s the view that counts!
(3-5 Geometry, Navigations)
When you have a 3-D shape, what do you
see when you look at eye level from the
front, then from above, and then at eye
level from the side?
How could you represent the shape so that
someone else might be able to build it?
It’s the view that counts!
(3-5 Geometry, Navigations)
Using three linking blocks, draw on grid
paper a two-dimensional representation of
the front, side, and top views of your
building. Label the views.
It’s the view that counts!
(3-5 Geometry, Navigations)
FRONT
SIDE
TOP
It’s the view that counts!
(3-5 Geometry, Navigations)
Using four linking blocks, draw on grid paper
a two-dimensional representation of the
front, side, and top views of your building.
Label the views.
Have your neighbor recreate your building
based on your views.
It’s the view that counts!
(3-5 Geometry, Navigations)
It’s the view that counts!
(3-5 Geometry, Navigations)
Transfer your drawing to a threedimensional view.
Isometric Explorations
(6-8 Geometry, Navigations)
Isometric Explorations
(6-8 Geometry, Navigations)
Isometric Explorations
(6-8 Geometry, Navigations)
Isometric Explorations
(6-8 Geometry, Navigations)
Volume

Cylinder

Cube
vs.
vs.
Cone
Square pyramid
Archimedes’ Puzzle
1
8
2
4
3
6
5
9
10
7
12
11
14
 http://mabbott.org/CMPUnitOrganizers.htm
Area of Circle

draw a square, and inscribe a circle in it, which means to draw the circle inside the square so that
the circle just touches each side of the square.

We can find the area of this square by first finding the area of the four smaller squares—each with
sides equal to r, the radius of the circle—and adding them together.




Notice that the sides of the square are twice as long as the radius of the circle. You could also find
the area of the square by multiplying the side times the side, or 2r x 2r, which also equals 4r2.
You can see that the area of the circle must be less than the area of four of the squares. But how
much less? We could make an educated guess and say that the area of the circle might be a little
bit larger than three of the smaller squares.
The actual number we're looking for, which is between 3 and 4, is the special number called pi,
represented by the Greek letter . Pi is approximately equal to 3.14.



The symbol you see here means "approximately equal to." Pi actually has an unending number of
decimal points, but 3.14 is usually close enough for our calculation purposes. Pi is the ratio
between the diameter and circumference of a circle.
The final formula for the area of a circle is shown here.
