Review #2

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Review #2

Grade 8

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Compare & Order Rational Numbers

Rational numbers are numbers that can be written as the ratio of two integers where zero is not the denominator.

Rational Numbers are in various forms: integers, percents, and positive and negative fractions and decimals

In order to compare rational numbers, rewrite all numbers so that they are in the same form: Either all decimals or all fractions with a common denominator

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Compare & Order Rational Numbers

To compare two positive fractions, find equivalent fractions that have a common denominator. Then compare the numerators to determine which fraction was smaller.

To compare a positive fraction and a positive decimal, find equivalent decimal for the fraction and then compare the digits in the two decimals that have the same place value.

To compare two negative fractions, find equivalent fractions that have a common denominator. Then compare the numerators; whichever numerator is closer to zero is the largest fraction.

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Irrational Numbers

Irrational numbers are numbers that cannot be written as the ratio of two integers. Examples π; √2

Square Root of a given number is a number that when multiplied by itself equals the given number. Example √16 = 4 The side length of a square is the square root of the area of

the square.

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Irrational Numbers

To estimate the value of an irrational number such as √6 Determine between which 2 consecutive numbers √6

would be located on a number line. √6 would be located between 2 and 3 since 22 is 4 and

32 is 9. 6 is closer to 4 than it is to 9; so √6 will be less than

halfway (2.5). A good estimate would for √6 would be 2.4

You can check the estimate by squaring it. 2.42 is 5.76 which is close to 6.

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1) Identify the irrational number.

2.3 , √-8, √25, 0.43433....., 8 9

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a) 0.43433.....

b) √-8

c) √169

d) 8 9

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2) Which is the rational number between 3 and 3 ?

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a) 51 80b) 25 29c) 125 129d) 5 11

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3) Which is the rational number between -3 and -1 ?

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a) -4 2b) -2 4c) 5 9d) -7 8

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4) Michael secured 83.68% marks; John secured 83 7 %marks 12

and Roger secured 83 14% marks. Who 19

secured the highest percentage of marks?

a) Johnb) Rogerc) Michael d) All of the above

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Pythagorean Theorem

The Pythagorean Theorem shows how the legs and hypotenuse of a right triangle are related.

legs

hypotenuse

In a right triangle , the two shortest sides are legs.

The longest side, which is opposite the right angle, is the hypotenuse.

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In words: In a right angled triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs.

In Symbols: a2 +b2 = c2 .

a

b

c

If we know the lengths of two sides of a right angled triangle, then Pythagoras' Theorem allows

us to find the length of the third side.

Pythagorean Theorem

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hypotenuse = 2 . shorter leglonger leg = shorter leg . √3

C

30º

60º

2s

A

B s

√3s

Using the Pythagorean theorem, we find that

The converse of Pythagorean Theorem allows you to substitute the lengths of the sides of a triangle into the

equation : c2 = a2 +b2

to check whether a triangle is a right triangle, if the Pythagorean equation is true the triangle is a right triangle.

30-60-90 Triangle

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1) One side of a right triangle is 6 cm and the length of its hypotenuse is 8 cm. Find the length of the other

side.

a) 7.3 cmb) 6.2 cmc) 4.9 cmd) 5.3

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2) One end of a 15 feet long wire is tied to the top of a pole and the other end is fixed on to the

ground at a distance of 9 feet from the foot of the pole. What is the height of the pole?

a) 12 ftb) 11 ftc) 10 ftd) 13 ft

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3) A bird leaves its nest and flies 12 kilometers west. The bird then flies 9 kilometers due north.

How far is the bird from the nest?

a) 16 km b) 14 km c) 15 km d) 17 km

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4) A painter places an 11 ft ladder against a house the base of the ladder is 3 ft from the house. How

high on the house does the ladder reach?

a) 8.6 ftb) 10.6 ftc) 9.3 ftd) 9.6 ft

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Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on

powers of the base number 10.

The second number is called the base. It must always be 10 in scientific notation. The base number 10 is always

written in exponent form. In the number 1.23 x 1011 the number 11 is referred to

as the exponent or power of ten.

Scientific notation

The number 123,000,000,000 in scientific notation is written as : 1.23 ×1011

The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

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Exponents

The "exponent" stands for how many times the thing is being multiplied. The thing that's being multiplied is called the "base". This process of using exponents is

called "raising to a power", where the exponent is the "power".

"53" is "five, raised to the third power".

Example 1: 2 x 2 x 2 x 2 x 2 = 25

i.e., 2 raised to the fifth power

Exponential notation is an easier way to write a number as a product of many factors.

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Whole numbers can be expressed in standard form, in factor form and in exponential form. Exponential notation makes it easier to write a number as a factor repeatedly. A number written in exponential form is a base raised to an exponent. The exponent tells us how many times the

base is used as a factor.

Write 103, 36, and 18 in factor form and in standard form.

Exponential

Form

Factor

Form

Standard

Form

103 10 × 10 × 10 1,000

36 3 ×3 ×3 × 3 × 3 ×3 729

18 1 ×1 × 1 ×1 × 1 × 1 × 1 × 1 1

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If a negative number is raised to an even power, the result will be positive.

(-2)4 = - 2 × - 2 × - 2 × - 2 = 16

If a negative number is raised to an odd power, the result will be negative.

(-2)5 = - 2 × - 2 × - 2 × - 2 × - 2 = -32

The negative number must be enclosed by parentheses to have the exponent apply to the negative term.

Note that (-2)4 = - 2 × - 2 × - 2 × - 2 = 16 and -24 = -(2 × 2 × 2 × 2) = -16

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Write in scientific notation: 35,800

Place the decimal point between the first two non-zero numbers, 3 and 5.

Since the 3 was in the ten-thousands place, the power of 10 is 104.

The number in the scientific notation is 3.58 × 104.

To change the number from scientific notation into standard notation, you can also count the number of times

the decimal point moved to determine the power of 10.

35,800 = 3 5 8 0 0 = 3.58 × 104

Decimal point moves 4 places to the left.

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To change the number from scientific notation into standard notation, begin with place value

indicated by the power of 10. Add zeroes as place holders when necessary.

0.0079 = 0. 0 0 7 9 = 7.9 × 10-3

Write in scientific notation: 0.0079

Decimal point moves 3 places to the right.

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1) The speed of sound in air is 331 m/s at 0oC. Write the speed of sound in air in scientific notation.

a) 3.31 × 105 m/sb) 3.31 × 104 m/sc) 3.31 × 103 m/sd) 3.31 × 102 m/s

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2) To write 0.18 ×10-6 in standard notation, how many times will the decimal point move?

a) 6 places to the right.b) 5 places to the right.c) 6 places to the left.d) 5 places to the right.

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3) Which number is the smallest?

a) 1.443 × 10-9

b) 1.443 × 10-8

c) 1.443 × 10-7

d) 1.443 × 10-11

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4) The mean distance of the Earth from the Moon is about 384,400 km. Write the distance in scientific notation.

a) 3.844 × 105 kmb) 3.844 × 104 kmc) 3.844 × 103 kmd) 3.844 × 102 km

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Surface area

The surface area of a solid figure is the sum of the areas of all faces of the figure.

The surface area of a rectangular solid is expressed in square units.

An area of study closely related to solid geometry is nets of a solid. Imagine making cuts along

some edges of a solid and opening it up to form a plane figure. The plane figure is called the net of

the solid.

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In general,

Surface Area of solid figures= 2 x area of the base +perimeter of the base x height

If,

B = area of the base

P = perimeter of the base

h = height

SA = Surface Area

Then,SA = 2B +Ph

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Cubes

A cube is a three-dimensional figure with all edges of the same length.

If s is the length of one of its sides, then SA = 2(s2) + (4s)s = 6s2

s

sss s

s

s s

s

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Rectangular prism

l w

h

l w

h

SA = 2B + Ph

SA = 2(lw) + (2l + 2w)h = 2(lw + lh + wh)

31CONFIDENTIAL 31

1) How much wrapping paper will be needed to cover the gift box shown below without any overlapping?

a) 5000 cm2

b) 5100 cm2

c) 5200 cm2

d) 5300 cm2

30 cm

40 cm

20 cm

32CONFIDENTIAL 32

2) Which of the following statement about the boxes is true?

a) Surface area of A > Surface area of Bb) Surface area of A = Surface area of Bc) Surface area of A = Surface area of Cd) Surface area of C < Surface area of B

A B C

4

4

4

2

6

53

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33CONFIDENTIAL 33

3) What is the surface area of a cube, if the area of each face of the cube is 16 ft2?

a) 64 ft2

b) 96 ft2

c) 128 ft2

d) 256 ft2

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4) What is the surface area of the given box?

a) 136 in2

b) 124 in2

c) 132 in2

d) 142 in2

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Great Job today!