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Slide 1
Areas of Polygons & Circumference of Circles
Slide 2
Define: Area
Slide 3
Area has been defined* as the following: a two dimensional
space measured by the number of non-overlapping unit squares or
parts of unit squares that can fit into the space Discuss... *State
of Arizona 2008 Standards Glossary
Slide 4
GEOBOARDS Geoboards are wonderful tools for exploring the
concept of area. To change the picture all we have to do is move
the geobands! If you do not have geoboards, use the grids to do the
explorations. Start by making as many different sized squares as
you can on your geoboard. Sketch them on the grids below. What is
the area of the smallest square? What is the area of the largest
square? Make as many different rectangles as you can that have an
area of 4 square units. Sketch them on the grids. Find their
perimeters. Are the perimeters all the same?
Slide 5
GEOBOARDS Make as many different rectangles as you can that
have an area of 4 square units. Sketch them on the grids. Find
their perimeters. Are the perimeters all the same?
Slide 6
Finding Triangles: Make as many different triangles as you can
that have an area of 2 square units. Sketch them on the grids.
Explain, in your own words, how your formed them.
Slide 7
Area Find the area of each polygon by counting unit
squares.
Slide 8
Areas of Irregular Figures Find the area of each of the
figures. Make sure to keep track of your work and/or the process
you took as you found the area.
Slide 9
Areas on a Geoboard Addition method divide an area into smaller
pieces and then add the areas. What is the area of this figure? 19
square units
Slide 10
A
Slide 11
36 complete squares
Slide 12
2 from 4 halves
Slide 13
36 complete squares 2 from 4 halves 1 from 1 x 3 triangle
Slide 14
36 complete squares 2 from 4 halves 1 from 1 x 3 triangle 3
from 1 x 7 triangle 36 + 2 + 1.5 + 3.5 = 43
Slide 15
B
Slide 16
Triangle Area = bh Base = 5 Height = 10 (5)(10) = 25
Slide 17
72 6-12.5-8-15-4-0.5-1.5 = 24.5
Slide 18
D
Slide 19
18 complete squares
Slide 20
1 from 1 x 2 triangle
Slide 21
18 complete squares 1 from 1 x 2 triangle 7 from 7 x 2 triangle
18 + 1 + 7 = 26
Slide 22
Rectangle method construct a rectangle encompassing the entire
figure and then subtract the areas of the unshaded regions. Area =
16 (3 + 1 + 1 + 1 + 1) = 9 E
Slide 23
F Find the area of the figure.
Slide 24
F The area of the hexagon equals the area of the surrounding
rectangle minus the sum of the areas of figures a, b, c, d, e, f,
and g.
Slide 25
Area and Perimeter Connections Consider a rectangle that has
length and width measurements that are whole numbers. Given the
below conditions determine the length and width measurements for
two examples. If it is not possible to create such a rectangle,
explain why. 1. The area is 30 square units. 2. The perimeter is 30
units. 3. The area is 25 square units. 4. The perimeter is 25
units. 5. The area is an even whole number 6. The perimeter is an
even whole number.
Slide 26
7. The area is an odd whole number. 8. The perimeter is an odd
whole number. 9. The area is a prime number. 10. The perimeter is a
prime number. 11. What generalizations can be made regarding area
and perimeter of rectangles? Consider only whole numbers in your
generalizations. 12. What generalizations can be made about the
relationship between area and perimeter of a rectangle? Consider
only whole numbers in your generalizations.
Slide 27
Finding Area by Dissection Area = length x width = lw 3 6 6 x 3
= 18 1. How do you compute the area of a rectangle? 2. Illustrate a
concrete method of finding the area of a rectangle.
Slide 28
Use the rectangle you created to find the formula for the area
of a triangle. If the formula for the area of the triangle is half
of the rectangle, why is the formula bh rather than lw?
Slide 29
Figure One a. Can you make a non-rectangular parallelogram with
these two pieces? b. Describe the process from part a. A right
triangle was cut from one end of the rectangle and slid to the
other side to create a non- rectangular parallelogram.
Slide 30
c. Based on your observation, write a sentence describing the
area of a parallelogram. d. Write a formula for the area of a
parallelogram. The area of the rectangle is equal to the area of
the parallelogram. The width of the rectangle is equal to the
height of the parallelogram and the length is equal to the base.
Area = base x height Area = bh
Slide 31
Figure Two Cut out both figure two shapes from your material
sheet. a.What are the two shapes? What word describes the
relationship between the two shapes? b. Put the two shapes together
to form a parallelogram. c. Describe the process from part b.
Trapezoids Congruent Two congruent trapezoids were put together by
rotating one of them 180 o to form a parallelogram
Slide 32
Figure Two continued d. Based on your observations, write a
sentence describing the area of one of these shapes. The area of
the trapezoid is half the area of the parallelogram (bh). The base
of the parallelogram is equal to the top + bottom of the trapezoid.
Area = (top + bottom) height Area = (a + b)h = (b 1 + b 2 )h Top
Bottom Bottom Top
Slide 33
Problem Solving Application. 3 ft 12 ft 4 ft 8 ft 10 ft 8 ft
Triangle: bh = (3 x 8) = 12 ft 2 Rectangle: lw = 10 x 8 = 80 ft 2
Parallelogram: bh = 12 x 4 = 48 ft 2 Pool Area = 12 + 80 + 48 =
140ft 2 Cost = 140 x $4.25 = $595 You have an unusually shaped pool
and you need to buy a pool cover. The pool cover cost $4.25 per
square foot. How much will it cost to cover your pool?
Slide 34
Find the areas: 20 cm 2 56 cm 2
Slide 35
Area of Polygons Triangle Rectangle Square Parallelogram
Trapezoid Odd Shape Polygon
Slide 36
Area of Polygons Triangle RectangleA = lw SquareA = s 2
ParallelogramA = bh TrapezoidA = (b 1 + b 2 )h Odd Shape
PolygonBreak into known polygons
Slide 37
How can you use these shapes to come up with the formula for
the area of a: rectangle, parallelogram, triangle, and
parallelogram?
Slide 38
What is a circle? Share definitions. A collection of points
equidistant from a given point
Slide 39
What does a compass do? Draw a circle with your compass.
Slide 40
Circumference What is a circumference? What is a diameter? The
total distance around a circle.
Slide 41
Diameter explorations Take one of the circular objects and
piece of string. Mark the length of the diameter on your piece of
string How many of those diameters fit around the circumference of
your circular object?
Slide 42
Understanding Circles Object measured CircumferenceDiameter
Circumference Diameter 3.14 = pi ( ) Locate at least 4 round
objects and measure the diameter and measure the circumference of
each. Record your results in the table below. Be sure to include
the units you used in the measuring process.
Slide 43
Circumference Pi = Circumference diameter Circumference = 2 r
Circumference = d
Slide 44
Circumference of a Circle Circle the set of all points in a
plane that are the same distance from a given point, the center.
Circumference the perimeter of a circle. Pi the ratio between the
circumference of a circle and the length of its diameter.
Slide 45
Find each of the following: a. The circumference of a circle
with radius 2 m. Leave answer in terms of pi b.The radius of a
circle with circumference 15 m 4 m 7.5 m
Slide 46
Which is the shorter route?
Slide 47
Discovering and Relating Area Formulas Using Dot Paper Area is
a spatial concept a covering of two-dimensional space. Complete the
tables What do you need to watch for with your students when doing
the triangle, parallelogram, and trapezoid? Generalize patterns in
the table Write the generalized formula for each
Slide 48
Alpha Shapes Sort the alpha shapes into two different
categories
Slide 49
Capture the Quadrilaterals You will need the quadrilaterals
from the alphashapes and a partner! Cut on the dotted lines. Make 2
piles of cards: (1) one pile contains all attributes referring to
angles and (2) the other pile contains all attributes referring to
sides. Piles should be placed upside down. Upon your turn, take one
card from each pile. You then look for all the quadrilateral
alphashapes that match both the attributes on the side and angle
cards. Those quadrilaterals that match both become captured by you.
It is than your opponents turn. He/she will follow the same
procedures. Once a quadrilateral is captured it may not be taken
unless the WILD CARD is used. The player(s) with the most captured
quadrilaterals at the end is the winner!
Slide 50
Use one set of Shapes spread them out in the middle of the
table. Group members take turns being the chooser. The chooser
chooses one of the Shapes while group members need to find the
shape that matches the shape the chooser choose. Group members are
only allowed to ask the chooser yes or no questions to help narrow
down the possibilities. Group members are not allowed to point to a
shape and ask, Is this the one? Also, group members are not allowed
to ask questions about the letter on the shape. Rather, they must
continue to ask questions that reduce the choices to one shape by
using different attributes of the Shapes. Once a group has an idea
what shape was chosen, they ask the chooser if they are correct.
What is My Shape?
Slide 51
Alpha Shapes and Area Using the centimeter grid paper, find the
area of shape W, you may need to estimate. Comparing Shapes Two Vs
What is the area of V? How can you use V to help find the area of
C?
Slide 52
Comparing Shapes 1) Compare the area of each piece to the areas
of W and Q. Sort the shapes into five piles, as shown: a) Area less
than W b) Area same as W c) Area greater than W but less than Q d)
Area same as Q e) Area greater than Q 2) Write about your results
and explain how you compared areas.
Slide 53
Class Discussion How did you decide what strategy to use to
find the area of each shape? Which area measurements are rough
estimates? Which ones are exact? Why? Did you find shapes that are
not congruent that have the same area?
Slide 54
Good Questions Each group will work through 3 - 5 of the
problems to come up with an answer key for the entire set of
problems *Questions from Good Questions for Math Teaching Grades 5
8 and Good Questions for Math Teaching Grades K 6.
Slide 55
Who Put the Tan in Tangram? Find the area of each of the shapes
from a set of tangrams Continue working through the packet from
Georgia Department of Education.