Download 2nd NW Content Review Notes

Standards of Learning
Content Review Notes
Grade 8 Mathematics
2nd Nine Weeks, 2016-2017
Mathematics
Content Review Notes
Grade 8 Mathematics: Second Nine Weeks
2016-2017
This resource is intended to be a guide for parents and students to improve content
knowledge and understanding. The information below is detailed information
about the Standards of Learning taught during the 2nd grading period and comes
from the Mathematics Standards of Learning Curriculum Framework, Grade 8
issued by the Virginia Department of Education. The Curriculum Framework in its
entirety can be found at the following website:
http://www.doe.virginia.gov/testing/sol/frameworks/mathematics_framewks/2009/framew
k_math8.pdf
SOL 8.15
The student will
a) solve multi-step linear equations in one variable on one and two sides of the
equation;
b) solve two-step linear inequalities and graph the results on a number line; and
c) identify properties of operations used to solve an equation.

A multi-step equation is defined as an equation that requires the use of more than one
different mathematical operation to solve (e.g., 2x + 3 = –4).
Example 1: 2x + 3 = - 5 Example 2: 19 + 3x = 5x + 4 Example 3: 4(x + 2) = 16

In an equation, the equal sign indicates that the value on the left is the same as the
value on the right.

To maintain equality, an operation that is performed on one side of an equation must be
performed on the other side.

The goal of any equation is to find the value of the variable. In order to do this, each side
must be completely simplified. Then undo any addition or subtraction by doing the
opposite operation. Next, undo any multiplication or division by doing the opposite
operation.
Example 1: 2x + 3 = - 5 Example 2: 19 + 3x = 5x + 4
-3
-3
- 5x –5x
2x
=-8
19 – 2x =
4
2
2
–19
–19
–2x =
–15
x=-4
–2
–2
x =
7
2
or
Example 3: 4(x + 2) = 16
4x + 8 = 16
–8
–8
4x
= 8
4
4
3
1
2
x
= 2

A two-step inequality is defined as an inequality that requires the use of two different
operations to solve (e.g., 3x – 4 > 9).
Example 1:

𝑥
3
–4>8
Example 2: 2x + 4 < 12
The same procedures that work for equations work for inequalities. When both
expressions of an inequality are multiplied or divided by a negative number, the inequality
sign reverses.
Example 1:
𝑥
3
𝑥
–4>8
Example 2: - 2x + 4 ≤ 12
+4 +4
> 12
3
𝑥
3 (3) > (12) 3
-4
-2x
-4
≤ 8
-2
Reverse the
inequality
symbol
-2
x > 36
Why?
x≥-4
Because the
expressions
are being
divided by a
negative
number.
x > 36 means
that any
number
greater than
36 is a possible
solution to this
inequality.
To graph the solution to example 1 (x > 36 )
located at the top of the page, place an “open”
(unshaded) circle on the number 4 and shade
to the right, including the arrow.
To graph the solution to example 2 (x ≥ –4 )
located at the top of the page, place an
“closed” (shaded) circle on the number –4 and
shade to the right, including the arrow.
–6
–5
–4
–3
–2
–1
0
1

The commutative property for addition states that changing the order of the addends
does not change the sum.
Example: 5 + 4 = 4 + 5

The commutative property for multiplication states that changing the order of the factors
does not change the product.
Example: 5 · 4 = 4 · 5

The associative property of addition states that regrouping the addends does not change
the sum.
Example: 5 + (4 + 3) = (5 + 4) + 3

The associative property of multiplication states that regrouping the factors does not
change the product.
Example: 5 · (4 · 3) = (5 · 4) · 3

Subtraction and division are neither commutative nor associative.

The distributive property states that the product of a number and the sum (or difference)
of two other numbers equals the sum (or difference) of the products of the number and
each other number.
Example 1: 5 · (3 + 7) = (5 · 3) + (5 · 7)
Example 2: 5 · (3 – 7) = (5 · 3) – (5 · 7)

Identity elements are numbers that combine with other numbers without changing the
other numbers.
The additive identity is zero (0).
The multiplicative identity is one (1).
*There are no identity elements for subtraction and division.

The additive identity property states that the sum of any real number and zero is equal to
the given real number.
Example:
The following demonstrates the additive identity property:
5+0=5

The multiplicative identity property states that the product of any real number and one is
equal to the given real number.
Example:

The following demonstrates the multiplicative identity property:
8·1=8
Inverses are numbers that combine with other numbers and result in identity elements.
Example 1: 5 + (–5) = 0
Example 2:
1
·5=1
5

The additive inverse property states that the sum of a number and its additive inverse
always equals zero.
Example: The following demonstrates the additive inverse property.
5 + (–5) = 0

The multiplicative inverse property states that the product of a number and its
multiplicative inverse (or reciprocal) always equals one.
Example:
The following demonstrates the multiplicative inverse property.
1
4· =1
4

Zero has no multiplicative inverse.

The multiplicative property of zero states that the product of any real number and zero is
zero.

Division by zero is not a possible arithmetic operation.
SOL Practice Items provided by the VDOE,
http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
Answers are located on the last page of the booklet.
SOL 8.15 (Multistep Equations and Inequalities)
1. Which of the following is
equivalent to the inequality
5x + 7 < 17 ?
4. What real number property of
multiplication is shown in this
equation?
2. What value of n makes this
equation true? 4n + 9 = 6
5. Which of the following
expressions is equivalent to
4.1(8.5 – 6.2) ?
3. What is the solution to
𝒏
– 4 > 10 ?
𝟐
6. What value of n makes the
equation true?
11
7. What is the value of x in the
following equation?
-4x + 2 = -14
12. Which is one value of the x that
makes the following true?
8. What value of w makes the
following true?
13. What value for n makes the
following sentence true?
9. Which property is used in the
following number sentence?
14. What value for w makes the
equation true?
10. Which of the following
equations illustrates the
multiplicative property of zero?
15. Which property is shown in the
following number sentence?
11. Which is false?
12
19. What value of x makes the
following statement true?
16. Which statement is false?
20.
Which is one of the solutions
to the following?
17.
Anne’s utility bills for three
months were $59, $67, and $33.
To add the utility bills mentally,
Anne thought
21.
22.
18.
What is the solution to
23.
Which number sentence
illustrates the commutative
property of multiplication?
If the number sentence is true,
then y is the –
What is one solution to
13
5x + 1 = 7? Your answer must be
in the form of an improper fraction.
27. Which graph only represents the
solutions?
24.
The first three steps of an
equation Justin is solving is
shown below:
What property justifies the work
between step 2 and step 3?
28. What value of x makes this
equation true?
25. What value of p makes this
equation true?
2p 
 3p  6
4
29. The steps used to solve an
equation are shown.
26. What is the solution?
6 ≥ ½x + 21
What property justifies the work
between Step 4 and Step 5?
14
34.
35.
36.
15
37.
This is the
Pythagorean Theorem
formula.
This is the
hypotenuse.
This is another
illustration and way of visualizing the
Pythagorean Theorem.
Ex
Fi
4 cm
3 cm
Th
SOL 8.10
The student will (a) verify the
Pythagorean Theorem; and (b) apply
the Pythagorean Theorem.
In a right triangle, the square of the length
of the hypotenuse equals the sum of the
squares of the legs (altitude and base).
This relationship is known as
the Pythagorean Theorem: a 2 + b
2 = c 2.
The Pythagorean Theorem is used to find
the measure of any one of the three
sides of a right triangle if the measures
of the other two sides are known.
Whole number triples that are the measures
of the sides of right triangles, such as
(3, 4, 5), (6,8,10), (9,12,15), and
(5,12,13), are commonly known as
Pythagorean triples.
16
SOL Practice Items provided by the
VDOE,
http://www.doe.virginia.gov/testing/sol/stand
ards_docs/mathematics/index.shtml
4. Which of the following equations is
represented by the figure?
Answers are located on the last page
of the booklet.
SOL 8.10 (Pythagorean Theorem)
1. Which group of three side lengths
could form a right triangle?
2. Mr. Malone plans to construct a
walkway through his rectangular
garden, as shown in the drawing.
5. What is the value of m in the right
triangle shown?
3. Three triangles are drawn in
rectangle PQRS.
6. The legs of a right triangle measure
9 inches and 12 inches. What is the
length of the hypotenuse of this
triangle?
17
7. Which correctly names the
hypotenuse of the triangle pictured?
10.
̅̅̅̅?
What is the measure of 𝑨𝑿
8. Which names one of the legs of the
triangle pictured?
11.
Triangle CAT was in Cedric’s
mathematics book.
9. A waterslide is one side of a right
triangle as shown.
12.
Dale drew triangle DOG with the
following measurements.
18
SOL 8.11
The student will solve practical area and perimeter problems involving composite
plane figures.
A polygon is a simple, closed plane figure with sides that are line segments.
Below are examples of different polyhedrons
The perimeter of a polygon is the distance around the figure.
The perimeter of this figure can be found by
adding all five sides together.
1 + 5 + 4 + 2 + 7 = 19
P = 19 units
The area of a rectangle is computed by multiplying the lengths of two adjacent sides (
A  lw ).
Example: Mr. Jones has a rectangular flower garden. What is the area of his
garden if the length is 7 ft and the width is 9 ft?
A = lw
l = 7 ft, w = 9 ft
A = (7)(9)
A = 63 ft²
19
The area of a triangle is computed by multiplying the measure of its base by the measure
1
of its height and dividing the product by 2 ( A  bh ).
2
1
A  bh
2
b = 3.7 cm, h = 2.4 cm
A = ½ (3.7)(2.4)
A = 4.44 cm²
The area of a parallelogram is computed by multiplying the measure of its base by the
measure of its height ( A  bh ).
A = bh
12
b = 12, h = 7
7
A = (12)(7)
A = 84
The area of a trapezoid is computed by taking the average of the measures of the two
1
bases and multiplying this average by the height [ A  h(b1  b2 ) ].
2
3
3
8
A
1
h(b1  b2 )
2
h = 3, b1 = 3, b2 = 8
A = ½(3)(3 + 8)
A = ½(3)(11)
A = 16.5
The area of a circle is computed by multiplying Pi times the radius squared ( A   r 2 ).
Example: What is the area of circle with a radius of 23?
A = πr²
r = 23
A = (π)(23)²
A = (π)(529)
A = 1661.9
20
The circumference of a circle is found by multiplying Pi by the diameter or multiplying Pi by
2 times the radius ( C   d or C  2 r ).
Example:
14
C = πd
C = 2πr
d = 14
r = 14/2 = 7
C = (π)(14)
C = 2(π)(7)
C = 43.98
C = 43.98
The area of any composite figure is based upon knowing how to find the area of the
composite parts such as triangles, rectangles and circles.
Area of Triangle
Area of Rectangle
A = ½ bh
A = lw
b = 4 cm, h = 4 cm
l = 4 cm, w = 2 cm
A = ½ (4)(4)
A = (4)(2)
A = 8 cm²
A = 8 cm²
Area of Figure = Area of Triangle + Area of Rectangle
Area of Figure = 8 cm² + 8 cm²
Area = 16 cm²
Area of Rectangle
A = lw
l = 20 cm, w = 14 cm
A = (14)(20)
A = 280 cm2
Area of Semi-circle
A=
𝜋𝑟 2
2
d = 14 cm, r = 7 cm
A=
A=
A=
(𝜋)(72 )
2
(𝜋)(49)
2
153.86
2
A = 76.93 cm2
Area of Figure = Area of Rectangle + Area of Semi-circle
Area of Figure = 280 cm2 + 76.93 cm2
Area = 356.93 cm2
21
15 in
Area of Rectangle
A = lw
12 in
l = 15 in, w = 12 in
A = (15)(12)
A = 180 in2
Area of Semi-circle
A=
2
d = 15 in – 9 in = 6 in,
r = 3 in
A=
(𝜋)(32 )
A=
9 in
𝜋𝑟 2
A=
2
(𝜋)(9)
2
28.26
2
A = 14.13 in2
Area of Figure = Area of Rectangle - Area of Semi-circle
Area of Figure = 180 in2 – 14.13 in2
Area = 165.87 in2
22
SOL Practice Items provided by the
VDOE,
http://www.doe.virginia.gov/testing/sol/sta
ndards_docs/mathematics/index.shtml
Answers are located on the last
page of the booklet.
3. Pablo has a large circular rug
on his square-shaped bedroom
floor. If the diameter of the rug
is equal to the length of the
bedroom floor, which is closest
to the area of the rug?
SOL 8.11 (Area and Perimeter of
Composite Plane Figures)
1. Travis is making a wall hanging
out of different colors of glass.
The shape of the wall hanging
is shown on the grid below.
Which is closest to the total
amount of glass needed to
make the wall hanging?
4. What is the minimum number
of the same-sized triangles as
the one above that would be
required to form the polygon
below?
2. Bob wants to paint a
rectangular wall that measures
16 ft by 9ft. The wall contains
a window with the dimensions
shown.
5. What is the area of the
parallelogram shown?
23
6. Katie is going to carpet her
living room floor and drew the
diagram shown. What is the
minimum number of square
feet of carpet she will need?
7. What is the total area of the
figure shown?
8. Leslie built a walkway around
a rectangular garden as shown.
The walkway is the same length
on all sides of the garden.
What is the perimeter of the
garden?
9. A composite figure is shown.
What is the total area of this
figure?
10. A rectangle as shown has a
length of 0.9 centimeters and a
width of 0.4 centimeters. A
circle is drawn inside that
touches the rectangle at two
points.
Which is closest to the total
area of the shaded region of
the rectangle
24
SOL 8.9
The student will construct a three-dimensional model, given the top, side,
and/or bottom views.
Three-dimensional models of geometric solids can be used to understand perspective
and provide tactile experiences in determining two-dimensional perspectives.
Three-dimensional models of geometric solids can be represented on isometric paper.
Example 1:
Given the views above, students should be able to construct a threedimensional model.
Example 2: A figure has the views shown.
Students should be able to identify the three dimensional model.
25
SOL Practice Items provided by the
VDOE,
http://www.doe.virginia.gov/testing/sol/sta
ndards_docs/mathematics/index.shtml
Answers are located on the last page
of the booklet.
3. A figure has the bottom and the
left-side views shown, and its
front view is shaded. Which
represents the figure?
SOL 8.9 (Three-Dimensional Figures)
1. Three different views of a
three-dimensional figure
constructed from cubes are
shown.
Which of the following figures
could these views represent?
4. Three different views of a three
dimensional figure are shown.
2. A three dimensional figure is
constructed from identical
cubes. Three views of the figure
are shown.
5. His shows three different views
of a three dimensional figure
constructed from cubes. Which
could be this figure?
Which of the following could be
the three dimensional figure?
26
6. Which three dimensional figure
in the position shown most
likely has the top view shown?
(top view)
7. This shows three different
views of a three-dimensional
figure made from cubes.
9. The front view of a threedimensional figure using
identical cubes is shown.
Identify each three-dimensional
figure that has this front view.
10. Which could represent the
front view of this figure?
Which could be a drawing of the
figure?
8. A figure has the views shown.
Which represents the figure?
11. Which three-dimensional
figure could be represented by
these three views?
27
Math Smarts!
Math + Smart Phone = Math Smarts!
Need help with your homework? Wish that your teacher could explain the math concept
to you one more time? This resource is for you! Use your smart phone and scan the QR
code and instantly watch a 3 to 5 minute video clip to get that extra help. (These videos
can also be viewed without the use of a smart phone. Click on the links included in this
document.)
Directions: Using your Android-based phone/tablet or iPhone/iPad, download any QR
barcode scanner. How do I do that?
1. Open Google Play (for Android devices) or iTunes (for Apple devices).
2. Search for “QR Scanner.”
3. Download the app.
After downloading, use the app to scan the QR code associated with the topic you need
help with. You will be directed to a short video related to that specific topic!
It’s mobile math help when you need it! So next time you hear, “You’re always
on that phone” or “Put that phone away!” you can say “It’s homework!!!”
Access this document electronically on the STAR website through
Suffolk Public Schools.
(http://star.spsk12.net/math/MSInstructionalVideosQRCodes.pdf)
PLEASE READ THE FOLLOWING:
This resource is provided as a refresher for lessons learned in class. Each link will
connect to a YouTube or TeacherTube video related to the specific skill noted under
“Concept.” Please be aware that advertisements may exist at the beginning of each
video.
28
Link
SOL
Solving Multistep Equations
8.15a
https://www.youtube.com/watch?v=HRRb-I1POHk
https://www.youtube.com/watch?v=F1azJEdfx5c
Solving and Graphing Multistep Inequalities
8.15b
https://www.youtube.com/watch?v=Fy344XYM0iU
Properties of Operations
https://www.youtube.com/watch?v=jAekG2q711o
8.15c
https://www.youtube.com/watch?v=awGWSBrN0as
Constructing three-dimensional models
8.9
https://www.youtube.com/watch?v=W-_z0UnA01g
8.10
Applying the Pythagorean Theorem
https://www.youtube.com/watch?v=QNl_yb8doRk
8.10
Verifying the Pythagorean Theorem
8.11
Solving area and perimeter problems involving composite plane
figures
https://www.youtube.com/watch?v=tFRCEdydcEk
https://www.youtube.com/watch?NR=1&v=uaj0XcLtN5c&feature=endscreen
QR Code
29
Vocabulary
SOL 8.15
additive identity
associative property
(x, +)
commutative property
(x, +)
distributive property
multiplicative identity
The sum of an addend and zero is zero.
a+0=0+a=a
The way in which three numbers are
grouped when they are added or
multiplied does not change their sum
or product.
The order in which
two numbers are
added or multiplied does not change
their sum or product.
To multiply a sum by a number, multiply
each addend of the sum by the number
outside the parenthesis.
The product of a factor and one is the
factor.
a·1=1·a=a
SOL 8.10
hypotenuse
The side opposite the right angle in a
right triangle
legs
The two sides of a right triangle that
form the right angle
Pythagorean Theorem
In a right triangle, the square of the
length of the hypotenuse c is equal to
the sum of the squares of the lengths
of the legs a and b. c2 = a2 + b2
Pythagorean triple
A set of three integers that satisfy the
Pythagorean Theorem
right angle
An angle that measure 90 degrees
right triangle
A triangle having one right angle
30
SOL 8.11
area
The amount of space taken up in a
plane by a figure
perimeter
The distance around a polygon
31
Practice Items Answer Key (2nd Nine Weeks)
SOL 8.15 (Multistep Equations and
Inequalities)
1. A
2. B
3. J
4. D
5. H
6. G
7. D
8. D
9. A
10. A
11. F
12. D
13. A
14. J
15. G
16. A
17. D
18. C
19. D
20. J
21. A
22. H
SOL 8.15 (Multistep Equations and
Inequalities) (continued)
23.
x=
6
5
24. Additive inverse property
−6
25. p =
11
26.
–30 ≥ x
27. D
28. x = –12 x = –12
29. Identity property for
multiplication
30.
n = –2
31. x = –1.24
32. k = –75
33. m = 6
34.
35.
36.
37. B
SOL 8.10 (Pythagorean Theorem)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
F
H
H
D
H
H
D
J
H
B
G
D
SOL 8.11 (Area and Perimeter)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
C
G
C
G
H
C
B
76 feet
D
B
32
SOL 8.9 (Three-Dimensional
Models)
1.
2.
3.
4.
5.
6.
7.
8.
9.
G
D
F
C
B
G
F
A
10. D
11. A