MATH 2160 1st Exam Review - Valdosta State University
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Transcript MATH 2160 1st Exam Review - Valdosta State University
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MATH 2160
Exam
Review
Geometry
and
Measurement
Problem Solving –
Polya’s 4 Steps
Understand the problem
Devise a plan
What goes into this step?
Why is it important?
Carry out the plan
What does this mean?
How do you understand?
What happens here?
What belongs in this step?
Look back
What does this step imply?
How do you show you did this?
Problem Solving
Polya’s 4 Steps
Understand the problem
Devise a plan
Carry out the problem
Look back
Which step is most important?
Why is the order important?
How has learning problem solving
skills helped you in this or another
course?
Problem Solving Strategies
Make a chart
Make a table
Draw a picture
Draw a diagram
Guess, test, and revise
Form an algebraic model
Look for a pattern
Try a simpler version of the problem
Work backward
Restate the problem differently
Eliminate impossible situations
Use reasoning
Geometry
Angles and congruency
Congruent– same size, same shape
Degree measure – real number
between 0 and 360 degrees that defines
the amount of rotation or size of an angle
Sum of the interior angles of any
polygon: (n – 2)180o where n is the
number of sides in the polygon
Geometry
Special angles
right angle – 90
acute angle – 0< angle < 90
obtuse angle – 90< angle < 180
Sum of the angles
Triangle = 180o
Quadrilateral = 360o
Pentagon = 540o
Etc.
Geometry
Circles
circle – special simple closed curve
where all points in the curve are
equidistant from a given point in the
same plane – NOTE: Circles are NOT
polygons!
center – middle point of the circle
diameter – is a chord that passes
through the center of the circle
radius – line segment connecting the
center of the circle to any point on the
circle
Geometry
Polygons – made up of line
segments
Triangles – 3-sided polygons
Quadrilaterals – 4-sided polygons
n - gons – the whole number n
represents the number of sides for the
polygon: a triangle is a 3-gon; a
square is a 4-gon; and so on
Regular Polygons – polygon where
the all the line segments and all of the
angles are congruent
Geometry
Triangles
Union of three line segments formed
by three distinct non-collinear points
vertices – intersection points of line
segments forming the angles of the
polygon
sides – the line segments forming the
polygon
height – line segment from a vertex of a
triangle to a line containing the side of the
triangle opposite the vertex
Geometry
Triangles
equilateral – all sides and angles
congruent
isosceles – at least one pair of
congruent sides and angles
scalene – no congruent sides or
angles
right – one right angle
acute – all angles acute
obtuse – one obtuse angle
Geometry
Quadrilaterals
parallelogram – quadrilateral with two
pairs of parallel sides
opposite sides are parallel
opposite sides are congruent
rectangle – quadrilateral with four
right angles
a parallelogram is a rectangle if and
only if
it has at least one right angle
trapezoid – exactly one pair of
opposite sides parallel, but not
congruent
Geometry
Quadrilaterals
rhombus – quadrilateral with four
congruent sides
a parallelogram is a rhombus if and
only if
it has four congruent sides
square – quadrilateral with four right
angles and four congruent sides
a square is a parallelogram if and only if
it is a rectangle with four congruent sides
it is a rhombus with a right angle
Geometry
Pentominos
* Won't fold into an open box
f
p*
j
i*
n
Geometry
Pentominos
* Won't fold into an open box
t
u*
v*
Geometry
Pentominos
w
x
z
y
Geometry
Patterns with points, lines, and
regions
Where k is the number of lines or line
segments
k 1
Po int s x
n1
P = [k (k – 1)] / 2
Regions = lines + points + 1
R = k + P + 1 = [k (k + 1)] / 2 + 1
Geometry
Tangrams
Flips, slides, and turns
Communication
Maps
Conservation of Area
Piaget
If use all of the pieces to make a new
shape, both shapes have the same
area
Geometry
Polyhedron
Vertices
Edges
Faces
Should be able to draw ALL of the
following:
Sphere
Prisms – Cube, Rectangular,
Triangular
Cylinder
Cone
Pyramids – Triangular, Square
Measurement
5 ft
Rectangle
3 ft
Perimeter
P = 2l + 2w, where l = length and
w = width
Example: l = 5 ft and w = 3 ft
P rectangle
=
P
=
2(5 ft) + 2(3 ft)
P
P
=
=
10 ft + 6 ft
16 ft
2l + 2w
Measurement
5 ft
Rectangle
3 ft
Area
A = lw where l = length and w =
width
Example: l = 5 ft and w = 3 ft
A rectangle = lw
A
=
(5 ft)(3 ft)
A
=
15 ft2
Measurement
Square
3 ft
Perimeter
P = 4s, where s = length of a side
Example: s = 3 ft
P square
=
P
=
4(3 ft)
P
=
12 ft
4s
Measurement
Square
3 ft
Area
A = s2 where s = length of a side
Example: s = 3 ft
A square =
s2
A
=
(3 ft)2
A
=
9 ft2
Measurement
Triangle
Perimeter
P = a + b + c, where a, b, and c
are the lengths of the sides of the
triangle
Example: a = 3 m; b = 4 m; c = 5
m
P triangle
P
=
P
=
=
a+b+c
3m+4m+5m
12 m
Measurement
Triangle
4m
5m
3m
Area
A = ½ bh, where b is the base and
h is the height of the triangle
Example: b = 3 m; h = 4 m
A triangle =
A
=
A
=
½ bh
½ (3 m) (4 m)
6 m2
Measurement
3 cm
Circle
Circumference
C circle = d or C = 2r, where d =
diameter and r = radius
Example: r = 3 cm
C circle =
2r
C
=
2(3 cm)
C
=
6 cm
Measurement
3 cm
Circle
Area
A = r2, where r = radius
Example: r = 3 cm
r2
A circle
=
A
=
(3 cm)2
A
=
9 cm2
7 cm
Measurement
Rectangular Prism
5 cm
6 cm
Surface Area: sum of the areas of all of the
faces
Example: There are 4 lateral faces: 2 lateral
faces are 6 cm by 7 cm (A1= wh) and 2
lateral faces are 5 cm by 7 cm (A2 = lh).
There are 2 bases 6 cm by 5 cm (A3 = lw)
A1 = (6 cm)(7 cm) = 42 cm2
A2 = (5 cm)(7 cm) = 35 cm2
A3 = (6 cm)(5 cm) = 30 cm2
SA rectangular prism = 2wh + 2lh + 2lw
SA = 2(42 cm2) + 2(35 cm2) + 2(30 cm2)
SA = 84 cm2 + 70 cm2 + 60 cm2
SA = 214 cm2
Measurement
Rectangular Prism
7 cm
5 cm
Volume:
6 cm
V = lwh where l is length; w is width;
and h is height
Example: l = 6 cm; w = 5 cm; h = 7 cm
V rectangular prism = Bh = lwh
V
=
(6 cm)(5 cm)(7 cm)
V
=
210 cm3
Measurement
Cube
5 cm
Surface Area: sum of the areas of all
6 congruent faces
Example: There are 6 faces: 5 cm by
5 cm (A = s2)
SA cube = 6A = 6s2
SA = 6(5 cm)2
SA = 6(25 cm2)
SA = 150 cm2
Measurement
Cube
5 cm
Volume:
V = s3 where s is the length of a side
Example: s = 5 cm
V cube = Bh = s3
V
=
(5 cm)3
V
=
125 cm3
Measurement
7m
5m
Triangular Prism
Surface Area: sum of the areas of all of the
faces
Example: There are 3 lateral faces: 6 m by
7 m (A1= bl). There are 2 bases: 6 m for the
base and 5 m for the height (2A2 = bh).
A1 = (6 m)(7 m) = 42 m2
2A2 = (6 m)(5 m) = 30 m2
SA triangular prism = bh + 3bl
SA = 30 m2 + 3(42 m2)
SA = 30 m2 + 126 m2
SA = 156 m2
6m
Measurement
7m
5m
Triangular Prism
6m
Volume:
V = ½ bhl where b is the base; h is
height of the triangle; and l is length of
the prism
Example: b = 6 m; h = 5 m; l = 7 m
V triangular prism = Bh = ½ bhl
V
=
½ (6 m)(5 m)(7 m)
V
=
105 m3
3 ft
Measurement
12 ft
Cylinder
Surface Area: area of the circles plus
the area of the lateral face
Example: r = 3 ft; h = 12 ft
2rh +2r2
SA cylinder=
SA = 2 (3 ft)(12 ft) + 2 (3 ft)2
SA
SA
SA
=
=
=
72 ft2 + 2 (9 ft2)
72 ft2 + 18 ft2
90 ft2
3 ft
Measurement
12 ft
Cylinder
Volume of a Cylinder: V = r2h
where r is the radius of the base
(circle) and h is the height.
Example: r = 3 ft and h = 12 ft.
V cylinder =
Bh = r2h
V
=
(3 ft)2 (12 ft)
V
=
(9 ft2)(12 ft)
V
=
108 ft3
13 ft
12 ft
Measurement
5 ft
Cone
Surface Area: area of the circle plus
the area of the lateral face
Example: r = 5 ft; t = 13 ft
rt +r2
SA cone=
SA = (5 ft)(13 ft) + (5 ft)2
SA
SA
SA
=
=
=
65 ft2 + (25 ft2)
65 ft2 + 25 ft2
90 ft2
13 ft
12 ft
Measurement
5 ft
Cone
Volume: V = r2h/3 where r is the
radius of the base (circle) and h is the
height.
Example: r = 5 ft; h = 12 ft
V cone=
r2h/3
V
=
[(5 ft)2 12 ft ]/ 3
V
=
[(25 ft2)(12 ft)]/3
V
=
(25 ft2)(4 ft)
V
=
100 ft3
8 mm
Measurement
Sphere
Surface Area: 4r2 where r is the
radius
Example: r = 8 mm
SA sphere =
4r2
SA
=
4(8 mm)2
SA
=
4(64 mm2)
SA
=
256 mm2
6 mm
Measurement
Sphere
Volume of a Sphere: V = (4/3) r3
where r is the radius
Example: r = 6 mm
V sphere =
4r3/3
V
=
[4 x (6 mm)3]/3
V
=
[4 x 216 mm3]/3
V
=
[864 mm3]/3
V
=
288 mm3
Measurement
Triangular Pyramid
Square Pyramid
Test Taking Tips
Get a good nights rest before
the exam
Prepare materials for exam in
advance (scratch paper, pencil,
and calculator)
Read questions carefully and
ask if you have a question
DURING the exam
Remember: If you are prepared,
you need not fear