Download UNIT 3-EXPRESSIONS AND EQUATIONS

MATHEMATICS
GRADE 7
UNIT 3
EXPRESSIONS AND
EQUATIONS
1
VOCABULARY FOR FLASHCARDS
A.) Adjacent Angles: angles which lie next to each other and share a common vertex
and one side. In the diagram below, <ABC or < 1 is adjacent to
<CBD or < 2. They share vertex B and side 
 (the side is a
BC
ray).
A
C
1
B
2
D
B.) An Expression in Standard Form (description): (An expression that is in
expanded form where all like-terms have been collected is said to be in
standard form.)
C.) An Expression in Factored Form: (middle school description) (An
expression that is a product of two or more expressions is said to be in
factored form.)
D.) Angles on a line: The sum of the measures of all angles on a line is 180 degrees.
E.) Angles at a Point: The measure of all angles formed by the center vertex is 360
degrees.
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F.) Area/Rectangular Array/Tape Diagram Model: This is a model that can
be used to help visualize and solve word problems.
G.) Coefficient of the Term: (The number found by multiplying just the
numbers in a term together is called the coefficient of the term.)
H.) Complementary Angles: Two angles whose sum is 90 degrees.
I.) Opposite: Another term for opposite is the additive inverse. The sum
of a number and the additive inverse is 0.
J.) Reciprocal: Another term for reciprocal is the multiplicative inverse.
The product of a number and the multiplicative inverse is 1.
K.) Supplementary Angles: Two angles whose sum is 180 degrees.
L.) Vertical Angles: two non-adjacent pairs of angles formed by two intersecting
lines. Vertical angles must be equal in measurement. In the
diagram below, <1 and < 2 are a pair of vertical angles as well
as < 3 and < 4.
3
2
1
3
Expressions and Equations
A.) Additive Identity Property of Zero: The additive Identity Property of
Zero states that zero is the only number that when summed to another
number, the result is again that number.
B.) An Expression in Expanded Form (description) (An expression that is
written as sums (and/or differences) of products whose factors are
numbers, variables, or variables raised to whole number exponents/powers
is said to be in expanded form. A single number, variable, or a single
product of numbers and/or variables is also considered to be in expanded
form.)
C.) An Expression in Standard Form (description) (An expression that is in
expanded form where all like-terms have been collected is said to be in
standard form.)
D.) An Expression in Factored Form: (middle school description) (An
expression that is a product of two or more expressions is said to be in
factored form.)
E.) Area/Rectangular Array/Tape Diagram Model: This is a model that can
be used to help visualize and solve word problems.
F.) Coefficient of the Term: (The number found by multiplying just the
numbers in a term together is called the coefficient of the term.)
G.) Multiplicative Property of One: The multiplicative Property of One
states that one is the only number that when multiplied with another
number, results in that number again.
H.) Opposite: Another term for opposite is the additive inverse. The sum
of a number and the additive inverse is 0.
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I.) Reciprocal: Another term for reciprocal is the multiplicative inverse.
The product of a number and the multiplicative inverse is 1.
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COMBINING EXPRESSIONS HORIZONTALLY
Example 1: Find the sum by aligning the expression horizontally.
(6a + 4b – 10c) + (-3a + 5b + 5c)
6a + 4b – 10c – 3a + 5b + 5c
3a + 9b – 5c
Example 2: Find the difference by aligning the expression horizontally.
(6a + 4b – 10c) – (-3a + 5b + 5c)
6a + 4b – 10c + 3a – 5b – 5c
9a – b – 15c
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Translating Algebraic Expressions and Equations
+
-
Add
Difference
Sum
Minus
Plus
Remainder
Total
less than
increased by decreased by
subtract
from
x
:
=
multiply
product
times
double
triple
divide
quotient
remainder
equals
is equal to
Is
1. Six more than a number b
1. 6 + b
2. Eight decreased by a number m
2. 8 - m
3. The product of four and a number x
3. 4x
4. The quotient when a number z is divided by fourteen
4. z  14
5. The value in cents of k dimes
5. 10k
6. Twice a number p increased by ten
6. 2p +10
7. Twelve more than a number x is seventeen
7. 12 + x = 17
8. Thirty-two less than a number m is eighteen
8. m – 32 = 18
9. Mark has q quarters worth a total of $12.50
9. 25q = 1250
9. .25q = 12.50
(decimal form)
10. Mary has d dimes and p pennies worth a total of $12.50 10. 10d + 1p
10.
.10d + .01p = 12.50
(decimal form)
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USING AREA/RECTANGULAR ARRAY/TAPE DIAGRAM MODEL
Example 1: Sean and Landon share a sum of 54 baseball cards. If the ratio
of Sean’s cards to Landon’s cards is 2:4, how much does each
person have?
Sean:
Landon:
------------------------------54------------------------------6 units = 54
1 unit = 9
2 units = 18
4 units = 36
Sean has 18 baseball cards and Landon has 36 baseball cards.
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Example 2: Multiply 4(4 + 2)
4
----------4---------------2-----
4(4 + 2)
4(6)
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Example 3: Draw an area/rectangular array/tape diagram model to represent
the following sum.
(a + b) + (a + b)
2
a
b
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FOR PROCEDURES SOLVING ALGEBRAIC EQUATIONS-PLEASE REFER TO
UNIT 2 RATIONAL NUMBERS PART 2 NOTES
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ALGEBRAIC EQUATIONS INVOLVING CONSECUTIVE INTEGER, AGE,
MOTION PROBLEMS AND THE USE OF THE LET STATEMENT
Example 1: The sum of three consecutive integers is 33. Find the smallest
integer.
Let x = first integer
Let x + 1 = second integer
Let x + 2 = third integer
x + x + 1 + x + 2 = 33
3x + 3
= 33
-3
-3
𝟑𝒙
𝟑
=
𝟑𝟎
𝟑
x = 10
Therefore, the smallest integer is 10.
Example 2: The sum of two consecutive even integers is 42. Find the
numbers.
Let x = first even number
Let x + 2 = second even number
x + x + 2 = 42
2x + 2 = 42
-2
-2
𝟐𝒙
𝟐
=
𝟒𝟎
𝟐
First even integer
x = 20
x = 20
Therefore, the two integers are 20 and 22.
11
Second even integer
x + 2
(20) + 2
22
Example 3: Sean is 3 years older than Landon. If the sum of their ages is
63, how hold is each boy?
Let x = Landon’s age
Let x + 3 = Sean’s age
x + x + 3 = 63
2x + 3
= 63
-3
-3
𝟐𝒙 𝟔𝟎
=
𝟐
𝟐
x = 30
Landon’s age
x
30
Sean’s age
x + 3
(30) + 3
33
Example 4: Kenneth is twice as old as his brother. If Kenneth was 40 four
years ago, how old is Kenneth’s brother now?
Let x = Kenneth’s brother’s age
Let 2x = Kenneth’s age
2x – 4 = 40
+4 +4
𝟐𝒙 𝟒𝟒
=
𝟐
𝟐
x = 22
Kenneth’s brother’s age is 22
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Example 5: Mary rode her bike 80 miles in 5 hours. She rode at an average
speed of 15 miles per hour for “h” hours and an average rate of
speed of 20 miles per hour for the rest of the time. How long
did Mary ride at the slower speed?
Let h = time, in hours, Mary rode at 15 miles per hour
Mary
15 mph
Mary 20 mph
Rate
(mph)
15
Time
(hours)
H
Distance
(miles)
15h
20
5 – h
20(5 – h)
15h + 20(5 – h) = 80
15h + 100 – 20h = 80
-5h + 100 = 80
-100 -100
−𝟓𝒉 −𝟐𝟎
=
−𝟓
−𝟓
h = 4
h = time, in hours, Mary rode at the slower speed of 15 miles per hour
Mary rode for 4 hours at the slower speed.
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Solving And Graphing Solutions Of One Step, Two-Step and
Multi-Step Inequalities On A Number Line
Method:
1. Simplify the left side of the equal sign (if possible).
2. Simplify the right side of the equal sign (if possible).
Please Note: When simplifying, this could involve combining like
terms, distributive property, etc.
a) Distributive Property of Multiplication over Addition:
a(b + c) = ab + ac
b) Distributive Property of Multiplication over Subtraction:
a(b – c) = ab – ac
3. Gather all variables to one side of the equal sign by using
inverse operations.
4. Gather all constants to the opposite side of the variables by using inverse operations.
5. Isolate (get by itself) the variable by using inverse operations.
PLEASE NOTE: IF THE INVERSE OPERATION INVOLVES DIVISION
OR MULTIPLICATION OF A NEGATIVE NUMBER, THE INEQUALITY
SIGN NEEDS TO CHANGE DIRECTIONS.
6. Graph the solutions on a number line
a) an unshaded dot is used if the number is not included in the solution set
(symbols: < and >)
b) a shaded dot is used if the number is included in the solution set
(symbols: ≤ and ≥)
7. Check if asked
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Solving And Graphing Solutions Of One Step, Two-Step and
Multi-Step Inequalities On A Number Line Continued
Examples
Check
1. a + 5 < 10
a + 5 < 10
-5
(4) + 5 < 10
9 < 10
a
-5
< 5
Graph of Solutions
Check
-3a ≤ 12
-3(-3) ≤ 12
9 ≤ 12
Graph of Solutions
-5 -4 -3 -2-1 0 1 2 3 4 5
3. 6x + 5 < -7
-5
-5
6 x  12

6
6
3a 12

2.
3 3
a ≥ -4
Check
6x + 5 < -7
6(-3) + 5 < -7
-18 + 5 < -7
-13 < -7
-5 -4 -3 -2-1 0 1 2 3 4 5
4. 21 ≥ 7(m – 2)
21 ≥ 7m – 14
+14
+14
35 7 m

7
7
5 ≥ m
-5 -4 -3 -2-1 0 1 2 3 4 5
21 ≥ 7(2)
21 ≥ 14
x < -2
Graph of Solutions
Check
21 ≥ 7(m – 2)
21 ≥ 7(4 – 2)
Graph of Solutions
-5 -4 -3 -2-1 0 1 2 3 4 5
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Solving And Graphing Solutions Of One Step, Two-Step and
Multi-Step Inequalities On A Number Line Continued
5. -8x +10 < -10x + 16
+10x
+10x
2x +10 <
16
-10
-10
2x 6

2 2
Check
-8x + 10 < -10x + 16
-8(2) + 10 < -10(2) + 16
-16 + 10 < -20 + 16
- 6 < -4
x<3
Graph of Solutions
-5 -4 -3 -2-1 0 1 2 3 4 5
7
1
c
2
2
7
1
2
  2c  2 
 2 
2
6.
-7 >2c + 1
-1
-1
-8 > 2c
 8 2c

2
2
-4 > c
Graph of Solutions
-5 -4 -3 -2-1 0 1 2 3 4 5
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Solving And Graphing Solutions Of One Step, Two-Step and
Multi-Step Inequalities On A Number Line Continued
7. Mary works part time and sells books. She earns $50 a week plus $10 dollars for
each book she sells. Write an inequality for the number of books Mary needs to
sell to make at least $100. Then solve and graph the inequality on a number line.
Let x = number of books sold
50 + 10x  100
-50
-50
10 x 50

10 10
x5
Graph of Solutions
-5 -4 -3 -2-1 0 1 2 3 4 5
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8. Twenty-four times a number is less than 72. Solve the inequality and graph the
solution set on a number line.
Let x = the number
24x < 72
24 x 72

24 24
x<3
Graph of Solutions
-5 -4 -3 -2-1 0 1 2 3 4 5
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GEOMETRY

The word geometry is derived from two Greek words meaning “land measure.”

Geometry may be defined as the study of space and figures in space.

The study of geometry begins with the selection of basic words.
Basic Words:
1. Point
A point has no dimensions. It does not have length, width, or height. It has one
property and that is the property of position. A point can be described by its position.
We label a point with a capital letter.
A point is a mental concept. It can be represented by a dot on a piece of paper or by
a dot on a chalkboard. The dot is not a point. The dot represents the position of the
point. The dot is a little ink or small bit of chalk having physical dimensions.
 A = Point A
2. Line
A line has no height and no width. It does have the property of infinite length.
Every place on the line can be thought of as a point. Therefore, a line is an
infinite set of points. A line extends infinitely in both directions forever.
If we are given two points, only one straight line can exist that contains both
points. In other words, given two points, we can draw one and only one straight
line through both points. The two points that lie on the same line are said to be
collinear. Collinear points lie on the same line. Two points determine a line.
The symbol 
is used to indicate a line containing the points A and B.
AB
Line AB = 
= Line m
AB
A
B
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3. Line Segment
A finite portion of a line is called a line segment. A line segment has two
endpoints. The line segment with endpoints D and F can be represented by the
symbol DF , which is read, “line segment DF.”
D
F
4. Ray
An infinite portion of a line which has only one endpoint is called a ray. A
ray starts at its endpoint and extends forever in one direction. A ray is identified
by its endpoint and any other point on it. If the endpoint is represented by the
letter X and the letter P represents any other point on the ray, the ray may be
identified by the symbol 
 , which is read “Ray XP.” The first letter always
XP
represents the endpoint, which is also called the origin.
X
P
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Angles
5.) Angle: A figure formed by two rays with the same endpoint is called an angle. The rays
are called the sides and the common endpoint is called the vertex.
side
vertex
side
An angle is identified by using three letters, the letter that represents the
endpoint (or vertex), is the middle letter.
This angle below is identified as “Angle LOB” or “Angle BOL” or by the use of
the symbol for angle (< = < = Angle), < LOB or < BOL or < BOL or < LOB.
Sometimes a number is used to identify an angle. In this diagram, < 1 is the
same as < LOB.
O
L
1
B
Types of Angles:
6. Straight Angle: an angle measuring 180 degrees.
7. Right Angle: an angle measuring 90 degrees.
8. Acute Angle: an angle measuring less than 90 degrees, but more than 0 degrees.
9. Obtuse Angle: an angle measuring more than 90 degrees, but less than
degrees.
10. Complementary Angles: Two angles whose sum is 90 degrees.
11. Supplementary Angles: Two angles whose sum is 180 degrees.
Note: When two lines intersect, four pairs of supplementary angles are formed.
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12. Adjacent Angles: angles which lie next to each other and share a common vertex
and one side. In the diagram below, <ABC or < 1 is adjacent to
<CBD or < 2. They share vertex B and side 
 (the side is a
BC
ray).
A
C
1
B
2
D
13. Vertical Angles: two non-adjacent pairs of angles formed by two intersecting
lines. Vertical angles must be equal in measurement. In the
diagram below, <1 and < 2 are a pair of vertical angles as well
as < 3 and < 4.
3
1
2
4
22
14. Angles on a line: The sum of the measures of all angles on a line is 180 degrees.
15. Angles at a Point: The measure of all angles formed by the center vertex is 360
degrees.
Examples Of Complementary, Supplementary, And Vertical Angle Problems
<1 = (2x – 5)o
<2 = (3x + 5)o
1
2
Find <1 and <2
o
<1 + <2 = 90 (complementary)
(2x -5) + (3x + 5) = 90
<1 = 2x – 5
5 x 90

5
5
<2 = 3x + 5
2(18) – 5
3(18) + 5
36 – 5
54 + 5
31o
59o
x = 18
2.)
2
1
<1 = (x +32)o
<2 = (4x + 5)o
Find <1 and <2
<1 = <2 (vertical)
(x + 32) = (4x + 5)
-x
-x
<2 = 4x + 5
(9) + 32
4(9) + 5
41o
32 = 3x + 5
-5
<1 = x + 32
41o
-5
27 = 3x
3
3
9=x
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Circles
A circle is the set of all points in a plane that are a fixed distance from a fixed point. The fixed distance is called
the radius and the fixed point is called the center. It is very important to include the words “in a plane” in the
definition of a circle. Otherwise, the definition will be that of a sphere.
All points whose distance from the center is less than the radius are said to be in the interior of the circle and the
set of all such points plus the center is the interior of the circle. All points whose distance from the center is
greater than the radius are said to be exterior to the circle and the set of all such points is the exterior of the
circle.
This is contrary to much of the language commonly used on reference to circles. Very often when we refer to a
circle, we really refer to the circle plus it’s interior. The circle plus its interior forms a closed circular region. For
example, reference is often made to the area of a circle. A circle is a curved line. It has no dimensions except
length. It has no area. It is the region that is composed of the circle and its interior that has area. We doubt the
phrase “area of a circle” will continue to be used, but in using it, you should be fully aware of the more complete
mathematical description that is being abbreviated in this usage.
The same applies to polygons. A polygon is composed of various line segments. It has no area. It is the region that
is composed of the polygon plus its interior that has area.
A chord is a straight line segment whose endpoints are points on the circle. A chord which contains as one of its
points the center of a circle is called the diameter.
The radius of a circle is the distance from the center to the circle. It is also a line segment whose endpoints are
the center and a point on the circle. The word “radius” is used for both the distance and line segment.
A semicircle is half of a circle. It should be represented by an arc as shown in the adjoining diagram and not by
the region shown in this diagram. However, the
latter will continue to be referred to as a semicircle,
although it should be called a semicircular region.
A central angle is an angle whose vertex is the center of the circle. Its sides are rays, each containing a radius of
a circle.
An arc of a circle is the set of all points on a circle between two given points on the circle, including the given
points. An arc cannot be identified without ambiguity by just two given points, as there is no way of knowing to
which set of points on the circle we are referring. Arc AB could refer to either the top or the bottom part of the
circle. Three letters representing respectively an endpoint, a point on the arc, and its other endpoint, identify an
arc exactly. However, it is customary to name a minor arc by its two endpoints only, and this is proper if the
convention is agreed upon.
A
B



AB
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Finding Circumference and Area of a Circle
1.) Find the circumference of a circle with a diameter of 5 inches. Leave
your answer in terms of 𝝅.
A.) Formula
C0 = πd
B.) Substitute number
= π(5 inches)
C.) Answer (with proper units)
= 5π inches
2.) Find the area. Leave your answer in terms of π.
A.) Formula
A0 = πr2
B.) Substitute number
= π(2.5 inches)2
C.) Answer (with proper units)
= π(6.25 inches2)
= 6.25π inches2
3.) Using the diameter of 5 inches, find the circumference and area and
round to the nearest tenths place.
Circumference
C0 = πd
= π(5 inches)
= 15.70796327…… inches
= 15.7 inches
Area
A0 = πr2
= π(2.5 inches)2
= π(6.25 inches2)
= 19.63495408….. inches2
= 19.6 inches2
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Circle Problems On Finding Missing Radii and Diameters
1. Find the radius of a circle, given the area is 100cm2. Round to tenths place.
Ao =  r2
r 2


100cm 2

100cm 2

 r2
5.641895835…cm = r
5.6cm  r
2. Given the circumference is 249  in, find the radius.
Co =  d
242in


Co = 2  r
OR
242in 2r

2
2
d

242in = d
121in = r
121in = r
3. Find the diameter if the area is 64  in2.
Ao =  r2
64in 2


r 2

64in 2  r 2
8in = r
16in = d
26
Perimeter Of Polygons, Circumference Of A Circle, and Areas
A. Perimeter: Perimeter is the total outside measurement of a polygon. To find the
perimeter of a polygon, add up all the lengths of the sides together.
B. Circumference: Circumference is the total outside measurement of a circle. To
find the circumference of a circle, use the following formulas:
Circumference = π ·diameter
diameter
Circumference = 2·π·radius
π = 3.141592654.....
diameter = 2·radius
radius
C. Area Of A Circle: The formula for the area of a circle is the following:
Area(circle) = π (radius)²
diameter
radius
D. Area Of A Parallelogram: The formula for the area of a parallelogram is the
following:
height
base
Area (parallelogram) = base · height
27
E. Area Of A Square: The formula for the area of a square is the following:
side
Area(square) = (side)²
F. Area Of A Rectangle: The formula for the area of a rectangle is the following:
width
w
length
Area(rectangle) = length · width
G. Area Of A Trapezoid: The formula for the area of a trapezoid is the following:
base
height
base
Area(trapezoid) = ½ (base 1 + base 2) · height
H. Area Of A Rhombus: The formula for the area of a rhombus is the following:
height
base
Area(rhombus) = base · height
I. Area Of A Triangle: The formula for the area of a triangle is the following:
height
base
Area(triangle) = ½ · base • height
28
Identifying Bases And Faces (Lateral Surface) Of Three-Dimensional Shapes
1. Cube:
2. Rectangular Prism:
3. Triangular Prism:
4. Trapezoidal Prism:
5. Cylinder:
29