Mathematics Skills 2
By: Idania Dorta
v 1.0
Mathematics Skills 2
INSTRUCTIONS
Welcome to your Continental Academy course “Mathematics Skills 2”. It is
made up of 8 indi vidual l essons, as listed i n the Table of Contents. Each
lesson includes practice questions with answers. You will progress through
this course one lesson at a time, at your own pace.
First, study the lesson thoroughly. Then, complete the lesson reviews at the
end of the lesson and carefully check your answers. Sometimes, those
answers will contain inform ation t hat you will need on the graded lesson
assignments. When you are ready, complete the 10-question, multiple
choice lesson assignment. At the end of each lesson, you will find notes to
help you prepare for the online assignments.
All lesson assignments are open-book. Continue work ing on the lessons at
your own pace until you have finished all lesson assignments for this
course.
When you have completed and passed all lesson assignments for this
course, complete the End of Course Eamination.
If you need help understanding any part of the lesson, practice
questions, or this procedure:
ƒ Click on the “Send a Message” link on the left side of the home
page
ƒ Select “Academic Guidance” in the “To” field
ƒ Type your question in the field provided
ƒ Then, click on the “Send” button
ƒ
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You will receive a response within ONE BUSINESS DAY
Mathematics Skills 2
About the Author…
Mrs. Idania Dorta earned her Bachelor of Science Degree in Mathematics
Education from Florida International University and her Master of Science
Degree in Computer Applications in Education from Barry University. She worked
for the Dade County Public Schools System from 1993 until 2003, first as a
classroom teacher, then as a mathematics department head and finally as the
District Mathematics Educational Specialist. Since 2003, she has worked as an
independent Mathematics Consultant. Mrs. Dorta is a veteran of numerous
seminar presentations for educators and students alike. Idania makes her home
in Miami, Florida with her husband and child.
Mathematics Skills 2 MA20
Editor: Dr. Leon Kriston
Copyright 2008 Continental Academy
ALL RIGHTS RESERVED
The Continental Academy National Standard Curriculum Series
Published by:
Continental Academy
3241 Executive Way
Miramar, FL 33025
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Mathematics Skills 2
The purpose of this course is to provide experiences in problem
solving, reasoning, and connections in mathematics. The content
includes: geometry (plane/transformations/coordinate), measurement
(metric/customary), ratio/proportion, solving algebraic equations and
working with simple statistics and probability.
™ Students will compute fluently and make reasonable estimates
™ Student will understand meanings of operations and how they relate to one
another
™ Student will understand measurable attributes of objects and the units,
systems, and processes of measurement
™ Student will develop and evaluate inferences and predictions that are based
on data
™ Student will understand and apply basic concepts of probability
™ Student will know how to apply appropriate techniques, tools, and formulas to
determine measurements
™ Student will know how to select and use statistical methods
™ Student will know how to use coordinate geometry and other representational
systems
™ Student will know how to apply transformations and use symmetry
™ Students will know how to use algebraic symbols
™ Student will use representations to model and interpret physical, social, and
mathematical phenomena
™ Students will solve problems that arise in mathematics and in other contexts
™ Students will solve problems that arise in mathematics and in arguments
about its relationship
™ Student will use the language of mathematics to express mathematical ideas
precisely
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Mathematics Skills 2
TABLE OF CONTENTS
PAGE
LESSON 1: Geometry----------------------------------------------------------------- 7
LESSON 2: Ratios and Proportions--------------------------------------------- 31
LESSON 3: Measurement----------------------------------------------------------- 53
LESSON 4: Transformations and Coordinate Geometry----------------- 85
LESSON 5: Algebra------------------------------------------------------------------ 107
LESSON 6: Statistical Methods-------------------------------------------------- 127
LESSON 7: Probability------------------------------------------------------------- 153
LESSON 8: Basic calculator skills-------------------------------------------- 161
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Mathematics Skills 2
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Mathematics Skills 2
Lesson 1: Geometry
In this lesson, you learn how to name and classify angles, triangles,
polygons, and polyhedra by their measurement and characteristics.
Naming and Classifying Angles and Triangles
Before we begin to discuss angles, let us look at how angles are formed.
There are three fundamental terms in Geometry: Point, Line, and Plane.
A point represents a specific point in space. One can be drawn by simply
drawing a dot with a pencil. The purpose of a point is simply to show
position. Every point is named with a single capital letter.
This is read as “point A”.
•A
A line is made up of an indefinite number of points. A line is straight and
continues without end in both directions. A line is also named, but is done
by choosing any two points on the line.
A
K
This is read as “line AK” or AK. Notice that the line above and the notation
for the line both show a line with arrows going in both directions. This
indicates that the line continues indefinitely in both directions.
Since a line is indefinite in length, we sometimes use a part of a line. We
can use a ray or a line segment. A ray is a part of a line that begins at one
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Mathematics Skills 2
point, but extends indefinitely in one direction. A ray also has many points,
as does a line, but has one endpoint.
J
K
This is read as “ray JK” or JK.
A line segment is also a part of a line, but has two endpoints. Even though
it contains infinitely many points, it begins at one point and ends at another.
••••••••••••••
BN
This is read as “line segment BN” or BN.
Angles are made of two rays meeting at one endpoint. The endpoint is
called a vertex or the vertex of the angle. Angles are named by their
vertex. The rays are the sides of the angle.
A
B
C
This is read as “angle ABC (∠ABC)”, “angle CBA (∠CBA)”,
or “angle B” (∠B).
∠, Is the symbol for angle and can be used instead of the word angle.
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Mathematics Skills 2
*Note that in each instance the letter B (the vertex) is always in the center
when naming the angle with all three letters. Otherwise, we can name the
angle with simply the letter of the vertex.
Let us look at some examples.
Example 1:
What is the endpoint of GH?
Based on the notation above GH, we see that this is a ray,
beginning with endpoint G and continuing through H.
Therefore, the endpoint of GH is G.
Example 2:
Name all the angles you see below.
T
R
U
S
The illustration above contains several angles. They are
∠RST or ∠TSR, ∠TSU or ∠UST, and ∠USR or ∠RSU.
Note that we can name all three angles in two different ways,
but we cannot simply name the angle ∠S because there is
more than one angle here.
We measure angles just as we measure line segments. While line
segments are measured with a ruler in centimeters, millimeters, inches,
and feet, angles are measured with a protractor in degrees. A protractor is
a tool used to measure angles and it does so in degrees.
To measure an angle with a protractor you place the center point of the
protractor on the vertex of the angle.
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Mathematics Skills 2
Align the 0° line on the protractor with one side of the angle. Then read the
number of degrees that the ray points to.
Angles are classified by their measures.
Acute
Angle
Measures
less than 90°
Right Angle
Measures
90°
Straight Angle
Measures 180°
Example 3:
Obtuse Angle
Measures
more than 90°
and less than
180°
Classify the following angles by their measure.
∠ABC = 90°
∠GHI = 78°
∠XYZ = 117°
∠ABC is a right angle since it measures exactly 90°.
∠GHI is an acute angle since it measures less than 90°.
∠XYZ is an obtuse angle since it measures more than 90°,
but less than 180°.
Triangles are polygons with three sides and three angles. The sides are
made up of line segments and they have three vertices. Triangles are also
classified by name, as are angles. Triangles can be classified by the
degree of the angles as well as by the length of the sides.
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Classified by Angles
Acute Triangle
All angles in the
triangle are acute
Obtuse Triangle
One angle in the
triangle is obtuse
Right Triangle
One angle in the triangle
is a right angle
Classified by Sides
Equilateral Triangle
All sides are the same in
length, i.e., congruent
sides
Isosceles Triangle
At least two sides of
the triangle are equal
in length (congruent)
Scalene Triangle
None of the sides are
equal in length
(congruent)
*An equilateral triangle has three congruent sides and, as a result,
three congruent (equal) angles.
*An isosceles triangle has at least two congruent sides and, as a result,
at least two congruent (equal) angles.
*A scalene triangle has no congruent sides and therefore no angles are
congruent (equal) in measure.
*The sum of the measures of the three angles in every triangle is 180°.
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Try the next example on your own and check your answers with the book.
Example 4:
Find the measure of the missing angle in the triangle
below and classify the triangle.
62°
28°
Since the three angles in a triangle add up to 180°, the missing angle must
measure 90°.
because 62° + 28° = 90°
90° + ?° = 180°. ? = 180° – 90° = 90°
Therefore, the triangle above can be classified as a right triangle since it
has one angle that measures 90°.
Practice
Classify each angle as acute, obtuse, right, or straight.
1.
a. right
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b. acute
c. obtuse
d. straight
Mathematics Skills 2
2.
a. right
b. acute
c. Straight d. obtuse
Find the missing measure in each triangle.
Then classify the triangle as acute, right, or obtuse.
3.
56°
x°
63°
a. 61; right
b. 51; acute
c.61; acute
d. 65; obtuse
4.
60°
60°
a. 60; acute
x°
b. 90; right
c. 30; acute
d. 60; obtuse
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5.
x°
60°
a. 60; obtuse
b. 90; right
c.
30; acute
Classify each triangle by its angles and by its sides.
6.
31°
23°
a. acute, scalene
c. obtuse,, isosceles
Answers to practice:
1. c. obtuse
2. c. straight
3. c. 61°; acute
4. a. 60°; acute
5. d. 30°; right
6. b. obtuse, scalene
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b. obtuse, scalene
d. right, scalene
d. 30; right
Mathematics Skills 2
Naming and Classifying Two- and Three-Dimensional Shapes
A polygon is a closed figure with three or more sides. Polygons that have
all sides congruent are called regular polygons. In the same manner
polygons that have sides that are not congruent are called irregular
polygons.
Previously we discussed triangles, which are three-sided polygons.
Here, we will discuss other polygons, which are also two-dimensional
figures.
The first of the polygons we will discuss is a quadrilateral. A quadrilateral
is a polygon that has four sides, four angles, and therefore four vertices.
The sum of the angles of every quadrilateral is 360°.
There are several different types of quadrilaterals: square, rectangle,
rhombus, trapezoid, and parallelogram. Each of these is a quadrilateral, but
has a specific name based on its characteristics.
Square
All four sides are
congruent and
all four angles
are congruent.
Rectangle
Parallelogram
Opposite sides are
Opposite sides are
congruent and all four congruent/parallel & opposite
angles are congruent. angles are congruent.
Trapezoid
One pair of opposite sides
is parallel.
Rhombus
All four sides are congruent.
Opposite angles are equal.
Opposite sides are parallel.
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Mathematics Skills 2
Let us look at an example.
Example 1:
Name the given quadrilateral.
Find the missing angle.
Is the quadrilateral a square, rhombus, rectangle,
parallelogram, and/or a trapezoid?
3 cm
3 cm
G
105°
H
3 cm
75°
105°
J
I
3 cm
The quadrilateral above is named, like all polygons, by the letters of the
vertices.
This quadrilateral can be named as GHIJ or HIJG or JIHG.
The quadrilateral can be named any way as long as the letters are in
sequence.
In order to find the missing angle we can add the angles that are given and
subtract from 360° since the sum of measure of all the angles of a
quadrilateral is 360°. Therefore, 105° + 75° +105° = 285°.
360° - 285° = 75°.
Quadrilateral GHIJ is a parallelogram because opposite sides are parallel,
opposite sides are equal, and it is a rhombus because of these
characteristics as well as that all four sides are congruent (equal).
We have discussed polygons that are triangles and quadrilaterals. Let’s
discuss the remaining polygons with 5, 6, 7, 8, … sides.
Every polygon has the same amount of sides as it does angles and
vertices. For example, a triangle has 3 sides, 3 angles, and 3 vertices.
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Mathematics Skills 2
A quadrilateral has 4 sides, 4 angles, and 4 vertices. This is true for all
polygons.
Types of Polygons
Quadrilateral
4 sides
Triangle
3 sides
Pentagon
5 sides
Octagon
8 sides
Hexagon
6 sides
A polygon with seven sides is called a heptagon. A polygon with nine
sides is called a nonagon and a polygon with ten sides is called a
decagon. In general, a figure with n sides is called an n-gon.
A diagonal (a line segment drawn in a polygon from one non-consecutive
vertex to another) can be drawn from one vertex to form as many triangles
as possible.
You learned earlier that the angle sum of a triangle is 180°. In a
quadrilateral, we can draw a diagonal from one vertex to form two triangles,
and therefore the angle sum of 360°.
∠ sum = 180°
∠ sum = 180°
∠ sum = 180°
180°
180°
180°
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Mathematics Skills 2
In a pentagon, we can draw two diagonals from one vertex to form three
triangles. Therefore 180° x 3 triangles = 540°. The sum of the measures of
the angles in a pentagon is 540°.
In addition, we can continue this same pattern to find the sum of measures
of all polygons or simply use the
formula for finding the sum of the angles of all polygons (n – 2) x 180°
where n is the number of sides of the polygon.
To find the measure of each angle of a regular polygon, you can use the
formula above and divide by the number of sides.
(n – 2) x 180°
n
where n is the number of sides of the regular polygon.
Try the next example on your own and check your answers with the book.
Example 2: A dodecagon is a polygon with 12 sides.
What is the sum of the angles of a dodecagon?
What is the measure of each angle in the regular dodecagon?
To find the angle sum of a dodecagon we apply the formula:
(n
–2 ) x 180°.
Since a dodecagon has 12 sides, (12 – 2) x 180° = 1800°.
To find the measure of each angle we use the formula:
(n – 2) x 180° or simply divide 1800° by 12 sides = 150°.
n
Therefore, the angle sum of a dodecagon is 1800° and each angle in a
regular dodecagon is 150°.
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Mathematics Skills 2
The last shapes we would like to discuss in this lesson are threedimensional figures. These three-dimensional figures are sometimes
called solids. They are sometimes referred to as three-dimensional
because they have three dimensions: length, width, and height.
Some solids or three-dimensional figures are curved such as a cone, a
cylinder, and a sphere.
Cylinder
Sphere
Cone
Other solids have flat surfaces such as polyhedra. A polyhedron is a
three-dimensional figure whose flat surfaces are polygons.
-----TOP--------
Cube
Rectangular
Prism
----FACES------BOTTOM-Triangular
Prism
Pentagonal
Prism
If you notice, these polyhedra all have faces in the shape of polygons. The
faces of the cube are squares. The faces of the rectangular prism are
squares and rectangles. The faces of the triangular prism are triangles and
rectangles. And the faces of the pentagonal prism are pentagons and
rectangles. These polyhedron above are all categorized as prisms since
their two bases are the same shape and are parallel to one another.
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Mathematics Skills 2
The following polyhedra are a little different in that they have one base in
the shape of a polygon and all the faces of the polyhedron meet at one
point called the apex. These polyhedral are called pyramids.
Triangular
Pyramid
Square
Pyramid
Rectangular
Pyramid
Pentagonal
Pyramid
*Notice how all the pyramids are named by their bases.
Try the next example on your own and check your answers with the book.
Example 3:
Identify the polyhedron below as a prism or pyramid
and name the polyhedron.
The polyhedron above is a prism because the figure has two
parallel bases in the shape of pentagons. The figure is a
Pentagonal prism.
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Mathematics Skills 2
Practice
Determine whether each statement is sometimes, always, or never true.
1. A trapezoid is a quadrilateral.
a. sometimes
b. always
c. never
2. A quadrilateral is a rhombus.
a. sometimes
b. always
c. never
Determine whether each figure is a polygon. If it is, classify the polygon
and state whether it is regular. If it is not a polygon, explain why.
3.
a. triangle; regular
b. quadri lateral; non-regular
c. triangle; non-regular
d. not a polygon; one of the sides is
not a line segment
4.
a. quadri lateral; regular
b. pentagon; non-regular
c. quadri lateral; non-regular
d. not a polygon; the figure is not
closed
The single, small lines crossing the line segments (as above) indicate that all
such line segments are congruent (equal in length).
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Find the value of x in the quadrilateral below.
5.
x°
100 °
122 °
a. 126°
57 °
b. 81°
c. 96°
d. 261°
Identify each solid. Name the number of faces, edges, and vertices.
6.
a. rectangular prism, 6 faces, 12 edges, 8 vertices
b. cylinder, 4 faces, 8 edges, 10 vertices
c. rectangular prism, 5 faces, 12 edges, 7 vertices
d. cube, 6 faces, 10 edges, 8 vertices
Answers to practice:
1 b. always
1. a. sometimes
2. d. not a polygon; one of the sides is not a line segment
3. a. quadrilateral; regular
4. b. 81°
5. a. rectangular prism, 6 faces, 12 edges, 8 vertices
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LESSON 1 THINGS TO REMEMBER
Angles are classified by their measures.
Acute Angle
Measures
less than 90°
Right Angle
Measures 90°
Obtuse Angle
Measures more
than 90° and less
than 180°
Straight Angle
Measures 180°
Classify the following angles by their measure.
∠ABC = 90°
∠GHI = 78°
∠XYZ = 117°
∠ABC is a right angle since it measures exactly 90°.
∠GHI is an acute angle since it measures less than 90°.
∠XYZ is an obtuse angle since it measures more than 90°, but less than 180°.
Triangles are polygons with three sides and three angles. The sides are made
up of line segments and they have three vertices. Triangles are also classified
by name, as are angles. Triangles can be classified by the degree of the angles
as well as by the length of the sides.
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Mathematics Skills 2
Classified by Angles
Acute Triangle
All angles in the
triangle are acute
Obtuse Triangle
One angle in the
triangle is obtuse
Right Triangle
One angle in the triangle
is a right angle
Classified by Sides
Equilateral Triangle
All sides are the same in
length, i.e., congruent
sides
Isosceles Triangle
At least two sides of
the triangle are equal
in length (congruent)
Scalene Triangle
None of the sides
are equal in length
(congruent)
*An equilateral triangle has three congruent sides and, as a result, three
congruent (equal) angles.
*An isosceles triangle has at least two congruent sides and, as a result, at
least two congruent (equal) angles.
*A scalene triangle has no congruent sides and therefore no angles are
congruent (equal) in measure.
*The sum of the measures of the three angles in every triangle is 180°.
Find the measure of the missing angle in the triangle below and classify
the triangle.
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62°
28°
Mathematics Skills 2
Since the three angles in a triangle add up to 180°, the missing angle must
measure 90° because 62° + 28° = 90°
90° + ?° = 180°. ? = 180° – 90° = 90°
Therefore, the triangle on the previ ous page can be classified as a right triangle
since it has one angle that measures 90°
Naming and Classifying Two- and Three-Dimensional Shapes
A polygon is a closed figure with three or more sides. Polygons that have all
sides congruent are called regular polygons. In the same manner, polygons
that have sides that are not congruent are called irregular polygons.
Previously we discussed triangles, which are three-sided polygons. Here, we
will discuss other polygons, which are also two-dimensional figures.
The first of the polygons we will discuss is a quadrilateral. A quadrilateral is a
polygon that has four sides, four angles, and therefore four vertices.
The sum of the angles of every quadrilateral is 360°.
There are several different types of quadrilaterals: square, rectangle, rhombus,
trapezoid, and parallelogram. Each of these is a quadrilateral, but has a specific
name based on its characteristics.
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Mathematics Skills 2
Square
All four sides are
congruent and
all four angles
are congruent.
Parallelogram
Opposite sides are
congruent/parallel & opposite
angles are congruent.
Trapezoid
One pair of opposite sides
is parallel.
Example 1:
Rhombus
All four sides are congruent.
Opposite angles are equal.
Opposite sides are parallel.
Name the given quadrilateral.
Find the missing angle.
Is the quadrilateral a square, rhombus, rectangle, parallelogram, and/or a
trapezoid?
G
3 cm
105°
H
3 cm
3 cm
75°
J
105°
3 cm
I
The quadrilateral above is named, like all polygons, by the letters of the
vertices.
This quadrilateral can be named as GHIJ or HIJG or JIHG.
The quadrilateral can be named any way as long as the letters are in sequence.
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Mathematics Skills 2
In order to find the missing angle we can add the angles that are given and
subtract from 360° since the sum of measure of all the angles of a quadrilateral
is 360°. Therefore, 105° + 75° +105° = 285°.
360° - 285° = 75°.
Quadrilateral GHIJ is a parallelogram because opposite sides are parallel,
opposite sides are equal, and it is a rhombus because of these characteristics
as well as that all four sides are congruent (equal).
Every polygon has the same amount of sides as it does angles and vertices. For
example, a triangle has 3 sides, 3 angles, and 3 vertices. A quadrilateral has 4
sides, 4 angles, and 4 vertices. This is true for all polygons.
Types of Polygons
Quadrilateral
4 sides
Triangle
3 sides
Hexagon
6 sides
Pentagon
5 sides
Octagon
8 sides
A polygon with seven sides is called a heptagon. A polygon with nine sides is
called a nonagon and a polygon with ten sides is called a decagon. In general,
a figure with n sides is called an n-gon.
A diagonal (a line segment drawn in a polygon from one non-consecutive
vertex to another) can be drawn from one vertex to form as many triangles as
possible.
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Mathematics Skills 2
You learned earlier that the angle sum of a triangle is 180°. In a quadrilateral,
we can draw a diagonal from one vertex to form two triangles, and therefore the
angle sum of 360°.
∠ sum = 180°
180°
180°
∠ sum = 180°
∠ sum = 180°
180°
In a pentagon, we can draw two diagonals from one vertex to form three
triangles. Therefore 180° x 3 triangles = 540°. The sum of the measures of the
angles in a pentagon is 540°.
In addition, we can continue this same pattern to find the sum of measures of all
polygons or simply use the formula for finding the sum of the angles of all
polygons (n – 2) x 180°
where n is the number of sides of the polygon.
To find the measure of each angle of a regular polygon, you can use the
formula above and divide by the number of sides.
(n – 2) x 180°
n
where n is the number of sides of the regular polygon.
Try the next example on your own and check your answers with the book.
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Mathematics Skills 2
A dodecagon is a polygon with 12 sides.
What is the sum of the angles of a dodecagon?
What is the measure of each angle in the regular dodecagon?
To find the angle sum of a dodecagon we apply the formula:
(n
–2 ) x 180°.
Since a dodecagon has 12 sides, (12 – 2) x 180° = 1800°.
To find the measure of each angle we use the formula:
(n – 2) x 180° or simply divide 1800° by 12 sides = 150°.
n
Therefore, the angle sum of a dodecagon is 1800°
and each angle in a regular dodecagon is 150°.
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Lesson 2: Ratios and Proportions
A ratio compares two quantities. A ratio can be written in three different forms.
a to b
a:b
a
b
In any instance, the ratio must be simplified and written in lowest terms.
Example 1: A flowerpot contains 3 pink flowers, 5 yellow, 2 white, and 4
red flowers. Write each situation as a ratio in simplest form.
Pink flowers to white flowers:
3 to 2, 3:2, or 3/2
White flowers to red flowers:
2 to 4 = 1 to 2, 1:2, or ½
Yellow flowers to pink flowers: 5 to 3, 5:3, or 5/3
Notice that, even when a fraction is improper (the numerator is larger than the
denominator), we do not write the fraction or ratio as a mixed number. Keep in
mind that a ratio compares two quantities and if we were to write it as a mixed
number, it would not compare the two quantities.
Also note that once we write the ratio in simplest form, we can write it in all the
other forms already simplified. It is not necessary to write the ratio in all three
forms each time, but it is shown here to show that it can be written in any of the
three forms.
When ratios are equivalent, they form a proportion. You can check to see if
two ratios are equivalent by checking their cross products.
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Mathematics Skills 2
Look:
If we want to find out if the ratio of pink to white flowers is equivalent to yellow to
pink flowers, we simply set up each ratio and find their cross product. If their
cross products are equivalent, then the two ratios DO form a proportion.
Pink to White Flowers =
3
5 = Yellow to Pink Flowers
2
3
To determine their cross products, multiply the numerator of one ratio with the
denominator of the other ratio and vice versa and then determine if the two
products are equal.
2 x 5 = 10 and 3 x 3 = 9. These cross products are not equivalent; therefore,
these ratios do not form a proportion.
Let’s take a look at some examples.
Example 2:
Determine whether the following ratios form a proportion.
4
3
12
9
6
25
4
117
18
5
9
3
Let us take the first example. Let us find their cross product. 3 x 12 = 36 and 9
x 4 = 36.
Since both cross products are equivalent, the first example is a proportion.
Let’s take the second example. 25 x 4 = 100 and 117 x 6 = 702. Since these
cross products are not equal, these ratios do not form a proportion.
Let us see the last example. 5 x 9 = 45 and 3 x 18 = 54. Since these cross
products are not equal, these ratios do not form a proportion.
In the same way, you can use a proportion to find a missing quantity in a ratio.
To do this, you must set up two ratios with the information you have and what
you wish to find out.
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Mathematics Skills 2
Let us look at an example
Example 3: ABC Cellular charges $0.20 per minute if you talk past 300
minutes a month. Last month you talked for 365 minutes. How much do
you have to pay for the additional minutes of airtime?
If you notice, we are comparing two quantities. We know that one
additional minute costs $0.20. We want to find out how much 65
additional minutes cost. We can set up a proportion.
When setting up a proportion you must be careful to set up the
equivalent ratios in the same form. For example:
Number of minutes
Amount
1
=
65
$0.20
x
In each ratio, we set the number of minutes as the numerator and the dollar
amount in the denominator.
Now we can cross-multiply and divide (as we do with equivalent fractions) to
solve for x (the cost for 65 minutes).
$0.20 x 65 = 1.x = $13
x = $13/1 = $13
Therefore the cost for 65 minutes is $13.
We could also place the amount in the numerator and the number of
minutes in the denominator, but we would need to keep the second
ratio in that same order.
Amount
Number of minutes
$0.20
1
=
x
65
When we cross-multiply and divide, we obtain the same answer.
$0.20 x 65 = 1.x = $13
x = $13/1 = $13
Regardless of how you set up the ratios, the numerator and denominator must
remain consistent in both ratios to create equivalent fractions, i.e., a proportion.
33
Mathematics Skills 2
Congruent figures are figures that have the same shape and same size.
Similar figures are figures that also have the same shape, but not necessarily
the same size. Even though the size of the figures need not be exactly the
same, they do however need to be in proportion.
Let us look at an example.
Example 1:
Determine if the figures below are similar.
1 cm
1cm
2 cm
2 cm
1.5 cm
3 cm
The figures above are similar because they are the same shape (triangles) and
their sides are in proportion.
Smaller triangle -------1 cm = 1 cm
1 cm = 1.5 cm
Larger triangle ------- 2 cm
2 cm
2 cm
3 cm
When taking the cross product of each of the sides we find that the ratios are
equivalent. In this example, the sides of the larger triangle are twice the size of
the corresponding sides of the smaller triangle. Also the smaller is half the size
the larger. Moreover, this is true for each of the sides of the triangles.
*When checking to see if the sides are in proportion, we need to check all
the sides to make sure that they are in proportion. If the figures are
triangles, we must verify all three sides. If the figures are quadrilaterals,
we must verify all four sides. If the figures are pentagons, we verify all five
sides and so on.
34
Mathematics Skills 2
Lesson 2a Ratios, Proportions, and Problem Solving
In this lesson you learn how to write a ratio in a variety of ways. You also
learn how to use ratios to set-up and solve proportions. In addition, you
learn how to apply these concepts in real-world applications including
similar figures and scale factors.
A ratio compares two quantities. A ratio can be written in three different forms.
a to b
a:b
a
b
In any instance, the ratio must be simplified and written in lowest terms.
Let us look at some examples.
Example 1: A flowerpot contains 3 pink flowers, 5 yellow, 2 white, and 4
red flowers. Write each situation as a ratio in simplest form.
Pink flowers to white flowers:
3 to 2, 3:2, or 3/2
White flowers to red flowers:
2 to 4 = 1 to 2, 1:2, or ½
Yellow flowers to pink flowers: 5 to 3, 5:3, or 5/3
35
Mathematics Skills 2
Notice that even when a fraction is improper (the numerator is larger than the
denominator), we do not write the fraction or ratio as a mixed number. Keep in
mind that a ratio compares two quantities and if we were to write it as a mixed
number, it would not compare the two quantities.
Also note that once we write the ratio in simplest form, we can write it in all the
other forms already simplified. It is not necessary to write the ratio in all three
forms each time, but it is shown here to show that it can be written in any of the
three forms.
When ratios are equivalent, they form a proportion. You can check to see if
two ratios are equivalent by checking their cross products.
Look:
If we want to find out if the ratio of pink to white flowers is equivalent to yellow to
pink flowers, we simply set up each ratio and find their cross product. If their
cross products are equivalent, then the two ratios DO form a proportion.
Pink to White Flowers =
3
5 = Yellow to Pink Flowers
2
3
To determine their cross products, multiply the numerator of one ratio with the
denominator of the other ratio and vice versa and then determine if the two
products are equal.
2 x 5 = 10 and 3 x 3 = 9. These cross products are not equivalent; therefore,
these ratios do not form a proportion.
36
Mathematics Skills 2
Let’s take a look at some examples.
Example 2:
Determine whether the following ratios form a proportion.
4
3
12
9
6
25
4
117
18
5
9
3
Let us take the first example. Let us find their cross product. 3 x 12 = 36 and 9
x 4 = 36.
Since both cross products are equivalent, the first example is a proportion.
Let’s take the second example. 25 x 4 = 100 and 117 x 6 = 702. Since these
cross products are not equal, these ratios do not form a proportion.
Let us see the last example. 5 x 9 = 45 and 3 x 18 = 54. Since these cross
products are not equal, these ratios do not form a proportion.
In the same way, you can use a proportion to find a missing quantity in a ratio.
To do this, you must set up two ratios with the information you have and what
you wish to find out.
37
Mathematics Skills 2
Let us look at an example.
Example 3: ABC Cellular charges $0.20 per minute if you talk past 300
minutes a month. Last month you talked for 365 minutes. How much do
you have to pay for the additional minutes of airtime?
If you notice, we are comparing two quantities. We know that one additional
minute costs $0.20. We want to find out how much 65 additional minutes cost.
We can set up a proportion.
When setting up a proportion you must be careful to set up the
equivalent ratios in the same form. For example:
Number of minutes
Amount
1
=
$0.20
65
x
In each ratio, we set the number of minutes as the numerator and the dollar
amount in the denominator. Now we can cross-multiply and divide (as we do
with equivalent fractions) to solve for x (the cost for 65 minutes).
$0.20 x 65 = 1.x = $13
x = $13/1 = $13
Therefore the cost for 65 minutes is $13.
We could also place the amount in the numerator and the number of minutes in
the denominator, but we would need to keep the second ratio in that same
order.
Amount
$0.20
Number of minutes
1
=
x
65
When we cross-multiply and divide, we obtain the same answer.
$0.20 x 65 = 1.x = $13
x = $13/1 = $13
Regardless of how you set up the ratios, the numerator and denominator must
remain consistent in both ratios to create equivalent fractions, i.e., a proportion.
38
Mathematics Skills 2
Try the next example on your own and check your answers with the book.
Example 4:
An employee earns $100 for 8 hours of work.
How much would she earn for 36 hours of work?
First, let us set up the proportion.
$100
Amount earned
8
Hours of work
Cross-multiply $100 x 36 = 8 x
=x
36
x = $3600/8 = $450
The amount of money earned by the employee is $450.
* Notice how the $100 and the 8 hours are part of the same ratio since the
employee earns $100 in 8 hours and not in 36 hours. This is also important
when setting up a proportion; make sure that you compare the quantities that
relate to one another.
Practice
Write the ratio as a fraction in simplest form.
1. 75:300
a.
1
5
b.
1
4
c. 4 d.
1
3
Determine whether the ratios are equivalent.
2.
10
and
17
a. yes
30
68
b. no
3. Solve the proportion.
x
30
=
11
55
a. 6
b. 8
c. 20.2
d. 3
39
Mathematics Skills 2
Solve each problem.
4. Carmen can type 18 words in 27 seconds. How many words can she type in
60 seconds?
a. 25
b. 90
c. 40
d. 46
5. Josiah traveled 196 miles on 8 gallons of gasoline. At this rate, how many
gallons of gasoline would he need to travel 343 miles?
a. 9
b. 14
c. 21
d. 16.5
Answers to practice:
1. b.
1
4
Lesson 2b
2. b. no
3. a. 6
4. c. 40
5. b. 14
Similar Figures and Scale Factors
Congruent figures are figures that have the same shape and same size.
Similar figures are figures that also have the same shape, but not necessarily
the same size. Even though the size of the figures need not be exactly the
same, they do however need to be in proportion.
Let us look at an example.
Example 1:
Determine if the figures below are similar.
1 cm
1cm
1.5 cm
2 cm
2 cm
3 cm
The figures above are similar because they are the same shape (triangles) and
their sides are in proportion.
40
Mathematics Skills 2
Smaller triangle -------1 cm = 1 cm
1 cm = 1.5 cm
Larger triangle ------- 2 cm
2 cm
2 cm
3 cm
When taking the cross product of each of the sides we find that the ratios are
equivalent. In this example, the sides of the larger triangle are twice the size of
the corresponding sides of the smaller triangle. Also, the smaller is half the size
the larger. Moreover, this is true for each of the sides of the triangles.
*When checking to see if the sides are in proportion, we need to check all the
sides to make sure that they are in proportion. If the figures are triangles, we
must verify all three sides. If the figures are quadrilaterals, we must verify all
four sides. If the figures are pentagons, we verify all five sides and so on.
Try the next example on your own and check your answers with the book.
Example 2:
Determine if the figures below are similar.
2.5 in.
5 in.
7 in.
10 in.
Recall that in order for two figures to be similar, they must be the same shape
and their sizes (or dimensions) need to be in proportion to one another.
These two figures are the same shape (rectangles). Now let us verify that the
sides are in proportion. Because these are rectangles, we do not need to check
all 4 sides since opposite sides are equal (congruent). We only need to check
the two sides given.
2.5
in = 1
5 in
2
7 in ≠ 1
10 in
2
41
Mathematics Skills 2
The sides of the rectangles are not in proportion. If we take the cross products
of the two fractions, we find that they are not equal or in proportion to one
another.
2.5
in ≠ 7 in
5 in
10 in
5 x 7 = 35 and 2.5 x 10 = 25.
Therefore, these rectangles are not similar because although they are the same
shape, their sides are not in proportion to one another.
*Remember that, in order for the shapes to be similar, all sides need to be in
proportion, not just one pair of sides.
Scale Factors
A scale factor is a ratio that indicates the sizes of two similar figures.
In example 1 above we saw that the two triangles were similar and the larger
was twice the size of the smaller or the smaller half the size the larger. This
indicates that these triangles had a scale factor of 2.
When a figure is enlarged from the original size, the scale factor is more than
1. If the figure is reduced from its original size the scale factor is less than 1.
Moreover, when the two figures are congruent or the same size and the same
shape, the scale factor is 1.
42
Mathematics Skills 2
Let us look at an example.
Example 1:
What is the scale factor for the similar figures below?
9
3
9
3
2.5
2.5
4
7
7
1
In this example we do not need to check to see if the sides are in proportion
because the question TELLS us that the figures are similar. Therefore the sides
are in proportion. We know this because in order to find the scale factor, the
figures must be similar.
To find the scale factor, we decide which is the original figure. In this case, let’s
assume that first (smaller) is the original and the larger is the new figure.
Now let’s make a ratio of one pair of corresponding sides.
New figure
= 9 = 3
Original figure
3
So the scale factor for these similar figures is 3. We could have chosen any pair
of corresponding sides and still have arrived at the same answer. Take a look.
New figure = 12 = 3 or
Original figure 4
New figure
Original figure
= 7.5 = 3
2.5
43
Mathematics Skills 2
Practice
Determine whether each pair of polygons is similar.
1.
12
4
12
36
a. yes
b. no
2.
20
37°
37°
15
20
16
25
53°
53°
12
a. yes
b. no
Each pair of polygons is similar. Write a proportion to find each missing
measure. Then solve.
44
Mathematics Skills 2
3.
45.9
21.6
x
17
a.
8
b. 17.5
c. 9
d. 36.1
Solve
4. Coach Henderson is drawing a scale model of a basketball court on a
chalkboard so he can diagram plays during timeouts. The court is 94 feet
long and 50 feet wide. If the model of the basketball court is 5 inches wide,
how long is the model?
a. 12.1 in.
b. 2.7 in.
c. 9.4 in.
d. 7.8 in.
5. An architect’s blueprint for a house shows the master bedroom being 4.5
inches wide. If the scale used to create the blueprint is 0.25 in. = 1 ft, what is
the actual width of the master bedroom?
a. 14.5 ft
b. 21.8 ft
c. 4.5 ft
d. 18 ft
Answers to practice:
1.
a. yes
2.
a. yes
3.
a. 8
4.
c. 9.4 in
5.
d. 18 ft
45
Mathematics Skills 2
LESSON 2 THINGS TO REMEMBER
A ratio compares two quantities. A ratio can be written in three different forms.
a to b
a:b
a
b
In any instance, the ratio must be simplified and written in lowest terms.
Example 1: A flowerpot contains 3 pink flowers, 5 yellow, 2 white, and 4
red flowers. Write each situation as a ratio in simplest form.
Pink flowers to white flowers:
3 to 2, 3:2, or 3/2
White flowers to red flowers:
2 to 4 = 1 to 2, 1:2, or ½
Yellow flowers to pink flowers:
5 to 3, 5:3, or 5/3
Notice that even when a fraction is improper (the numerator is larger than the
denominator), we do not write the fraction or ratio as a mixed number. Keep in
mind that a ratio compares two quantities and if we were to write it as a mixed
number, it would not compare the two quantities.
Also note that once we write the ratio in simplest form, we can write it in all the
other forms already simplified. It is not necessary to write the ratio in all three
forms each time, but it is shown here to show that it can be written in any of the
three forms.
46
Mathematics Skills 2
When ratios are equivalent, they form a proportion. You can check to see if
two ratios are equivalent by checking their cross products.
Look:
If we want to find out if the ratio of pink to white flowers is equivalent to yellow to
pink flowers, we simply set up each ratio and find their cross product. If their
cross products are equivalent, then the two ratios DO form a proportion.
Pink to White Flowers =
5 = Yellow to Pink Flowers
3
3
2
To determine their cross products, multiply the numerator of one ratio with the
denominator of the other ratio and vice versa and then determine if the two
products are equal.
2 x 5 = 10 and 3 x 3 = 9. These cross products are not equivalent; therefore,
these ratios do not form a proportion.
Let’s take a look at some examples.
Example 2: Determine whether the following ratios form a proportion.
3
4
12
9
25
6
4
117
18
5
9
3
Let us take the first example. Let us find their cross product.
3 x 12 = 36 and 9 x 4 = 36.
Since both cross products are equivalent, the first example is a proportion.
Let’s take the second example. 25 x 4 = 100 and 117 x 6 = 702. Since these
cross products are not equal, these ratios do not form a proportion.
Let us see the last example. 5 x 9 = 45 and 3 x 18 = 54. Since these cross
products are not equal, these ratios do not form a proportion.
47
Mathematics Skills 2
In the same way, you can use a proportion to find a missing quantity in a ratio.
To do this, you must set up two ratios with the information you have and what
you wish to find out. Let us look at an example.
Example 3:ABC Cellular charges $0.20 per minute if you talk past 300
minutes a month. Last month you talked for 365 minutes. How much do
you have to pay for the additional minutes of airtime?
If you notice, we are comparing two quantities. We know that one additional
minute costs $0.20. We want to find out how much 65 additional minutes cost.
We can set up a proportion. When setting up a proportion you must be careful
to set up the equivalent ratios in the same form. For example:
Number of minutes
Amount
1
=
$0.20
65
x
In each ratio, we set the number of minutes as the numerator and the dollar
amount in the denominator.
Now we can cross-multiply and divide (as we do with equivalent fractions) to
solve for x (the cost for 65 minutes).
$0.20 x 65 = 1.x = $13
x = $13/1 = $13
Therefore the cost for 65 minutes is $13.
We could also place the amount in the numerator and the number of minutes in
the denominator, but we would need to keep the second ratio in that same
order.
Amount
$0.20
Number of minutes
1
=
x
65
When we cross-multiply and divide, we obtain the same answer.
$0.20 x 65 = 1.x = $13
x = $13/1 = $13
Regardless of how you set up the ratios, the numerator and denominator must
remain consistent in both ratios to create equivalent fractions, i.e., a proportion.
48
Mathematics Skills 2
Congruent figures are figures that have the same shape and same size.
Similar figures are figures that also have the same shape, but not necessarily
the same size. Even though the size of the figures need not be exactly the
same, they do however need to be in proportion.
Let us look at an example.
Example 1:
Determine if the figures below are similar.
1 cm
1cm
2 cm
2 cm
1.5 cm
3 cm
The figures above are similar because they are the same shape
(triangles) and their sides are in proportion.
Smaller triangle -------1 cm = 1 cm
1 cm = 1.5 cm
Larger triangle ------- 2 cm
2 cm
2 cm
3 cm
When taking the cross product of each of the sides we find that the ratios are
equivalent. In this example, the sides of the larger triangle are twice the size of
the corresponding sides of the smaller triangle. Also, the smaller is half the size
the larger. Moreover, this is true for each of the sides of the triangles.
*When checking to see if the sides are in proportion, we need to check all the
sides to make sure that they are in proportion. If the figures are triangles, we
must verify all three sides. If the figures are quadrilaterals, we must verify all
four sides. If the figures are pentagons, we verify all five sides and so on.
49
Mathematics Skills 2
Example 2: Determine if the figures below are similar.
5 in.
2.5 in.
7 in.
10 in.
Recall that, in order for two figures to be similar, they must be the same shape
and their sizes (or dimensions) need to be in proportion to one another.
These two figures are the same shape (rectangles). Now let us verify that the
sides are in proportion.
Because these are rectangles, we do not need to check all 4 sides since
opposite sides are equal (congruent). We only need to check the two sides
given.
2.5
in = 1
5 in
2
7 in ≠ 1
10 in
2
The sides of the rectangles are not in proportion. If we take the cross products of
the two fractions, we find that they are not equal or in proportion to one another.
2.5
in ≠ 7 in
5 in
10 in
5 x 7 = 35 and 2.5 x 10 = 25.
Therefore, these rectangles are not similar because although they are the same
shape, their sides are not in proportion to one another.
*Remember that, in order for the shapes to be similar, all sides need to be in
proportion, not just one pair of sides.
50
Mathematics Skills 2
Scale Factors
A scale factor is a ratio that indicates the sizes of two similar figures.
In Example 1 above we saw that the two triangles were similar and the larger
was twice the size of the smaller or the smaller half the size the larger. This
indicates that these triangles had a scale factor of 2.
When a figure is enlarged from the original size, the scale factor is more than
1. If the figure is reduced from its original size the scale factor is less than 1.
Moreover, when the two figures are congruent or the same size and the same
shape, the scale factor is 1. Let us look at an example.
Example 1:
What is the scale factor for the similar figures below?
9
3
9
3
2.5
2.5
4
7.5
7.5
1 sides are in proportion
In this example, we do not need to check to see if the
because the question TELLS us that the figures are similar. Therefore the sides
are in proportion. We know this because in order to find the scale factor, the
figures must be similar.
To find the scale factor, we decide which is the original figure. In this case, let’s
assume that first (smaller) is the original and the larger is the new figure.
51
Mathematics Skills 2
Now let’s make a ratio of one pair of corresponding sides.
New figure
= 9 = 3
Original figure
3
So the scale factor for these similar figures is 3. We could have chosen any pair
of corresponding sides and still have arrived at the same answer. Take a look.
New figure
Original figure
52
= 12 = 3 or
4
New figure
Original figure
= 7.5 = 3
2.5
Mathematics Skills 2
Lesson 3: Measurement
In this lesson you will learn how to convert units of measure within the
same system. You also learn how to calculate the perimeter and area of
several polygons as well as find the volume of three-dimensional figures.
The Metric System
The most common system of measurement in the world is the metric system.
The metric system of measuring is based on powers of ten, i.e., 10, 100 or 102
or ten to the second power, 1000 or 103 or ten to the third power, and so on..
This system is most commonly used around the world because it is easy to
multiply and divide using powers of ten.
The basic measures include: distance, volume, and weight. Using the metric
system, distance is measured in meters, volume is measured in liters, and
weight is measured in grams.
Prefixes in the metric system are the same whether you are measuring
distance, volume, or weight. They are:
milli- meaning one thousandth (1/1000)
centi- meaning one hundredth (1/100)
kilo- meaning one thousand (1000)
The following are Metric Equivalents to help you convert within the metric
system.
53
Mathematics Skills 2
Distance
1km = 1000 m
= 100,000 cm
= 1,000,000 mm
Weight
Volume
1kL = 1000 L
1kg = 1000 g
= 100,000 cL
= 100,000 cg
= 1,000,000 mL
= 1,000,000 mg
1 m = 100 cm
= 1000 mm
1L
1 cm = 10 mm
1 cL = 10 mL
1 cg = 10 mg
1 mL = 0.1 cL
= 0.001 L
1 mg = 0.1 cg
= 0.001 g
1 cm = 0.01 m
1 cL = 0.01 L
1cg = 0.01 g
1 m=0 001 km
1L
1g
1 mm = 0.1 cm
=
0.001 m
= 100 cL
= 1000 mL
= 0.001 kL
1g
= 100 cg
= 1000 mg
= 0.001 kg
When converting from a large unit of measure to a smaller unit of measure
you will multiply by the amount of smaller units within that larger unit.
When converting from a small unit of measure to a larger unit of measure you
will divide by the amount of smaller units within that larger unit.
For this lesson, Kilo- is the largest of the units and milli- is the smallest.
Let’s take a look at some examples.
Example 1:
Convert 7.36 km = ____ m.
In this example we are converting a large unit (kilo) into a smaller unit (meter),
therefore we multiply.
We are going to multiply 7.36 x 1000 because there are 1000 m in 1 kilometer.
7.36 x 1000 = 7360. Therefore, 7.36 km = 7,360 m.
54
Mathematics Skills 2
Example 2:
Convert 80 L = ____ kL.
In this example we are converting a smaller unit (Liter) into a larger unit (kilo),
therefore we divide.
We are going to divide 80 by 1,000 because there are 1000 L in 1 kiloliter.
80 ÷ 1000 = 0.08. Therefore, 80 L = 0.08 kL.
Example 3:
Convert 60,000 cg = ____kg.
In this example we are converting from a smaller unit (centi) to a larger unit
(kilo), therefore we divide.
First, to convert from centigrams to grams we divide by 100 since there are 100
centigrams in 1 gram. Then we divide again by 1000 because there are 1000 g
in 1kg.
First 60,000 ÷ 100 = 600
Then 600 ÷ 1000 = 0.6 kg
Therefore, 60,000 cg = 0.6 kg.
Try the next two examples on your own and check your answers with the book.
Example 4:
Convert 325 g = ____ mg.
Since grams are a larger unit of measure than milligrams, we multiply. We
multiply 325 x 1000 because there are 1000 mg in 1 g.
325
x 1000
= 325,000. Therefore, 325 g = 325,000 mg.
Example 5:
Convert 2.67 kL = ____ mL.
Since a kiloliter is a larger unit than a milliliter, we multiply.
First, to convert from kiloliter to liter we multiply by 1000 since there are 1000 L
= 1 kL. Then we multiply again by 1000 because there are 1000 mL = 1 L.
First 2.67 x 1000 = 2670
Therefore,
Then 2670 x 1000 = 2,670,000.
2.67 kL = 2,670,000 mL.
55
Mathematics Skills 2
Practice
Convert the following units of measure.
1. 5.3 m = ___ mm
a. 0.0053
b.
0.053
c.
53 d.
5300
0.42
c.
4.2 d.
420
6,780
c.
678,000 d.
6,780,000
0.008
c.
0.08
d.
0.8
432.6
d.
4326
2. 0.0042 kg = ___ g
a.0.042
b.
3. 6.78 km = ___ cm
a. 678
b.
4. 8 mL = ___ L
a. 0.0008
b.
5. How many millimeters are in 432.6 cm?
a. 4.326 b.
Answers to practice:
1. d. 5300
2. c. 4.2
3. c. 678,000
4. b. 0.008
5. d. 4326
56
43.26
c.
Mathematics Skills 2
The Customary System
In the United States, we use the customary system of measurement. This is
not based on powers of ten like the metric system. You will notice that the
customary system uses fractions, while the metric system uses only decimals.
The basic measures also include distance, volume, and weight, but the
customary system measures distance in inches, feet, yards, and miles; volume
is measured in fluid ounces, cups, pints, quarts, and gallons; and weight is
measured in ounces, pounds, and tons.
There are no convenient prefixes that can be used among the basic measures,
so you will need to refer to the table below when you practice converting among
units.
The following measurements will help you convert within the customary system.
Distance (length)
Volume
12 inches (in) = 1 foot (ft)
3 ft = 1 yard (yd)
36 in. = 1 yd
5280 ft = 1 mile (mi)
1760 yd = 1 mi
6076 ft =1 nautical mile
8 fluid ounces (fl oz) =1 cup (c)
2 c = 1 pint (pt)
2 pt = 1 quart (qt)
4 qt = 1 U. S. gallon (gal)
Weight
16 ounces (oz) =1 pound (lb)
2000 lb = 1 ton (T)
As in the metric system, when converting from a large unit of measure to a
smaller unit of measure you will multiply by the amount of smaller units within
that larger unit.
When converting from a small unit of measure to a larger unit of measure you
will divide by the amount of smaller units within that larger unit.
Then units above are listed in order from smallest to greatest in each measure.
57
Mathematics Skills 2
Let’s take a look at some examples.
Example 1:
Convert 4ft = ____ in.
Since feet is a larger unit of measure than inches and 12 in = 1ft, we will need to
multiply 4 x 12 to find out how many inches there are in 4 ft. 4 x 12 = 48.
Therefore, 4 ft = 48 in.
Example 2:
Convert 496 oz = ____ lb.
An ounce is a smaller unit of measure than a pound, therefore we will divide 496
by 16 since there are 16 oz = 1 lb. 496 ÷ 16 = 31. Therefore, 496 oz = 31 lb.
Example 3:
Convert 9 gal = ____ qt.
A gallon is a larger unit of measure than a quart, therefore we will multiply.
Since 4 qt = 1 gal. 9 x 4 = 36. Therefore, 9 gal = 36 qt.
Try the next two examples on your own and check your answers with the book.
Example 4:
Convert 10,560 ft = ___ mi.
A foot is a smaller unit of measure than a mile, therefore we will divide 10,5600
by 5,280,since 5,280 ft = 1 mi. 10,560 ÷ 5,280 = 2, therefore, 10,560 ft = 2 mi.
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Mathematics Skills 2
Practice
Convert the following units of measure.
1. 219
=
ft ___ yd
a.
2. 4 ft = ___ in
3. 3.5 =
gal ___ qt
14
b. 192
c.
4. 48 qt = ___ c
73
d. 48
e. 12
5. To change 16 quarts into gallons, you should
a. multiply by 2
b. multiply by 4
c. divide by 2
d. divide by 4
Answers to practice:
1.
c. 73
2.
d. 48
3.
a. 14
4.
b. 192
5.
d. divide by 4
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Mathematics Skills 2
Perimeter, Area, and Volume
Perimeter
The perimeter of a polygon is the sum of the lengths of the sides of the polygon.
The perimeter is often described as the ‘distance around’ the polygon. Keep in
mind that a polygon is simply a closed shape or figure with 3 or more sides. The
sides are line segments that are connected by vertices.
To find the perimeter of any polygon simply add all the sides (line segments).
Let’s take a look at some examples.
Example 1:
Find the perimeter of the triangle below.
13 in
12 in
17 in
To find the perimeter of this triangle we simply add all the sides.
12 in + 13 in + 17 in = 42 in. Therefore, the perimeter is 42 in.
Note that we have 3 addends in the example above because a triangle is a
polygon with 3 sides.
Example 2:
Find the perimeter of the polygon below.
5 cm
6.5 cm
6.5 cm
7 cm
7 cm
5 cm
To find the perimeter of the hexagon we simply add all 6 sides.
5 cm + 6.5 cm + 7 cm + 5 cm + 7 cm + 6.5 cm = 37 cm.
Therefore, the perimeter is 37 cm.
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Mathematics Skills 2
Try the next example on your own and check your answers with the book.
Example 3:
Find the perimeter of polygon below.
20 mm
8 mm
7 mm
8 mm
20 mm
To find the perimeter of the pentagon we simply add all 5 sides.
20 mm + 8 mm + 8 mm + 20 mm + 7 mm = 63 mm.
The one shape we did not discuss when finding the perimeter of polygons is a
circle. This is because a circle is not a polygon. Remember that a polygon is a
figure with 3 or more sides and a circle, as you know, does not have any sides.
Nonetheless, we can still find its perimeter. We name the perimeter of a circle
its circumference. The circumference of a circle is the distance around the
circle. And it is calculated by multiplying 2 x the radius of the circle x π or the
diameter x π.
Circumference = 2 x radius x π or diameter x π
C=2rπ
or
dπ
Radius
Diameter
If you notice, from the circle above, the diameter is twice the radius. π is a
symbol used in the measurement of a circle and is read as pi. π describes the
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Mathematics Skills 2
ratio of every circle’s circumference (or distance around) to its diameter and
has a value of 3.14… or 22/7.
Let’s take a look at an example.
Example 4:
What is the circumference of the circle below?
10 cm
The circle above has a diameter of 10 cm or a radius of 5 cm. We know this
because the radius is half the diameter or the diameter is twice the radius.
Then all we do is use the formula C = dπ = 10 x π = 10 x 3.14 = 31.4 cm.
We can also calculate the circumference using the radius.
C = 2 x 5 x π = 31.4 cm.
*Note that circumference, like perimeter, expresses units in linear form (not
squared).
Area
Area measures the size of a surface. Area is usually measured by the number
of square units of the same size that fit into the figure. Area is always
expressed in square units.
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Mathematics Skills 2
Let’s take a look at an example.
Example 5:
What is the area of the rectangle below?
To calculate the area of the rectangle above, we count the amount of unit
squares. There are a total of 12 unit squares; therefore, the area of the
rectangle is 12 square units.
Notice that the area is given in square units because we don’t know the exact
measurement of each square. If the measurement of each square would be in
2
inches, then the area of the rectangle would be 12 inches squared or 12 in . If
the measurement of each square were in centimeters, then the area of the
2
rectangle would be 12 cm .
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Mathematics Skills 2
You will notice that, instead of counting the amount of squares the rectangle
contains, we can also count how many unit squares wide and how many unit
squares long the rectangle contains and simply multiply. The rectangle in the
first example contains 4 unit squares wide and 3 unit squares long; therefore 4 x
3 = 12 unit squares.
From this we derive the formula for finding the area of a rectangle.
Area = Length x Width
Length
A=LxW
Width
If the rectangle is a square, the length and width are the same. Therefore, the
formula for calculating the area of a square is:
Area = Side x Side = s2
side
A = s x s = s2
side
where s is the measurement of each side.
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Mathematics Skills 2
Finding the area of a parallelogram is similar to finding the area of a rectangle.
The area of a parallelogram is:
Area = Base x Height
A=BxH
Height
Base
The base is always perpendicular to the height of the parallelogram.
Perpendicular is denoted by a small square, which means that the base and the
height make a 90° angle.
If you cut a parallelogram in half along the diagonal, you would have two
triangles with equal bases and equal heights.
B
H
H
H
B
B
Therefore, it makes sense to say that we can also cut the formula for the area of
a parallelogram in half to obtain the formula for the area of the triangle.
The area of a triangle is:
Area = Base x Height or A = ½ (BH)
2
B
H
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Mathematics Skills 2
*Note that, just as in the parallelogram; the height is always perpendicular to the
base.
The last polygon that we will discuss in this lesson is the trapezoid. A trapezoid
has two bases. We can model the trapezoid in the same way we did the
parallelogram to see how the formula for area is derived.
B1
B1
H
H
H
B2
B2
Therefore, it makes sense to say that we can take the formula for the area of a
triangle and include the additional base. We do not need to include the height
twice since there is only one height for the trapezoid.
The area of a trapezoid is:
Area = Height (Base 1 + Base 2) or A = ½h(b1 + b2)
2
B1
H
B2
*The subscript of 1 and 2 on B1 and B2 are read as “b sub 1” and “b sub 2”.
The area of a circle is found by using the formula:
Area = r2π
Radius
Diameter
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Mathematics Skills 2
If given an example when the diameter is given and not the radius, then you
must find the radius first and later apply the formula.
Let’s take a look at some examples.
Example 6:
Find the area of a triangle with a height of 4 cm and a
base of 7 cm.
To calculate the area of the triangle we use the formula
A = ½ (BH).
A = ½ (4 x 7) = ½ (28) = 14 cm2
Recall that area is always expressed in square units.
It is a common MISTAKE to calculate the area of a triangle and
forget to divide by 2 (or multiply by ½).
Example 7:
Find the area of a trapezoid with bases 13 yd and 15 yd
and a height of 17 yd.
The formula for calculating the area of a trapezoid is A = ½h(b1+b2).
Using the formula, 13 and 15 are the two bases and 17 is the height.
Therefore,
=
A
A
A = ½ (17)(13 + 15)
½ (17)(28)
= ½ (476)
= 238.
2
The area of trapezoid is 238 yd .
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Mathematics Skills 2
Example 8:
What is the area of a circle with a diameter of 18 in?
The formula for finding the area of a circle is A = r
π. In this example we are
2
given the diameter of the circle, therefore we must first find the radius to be able
to apply the formula. The radius of the circle is 9 in (1/2 of 18 in.).
Let’s apply the formula.
A = r2π = (9 in)2 (π) = (81 in2)(π) = 254.34 in2.
area of the circle is 254.34 in2.
The
*It is common to leave the answer in terms of π. Therefore, an equivalent
answer would be 81π in2.
Volume
Volume measures the space inside a three-dimensional figure. Volume is
usually measured by the number of cubic units of the same size that fit into the
three-dimensional figure. Volume is always expressed in cubic units.
Let’s take a look at an example.
Example 9:
What is the volume of the rectangular prism below?
To calculate the volume of the rectangular prism above, we count the
number of cubes. There is a total of 8 cubes; Therefore, the volume of the
rectangular prism is 8 cubic units.
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Mathematics Skills 2
*Notice that the volume is given in cubic units because we don’t know the exact
measurement of each cube. If the measurement of each cube would be in
inches, then the volume of the rectangular prism would be 8 inches cubed or
8 in3. If the measurement of each cube were in centimeters, then the volume of
the rectangular prism would be 8 cm3.
You will notice that, instead of counting the amount of cubes the rectangular
prism contains, we can also count how many unit cubes wide, how many cubic
units long, and how many cubic units high the rectangular prism is and simply
multiply. The rectangular prism in the previous example is 2 cubic units wide, 4
cubic units high, and 1 cubic unit deep; therefore 2 x 4 x 1 = 8 cubic units.
From this we derive the formula for finding the volume of a rectangular prism.
Volume = Length x Width x Height
Height
V=lxwxh
Length
Width
If the rectangular prism is a cube where the length, width, and height are the
same, the formula for calculating the volume of a cube is:
Volume = Side x Side x Side = s3
Side
V = s x s x s= s3
Side
Side
where s is the measurement of each side.
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Mathematics Skills 2
A cube is also a rectangular prism with the length, width, and height of equal
measure. We find the volume of all prisms the same way. The volume of any
prism also can be found by multiplying the area of the base x the height.
This is exactly how we find the volume of a cylinder. Multiply the area of a
circle which is A= r
π
2
by the height. Therefore, the formula for the volume of
a cylinder is:
Volume = (r2π)(height) or V = r2πh
Radius
V = r2πh
Height
The volume of a pyramid and a cone are somewhat different in that, in
addition to multiplying the area of the base by the height, you must also multiply
the product by ⅓ or Volume =
70
⅓ (Height)(Area of the Base).
Mathematics Skills 2
Let’s take a look at some examples.
Example 10:
Find the volume of each rectangular prism below.
10 in
10 in
8.2 in
20 in
10 in
7.5 in
To calculate the volume of the triangular prism above, we
find the area of the base, which is a triangle and then multiply by the height.
Area of the base=½ (8.2 in)(7.5 in)=30.75 in2 .
Then multiply by the area (30.75 in2) by the height (20 in) = 615.
3
Therefore, the volume of the triangular prism is 615 in .
To calculate the volume of the rectangular prism (cube), apply the formula
for volume of a cube.
3
V = L x W x H = 10 in x 10 in x10 in = 1,000 in .
3
*Keep in mind that volume is always expressed in cubic units, i.e., in .
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Mathematics Skills 2
Example 11:
Find the volume of each solid below.
41 in.
25 cm
18 in.
61 cm
22 cm
24 in.
To calculate the volume of the pyramid above, we
find the area of the base, which is a rectangle, multiply by the height,
and then divide by 3 (or multiply by ⅓).
2
Area of the base = L x W = (24 in)(18 in) = 432 in
2
3
Multiply area of base (432 in ) by height (41 in) = 17,712 in .
Now we can divide by 3 = 17,712 ÷ 3 = 5,904 in3.
Therefore, the volume of the pyramid is 5,904
in3.
To calculate the volume of the cone, we find the area of the base, which is a
circle, multiply by the height, and then divide by 3 (or multiply by ⅓).
Area of the base = r
π = (11 cm) 2 x (3.14) = 379.94 cm2
2
3
Multiply area of the base (379.94 cm2) by height (25 cm) = 9498.5 cm .
Now we can divide by 3 9498.5 ÷ 3 = 3166.17 cm3.
3
Therefore the volume of the cone is 3166.17 cm .
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Mathematics Skills 2
Practice
1. Find the area to the nearest tenth. Use 3.14 for π
a. 379.9 yd2
b. 168.9 yd2
c. 95 yd2
11 yd
d. 34.5 yd2
2. Find the area of this triangle. Round to the nearest tenth if necessary.
a. 80 ft2
b. 160 ft2
c. 24 ft2
2020ftft
d. 76 ft2
8 ft
8 ft
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Mathematics Skills 2
Find the volume of each solid. Round to the nearest tenth if necessary.
3.
a. 18.3 ft3
b. 100.8 ft3
c. 201.6 ft3
4 4ftft
d. 256 ft3
6.3 ftft
6.3
88ftft
4.
a. 1046.4 cm3
b. 731.8 cm3
5.5 cm
c. 266.1 cm3
7.7
7.7
cm cm
74
d. 133 cm3
Mathematics Skills 2
Find the volume of each solid. Round to the nearest tenth if necessary.
5.
a. 942.5 yd3
b. 1256.6 yd3
c. 251.3 yd3
d. 314.2 yd3
12 yd
5 yd
6.
a. 300 ft3
b. 80 ft3
c. 110 ft3
d. 100 ft3
10 ft
55ft ft
6 ft
6 ft
Find the circumference of the circle below. Round to the nearest tenth.
7.
a. 110.6 mm
b. 55.3 mm
17.6 mm
c. 27.6 mm
d. 35.2 mm
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Mathematics Skills 2
Find the perimeter of each polygon below.
8.
3.5 yd.
3.5 yd.
a. 14 yd
b. 18 yd
3.5 yd.
c. 600.25 yd
d. 600.25 yd2
4 yd.
18.5 cm
9.
16 cm
3.5 yd.
a.
60.5
b. 74.5 cm
16 cm
14 cm
c. 259 cm
d. 259 cm2
10 cm
10.
1 in.
2 in.
1 in.
a. in
1.5
b.
1 in.
1.5 in.
1 in.
2 in.
6 in.
9 in.
c. 11 in.
d.
15 in.
Answers to practice:
76
1. c. 95 yd2
6.d. 100 ft3
2. a. 80 ft2
7. a. 110.6 mm
3. c. 201.6 ft3
8. b. 18 yd
4. b. 731.8 cm3
9. a. 60.5 cm
5. . 314.2 yd3
10. c. 11 in.
Mathematics Skills 2
LESSON 3 THINGS TO REMEMBER
The Metric System
The most common system of measurement in the world is the metric system.
The metric system of measuring is based on powers of ten, i.e., 10, 100 or 102
or ten to the second power, 1000 or 103 or ten to the third power, and so on.
This system is most commonly used around the world because it is easy to
multiply and divide using powers of ten.
The basic measures include: distance, volume, and weight. Using the metric
system, distance is measured in meters, volume is measured in liters, and
weight is measured in grams.
Prefixes in the metric system are the same whether you are measuring
distance, volume, or weight. They are:
milli- meaning one thousandth (1/1000)
centi- meaning one hundredth (1/100)
kilo- meaning one thousand (1000)
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Mathematics Skills 2
The following are Metric Equivalents to help you convert within the metric
system.
Distance
1km = 1000 m
= 100,000 cm
= 1,000,000 mm
Weight
Volume
1kL = 1000 L
1kg = 1000 g
= 100,000 cL
= 100,000 cg
= 1,000,000 mL
= 1,000,000 mg
1 m = 100 cm
= 1000 mm
1 cm = 10 mm
1L
= 100 cL
= 1000 mL
1 cL = 10 mL
1g
1 mL = 0.1 cL
= 0.001 L
1 mg = 0.1 cg
= 0.001 g
1 cm = 0.01 m
1 cL = 0.01 L
1cg = 0.01 g
1 m=0 001 km
1L
1g
1 mm = 0.1 cm
=
0.001 m
= 0.001 kL
= 100 cg
= 1000 mg
1 cg = 10 mg
= 0.001 kg
When converting from a large unit of measure to a smaller unit of measure
you will multiply by the amount of smaller units within that larger unit.
When converting from a small unit of measure to a larger unit of measure you
will divide by the amount of smaller units within that larger unit.
For this lesson, Kilo- is the largest of the units and milli- is the smallest.
Let’s take a look at some examples.
Example 1:
Convert 7.36 km = ____ m.
In this example we are converting a large unit (kilo) into a smaller unit (meter),
therefore we multiply.
We are going to multiply 7.36 x 1000 because there are 1000 m in 1 kilometer.
7.36 x 1000 = 7360. Therefore, 7.36 km = 7,360 m.
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Mathematics Skills 2
Example 2:
Convert 80 L = ____ kL.
In this example we are converting a smaller unit (Liter) into a larger unit (kilo),
therefore we divide.
We are going to divide 80 by 1,000 because there are 1000 L in 1 kiloliter.
80 ÷ 1000 = 0.08. Therefore, 80 L = 0.08 kL.
Try the next two examples on your own and check your answers with the book.
Example 3:
Convert 2.67 kL = ____ mL.
Since a kiloliter is a larger unit than a milliliter, we multiply.
First, to convert from kiloliter to liter we multiply by 1000 since there are 1000 L
= 1 kL. Then we multiply again by 1000 because there are 1000 mL = 1 L.
First 2.67 x 1000 = 2670
Therefore,
Distance (length)
12 inches (in) =
1 foot (ft)
3 ft = 1 yard (yd)
36 in. = 1 yd
5280 ft = 1 mile (mi)
1760 yd = 1 mi
6076 ft =
1 nautical mile
Then 2670 x 1000 = 2,670,000.
2.67 kL = 2,670,000 mL.
Weight
Volume
8 fluid ounces (fl oz) =
16 ounces (oz) =
1 cup (c)
1 pound (lb)
2 c = 1 pint (pt)
2000 lb = 1 ton (T)
2 pt = 1 quart (qt)
4 qt = 1 U. S. gallon (gal)
Convert 4ft = ____ in.
Since feet is a larger unit of measure than inches and 12 in = 1ft, we will need to
multiply 4 x 12 to find out how many inches there are in 4 ft. 4 x 12 = 48.
Therefore, 4 ft = 48 in.
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Mathematics Skills 2
Convert 496 oz = ____ lb.
An ounce is a smaller unit of measure than a pound, therefore we will divide 496
by 16 since there are 16 oz = 1 lb. 496 ÷ 16 = 31. Therefore, 496 oz = 31 lb.
Perimeter
The perimeter of a polygon is the sum of the lengths of the sides of the polygon.
The perimeter is often described as the ‘distance around’ the polygon. Keep in
mind that a polygon is simply a closed shape or figure with 3 or more sides. The
sides are line segments that are connected by vertices.
To find the perimeter of any polygon simply add all the sides (line segments
Find the perimeter of the triangle below.
12 in
13 in
17 in
To find the perimeter of this triangle we simply add all the sides.
12 in + 13 in + 17 in = 42 in. Therefore, the perimeter is 42 in.
Area
Area measures the size of a surface. Area is usually measured by the number
of square units of the same size that fit into the figure. Area is always
expressed in square units.
Find the area of a triangle with a height of 4 cm and a base of 7 cm.
To calculate the area of the triangle we use the formula
A = ½ (BH).
A = ½ (4 x 7) = ½ (28) = 14 cm2
Recall that area is always expressed in square units.
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Mathematics Skills 2
It is a common MISTAKE to calculate the area of a triangle and forget to
divide by 2 (or multiply by ½).
The circumference of a circle is the distance around the circle. And it is
calculated by multiplying 2 x the radius of the circle x π or the diameter x π.
Circumference = 2 x radius x π or diameter x π
C=2rπ
or
dπ
Radius
Diameter
If you notice, from the circle above, the diameter is twice the radius. π is a
symbol used in the measurement of a circle and is read as pi. π describes the
ratio of every circle’s circumference (or distance around) to its diameter and
has a value of 3.14… or 22/7.
Find the area of a trapezoid with bases 13 yd and 15 yd and a height of 17
yd.
The formula for calculating the area of a trapezoid is A = ½h(b1+b2). Using the
formula, 13 and 15 are the two bases and 17 is the height.
Therefore,
=
A
A
A = ½ (17)(13 + 15)
½ (17)(28)
= ½ (476)
= 238.
The area of trapezoid is 238 yd2.
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Mathematics Skills 2
What is the circumference of the circle below?
10 cm
The circle above has a diameter of 10 cm or a radius of 5 cm. We know this
because the radius is half the diameter or the diameter is twice the radius.
Then all we do is use the formula C = dπ = 10 x π = 10 x 3.14 = 31.4 cm.
We can also calculate the circumference using the radius.
C = 2 x 5 x π = 31.4 cm.
Volume
Volume measures the space inside a three-dimensional figure. Volume is
usually measured by the number of cubic units of the same size that fit into the
three-dimensional figure. Volume is always expressed in cubic units.
the volume of a cylinder. Multiply the area of a circle which is A= r2π by the
height. Therefore, the formula for the volume of a cylinder is:
Volume = (r2π)(height) or V = r2πh
●Radius
V = r2πh
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Height
Mathematics Skills 2
The volume of a pyramid and a cone are somewhat different in that, in
addition to multiplying the area of the base by the height, you must also multiply
the product by ⅓ or Volume = ⅓ (Height)(Area of the Base).
Find the volume of each rectangular prism below.
10 in
8.2 in
20 in
10 in
10 in
7.5 in
To calculate the volume of the triangular prism above, we find the area of the
base, which is a triangle and then multiply by the height.
Area of the base=½ (8.2 in)(7.5 in)=30.75 in2 .
Then multiply by the area (30.75 in2) by the height (20 in) = 615.
Therefore, the volume of the triangular prism is 615 in3.
To calculate the volume of the rectangular prism (cube), apply the formula
for volume of a cube.
V = L x W x H = 10 in x 10 in x10 in = 1,000 in3.
*Keep in mind that volume is always expressed in cubic units, i.e., in3.
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Mathematics Skills 2
Example: Find the volume of each solid below.
41 in.
25 cm
61 cm
18 in.
24 in.
22 cm
To calculate the volume of the pyramid above, we
find the area of the base, which is a rectangle,
multiply by the height,
and then divide by 3 (or multiply by ⅓).
Area of the base = L x W = (24 in)(18 in) = 432 in2
Multiply area of base (432 in2) by height (41 in) = 17,712 in3.
Now we can divide by 3 = 17,712 ÷ 3 = 5,904 in3.
Therefore, the volume of the pyramid is 5,904 in3.
To calculate the volume of the cone, we
find the area of the base, which is a circle,
multiply by the height,
and then divide by 3 (or multiply by ⅓).
Area of the base = r2π = (11 cm) 2 x (3.14) = 379.94 cm2
Multiply area of the base (379.94 cm2) by height (25 cm) = 9498.5 cm3.
Now we can divide by 3 9498.5 ÷ 3 = 3166.17 cm3.
Therefore the volume of the cone is 3166.17 cm3.
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Mathematics Skills 2
Lesson 4: Transformations and Coordinate Geometry
In this lesson, you learn about the Cartesian Coordinate Plane. You learn
how to graph points and identify ordered pairs. You also learn about
Transformations in Geometry. Transformations take place when a figure
on a plane is moved. A figure can move in a variety of ways; it can be
reflected over a line of symmetry, rotated about a point of origin, or
translated from one location to another.
Coordinate Geometry
You can use a coordinate plane to graph and locate points on a plane. A
coordinate plane is made up of two axes: the x-axis and the y-axis. The two
axes meet at a point called the origin. This is the zero point of both axes. The
two axes meet to form 90° angles at this point of origin.
Quadrant II
Quadrant I
Origin
(0, 0)
Quadrant III
x-axis
Quadrant IV
y-axis
To the right of the origin, on the x-axis, are the positive numbers and to the
left of the origin are the negative numbers (just as in a number line).
Above the origin on the y-axis are the positive numbers and below the
origin on the y-axis are the negative numbers.
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Mathematics Skills 2
You can locate or graph points on the coordinate plane by using ordered pairs.
Ordered pairs are used in determining a specific location on the Cartesian
Coordinate Plane. An ordered pair first tells the location of the point on the xaxis and then on the y-axis;
(x, y). This means that, when locating a point, you look at the first value in the
ordered pair – the x-coordinate – and then at the second – the y-coordinate.
A coordinate plane is separated into four quadrants as you can see above.
Quadrant I contains points where the x-coordinate is positive
and the y-coordinate is positive (example, 5,8).
Quadrant II contains points where the x-coordinate is negative
but the y-coordinate is positive (example, -5, 8).
Quadrant III contains points where both the x- and y-coordinate are negative
(example, -5, -8).
Quadrant IV contains points where the x-coordinate is positive
but the y-coordinate is negative (example, 5, -8).
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Mathematics Skills 2
Let’s take a look at some examples.
Example 1:
Answer the following questions below using the graph.
1.
What are the coordinates for point U? (5, 6)
2.
What point has ordered pair (2, 8)?
3.
What is the ordered pair for point S? (5, 8)
4.
Which point has coordinates (5, 3)?
Point R
Point V
To answer question 1, we begin at the origin and count how many lines to the
right (the x-coordinate) and then how many lines up (the y-coordinate) (5, 6) to
get to Point U.
To answer question 2, we begin at the origin and count 2 lines to the right
(since 2 is positive) and 8 lines up (since 8 is positive) and see Point R at this
location.
To answer question 3, we begin at the origin and count how many lines to the
right (the x-coordinate) and then how many lines up (the y-coordinate) (5, 8) to
get to Point S.
To answer question 4, we begin at the origin and count 5 lines to the right
(since 5 is positive) and 3 lines up (since 3 is positive) and see Point V at this
location.
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Mathematics Skills 2
Example 2:
Answer the following questions below using the graph.
.
Which point has coordinates (
.
–
2, 4)?
Point Q
( -2, -4)
Which are the coordinates of Point X?
This example is a little different because it involves all four quadrants, i.e.,
positive and negative values for the x- and y-coordinates, but we locate the
points and ordered pairs in exactly the same way.
To answer the first question, we begin at the origin and travel 2 lines to the left
(since 2 is negative) and then 4 lines up (since 4 is positive).
To answer the second question, we locate Point X. Then we begin at the origin
-
-
and travel left 2 lines ( 2) and then travel down 4 spaces ( 4). Therefore, the
ordered pair is (
88
-
2, -4).
Mathematics Skills 2
*Notice that, in each example, we always travel right or left first –
depending if the x-coordinate is positive or negative and then we travel up
or down – depending if the y-coordinate is positive or negative.
Practice
Use the coordinate grid to answer the questions.
–
–
1. Which point has coordinates ( 4, 3)?
a. J
b. K
c. N
d. Q
c. (2, 3)
d. (3, 2)
2. What are the coordinates of point P?
–
–
a. ( 2, 3) b.
-
( 2, 3)
3. What are the coordinates of point L?
–
–
a. ( 1, 2) b.
-
-
( 2, 1)
–
c. ( 1, 2)
–
d. ( 2, 1)
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4. Which statement best describes the points in Quadrant I?
a. The x-coordinate of each point is positive, and the y-coordinate is negative.
b. The x-coordinate of each point is negative, and the y-coordinate is positive.
c. The x- and y-coordinates of each point are negative.
d. The x- and y-coordinates of each point are positive.
5. Which point(s) are located in Quadrant III?
a. point
J
b. points K & L
c. points M & N
d. points P & Q
Answers to practice:
1. b. K
2. c. (2, 3)
3. a. (-1, -2)
4. d. The x- and y-coordinates of each point are positive.
5. b.2 points K
&L
Lesson
Reflections,
Rotations, and Translations
Lesson 4b
Reflections, Rotations, and Translations
Reflections
A reflection is one kind of a transformation. A reflection is the mirror image of
a point or a figure flipped over a line. This line is called the line of symmetry.
The point or figure is the same distance away from the line as is its reflection.
Think of standing in front of a mirror and imagine seeing the same object on the
other side of the mirror. In this case, the mirror is the line or plane of symmetry.
The object and its reflection are equal distances away from the mirror.
So that we don’t confuse the object and its reflection, we have different names
for each. For example, if an object of reflection is ΔABC then its reflection
would be named ΔA’B’C’. This notation (‘) is read as “prime”. When reflecting
a point or a polygon you can verify it is an exact image because both the image
and its reflection must be the same shape and the same size (congruent).
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Mathematics Skills 2
In order to verify that it has been reflected correctly, you can select several
points or vertices (in a polygon) of the image and count to see if the image and
its reflection are the same distance away from the line of symmetry.
Let’s take a look at an example.
Example 1: What is the line of symmetry about which the rectangle is
reflected? Explain how you can tell that the figure has been correctly
reflected.
y
5
4
3
2
B’
A
A’
B
1
–5
–4
–3
–2
–1
–1
1
2
3
4
5
x
–2
–3
–4
C’
D’
–5
D
C
The line of symmetry about which the rectangle has been reflected is the y-axis.
Rectangle ABCD has been correctly reflected because it is congruent to its
image A’B’C’D’ and each of the vertices in rectangle ABCD is the same distance
away from each of the corresponding vertices in rectangle A’B’C’D’.
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Mathematics Skills 2
In order to make a reflection of an image, simply fold your paper along the line
of symmetry. Then trace the figure onto the other side of the paper. Make sure
to label each part correctly as the image and the reflection of the image.
Try this next example on your own and check your answer with the book.
Example 2:
Reflect the image below along the line of symmetry.
y
6
5
B
4
C
3
2
1
A
–6
–5
–4
–3
–2
–1
–1
–2
D
1
2
3
4
A’
5
D’
6
x
–3
–4
B’
–5
C’
–6
In order to reflect this trapezoid along the line of symmetry, we can fold the
paper at the line of symmetry (which happens to be the x-axis). Then trace the
figure onto Quadrant IV. Keep in mind that AD is one line above the x-axis
therefore; A’D’ must be one line below the x-axis.
Also, B and C are 4 lines above the x-axis.
Therefore; B’ and C’ must be 4 lines below the x-axis.
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Rotations
A rotation or turn is a transformation that turns an object about a fixed point.
When the top of the figure turns to the right, the figure is turned clockwise.
When the top of the figure turns to the left, the figure is turned
counterclockwise.
When rotating a figure, we need to know three things:
1. the direction the figure is being rotated (clockwise or counterclockwise)
2. the point at which it is being rotated from
3. the degrees it is being rotated at (90°, 180°, 270°,…)
If a figure can be rotated less than 360°, and the rotation exactly matches the
original image, the figure is said to have rotational symmetry.
We can also specify the degree of rotational symmetry a figure has, i.e., a figure
can have 90° rotational symmetry, 180° rotational symmetry, and so on.
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Mathematics Skills 2
Let’s take a look at an example.
Example 3:
Find the coordinates of ΔABC after a 90°
counterclockwise rotation about the origin.
C’
B’
y
B (4,5)
5
B(4, 5)
4
3
C (5,2)
2
A’ 1
C(5, 2)
A(0, 0)
–5
–4
–3
–2
–1
–1
1
2
3
4
5
x
–2
–3
–4
–5
Recall that, when rotating a figure, we need three things: the direction, the
degrees, and the point at which we are rotating. We have all three. We need
to rotate the triangle 90° counterclockwise about the origin and then give the
new coordinates for each vertex.
To rotate the figure, it is easiest to draw the figure on your own paper. Then
rotate the paper 90° counterclockwise. Then trace the figure onto the original
sheet. We know that the figure will be on Quadrant II because we are rotating
90° counterclockwise about the origin. After the rotation we see that the
coordinates of ΔA’B’C’ are: A’ (0, 0), B’ ( -5, 4), C’ ( -2, 5).
*Note the similarities and differences in the coordinates of ΔABC and ΔA’B’C’.
A(0, 0) —> A´(0, 0)
–
B(4, 5) —> B´( 5, 4)
–
C(5, 2) —> C´( 2, 5)
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The x- and y-coordinates switched position and the x-coordinate is negative in
the rotated figure because the rotated figure lies in Quadrant II where the xcoordinate is negative.
What do you think would happen if we were to rotate the figure another
90°?
If we were to rotate the triangle another 90° the coordinates would return to
what they were in Quadrant 1 with the exception that both the x- and ycoordinates would be negative (since both the x and y are negative in Quadrant
III).
Translations
A translation is a transformation that slides an object without flipping
(reflecting) or turning (rotating). The slide is either up or down, right or left, or
diagonally. An object can be translated up and right, down and left, down and
right, or simply down or right.
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Mathematics Skills 2
Let’s take a look at an example.
Example 4: If triangle RST is translated 1 line right and 2 lines up,
what are the coordinates of the new figure?
y
6
5
S
4
3
2
1
–6
–5
–4
–3
–2
–1
–1
1
2
3
4
5
6
x
–2
R
T
–3
–4
–5
–6
To translate ΔRST 1 line right and 2 lines up we must shift all three vertices of
the triangle. When we do so, both the x- and y-coordinates shift. When shifting
1 line right, only the x-coordinates are changed because it is the only coordinate
that moves in that direction. Likewise, when shifting 2 lines up, only the ycoordinates are changed because it is the only coordinate that moves up and
down.
Therefore, the coordinates of the vertices of ΔRST are:
-
-
-
R ( 3, 2), S (0, 5), and T (3, 2).
After translating the new coordinates are:
-
R’ ( 2, 0), S’ (1, 7), and T’ (4, 0).
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Mathematics Skills 2
*Note that all we did was add 1 line to all the x-coordinates and add 2 lines
to the y-coordinates. In each instance we added because the two
translations were going in the positive direction: right and up. If it had
been left and down then we would have subtracted the amount of lines
designated by the move.
Practice
Identify the letter of the choice that best completes the statement or answers the
question.
1. Which of the following shows a reflection of the figure over the line?
a.
b.
c.
d.
2. Which type of transformation moved ΔABC to ΔA’B’C’?
a. 90
b. 180° rotation
c. reflection across both axes
d. translation
.
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3. If you translate the quadrilateral 5 lines right and 4 lines down, what will
be the coordinates of D’?
–
b. (3, 4)
a. (4, 1)
–
–
c. (4, 3)
d. ( 4,3)
4. Which grid shows a 90° counterclockwise rotation of the triangle DEF
about the origin?
a.
b.
c.
d.
Find the coordinates of the vertices of a quadrilateral after a reflection
over the y- axis.
–
–
–
5. quadrilateral ABCD with vertices A ( 6, 8), B ( 3, 0), C (1, 7), D ( 6, 4)
–
–
–
–
–
–
a. A´ ( 6, 8), B´ ( 3, 0), C´ (1, 7), D´ ( 6, 4)
–
b. A´ (6, 8), B´ (3, 0), C´ ( 1, 7), D´ (6, 4)
–
c. A´ (6, 8), B´ (0, 3), C´ (1, 7), D´ ( 6, 4)
–
–
–
d. A´ (8, 6), B´ (0, 3), C´ (7, 1), D´ (4, 6)
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Mathematics Skills 2
Answers to practice:
1. a.
2. d. translation
-
3. c. (4, 3)
4. b.
–
5. b. A´ (6, 8), B´ (3, 0), C´ ( 1, 7), D´ (6, 4)
LESSON 4 THINGS TO REMEMBER
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Mathematics Skills 2
LESSON 4 THINGS TO REMEMBER
Coordinate Geometry
You can use a coordinate plane to graph and locate points on a plane. A
coordinate plane is made up of two axes: the x-axis and the y-axis. The two
axes meet at a point called the origin. This is the zero point of both axes. The
two axes meet to form 90° angles at this point of origin.
Quadrant II
Quadrant I
Origin
(0, 0)
Quadrant III
x-axis
Quadrant IV
y-axis
To the right of the origin, on the x-axis, are the positive numbers and to the
left of the origin are the negative numbers (just as in a number line).
Above the origin on the y-axis are the positive numbers and below the
origin on the y-axis are the negative numbers.
You can locate or graph points on the coordinate plane by using ordered pairs.
Ordered pairs are used in determining a specific location on the Cartesian
Coordinate Plane. An ordered pair first tells the location of the point on the xaxis and then on the y-axis;
(x, y). This means that, when locating a point, you look at the first value in the
ordered pair – the x-coordinate – and then at the second – the y-coordinate.
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Mathematics Skills 2
A coordinate plane is separated into four quadrants.
Quadrant I contains points where the x-coordinate is positive
and the y-coordinate is positive (example, 5,8).
Quadrant II contains points where the x-coordinate is negative
but the y-coordinate is positive (example, -5, 8).
Quadrant III contains points where both the x- and y-coordinate are negative
(example, -5, -8).
Quadrant IV contains points where the x-coordinate is positive
but the y-coordinate is negative (example, 5, -8).
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Mathematics Skills 2
Answer the following questions below using the graph.
1. What are the coordinates for point U? (5, 6)
2. What point has ordered pair (2, 8)?
Point R
3. What is the ordered pair for point S? (5, 8)
4. Which point has coordinates (5, 3)?
Point V
To answer question 1, we begin at the origin and count how many lines to the
right (the x-coordinate) and then how many lines up (the y-coordinate) (5, 6) to
get to Point U.
To answer question 2, we begin at the origin and count 2 lines to the right
(since 2 is positive) and 8 lines up (since 8 is positive) and see Point R at this
location.
To answer question 3, we begin at the origin and count how many lines to the
right (the x-coordinate) and then how many lines up (the y-coordinate) (5, 8) to
get to Point S.
To answer question 4, we begin at the origin and count 5 lines to the right
(since 5 is positive) and 3 lines up (since 3 is positive) and see Point V at this
location.
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Mathematics Skills 2
Reflections
A reflection is one kind of a transformation. A reflection is the mirror image of
a point or a figure flipped over a line. This line is called the line of symmetry.
The point or figure is the same distance away from the line as is its reflection.
Think of standing in front of a mirror and imagine seeing the same object on the
other side of the mirror. In this case, the mirror is the line or plane of symmetry.
The object and its reflection are equal distances away from the mirror.
So that we don’t confuse the object and its reflection, we have different names
for each. For example, if an object of reflection is ΔABC then its reflection
would be named ΔA’B’C’. This notation (‘) is read as “prime”. When reflecting
a point or a polygon you can verify it is an exact image because both the image
and its reflection must be the same shape and the same size (congruent).
In order to verify that it has been reflected correctly, you can select several
points or vertices (in a polygon) of the image and count to see if the image and
its reflection are the same distance away from the line of symmetry.
In order to make a reflection of an image, simply fold your paper along the line
of symmetry. Then trace the figure onto the other side of the paper. Make sure
to label each part correctly as the image and the reflection of the image.
Rotations
A rotation or turn is a transformation that turns an object about a fixed point.
When the top of the figure turns to the right, the figure is turned clockwise.
When the top of the figure turns to the left, the figure is turned
counterclockwise.
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Mathematics Skills 2
When rotating a figure, we need to know three things:
1) the direction the figure is being rotated (clockwise or counterclockwise); 2)
the point at which it is being rotated from, 3) the degrees it is being rotated at
(90°, 180°, 270°,…). If a figure can be rotated less than 360°, and the rotation
exactly matches the original image, the figure is said to have rotational
symmetry.
We can also specify the degree of rotational symmetry a figure has, i.e., a figure
can have 90° rotational symmetry, 180° rotational symmetry, and so on.
Example 3: Find the coordinates of ΔABC
after a 90° counterclockwise rotation about
the origin.
Recall that, when rotating a figure, we need three things: the direction, the
degrees, and the point at which we are rotating. We have all three. We need
to rotate the triangle 90° counterclockwise about the origin and then give the
new coordinates for each vertex. To rotate the figure, it is easiest to draw the
figure on your own paper. Then rotate the paper 90° counterclockwise. Then
trace the figure onto the original sheet. We know that the figure will be on
Quadrant II because we are rotating 90° counterclockwise about the origin.
After the rotation we see that the coordinates of ΔA’B’C’ are: A’ (0, 0),
B’ ( -5, 4), C’ ( -2, 5).5).
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Mathematics Skills 2
*Note the similarities and differences in the coordinates of ΔABC and ΔA’B’C’.
A(0, 0) —> A´(0, 0)
B(4, 5) —> B´( –5, 4)
C(5, 2) —> C´( –2, 5)
The x- and y-coordinates switched position and the x-coordinate is negative in
the rotated figure because the rotated figure lies in Quadrant II where the xcoordinate is negative.
What do you think would happen if we were to rotate the figure another
90°?
If we were to rotate the triangle another 90° the coordinates would return to
what they were in Quadrant 1 with the exception that both the x- and ycoordinates would be negative (since both the x and y are negative in Quadrant
III).
Translations
A translation is a transformation that slides an object without flipping
(reflecting) or turning (rotating). The slide is either up or down, right or left, or
diagonally. An object can be translated up and right, down and left, down and
right, or simply down or right.
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Lesson 5: Algebra
In this lesson you learn how to write algebraic expressions and equations.
In addition, you will learn how to set-up and solve one- and two-step
equations.
Writing Algebraic Expressions and Equations
An algebraic expression consists of a number, a variable, or a number and
variable with operational symbols. A variable is a symbol, usually a letter, which
represents an unknown quantity. The term variable is used because its value
can vary from problem to problem. This means that the value of the variable can
be one quantity in one question and then another quantity in another.
We know the operational symbols to be + (addition), - (subtraction),
x (multiplication), and ÷ (division). In algebra we will often see
multiplication denoted by a dot (•) or simply a number next to a letter (5n
“five times n”). Likewise, we will often see division denoted as a fraction (5/n
“five divided by n”).
The following are examples of some expressions:
5+x
five plus the value of x
7n – 2
seven times the value of n minus 2
2y + 1/y
two times y plus one divided by y
Translating and Writing Algebraic Expressions
Some of the expressions above show addition, subtraction, multiplication,
division, or a combination of these. You may also have to translate from words
to algebraic expressions or algebraic expressions into words. Here is a guide to
help you translate these expressions.
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Mathematics Skills 2
Addition
Subtraction
Multiplication
ƒ More than
ƒ Less than
ƒ Product of
ƒ Plus
ƒ Subtracted from ƒ Times
ƒ Greater than
ƒ Minus
ƒ Multiplied by
ƒ The sum of
ƒ Less
ƒ Twice
ƒ Increased by
ƒ The difference
ƒ Half of
Division
ƒ Quotient of
ƒ Divided by
ƒ Decreased by
*Caution: you must be careful in subtraction when using “less than” and
“subtracted from” as the terms must be written in the correct order.
Let’s take a look at some examples.
Example 1:
a
Write an algebraic expression for each phrase.
The sum of a number and seven:
x+7
4 more than a number:
n+4
number decreased by 8:
6 less than a number z
the
one-third
product of 14 and a:
of a number n:
the quotient of a number divided by 9:
w-8
z-6
14a
⅓ n
s/9
*Note that, when a variable is not given, we can write any letter or symbol of
our choice to represent the unknown number.
The expressions in the example above all contain one operation. We can also
write expressions with more than one operation.
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Let’s take a look at some examples.
Example 2:
Write an algebraic expression for each phrase.
12 less than the product of 3 and a number:
twice
the difference between a number and 17:
3s - 12
2(x – 17)
5 times the sum of a number and 1:
5(f + 1)
2 subtracted from the quotient of 4 divided by x:
4/x - 2
* Note “less than” and “subtracted from” were written in reverse order of
appearance because they were being subtracted from the second term. Also,
the reason parenthesis were necessary in the 2nd and 3rd example was
because the multiplication takes place after the difference and the sum, (i.e.,
to express order of operations).
The same way that we translate phrases into algebraic expressions, we can
also translate sentences into algebraic equations. The only difference is that an
equation ALWAYS includes an equivalence symbol (equal sign).
Example 3:
Write an algebraic equation for the sentence:
5 less than 4 times a number is 4 more than twice the number.
This sentence is an equation because we note the word “is”. The word “is”
means =. It separates what is on the left side of the equal sign and what is on
the right side. Now all we have to do is translate the phrase before the word “is”
and place it before the = sign in our equation. Then we translate the phrase
after the word “is” and place it to the right of the = sign.
5 less than 4 times a number ⇒ 4n – 5
4 more than twice the number ⇒ 2n + 4
Therefore the equation equivalent to the sentence above is: 4n – 5 = 2n.+ 4
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Mathematics Skills 2
*Note that we used the same variable to represent the same number in the
equation. This is because we are referring to the same unknown number in the
sentence.
Several different words or phrases are used to represent the equal (=) sign.
They are:
The same as
Equals
=
Is equal to
Is
The result is
Try the next examples on your own and check your answer with the book.
Example 4:
Write an algebraic equation for the sentence:
1 added to the quotient of a number divided by 6
is the same as 9 less than the number.
First locate the equivalence of the equation and then let’s identify the left and
right side.
“Is the same as” describes the equal sign in this example.
This means that the phrase before it is the left side of the equation and the
phrase after it is the right side.
1 added to the quotient of a number and 6 ⇒ y/6 + 1
9 less than the number ⇒ y - 9
Therefore the equation equivalent to the sentence above is: y/6 + 1 = y - 9.
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Mathematics Skills 2
Example 5:
Write an algebraic equation for the sentence:
If 7 is subtracted from twice a number, the result is 15.
First locate the equivalence of the equation and then let’s designate the left and
right side.
“The result is” describes the equal sign in this example.
This means that the phrase before it is the left side of the equation and the
phrase after it is the right side.
If 7 is subtracted from a twice a number ⇒ 2z - 7
15 ⇒ 15
Therefore the equation equivalent to the sentence above is: 2z – 7 = 15.
Practice
Write each phrase as an algebraic expression.
1. Maria’s allowance divided by 4.
a.
a + 4 b.
a - 4 c.
4/a
d. a/4
n/13
d. 13/n
2. a number less 13
a.
13 - n
b. n - 13 c.
Write each sentence as an algebraic equation.
3. The sum of 5 and a number equals 14.
a.
n – 5 = 14
b. 5 – n = 14 c. n + 5 = 14 d.
n +14 = 5
4. The product of k and -9 equals 17.
a.
-
k/ 9 = 17
–
b. 9/k = 17
c. -9k = 17 d.
9k = 17
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Mathematics Skills 2
5. Three more than the product of z and 7 equals 12.
a.
b. 7 + 3z = 12
7z + 3=12
c. 12 + 7z = 3
d. 12 + 3z = 7
Answers to practice:
1.
d. a/4
2.
b. n – 13
3.
c. n + 5 = 14
4.
c. 9k = 17
5.
a. 3 + 7z = 12
–
Evaluating Expressions and Formulas
When you wish to find out the value of an expression, you evaluate it.
In order to evaluate an expression, you need to know the value of the variable.
Then you can substitute the value of the variable in the expression. Then use
the “order of operations” to find the value of the expression.
Let’s take a look at an example.
Example 1:
Evaluate the expressions below.
9x – 14,
if x = 4
n/3 + 2n – 5,
if n = 12
5a + 7 + a2,
if a = 3
-
To evaluate each expression, substitute the value of the variable given and use
order of operations to simplify the expression.
9x – 14 = 9(4) – 14 = 36 – 14 = 22
n/3 + 2n – 5 = (12)/3 + 2(12) – 5 = 4 + 24 – 5 = 23
-
-
-
5a + 7 + a2 = 5( 3) + 7 + ( 3) 2 = 15 + 7 + 9 = 1
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Mathematics Skills 2
Evaluating formulas is done in exactly the same way as evaluating
expressions. The only difference is that formulas are usually equations and not
expressions. Either way, you substitute the given value and solve for the
remaining unknown value.
Let’s take a look at an example.
Example 2:
The formula for finding the perimeter of a rectangle is:
P = 2W + 2L. Find the perimeter of a rectangle with a
length of 2 feet and a width of 7 feet.
Let’s use the formula and substitute the values for the length and the width and
then solve for the missing variable.
P = 2W + 2L
P = 2(7 ft) + 2(2 ft) = 14 ft + 4 ft = 18 ft
Therefore, the perimeter of the rectangle is 18 ft.
Try the next example on your own and check your answer with the book.
Example 3:
The formula for finding the distance traveled is: D = RT
or distance = rate x time.
Find the distance traveled if a person runs at 6 miles/hour (mi/hr)
for 1 ½ hours.
Apply the formula and substitute the values that are given.
Then evaluate the formula.
D = RT
D = (6 mi/hr)(1.5 hr) = 9 miles
Therefore, the person ran for 9 miles.
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Practice
Evaluate each expression.
1. (28
+ 5) x 3
a. 102
2. 7
c. 99
d. 66
c. 2.8
d. 54
c. 14
d. 20
c. 44
d. 150
c. 69
d. 37
÷ 7 + 8 x 6
a. 49
3.
b. 69
b. 51
8 + (4 + 8 ÷ 4)3
a. 224
b. 2,744
4. 5d + 3c if d = 7 and c = 3
a. 114
b. 26
5. 2m2 + 5 if m = 4
a. 42
b. 27
Answers to practice:
1. c. 99
2. a. 49
3. a. 224
4. c. 44
Solving
One-Step Equations
5. d.37
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An equation is a mathematical sentence that has an equal sign (=) and shows
that two expressions are equivalent. An equation can be true or false. For
example:
8 + 4 = 12 is a true statement.
3 – 1 = 1 is a false statement.
An equation that contains a variable can also be true or false depending on the
value of the variable. When an equation has a variable, we solve the equation to
find the value of variable that makes the equation a true statement.
For example:
8 + x = 12
We want to find out the value of the variable x that will make this statement
true. Adding 4 to 8 will give us a sum of 12. Therefore, the only value that we
can use to substitute for x to make a true statement is 4.
We can also solve equations for the variable by performing the inverse
operation. The inverse operation of addition is subtraction and the inverse of
subtraction is addition. The inverse of multiplication is division and the inverse of
division is multiplication.
Let’s see how we could use the inverse operation to solve an equation for the
variable. When solving an equation, we need to isolate the variable on one side
of the equation. Remember that we are looking for a value for the variable that
makes a true statement.
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Mathematics Skills 2
For example:
y+7
6=
1
In order to isolate the variable in this example, we need to take the inverse of
adding 6. The inverse of adding 6 is to subtract 6. To keep the equation in
balance we must subtract 6 from both sides of the equation.
y + 6 – 6 = 17 - 6
Therefore,
y + 0
= 11
y = 11
The only value that would make this equation a true statement is 11.
Let’s take a look at some examples.
Example 1:
Solve the following equations for the given variables.
a–8=1
r + 20 = 35
a - 8 + 8 = 1+8
r + 20 - 20 = 35 - 20
a + 0 = 9
r + 0 = 15
a =9
r = 15
In each of the examples, we performed the inverse operation in order to solve
the equations.
These equations are examples of one-step equations. We call these equations
one-step equations because we only perform one-step in order to solve for the
value of the variable.
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Mathematics Skills 2
Try the next example on your own and check your answer with the book.
Example 2:
Solve the following equations for the given variables.
14n = 28
z = 9
3
These equations are also one-step equations.
The first is a multiplication equation (because we see that 14 and n are being
multiplied).
The second equation is a division equation (because z is being divided by 3).
In each of the examples we perform the inverse operation in order to solve the
equations. To solve the multiplication equation we divide and to solve the
division equation we multiply. Let’s see.
14n = 28
14
14
3•z = 9•3
3
In the first equation, we divide by 14 since 14 is the number that is being
multiplied by the variable. (Remember we try to isolate the variable). And in the
2nd equation we multiply by 3 since 3 is the number that is dividing the variable.
In the first equation, the value of n is 2
(28 ÷ 14 = 2).
In the 2nd equation the value of z is 27
(3 x 9 = 27).
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Mathematics Skills 2
Practice
Solve each equation below.
1. m + 9 = -2
a. 11
2.
–
–
7
b. 4 c.
–
–
10
17
c. 1 d.
–
1
b. 11 c.
7 d.
7=3+w
a. 4
10 d.
3. a – 9 = 8
–
a. 17 b.
4. 11b = 55
a. 605
b. 5
c. 66
d. 44
5. y/8 = 4
-
a. 2 b.
2
-
c. 32 d.
Answers to practice:
1. b. -11
2. a. 4
3. b. 17
4. b. 5
5. d. 32
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Mathematics Skills 2
Solving Two-Step Equations
Solving two-step equations is much like solving one-step equations in that you
must perform the inverse operation in order to solve for the value of the
variable. The only difference though is that two-step equations require two
steps in order to find the value of the variable that will make the sentence true.
You will notice that, when solving two-step equations, the variable is being
added or subtracted and multiplied or divided. Therefore you must perform the
inverse of addition or subtraction and later perform the inverse operation of
multiplication or division.
Let’s take a look at an example.
Example 1:
Solve the following equation for the given variable.
6x + 11 = 29
Notice that there are two operations here: addition and multiplication by the
variable. Our task is to isolate the variable.
We must first do so by performing the inverse of the addition.
6x + 11 - 11 = 29 - 11
6x + 0 = 18
6x = 18
Now we can perform the inverse of multiplication.
6x = 18
66
x=3
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Mathematics Skills 2
Try the next examples on your own and check your answers with the book.
Example 2:
Solve the following equation for the given variable.
y – 3 = 7
5
This is also a two-step equation because we see subtraction and division. Our
task is to isolate the variable.
We must first do so by performing the inverse of the subtraction.
y–3 + 3 = 7 + 3
5
y + 0 = 10
y = 10
5
5
Now we can perform the inverse of division.
5 • y = 10 • 5
5
y = 50
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Mathematics Skills 2
Example 3:
Solve the following equation for the given variable.
2n + 15 = 1
This is also a two-step equation because we see addition and multiplication.
Our task is to isolate the variable.
We must first do so by performing the inverse of the addition.
2n + 15 = 1
2n + 15 - 15 = 1 - 15
-
2n + 0 = 14
-
2n = 14
*Recall the rules for integers that state that, when adding two integers with
unlike signs, you must subtract and take the sign of the integer with the greatest
absolute value.
Now we can perform the inverse of multiplication.
2n = - 14
2
2
-
n = 7
*Recall the rules for integers that state that, when dividing integers with different
signs, the quotient is negative.
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Mathematics Skills 2
Practice
Solve each equation. Check your solution.
1.
2y – 8 = 14
a. 11 b.
2.
22 c.
44 d.
3
15 = 3y - 3
a. 6
3.
b. 4
c. 18
d. 54
b. 0
c. 1
d. 225
b. 6
c. 3 d.
15a + 49 = 64
a. 7
4.
8
15
3f + 13 = 4
a. 3
-
-
27
+ 15 = 16
5. b
2
a. 1
b.
2
c.
Answers to practice:
122
1.
a. 11
2.
a. 6
3.
c. 1
4.
c. 3
5.
b. 2
-
16 d.
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Mathematics Skills 2
LESSON 5 THINGS TO REMEMBER
Writing Algebraic Expressions and Equations
An algebraic expression consists of a number, a variable, or a number and
variable with operational symbols. A variable is a symbol, usually a letter, which
represents an unknown quantity. The term variable is used because its value
can vary from problem to problem. This means that the value of the variable can
be one quantity in one question and then another quantity in another.
We know the operational symbols to be + (addition), - (subtraction), x
(multiplication), and ÷ (division). In algebra we will often see multiplication
denoted by a dot (•) or simply a number next to a letter (5n “five times n”).
Likewise, we will often see division denoted as a fraction (5/n “five
divided by n”).
The following are examples of some expressions:
5+x
five plus the value of x
7n – 2
seven times the value of n minus 2
2y + 1/y
two times y plus one divided by y
Translating and Writing Algebraic Expressions
Some of the expressions above show addition, subtraction, multiplication,
division, or a combination of these. You may also have to translate from words
to algebraic expressions or algebraic expressions into words. Here is a guide on
the next page to help you translate these expressions.
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Mathematics Skills 2
Addition
Subtraction
Multiplication
Division
ƒ More than
ƒ Plus
ƒ Greater than
ƒ The sum of
ƒ Increased by
ƒ Less than
ƒ Product of
ƒ Quotient of
ƒ Subtracted from ƒ Times
ƒ Divided by
ƒ Minus
ƒ Multiplied by
ƒ Less
ƒ Twice
ƒ The difference
ƒ Half of
ƒ Decreased by
*Caution: you must be careful in subtraction when using “less than” and
“subtracted from” as the terms must be written in the correct order.
Write an algebraic expression for each phrase.
4 more than a number:
a
number decreased by 8:
the quotient of a number divided by 9:
n+4
w-8
s/9
*Note that, when a variable is not given, we can write any letter or symbol of
our choice to represent the unknown number.
The expressions in the example above all contain one operation. We can also
write expressions with more than one operation.
Write an algebraic expression for each phrase.
twice the difference between a number and 17:
2(x – 17)
2 subtracted from the quotient of 4 divided by x:
4/x - 2
*Note “less than” and “subtracted from” were written in reverse order of
appearance because they were being subtracted from the second term.
Also, the reason parenthesis were necessary in the first example was
because the multiplication takes place after the difference and the sum,
(i.e., to express order of operations).
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Mathematics Skills 2
The same way that we translate phrases into algebraic expressions, we can
also translate sentences into algebraic equations. The only difference is that an
equation ALWAYS includes an equivalence symbol (equal sign).
Write an algebraic equation for the sentence:
5 less than 4 times a number is 4 more than twice the number.
This sentence is an equation because we note the word “is”. The word “is”
means =. It separates what is on the left side of the equal sign and what is on
the right side. Now all we have to do is translate the phrase before the word “is”
and place it before the = sign in our equation. Then we translate the phrase
after the word “is” and place it to the right of the = sign.
5 less than 4 times a number ⇒ 4n – 5
4 more than twice the number ⇒ 2n + 4
Therefore the equation equivalent to the sentence above is:
4n – 5 = 2n.+ 4
*Note that we used the same variable to represent the same number in the
equation. This is because we are referring to the same unknown number in the
sentence.
Several different words or phrases are used to represent the equal (=) sign.
They are:
The same as
Equals
=
Is equal to
Is
The result is
Write an algebraic equation for the sentence:
1 added to the quotient of a number divided by 6
is the same as 9 less than the number.
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Mathematics Skills 2
First locate the equivalence of the equation and then let’s identify the left and
right side.
“Is the same as” describes the equal sign in this example.
This means that the phrase before it is the left side of the equation and the
phrase after it is the right side.
1 added to the quotient of a number and 6 ⇒ y/6 + 1
9 less than the number ⇒ y - 9
Therefore the equation equivalent to the sentence above is:
y/6 + 1 = y - 9.
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Mathematics Skills 2
Lesson 6: Statistical Methods
In this lesson, you learn how to collect data and display it in its most
natural form. Also you learn how to analyze the data by finding the
measures of central tendency and measure of dispersion.
Collecting and Organizing Data
The most common method to collect data is by taking a survey. A survey is a
question or series of questions designed to collect data about a group of people.
These people are what we call a population. If the population is too large, then
only a sample of the population is surveyed. A sample is a group of people that
represents the population and is selected at random.
In a survey, the population consists of the people about whom information is
requested. The sample consists of the people in the population that are actually
studied.
Let’s take a look at an example.
Example 1: Seven hundred students in a city were asked to
name their favorite movie. What is the population and the
sample? What is the size of the sample?
The population is the students in the city. The sample is the students surveyed.
The sample size is 700.
When choosing a sample you want to make sure to choose a sample that is
representative of the population.
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Mathematics Skills 2
You also want to be certain it is a random sample. A random sample means
that each person has the same chance of being chosen.
Let’s take a look at an example.
Example 2: A manufacturer selects every 25th microwave oven from its
warehouse to find out if the microwave oven works properly.
This is a random sample because every 25th microwave oven is chosen. What
would not be a random sample is if the first 5 microwaves were chosen and rest
were not inspected.
Example 3: A group of students who entered a math competition was
asked if they like mathematics.
This is not a random sample since these students have an inclination to like
mathematics (since they are participating in a mathematics competition). A
random sample would be to ask every couple of students in a class if they like
mathematics.
When questions are written for a survey, they need to be written in such a way
that they are not biased. In other words, the questions cannot influence the
answer in any way. Look at the following questions and decide which is a good
question for a survey.
What flavor of ice cream do you like?
What is your favorite food?
The first question is not appropriate for a food survey because it is biased. The
question is biased because it assumes that the person likes ice cream, which
may or may not be the case. The second question is a better question for a food
survey because it is not making an assumption on what the person likes.
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Mathematics Skills 2
Once you:
¾ survey a group of people from a population
¾ are certain your sample is random
¾ the questions are not biased
you can organize the data in a table and use tally marks to represent the
number of responses.
Let’s take the question above, “What is your favorite food?”
We can make a table to organize the responses by listing the categories in one
column and the tally marks in another.
Favorite Food
Number of
responses
Hamburger
⏐⏐⏐⏐
Cheeseburger
⏐⏐⏐⏐
Pizza
⏐⏐⏐⏐ ⏐
Chicken Fingers
⏐⏐⏐⏐ ⏐⏐
Salad
⏐⏐⏐
You can tell many things from the table above.
For example:
ƒ the total number of people that responded to the survey
ƒ the amount of people that like hamburger, cheeseburger, pizza, chicken
fingers, or salad
ƒ how many more people prefer one food over another
Each tally mark represents one response. In the case of pizza and chicken
fingers, we see that there is a diagonal tally mark. This diagonal tally mark is
used on every fifth response to mark off a set of five.
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Mathematics Skills 2
Use the table above to answer the questions.
Example 4:
How many people chose pizza?
6
How many people chose salad?
3
How many more people chose chicken fingers than cheeseburger?
3
How many people participated in the survey?
24
Later in this lesson we will see how we can use the information in this table to
make graphs.
Practice
Several students went bowling after school. Their scores are shown in the tabl e
below.
Score
Tally
Under 100
100–109
110–119
120–129
130–139
At least 140
1.
How many students went bowling?
a. 12
2.
c. 35
d.
28
Which is the most common range of scores?
a. 100–109
130
b. 2
b. 110–119
c.
120–129
d. 130–139
Mathematics Skills 2
3.
How many students scored 120–129 points?
a.
4.
b. 4
c. 7
d. 6
How many students scored at least 110 points?
a.
5.
5
12
b. 19
c. 14
d. 22
If these same students were to bowl another game, how many scores above
139 would you expect to see?
a.
3
b. 2
c. 1
d. 4
Answers to practice:
1.
c. 35
2.
b. 110-119
3.
a. 5
4.
d. 22
5.
b. 2
Displaying Data in a Variety of Forms
Once you have gathered data, you can display it in a variety of forms. Data can
be displayed using a bar graph, a line graph, a circle graph, or a stem-andleaf plot to name only a few ways. Let’s take a look at each of these graphs,
how to create them, and when it’s best to use each one.
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Mathematics Skills 2
Bar Graph
A bar graph is a way to compare and display numerical data. In this type of
graph, either horizontal or vertical bars are used to display data. You will need a
horizontal axis and a vertical axis (as in an x- and y-axis). You will also need to
decide what data you want to display on which axis.
To make a bar graph:
ƒ You will need to choose what you would like to display along the horizontal
axis and create a scale for the vertical axis (the scale is what measures the
height of the bars).
ƒ For each category on the horizontal axis, you will need to draw a bar of the
appropriate height along the vertical axis.
ƒ You will also need to make a title for the graph.
Number of
Favorite Food
responses
Hamburger
⏐⏐⏐⏐
Cheeseburger
⏐⏐⏐⏐
Pizza
⏐⏐⏐⏐ ⏐
Chicken Fingers
⏐⏐⏐⏐ ⏐⏐
Salad
⏐⏐⏐
Let’s use this table to create a bar graph.
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Mathematics Skills 2
Example 1:
Remember, we can use a bar graph for these data because we are comparing
and displaying numerical data.
To make a bar graph, we first have to decide what we want to place on the
horizontal axis. It may be a good idea to place the food items across the
horizontal axis and make an appropriate scale on the vertical axis.
When making a scale for the vertical axis, we need to make sure we use
appropriate intervals according to the data in the table. The divisions need to be
equally spread on the vertical axis. For example, we can choose to do the
intervals in either 2’s or 3’s or 5’s or 10’s.
Let’s take a look at the numbers in our data set. They are: 4, 4, 6, 7, and 3.
Since these numbers are rather small and the majority are even, we can choose
to do the intervals in 2’s.
We could also choose to do it 3’s because it too would be appropriate.
Whichever interval we decide to go with is fine, as long as we keep the
intervals consistent. Let’s take a look at the graph.
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Mathematics Skills 2
Favorite Foods
8
6
4
2
hamburger
cheeseburger pizza
chicken
fingers
salad
Interpreting the data on the bar graph is similar to reading the information off the
chart or table. By looking at the graph we can note several findings:
ƒ Chicken fingers is the most popular food
ƒ Salad is the least popular food
ƒ Hamburgers and cheeseburgers are equally liked
Six people of the sample population prefer pizza
And so on…
Line Graphs
A line graph is used to display data that change over a period of time. In a line
graph you also need horizontal and vertical axes as well as decide what data
you would like to display on what axis.
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Mathematics Skills 2
To make a line graph:
ƒ Choose what you would like to display along the horizontal axis and create a
scale for the vertical axis.
ƒ For each category on the horizontal scale, draw a point at the appropriate
height along the vertical axis.
ƒ After you have plotted all the points according to the data, draw a line
connecting, the points on the graph (as in “connect the dots”).
ƒ Title the graph.
Notice that we cannot use the data from the table of “Favorite Foods” to create
a line graph because those data do not change over a period of time.
Let’s take a look at an example using the data provided in the table.
Points Scored
Game
Points
1 12
29
3 14
4 15
5 18
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Mathematics Skills 2
Example 2:
We can use a line graph for these data because we are displaying data over a
period of time. (The time period in this example is a progression of games from
game 1 to game 5). To make a line graph, we first have to decide what we want
to place on the horizontal axis. It may be a good idea to place the game number
across the horizontal axis and make an appropriate scale on the vertical axis.
second have to make a scale for the vertical axis. We need to make sure we
use appropriate intervals according to the data in the table. The divisions need
to be equally spread on the vertical axis. For example, we can choose to do the
intervals in 2’s or 3’s or 5’s or 10’s.
Let’s take a look at the numbers in our data set. They are: 12, 9, 14, 15, and 18.
We can choose to do the intervals in 3’s. Let’s take a look at the graph.
18
15
Points
Scored
12
9
6
3
1
2
3
Games
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4
5
Mathematics Skills 2
Interpreting the data on the line graph is similar to reading the information off a
bar graph. By looking at the graph we can note several findings:
ƒ The most points were scored during the fifth (last) game
ƒ The least points were scored during the second game
ƒ The difference in points scored between the 2nd and 5th game were 9
points
ƒ There was little change in the amount of points between the 3rd and 4th
games
ƒ And so on…
Circle Graphs
A circle graph displays portions and how these portions compare to the whole
data set.
The data are often given in percentages to show the portion out of 100%. The
larger the portion, the wider the area displayed on the graph.
To make a circle graph:
ƒ If percentages are not given, find the percent of the whole each part is.
ƒ Multiply each percentage by 360° (the number of degrees in a circle) to
determine the measure of the angle each portion will be.
ƒ Draw a circle and each angle that represents each portion of the graph.
ƒ Title the graph.
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Mathematics Skills 2
Let’s take a look at an example using the data provided.
Example 3:
The information below describes how Marie must structure
her budget for her monthly expenses.
Rent
40%
Food
16%
Car Payment
14%
Entertainment
10%
Savings
8%
Utilities
12%
™ Multiply each percentage by 360° to determine the measure of each angle.
138
Rent
40% = 0.4 x 360° = 144°
Food
16% = 0.16 x 360° = 57.6°
Car Payment
14% = 0.14 x 360° = 50.4°
Entertainment
10% = 0.1 x 360° = 36°
Savings
8% = 0.08 x 360° = 28.8°
Utilities
12% = 0.12 x 360° = 43.2°
Mathematics Skills 2
™ Then use a protractor to draw each angle.
™ Label each part of the circle corresponding to each part.
™ Title the graph.
Marie’s Budget
Interpreting the data on the circle graph is similar to reading the information off a
bar or line graphs. By looking at the graph we can note several findings:
ƒ The majority of Marie’s budget is dedicated to her rent
ƒ The least amount of her money goes to savings
ƒ If we know how much money Marie made on a monthly basis, we could
calculate how much money Marie spends on each category by
multiplying the amount she earns by the percentage
(i.e., $500 (monthly salary) x 0.4 (40%) = $200 is spent on rent)
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Mathematics Skills 2
Stem-and-Leaf Plots
A stem-and-leaf plot is yet another way to display data similar to a bar graph,
table, or even a line graph. The numbers in the data set are separated into
stems and leaves. The stem of the number is the digit in the tens place and
larger and the leaf is the digit in the ones place.
To make a stem-and-leaf plot:
ƒ List the numbers in the data set in order from least to greatest
ƒ Categorize the data by the same stem (tens digit)
ƒ List the number in the ones place behind each stem
ƒ You will need a description of what the data represent.
ƒ You will need a key so that the person interpreting the graph knows how to
read the data.
Let’s take a look at an example using the data provided.
Example :
The following data represent the height (in inches)
of 20 members of Mr. Taylor’s class.
57
53
52
61
67
70
62
65
58
55
71
64
66
69
64
62
60
59
58
65
Let’s list the numbers in order from least to greatest and categorize them by the
same stem (tens place).
52 53 55 57 58 58 59
60 61 62 62 64 64 65 65 66 67 69
70 71
The stems in this example are 5, 6, and 7 (representing the data in the 50’s,
60’s, and 70’s).
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Mathematics Skills 2
Now we can place the stems on one side of a vertical (or horizontal line) and
the leaves corresponding to each stem on the other side of each stem.
Take a look at the stem-and-leaf plot based on these data.
5
2 3 5 7 8 8 9
6
0 1 2 2 4 4 5 5 6 79
7
0 1
Note the heights that appear twice in the data set; see how the leaves also
appear twice, i.e., 58.
When creating a stem-and-leaf plot, describe the data the numbers represent.
Show a key to tell the interpreter how to read the data.
For example, 5⏐7 means 57 inches.
*Note that, if the numbers in the data set would be 123, 164, 148, etc.
¾ the stem would be 12 and the leaf 3 for 123,
¾ the stem 16 and the leaf 4 for 164,
and so on.
Interpreting the data on a stem-and-leaf plot is similar to reading the information
off a bar, line, or circle graph. By looking at the plot we can note several
findings:
ƒ The most common height among the members of Mr. Taylor’s class
ƒ The least common height among the members of Mr. Taylor’s class
ƒ How many students measure more than five feet (60 inches) - 12
ƒ How many students are not yet five feet - 7
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Mathematics Skills 2
Practice
Select the choice that best answers the question.
1.
Ms. Garcia’s Math class has 12 students. The final averages of these
students are displayed on the stem-and-leaf plot at the right. Which is the
grade the majority of students in the class have?
Average Grade
Grading
Scale
90-100 A
Stem Leaf
9
8
7
6
5
80-89 B
70-79 C
60-69 D
1 3 4 7 8
2 5 6 9
5 7
9
0-59 F
a. A
b. B
c.
C
d.
F
2. Florida is most famous for its oranges, but tangerines, limes, and grapefruit
are also grown in the state. The graph shows how the production of each fruit
compared during one season. What information is missing from the graph?
a. the title of the graph
b.
scale explaining what percents mean
c. c. key telling what each section represents
d. a data table
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Mathematics Skills 2
3. Ms. Townsend has two savings accounts. She monitors the values of the two
accounts over a three-month period. What was the difference in value
between on Account 1 and Account 2 on week #7?
a. $750
b. $1,880
c. $1,950
d. $2,000
4. The families living in the new development by Shoma Homes were asked
which improvements were needed in their community. Which improvement
was most important to the people living in this development?
a.
add sidewalks
b. trim trees/bushes
c.
improve roads
d.
add hike/bike trails
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Mathematics Skills 2
Answers to practice:
1. a. A
2. c. a key telling what each section represents
3. a. $750
4. c. improve roads
Measures of Central Tendency
Measures of central tendency describe an entire set of data with one number.
The measures of central tendency we will discuss are mean, median, and
mode.
The mean of a data set represents the average of the set of data. It is found by
dividing the sum of all the numbers in the set of data by the amount of numbers.
The mode of a data set is the number that appears the most often in the set of
data. If all numbers in the data set appear an equal amount of times, then we
say that there is “no mode”. We cannot use “zero” because 0 represents a
number that may or may not appear in the data set.
The median of a data set is the middle value of the set of data. When choosing
the median, the data must be in numerical order, be it from least to greatest or
from greatest to least. The median of a set of data is easy to find when you
have an odd number of data in your data set. When you have an even number
of data, locate the two middle values and find the mean or average of these.
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Mathematics Skills 2
Let’s take a look at some examples.
Example 1: Find the mean, median, and mode of the data set below.
29 33
28
30
25
24
21
9
11
13
15
In order to find the mean of the data set we add all the
numbers and divide by 11 (there are 11 numbers in our data set).
29 + 33 + 28 + 30 + 25 + 24 + 21 + 9 + 11 + 13 + 15 = 238
238 ÷ 11 = 21.6 is the mean.
In order to find the median, we must first organize the numbers
in order from least to greatest or greatest to least.
9
11 13 15 21 24 25 28 29 30 33
Since there is an odd number of items in our data, set it is easy to find the
middle value. The median is 24.
To find the mode we see which of the numbers in our data set appears most
often. By looking at our data we see that all numbers appear once therefore
there is no mode for this set of data.
Try the next example on your own and check your answer with the book.
Example 2:
Find the mean, median, and mode of the set of data.
50
80
90
50
40
30
50
80
70
10
First let’s find the mean by calculating the sum of all the numbers in the data set
and dividing that sum by 10 (there are 10 items in the data
set). 50 + 80 + 90
+ 50 + 40 + 30 + 50 + 80 + 70 + 10 = 550 ÷ 10 = 55
Therefore, the mean of the data set is 55.
Now let’s find the median. Recall from the last example that, in order to find the
median, we must first place the numbers in numerical order.
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Mathematics Skills 2
Let’s order them from greatest to least.
90 80 80 70 50 50 50 40 30 10
Since we have an even number of items in our data set, we must
choose
the two middle values and find their average. The two middle values are 50 and
50. Since the two middle values are exactly the same the median is 50.
To find the mode we select the number that appears most often in the data set.
The number that appears most (3 times) is 50.
Therefore, the mode of the data set is 50.
(Even though 80 appears more than once, it appears only twice and
therefore less often than 50).
A measure of dispersion is also used to describe a data set by only one
number. The measure of dispersion we will discuss in this lesson is the range.
The range of a set of numbers refers to the difference between the highest and
lowest values in the data set. To find the range of a set of data, simply subtract
the lowest value from the highest.
Example 3:
Find the range of both data sets above.
The data set in example 1 was:
29
33
28
30
25
24
21
9
11
13
15
To find the range, we subtract the lowest value (9) from the
highest (33), 33 – 9 = 24. Therefore,
the range of the set of data in example 1is 24.
The data set in example 2 was:
50
80
90
50
40
30
50
80
70
To find the range subtract 90 – 10 = 80. Therefore,
the range of the set of data in example 2 is 80.
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10
Mathematics Skills 2
Practice
Find the mean for this set of data.
1. 14, 13, 9, 13, 16
a. 14
b. 12
c. 13
d. 17
Find the median for this set of data.
2. 21, 41, 21, 29, 32, 25
a. 27
b. 25
c. 34
d. 28
Find the mode for this set of data.
3. 18, 33, 38, 17, 33
a. 33
b. 18
c. none
d. 17
Find the mode for this set of data.
4. 64, 59, 67, 50, 59, 54, 59, 56
a. 67
b. none
c. 59
d. 54
Find the range for this set of data.
5. 41, 40, 34, 48, 57, 48, 39, 34
a. 19
b. 23
c. 57
d. 48
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Mathematics Skills 2
6. How can you find the mode from the bar graph?
a. find the average of the values on a calculator
b. find the most bars that are the same height
c. find the bar that represents the middle value
d. find the difference between the tallest bar and the shortest bar
Answers to practice:
1.
c. 13
2.
a. 27
3.
a. 33
4.
c. 59
5.
b. 23
6.
b. the most bars that are the same height
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Mathematics Skills 2
LESSON 6 THINGS TO REMEMBER
Collecting and Organizing Data
The most common method to collect data is by taking a survey. A survey is a
question or series of questions designed to collect data about a group of people.
These people are what we call a population. If the population is too large, then
only a sample of the population is surveyed. A sample is a group of people that
represents the population and is selected at random.
In a survey, the population consists of the people about whom information is
requested. The sample consists of the people in the population that are actually
studied.
Example 1: Seven hundred students in a city were asked to name their
favorite movie. What is the population and the sample? What is the size of
the sample?
The population is the students in the city. The sample is the students surveyed.
The sample size is 700.
We can make a table to organize the responses by listing the categories in one
column and the tally marks in another.
Favorite Food
Number of
responses
Hamburger
⏐⏐⏐⏐
Cheeseburger
⏐⏐⏐⏐
Pizza
⏐⏐⏐⏐ ⏐
Chicken Fingers ⏐⏐⏐⏐ ⏐⏐
Salad
⏐⏐⏐
You can tell many things from the table above.
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Mathematics Skills 2
For example:
ƒ the total number of people that responded to the survey
ƒ the amount of people that like hamburger, cheeseburger, pizza, chicken
fingers, or salad
ƒ how many more people prefer one food over another
Each tally mark represents one response. In the case of pizza and chicken
fingers, we see that there is a diagonal tally mark. This diagonal tally mark is
used on every fifth response to mark off a set of five.
Use the table above to answer the questions.
Example:
How many people chose pizza? 6
How many people chose salad? 3
How many more people chose chicken fingers than cheeseburger? 3
How many people participated in the survey? 24
Displaying Data in a Variety of Forms
Once you have gathered data, you can display it in a variety of forms. Data can
be displayed using a bar graph, a line graph, a circle graph, or a stem-andleaf plot to name only a few ways. Let’s take a
Measures of Central Tendency
Measures of central tendency describe an entire set of data with one number.
The measures of central tendency we will discuss are mean, median, and
mode.
The mean of a data set represents the average of the set of data. It is found by
dividing the sum of all the numbers in the set of data by the amount of numbers.
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Mathematics Skills 2
The mode of a data set is the number that appears the most often in the set of
data. If all numbers in the data set appear an equal amount of times, then we
say that there is “no mode”. We cannot use “zero” because 0 represents a
number that may or may not appear in the data set.
The median of a data set is the middle value of the set of data. When choosing
the median, the data must be in numerical order, be it from least to greatest or
from greatest to least. The median of a set of data is easy to find when you
have an odd number of data in your data set. When you have an even number
of data, locate the two middle values and find the mean or average of these.
Example: Find the mean, median, and mode of the set of data.
50
80
90
50
40
30
50
80
First let’s find the mean by calculating the sum of all the
70
10
numbers in the data
set and dividing that sum by 10 (there are 10 items in the data set). 50 + 80 + 90
+ 50 + 40 + 30 + 50 + 80 + 70 + 10 = 550 ÷ 10 = 55
Therefore,
the mean of the data set is 55.
Now let’s find the median. Recall from the last example that, in order to find the
median, we must first place the numbers in numerical order. Let’s order them
from greatest to least.
90 80 80 70 50 50 50 40 30 10
Since we have an even number of items in our data set, we must choose the
two middle values and find their average. The two middle values are 50 and 50.
Since the two middle values are exactly the same the median is 50. To find the
mode we select the number that appears most often in the data set. The number
that appears most (3 times) is 50. Therefore, the mode of the data set is 50.
(Even though 80 appears more than once, it appears only twice and therefore
less often than 50).
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Mathematics Skills 2
A measure of dispersion is also used to describe a data set by only one
number. The measure of dispersion we will discuss in this lesson is the range.
The range of a set of numbers refers to the difference between the highest and
lowest values in the data set. To find the range of a set of data, simply subtract
the lowest value from the highest.
The data set in Example 1 was:
50
80
90
50
40
30
50
80
70
10
To find the range subtract 90 – 10 = 80. Therefore, the range of the set of
data in example 2 is 80.
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Mathematics Skills 2
Lesson 7: Probability
In this lesson, you learn about probability and how to compute simple
probability.
Simple Probability
Probability measures the chance that an event will occur. The event is the
outcome that you are looking for. The chance is represented by a number.
Often times it is written as a fraction. It can also be a decimal, percent, or even
a ratio. The probability of an event will be a number between 0 and 1. If the
numbers 0 or 1 are used to describe a probability, this means that the event has
zero chance of occurring (0) or 100% chance of occurring (1).
For example, the probability that you will win the lottery is 0 if you don’t play.
Likewise, the probability of drawing a red marble from a bag that has only red
marbles is 100% or 1.
A probability can be expressed as:
P(event) = number of ways the event can occur
number of possible outcomes
Let’s take a look at some examples.
Example 1:
When tossing a coin, what is the probability of it landing
with its “head” side facing up?
P(heads) = number of ways the event can occur = 1 side
number of possible outcomes
2 sides
There is only one way this event can occur because there is only one heads in a
coin. The number of possible outcomes is 2 because a coin has only two sides;
heads and tails. Therefore, the probability of getting a toss of heads is ½.
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Mathematics Skills 2
*Note that, when a probabi lity is expressed as a fraction
or ratio, it must be
reduced to its lowest terms.
Example 2:
In a bag there are 2 red, 3 yellow, 4 green, 6 blue, and 9
purple marbles. Find each probability below.
P(red) =
2 red marbles = 1
24
total marbles 12
P(yellow) = 3 yellow marbles = 1
24 total marbles
P(blue) =
6 = 1
24
4
P(green or purple) = 4 + 9
24
*When finding the probability
8
=
24
of two events at one
13
24
time, simply add the
probabilities.
Try the next example on your own and check your answers with the book.
Example 3:
When rolling a cube with a different number of dots (1 – 6)
on each face, what is the probability of rolling a
number of dots greater than 3?
In a number cube, there are 6 faces.
There are 3 numbers that are greater than 3: 4, 5, and 6.
P(greater than 3) =
3=1
6
Therefore
154
2
the probability of rolling a number greater than 3 is ½.
Mathematics Skills 2
Practice
Identify the letter that best answers the question.
The spinner shown is spun once. Find each probability. Write each answer as a
fraction, a decimal, and a percent.
1.
2.
P(3, 6, or 8).
a.
1
2 = 0.5 = 50%
b. 1, 1.0, 100%
c.
5
8 = 0.625 = 62.5%
d.
3
8 = 0.375 = 37.5%
P(less than 3).
a.
5
8 = 0.625 = 62.5%
b. 1, 1.0, 100%
c.
1
4 = 0.25 = 25%
d.
1
8 = 0.125 = 12.5%
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Mathematics Skills 2
A 6-sided number cube is rolled. Find the probability of each event. Write each
answer as a fraction, a decimal, and a percent.
3.
4.
P(5).
a. 1, 1.0 , 100%
b.
1
6
= 0.16 = 16.6%
c. 0, 0.0, 0%
d.
1
8
= 0.125 = 12.5%
b.
1
5 = 0.16 = 16.6%
d.
1
3 =
P(less than 4).
a.
2
3
c.
1
2
= 0.6 = 66.6%
= 0.5 = 50%
0.3 = 33.3%
A drawer contains 3 blue ri bbons, 4 red ribbons, and 3 green ri bbons. A ri bbon
is randomly chosen from the drawer,
replaced, and another ribbon is chosen.
Find each probability.
5.
156
P(blue or red)
a.
3
10
b.
7
20
c.
3
25
d.
7
10
Mathematics Skills 2
6.
P(blue or green)
3
10
b.
9
100
c. 0
d.
3
5
a.
Answers to practice:
1. d.
3
8
= 0.375 = 37.5%
2. c.
1
4
= 0.25 = 25%
3. b.
1
6
= 0.16 = 16.6%
4. c.
5. d.
6. d.
1
2
= 0.5 = 50%
7/10 = 0.7 = 70%
3/5 = 0.6 = 60%
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Mathematics Skills 2
LESSON 7 THINGS TO REMEMBER
Probability measures the chance that an event will occur. The event is the
outcome that you are looking for. The chance is represented by a number.
Often times it is written as a fraction. It can also be a decimal, percent, or even
a ratio. The probability of an event will be a number between 0 and 1. If the
numbers 0 or 1 are used to describe a probability, this means that the event has
zero chance of occurring (0) or 100% chance of occurring (1).
For example, the probability that you will win the lottery is 0 if you don’t play.
Likewise, the probability of drawing a red marble from a bag that has only red
marbles is 100% or 1.
A probability can be expressed as:
P(event) = number of ways the event can occur
number of possible outcomes
Example 1:
When tossing a coin, what is the probability of it landing
with its “head” side facing up?
P(heads) = number of ways the event can occur = 1 side
number of possible outcomes
2 sides
There is only one way this event can occur because there is only
one heads in a coin. The number of possible outcomes is 2 because
a coin has only two sides; heads and tails.
Therefore, the probability of getting a toss of heads is ½.
*Note that, when a probabi lity is expressed as a fraction
reduced to its lowest terms.
158
or ratio, it must be
Mathematics Skills 2
Example 2:
In a bag there are 2 red, 3 yellow, 4 green, 6 blue, and 9
purple marbles. Find each probability below.
P(red) =
2 red marbles = 1
24
total marbles 12
P(yellow) = 3 yellow marbles = 1
24 total marbles
P(blue) =
6 = 1
24
4
P(green or purple) = 4 + 9
24
*When finding the probability
8
=
24
of two events at one
13
24
time, simply add the
probabilities.
Example 3:
When rolling a cube with a different number of dots (1 – 6)
on each face, what is the probability of rolling a number of dots greater
than 3?
In a number cube, there are 6 faces.
There are 3 numbers that are greater than 3: 4, 5, and 6.
P(greater than 3) =
3=1
6
Therefore
2
the probability of rolling a number greater than 3 is ½.
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Mathematics Skills 2
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Mathematics Skills 2
Lesson 8: Basic Calculator Skills
In this Lesson, you will learn how to use a four-function calculator. You
learn how to use these four functions and other special keys on the
calculator as well.
How to Use Your Calculator
Calculators are useful when computing mathematical problems when mental
math is not as feasible. Below let’s take a look at a diagram of a four-function
calculator, its functions, and what each function performs.Display
On/Clear
Turns the
calculator on and
clears the display.
Solar Panel
Memory Recall
Displays what is
in memory.
Memory Minus
Subtracts the amount
on the display from what
is in memory.
Memory Plus
Adds the amount on the
display to what is in
memory.
on/c
±
MRC
√
Change Sign
Changes the sign
of the number on
display.
M+
M-
7
8
9
x
4
5
6
-
Divide
1
2
3
+
Multiply
0
.
%
Decimal Point
Percent
÷
=
Answer
Square Root
Takes the square
root of the number
on display.
Subtract
Add
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Mathematics Skills 2
Basic Operations
Adding, subtracting, multiplying, and dividing are clear-cut. Simply punch-in the
numbers you wish to perform (and the operations) in the order that they appear
and press the equal sign.
Example 1:
Perform the following operations.
167 + 29
291
– 273
45 x 97
75 ÷ 4
To perform the above operations, simply punch in the numbers
and operations in the order they appear.
167
+
29
291
-
273
=
18
45
x
97
=
4,365
75
÷
4
=
18.75
=
196
Negative Numbers
When performing operations with negative numbers, simply punch-in the
number first then press the
±
key. This will convert a positive number into
a negative. Then you can continue to perform the operations as you did above
with the basic keys.
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Mathematics Skills 2
Example 2:
Perform the following operations with the change sign key.
-15 + -29
18.47 – (-50)
(-4) x (-27)
75 ÷ 4 + 6
To perform the above operations, simply punch in the numbers, then
the change sign key when necessary, and the operations in the
order
±
15
18.47
4
75
they appear.
-
±
÷
+
29
± =
50
±
=
x
27
±
4
+
6
-
44
68.47
=
=
108
24.75
Memory
The memory keys on your calculator are useful when performing several
operations at one time.
MRC
This button displays what is in memory. If you press it twice, you can
clear what is in memory.
M+
This button adds the amount on the display to what is in memory.
M-
This button subtracts the amount on the display from what is in
memory.
When the calculator holds a value in memory, the display looks like
this: M
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Mathematics Skills 2
It will also display the number amount that it holds in memory. Performing any
operation on your calculator will not affect what is in memory unless you use
any of the memory keys.
Example 3: Solve the problem below using your calculator’s memory
keys.
7 + 14 x 5 – 162
Using order of operations, we must compute the exponents first.
16 x 16
14 x
7
On/C
MRC
MRC
5
M+
M-
M+
M
256
M
70
M
7
M
MRC
-
179
Notice that, when we multiplied 16 x 16, we used the memory minus key
because 162 was being subtracted from the rest of the expression. As we
computed each part of the expression, we used the memory plus and minus
depending on what was performed in the problem. Once we were done
computing all the parts, we pressed memory recall so that the calculator could
display what was in memory.
In most calculators, the memory keys work the same way. Please check your
manual if your memory keys work different.
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Mathematics Skills 2
Special Keys
Some of the basic four-function calculators have special keys that perform
certain functions. The calculator displayed above contains two special keys: the
square root key and the percent key.
√
The square root key will calculate the square root of the number on
display.
%
The percent key converts the number on the display to the decimal
expression of the percent.
Example 4:
Solve each problem below using the special keys on the
calculator.
8 + √70
20% of 65
In order to calculate the first problem, simply press the following keys
on the calculator:
8 + 70 √
=
16.3666
To calculate the second problem press the following keys:
65 x 20 %
The result is 13.
*Note that, in the first problem, we had to press the equal’s key to obtain the
final answer.
In the second problem, recall that the word “of” represents multiplication. Note
that we had to input the amount first and the percentage last. This is done so
that the calculator can convert the percentage into a decimal and then multiply
by the amount. Keep this in mind when multiplying percentages. (You may not
even need this key if you know how to convert a percentage to a decimal.)
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Mathematics Skills 2
Practice
Calculate the value of each expression using a calculator.
–
225 – 17 x 33
1.
–
a. 786
b.
–
7,986
c. 786
d. 7,986
b. 78
c. 118.14435
d.131.85565
b. 408.16325
c. 2,800
- √47
2. 125
a. 6.8556546
3. 7% of 200
a. 14
d. 2,857.1428
4. √804 ÷ 17.35 + 620
a. 0.0444887
5. 15
621.63428 d.
666.34005
÷ 50 + 13
a. –13.3
Answers to practice:
-
1. a. 786
2. c. 118.14435
3. a. 14
4. c. 621.63428
5. d. 13.3
166
b. 1.2616712 c.
b. -0.2380952
c. 0.2380952
d. 13.3
Mathematics Skills 2
LESSON 8 THINGS TO REMEMBER
Basic Operations
Adding, subtracting, multiplying, and dividing are clear-cut. Simply punch-in the
numbers you wish to perform (and the operations) in the order that they appear
and press the equal sign.
Example 1:
Perform the following operations.
167 + 29
291 – 273
45 x 97
75 ÷ 4
To perform the above operations, simply punch in the numbers
and operations in the order they appear.
167
+
29
291
-
273
=
18
45
x
97
=
4,365
75
÷
4
=
18.75
=
196
Negative Numbers
When performing operations with negative numbers, simply punch-in the
number first then press the
±
key. This will convert a positive number into
a negative. Then you can continue to perform the operations as you did above
with the basic keys.
167
Mathematics Skills 2
Example 2:
Perform the following operations with the change sign key.
-15 + -29
18.47 – (-50)
(-4) x (-27)
75 ÷ 4 + 6
To perform the above operations, simply punch in the numbers, then
the change sign key when necessary, and the operations in the
order
±
15
18.47
4
75
they appear.
-
±
÷
+
29
± =
50
±
=
x
27
±
4
+
6
-
44
68.47
=
=
108
24.75
Memory
The memory keys on your calculator are useful when performing several
operations at one time.
MRC
This button displays what is in memory. If you press it twice, you can
clear what is in memory.
M+
This button adds the amount on the display to what is in memory.
M-
This button subtracts the amount on the display from what is in
memory.
When the calculator holds a value in memory, the display looks like
M
168
this:
Mathematics Skills 2
It will also display the number amount that it holds in memory. Performing any
operation on your calculator will not affect what is in memory unless you use
any of the memory keys.
Example 3: Solve the problem below using your calculator’s memory
keys.
7 + 14 x 5 – 162
Using order of operations, we must compute the exponents first.
16 x 16
15 x
7
On/C
MRC
MRC
5
M+
M-
M+
M
256
M
70
M
7
M
MRC
-
179
Notice that, when we multiplied 16 x 16, we used the memory minus key
because 162 was being subtracted from the rest of the expression. As we
computed each part of the expression, we used the memory plus and minus
depending on what was performed in the problem. Once we were done
computing all the parts, we pressed memory recall so that the calculator could
display what was in memory.
In most calculators, the memory keys work the same way. Please check your
manual if your memory keys work different.
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Mathematics Skills 2
Special Keys
Some of the basic four-function calculators have special keys that perform
certain functions. The calculator displayed above contains two special keys: the
square root key and the percent key.
√
The square root key will calculate the square root of the number on
display.
%
The percent key converts the number on the display to the decimal
expression of the percent.
Example 4:
Solve each problem below using the special keys on the
calculator.
8 + √70
20% of 65
In order to calculate the first problem, simply press the following keys
on the calculator:
8 + 70 √
=
16.3666
To calculate the second problem press the following keys:
66 x 20 %
The result is 13.
*Note that, in the first problem, we had to press the equal’s key to obtain the
final answer.
In the second problem, recall that the word “of” represents multiplication. Note
that we had to input the amount first and the percentage last. This is done so
that the calculator can convert the percentage into a decimal and then multiply
by the amount. Keep this in mind when multiplying percentages. (You may not
even need this key if you know how to convert a percentage to a decimal.)
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Mathematics Skills 2
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Mathematics Skills 2
172