Transcript Algebra I - Denise Kapler

Unit 6
A. Perimeter, Area,
Surface Area and
Volume Formulas
B. Changing Dimensions
C. Geometric Probability
Agenda
?
Developing Perimeter and Area Formulas
How do you develop the formulas for area and perimeter of a
parallelogram from those of a rectangle?
I know
For a rectangle:
Perimeter = 2w + 2l
Area = lw
For a parallelogram:
Perimeter = 2b + 2h
Area = bh
Word
Area = size of a surface
April 2015
Agenda
?
Developing Perimeter and Area Formulas
How do you develop the formulas for area and
perimeter of a triangle from those of a rectangle?
I know
For a triangle:
Perimeter = sum of side lengths
Area = ½ bh
For a parallelogram:
Perimeter = 2b + 2h
Area = bh
Directions: make a copy of the
triangle and rotate it about a
vertex. It forms a parallelogram.
Word
Height and base are often used
synonymously with width and length.
April 2015
Area of Trapezoid
A = ½ h(b1 + b2)
By consdiering this
construction, you should be
able to see why the formula
for area of a trapezoid
includes multiplying by ½.
Area of a circle
Circumference = 2 pi r
Area of a circle
Area of a circle
What happens if we increase the
number of wedges to infinity?
s = length of one side of polygon
Area of Regular Polygon
Agenda
?
Composite Figures
Which properties of composite figures allow
you to use the Area Addition Postulate?
I know
The area of a composite figure is the sum of the
areas of its non-overlapping parts
Word
Perimeter – distance around the outer edge of a figure
Area is the size of a surface
April 2015
Agenda Effects of Changing Dimensions Proportionally
? What happens to the area of a triangle if only the base
is multiplied by 2? If the height is multiplied by 2?
I know
One dimension multiplied by “a” result in
the area changing by a factor of a.
Word
Dimension - measurement of length in one direction.
April 2015
Dimension - measurement of length in
one direction
In Physics dimension can also mean any physical
measurement such as length, time or mass.
Agenda Effects of Changing Dimensions Proportionally
? What happens to the area of a triangle if all dimensions
are multiplied by 3?
I know
All dimension multiplied by “a” result in the
area changing by a factor of a2.
Word
Note: If change all dimensions by the same factor,
it is called a scale factor.
April 2015
Agenda Effects of Changing Dimensions Proportionally
? What happens to the perimeter if each side length of
a polygon is multiplied by 7? To the area?
I know
All dimensions multiplied by “a” result in the
perimeter or circumference changing by a factor
of “a”.
Given n = 5 and s = 6
Word Polygon – many
April 2015
Agenda
?
Geometric Probability
How can Geometric Probability be used to predict
real world outcomes?
I know
When considering geometric probability, one
needs to consider: 1. Sample Space (possible
outcomes), 2. Event (set of outcomes), and 3.
Probability. Models most often used in this study
are length, angle measure, and area.
What is the probability of landing on green?
Word Probability – how
likely something is to happen
April 2015
Probability is always between 0 and 1
Math Review April 9, 2015
1. Find area of a
trapezoid REMN.
Trapezoid ~
4-sided flat shape
with straight sides
that has a pair of
opposite sides
parallel.
2. Find area of
rhombus
where d1=8
and d2=12.
Rhombus - flat
shape with 4 equal
straight sides.
Opposite sides are
parallel, and
opposite angles are
equal (it is a
Parallelogram). And
the diagonals "p"
and "q" of
a rhombus bisect
each other at right
angles.
3. Sketch and find
area of composite
figure
Composite figures are
just a combination of
simpler figures in disguise!
Add up the areas of the
simple figures to find the
area of the entire figure.
Area of Trapezoid
Area of Kite and Rhombus
Math Review April 13, 2015
1. Find area of a
trapezoid PQRS.
Trapezoid ~
4-sided flat shape
with straight sides
that has a pair of
opposite sides
parallel.
2.
Find area of the
square and each
side length.
Square - flat shape with 4
equal straight sides.
Opposite sides are parallel,
and opposite angles are
equal (it is a
Parallelogram). And the
diagonals bisect each other
at right angles and are
congruent.
3. Sketch and find area of
composite 2-D figure: A
rectangle house (24x8’)
with an isosceles
triangle roof (h=6’).
Composite figures are
just a combination of
simpler figures in disguise!
Add up the areas of the
simple figures to find the
area of the entire figure.
Math Review April 14, 2015
1. Find perimeter
of parallelogram.
2a. Find area of
rectangle where
l = 8cm and w=3cm.
2b. What is the
change in area if the
l and w are each
multiplied by 5?
Parallelogram ~
4-sided flat shape
with 2 pairs of
opposite side and
opposite angles
are congruent.
Rectangle - flat shape with
2 pairs of parallel sides and
four right angles. Opposite
sides are parallel, and
opposite angles are equal (it
is a Parallelogram). And the
diagonals are congruent.
3. Sketch and find
area of
composite 2-D
figure: Ice cream
cone with height
of 10 cm and
diameter of 6 cm.
Composite figures are
just a combination of
simpler figures in disguise!
Add up the areas of the
simple figures to find the
area of the entire figure.
Math Review April 15, 2015
1a. Find perimeter
given trapezoid.
1b. What is the
change, if each
side length is
multiplied by 2?
Trapezoid ~
4-sided flat shape with
straight sides that has
a pair of opposite
sides parallel.
2a. Find area of
circle with d=10cm.
2b. What is the
change in area if the
d is multiplied by 5?
Circle-
locus of all
points a given distance
from the center of the
circle.
3. Sketch and find
area of composite
figure
Composite figures are
just a combination of
simpler figures in disguise!
Add up the areas of the
simple figures to find the
area of the entire figure.
Math Review April 20, 2015
1. Find the
probability
of landing
on blue?
Area - size of a
surface
What is the probability
of breaking a piece of
10" linguine into two
parts and having one
part be more than 8"?
2.
Line segment -
part of
a line that connects two
endpoints
3. Sketch circle C with
r=14m. What is the
probability of landing
in the blue sectors?
Circle - 360°
2. In the diagram, the green points represent possible breaks for successful outcomes. In
break that occurs in the red area is successful. Therefore the P(randomly breaking a 10"
piece of linguine into two parts having one part be more than 8") = 2+2 /10 = 2/5
Math Review April 21, 2015
1a. Find the
circumference of
the circle.
1b. What is the
change in
circumference if
the radius is
multiplied by 5?
Circumference ~
Distance around
the circle
2.
Find perimeter
and area of figure.
Semicircle - a onedimensional locus of
points that forms half of
a circle.
3. Sketch and find area
of composite figure.
Handle width = 2cm
Composite figures are
just a combination of
simpler figures in disguise!
Add up the areas of the
simple figures to find the
area of the entire figure.
Math Review April 21, 2015
1a. Find the
circumference of
the circle.
1b. What is the
change in
circumference if
the radius is
multiplied by 5?
Circumference ~
Distance around
the circle
2.
Find perimeter
and area of figure.
Semicircle - a onedimensional locus of
points that forms half of
a circle.
3. Sketch and find area
of composite figure.
Handle width = 0.5cm
Composite figures are
just a combination of
simpler figures in disguise!
Add up the areas of the
simple figures to find the
area of the entire figure.
Math Review April 22, 2015
1. Find the area
of the
parallelogram.
Area ~
Size of a
surface. Two
dimensional.
2.
Find the grey
shaded area.
Semicircle - a onedimensional locus of
points that forms half of
a circle.
3. What is the area
of JKL if the
coordinates of J,
K, and L are
J(0, 0)
K(0, 3)
L(4, 0) ?
Coordinate Plane The plane containing the
"x" axis and "y" axis
Math Review April 23, 2015
1. A regular hexagon
is circumscribed about
a circle. The circle has
a radius of 9 feet.
Find the area of the
hexagon to the
nearest tenth.
Hexagon ~
Six-sided
polygon
2. What is the
probability that a dart
thrown at the target
shown will land in the
shaded area? To the
nearest hundredth of a
percent.
Semicircle - a onedimensional locus of
points that forms half of
a circle.
3. A point is chosen
randomly on line
segment AD . What
is the probability
that the point is on
line segment BC ?
Coordinate Plane The plane containing the
"x" axis and "y" axis
Math Review April 27, 2015
1. Find the area of the
square.
Square ~
Four-sided
regular polygon
with at least one
right angle
2. The radius of the
circle circumscribed
around the regular
hexagon is 10
centimeters. Find the
area of the shaded
part of the figure to the
nearest tenth.
Circle – locus of all
points that are at an
equal distance from a
center point.
3. Find the area of
the polygon with
vertices D(4, 1),
E(2, 4), F(-3, 2),
and G(0, -4).
Coordinate Plane The plane containing the
"x" axis and "y" axis