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Standards of Learning Content Review Notes

Grade 8 Mathematics

2nd Nine Weeks, 2018-2019

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Mathematics Content Review Notes

Grade 8 Mathematics: Second Nine Weeks 2018-2019

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.

8.18 The student will solve multistep linear inequalities in one variable with the variable on one or both sides of the inequality symbol, including practical problems, and graph the solution on a number line.

Apply properties of real numbers and properties of inequality to solve multistep linear inequalities (up to four steps) in one variable with the variable on one or both sides of the inequality. Coefficients and numeric terms will be rational. Inequalities may contain expressions that need to be expanded (using the distributive property) or require collecting like terms to solve.

Graph solutions to multistep linear inequalities on a number line.

Write verbal expressions and sentences as algebraic expressions and inequalities.

Write algebraic expressions and inequalities as verbal expressions and sentences.

Solve practical problems that require the solution of a multistep linear inequality in one variable.

Identify a numerical value(s) that is part of the solution set of a given inequality.

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 – 4 > 8 Example 2: - 2x + 4 ≤ 12

+4 +4 - 4 - 4

𝑥

3 > 12 -2x ≤ 8

-2 -2

3 (𝑥3

) > (12) 3

x > 36 x ≥ - 4

Reverse the

inequality symbol

Why?

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.

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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

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Practice Problems (Answers can be found in the back of the booklet) SOL 8.18 Solving and Graphing Inequalities 1. Which of the following is equivalent to the inequality

5x + 7 < 17 ?

2. What is the solution to 𝒏

𝟐 – 4 > 10 ?

A. n > 7 B. n > 12 C. n > 24 D. n > 28

3. Which is one value of x that makes the following true? 7x + 3 > 17

A. 0 B. 1 C. 2 D. 3

4. Which is one of the solutions to the following? 2x + 4 < 12

A. 6 B. 5 C. 4 D. 3

5. Which graph only represents the solutions?

−𝟑𝒂 + 𝟏

𝟐 ≤ 𝟖

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6. What is the solution?

6 ≥ ½x + 21 Pick the appropriate number and symbol from each column and place in

boxes.

7. Graph the solution to -5x - 7 ≤ −𝟏𝟐 𝐨𝐧 𝐚 𝐧𝐮𝐦𝐛𝐞𝐫 𝐥𝐢𝐧𝐞.

8. 9. Brianna has some video games. Sam has twice as many video games as Brianna. Together they have more than 24 video games. How many games could Brianna have?

A. x < 8 B. x > 8 C. x ≤ 𝟖 D. x ≥ 𝟖 10. Mrs. Jones charges $25 per hour for tutoring services plus a one- time fee of $50 for supplies. How many tutoring sessions can a student attend if the parent will spend, at most, $1,500?

B. t < 58 B. t > 58 C. t ≤ 𝟓𝟖 D. t ≥ 𝟓𝟖

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SOL 8.9 The student will (a) verify the Pythagorean Theorem; and (b) apply the Pythagorean Theorem.

Verify the Pythagorean Theorem, using diagrams, concrete materials, and measurement. (a)

Determine whether a triangle is a right triangle given the measures of its three sides. (b)

Determine the measure of a side of a right triangle, given the measures of the other two sides. (b)

Solve practical problems involving right triangles by using the Pythagorean Theorem. (b)

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. This is another illustration and way of visualizing the Pythagorean Theorem.

This is the hypotenuse.

This is the altitude.

This is the base.

This is the Pythagorean Theorem

formula.

Example:

Find the length of the hypotenuse (side c).

42 + 32 = 𝑐2 OR √42 + 32 = 𝑐

√16 + 9 = 𝑐

√25 = 𝑐

5 = 𝑐

The length of the hypotenuse is 5 cm.

3 cm

4 cm

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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.

Practice Items, Answers are located on the last page of the booklet.

SOL 8.9 (Pythagorean Theorem) 1. Which group of three side lengths could form a right triangle?

A. 5,12,13 B. 7,11,14 C. 15,20, 22 D. 18, 34, 39 2. Mr. Malone plans to construct a walkway through his rectangular garden, as

shown in the drawing.

Which is closest to the value of W?

A. 22 ft B. 21 ft C. 15 ft D. 11 ft

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3. Which of the following equations is represented by the figure?

4. What is the value of m in the right triangle shown? 5. The legs of a right triangle measure 9 inches and 12 inches. What is the length

of the hypotenuse of this triangle?

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6. Which correctly names the hypotenuse of the triangle pictured? 7. Which names one of the legs of the triangle pictured?

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8. A waterslide is one side of a right triangle as shown. 9. What is the measure of 𝑨𝑿̅̅ ̅̅ ?

10. What is the length of a garden hose that is stretched diagonally, corner to corner, across a yard that measures 64 meters long and 48 meters wide? Round to the nearest meter.

A. 6400 meters B. 80 meters C. 55 meters D. 112 meters

11. The diagonal of a TV screen is 36 inches. The screen is 25 inches wide. How high is the screen?

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SOL 8.10 The student will solve area and perimeter problems, including practical problems,

involving composite plane figures.

Subdivide a plane figure into triangles, rectangles, squares, trapezoids, parallelograms, and semicircles.

Determine the area of subdivisions and combine to determine the area of the composite plane figure.

Subdivide a plane figure into triangles, rectangles, squares, trapezoids, parallelograms, and semicircles. Use the attributes of the subdivisions to determine the perimeter of the composite plane figure

Apply perimeter, circumference, and area formulas to solve practical problems involving composite plane

figures.

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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 area of a rectangle is computed by multiplying the lengths of two adjacent sides ( lwA ).

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

The perimeter of this figure can be found by adding all five sides together.

1 + 5 + 4 + 2 + 7 = 19

P = 19 units

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The area of a triangle is computed by multiplying the measure of its base by the measure

of its height and dividing the product by 2 ( bhA2

1 ).

The area of a parallelogram is computed by multiplying the measure of its base by the measure of its height ( bhA ).

The area of a trapezoid is computed by taking the average of the measures of the two

bases and multiplying this average by the height [ )(2

121 bbhA ].

3

8

12

bhA2

1

b = 3.7 cm, h = 2.4 cm

A = ½ (3.7)(2.4)

A = 4.44 cm²

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A = bh

b = 12, h = 7

A = (12)(7)

A = 84

3 )(

2

121 bbhA

h = 3, 1b = 3, 2b = 8

A = ½(3)(3 + 8)

A = ½(3)(11)

A = 16.5

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The area of a circle is computed by multiplying Pi times the radius squared ( A 2r ).

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

The circumference of a circle is found by multiplying Pi by the diameter or multiplying Pi by 2 times the radius ( C d or 2C r ).

Example:

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.

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C = πd

d = 14

C = (π)(14)

C = 43.98

C = 2πr

r = 14/2 = 7

C = 2(π)(7)

C = 43.98

A = ½ bh

b = 4 cm, h = 4 cm

A = ½ (4)(4)

A = 8 cm²

A = lw

l = 4 cm, w = 2 cm

A = (4)(2)

A = 8 cm²

Area of Triangle

Area of Figure = Area of Triangle + Area of Rectangle

Area of Rectangle

Area of Figure = 8 cm² + 8 cm²

Area = 16 cm²

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Area of Rectangle A = lw

l = 20 cm, w = 14 cm A = (14)(20) A = 280 cm2

Area of Semi-circle

A = 𝜋𝑟2

2

d = 14 cm, r = 7 cm

A = (𝜋)(72)

2

A = (𝜋)(49)

2

A = 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

15 in

12 in

Area of Rectangle A = lw

l = 15 in, w = 12 in A = (15)(12) A = 180 in2

Area of Semi-circle

A = 𝜋𝑟2

2

d = 15 in – 9 in = 6 in, r = 3 in

A = (𝜋)(32)

2

A = (𝜋)(9)

2

A = 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

9 in

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SOL Practice Items -Answers are located on the last page of the booklet. SOL 8.10 (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?

2. Bob wants to paint a rectangular wall that measures 16 ft by 9ft. The wall contains a window with the dimensions shown.

If Bob does not paint the window, what is the total shaded area he will paint? F. 144 sq ft G. 104 sq ft H. 50 sq ft J. 40 sq ft

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?

4. What is the area of the parallelogram shown?

5. 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?

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6. What is the total area of the figure shown?

7. 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?

8. A composite figure is shown. What is the total area of this figure?

9. 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

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SOL 8.8 The student will construct a three-dimensional model, given the top, side, and/or bottom views.

Construct three-dimensional models, given the top or bottom, side, and front views.

Identify three-dimensional models given a two-dimensional perspective.

Identify the two-dimensional perspective from the top or bottom, side, and front view, given a three-dimensional model.

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.

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Example 1:

Given the views above, students should be able to construct a three-dimensional model.

Example 2: A figure has the views shown.

Students should be able to identify the three dimensional model.

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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 8.8 (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?

2. A three dimensional figure is constructed from identical cubes. Three views of the figure are shown.

Which of the following could be the three dimensional figure?

3. A figure has the bottom and the left-side views shown, and its front view is shaded. Which represents the figure?

4. Three different views of a three dimensional figure are shown.

5. His shows three different views of a three dimensional figure constructed from cubes. Which could be this figure?

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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. Which could be a drawing of the figure?

8. A figure has the views shown. Which represents the figure?

9. The front view of a three-

dimensional 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?

11. Which three-dimensional figure could be represented by these three views?

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8.6 The student will a) solve problems, including practical problems, involving volume and

surface area of cones and square-based pyramids; and b) describe how changing one measured attribute of a rectangular prism

affects the volume and surface area.

Distinguish between situations that are applications of surface area and those that are applications of volume. (a)

Determine the surface area of cones and square-based pyramids by using concrete objects, nets, diagrams and formulas. (a)

Determine the volume of cones and square-based pyramids, using concrete objects, diagrams, and formulas. (a)

Solve practical problems involving volume and surface area of cones and square-based

pyramids. (a)

Describe how the volume of a rectangular prism is affected when one measured attribute

is multiplied by a factor of 1

4,

1

3,

1

2, 2, 3, or 4. (b)

Describe how the surface area of a rectangular prism is affected when one measured

attribute is multiplied by a factor of 1

2 or 2. (b)

A rectangular prism can be represented on a flat surface as a net that contains six rectangles, two that have measures of the length and width of the base, two that have measures of the length and height, and two that have measures of the width and height.

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The surface area of a rectangular prism is the sum of the areas of all six faces (SA = 2lw + 2lh + 2wh).

Example: Carl is covering a rectangular prism-shaped box with cloth.

l = 8 in. w = 12 in. h = 2 in. What is the minimum amount of cloth Carl needs to cover the entire box?

SA = 2lw + 2lh + 2wh

SA = 2(8)(12) + 2(8)(2) + 2(12)(2)

SA = 2(96) + 2(16) + 2(24)

SA = 192 + 32 + 48

SA = 272 inches2

The volume of a rectangular prism is computed by multiplying the area of the base, B,

(length times width) by the height of the prism (V = lwh).

Example: Joseph is filling a box with peanuts.

l = 10 cm w = 25 cm h = 20 cm

If the box is empty, what is closest to the amount of peanuts the box will hold? V = lwh

V = (10)(25)(20)

V = (10)(500)

V = 5,000 cm3

25 cm

10 cm

20 cm

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When one attribute of a prism is changed through multiplication or division, the volume increases or decreases by the same factor.

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Example 1: Find the volume of a cone that has a radius of 4 centimeters and a height of 9 centimeters.

V = 1

3 r 2h

V = 1

3 (42)(9)

V = 1

3 (3.14)(16)(9)

V = 1

3 (452.16) OR 452.16 ÷ 3

V = 150.72 cm3

Example 2: Find the surface area of the following cone:

S. A. = 𝜋 𝑟2 + 𝜋 𝑟 𝑙

= 𝜋 ∙ 52 + 𝜋 ∙ 5 ∙ 13

= 282.74 m

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A square based pyramid is a three-dimensional geometric figure with a square base and

four triangular sides that connect at one point. Example:

What is the surface area of a pyramid with dimensions shown?

SA = 2

1lp + B

SA = 2

1(7)(12) + 9

SA = 2

1(84) + 9 OR 84 ÷ 2 + 9

SA = 42 + 9 SA = 51 feet2

7 ft

5 ft

3 ft

3 ft

The perimeter is found by adding

the length of each side of the base: 3 + 3 +3 + 3 = 12 ft

The area of the base is found by

multiplying the length times the width of the base:

(3)(3) = 9 ft

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The volume of a pyramid is 1

3 Bh, where B is the area of the base and h is the height

(altitude).

V = 1

3 Bh

V = 1

3 (9)(5)

V = 1

3 (45) OR 45 ÷ 3

V = 15 feet

7 ft

5 ft

3 ft

3 ft

The area of the base is found by multiplying the length times the

width of the base: (3)(3) = 9 ft

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Practice Items (Answers can be found in the back of the booklet.)

Volume and Surface Area, SOL 8.7

1. What is the surface area of a rectangular prism with the dimensions shown?

2. The radius of the base of a cone is 4 inches. The slant height of the cone is 6 inches. Which is closest to the surface area of the cone?

3. What is the volume of a square-based pyramid with base side lengths of 16 meters, a slant height of 17 meters, and height of 15 meters?

4. What is the surface area of the rectangular prism shown?

5. What is the surface area of a cube with the measurements shown?

6. Which is the closest to the surface are of a cone with the dimensions shown?

7. The volume of a square-based pyramid is 588 cubic inches. The height of this pyramid is 9 inches. What is the area of the base of this pyramid? A. 196 sq in. B. 65 sq in. C. 49 sq in. D. 14 sq in.

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8. Brian purchased a trophy in the shape of a square pyramid for the most valuable player on his lacrosse team. The trophy had a slant height of 4 inches and each side of its base measured 4 inches. Brian wanted to engrave on the four sides of the trophy, but not on the base of the trophy. How many square inches of the trophy were available for engraving?

9. Megan wrapped a present inside a cube-shaped box. The box had an edge length of 4 inches. How many square inches of paper were needed to wrap the box, if there was no overlap?

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10. Anna built a prism (Prism A) in the shape of a cube out of wood. The side length of the cube measured 18 inches in length. Anna built another prism (Prism B) with the same dimensions as the cube, except she doubled its height.

How does the volume of the two prisms compare?

How does the surface area of the two prisms compare?

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8.5 The student will use the relationships among pairs of angles that are supplementary angles and complementary angles to determine the measure of unknown angles.

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1. What are the complementary angles for each diagram?

2. What pairs of angles are supplementary?

34°

27°

78°

115° 98° 82° 65°

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Practice Items Answer Key (2nd Nine Weeks) SOL 8.18 (Multistep inequalities)

1. A 2. D 3. D 4. D 5. D 6. -30 ≥

7.

8.

9. B 10. C

SOL 8.9 (Pythagorean Theorem) 1. A 2. C 3. D 4. H 5. H 6. D 7. J 8. H 9. B 10. B 11. 25.9 inches

SOL 8.10 (Area and Perimeter) 1. C 2. G 3. C 4. H 5. C 6. B 7. 76 feet 8. D 9. B

SOL 8.8 (Three-Dimensional Models) 1. G 2. D 3. F 4. C 5. B 6. G 7. F 8. A 9. 10. D 11. A

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SOL 8.6 (Volume and Surface Area) Answers 1. D 2. H 3. 1280 4. 158 5. B 6. 282.74 7. A 8. 32 9. 96 in2 10. a) The volume of Prism B is twice the volume of Prism A b) The surface area of Prism B is greater than the surface area of Prism A. The surface area of the four sides of

Prism B are twice the surface area of the four sides of Prism A, and the surface are of the two bases of each prism are the same.

8.5 Complementary and Supplementary

1. a. 56° b. 63° c. 12°

2. 115° and 65°, 98° and 82°