Quarter 2 Benchmark Review 8th Grade

Quarter 2 Benchmark Review 8th Grade

Part One: Non-Calculator

Rational Versus Irrational Numbers

• Rational numbers can be made into a simple fraction: Integers, Whole Number, Fractions, Decimals that end, Decimals that repeat, Perfect squares

• Examples: 0, -4 (integers), 10, 100 (whole numbers), 1/4, 5/6 (fractions), 2.5, 1.125 (decimals that end), 2.33333... (decimals that repeat),  (perfect squares)

   $\sqrt{16},\ \sqrt{36}$  

Rational Versus Irrational Numbers

• Irrational numbers are numbers that cannot be made into a simple fraction: Decimals that continue forever and do not repeat, square roots that are not perfect squares, π

   

• Examples: 2.4562391... , 

   $2\pi\ ,\ \sqrt{13}$  

Multiple Choice

Which of these numbers is IRRATIONAL?

• $\sqrt{25}$

• 2.14523586467...

• 1.1212121...

• -18

Multiple Choice

Is  rational or irrational and why?

 $\sqrt{144}$  

• Irrational because it is a perfect square that can be simplified to 12, which is irrational

• Irrational because all square roots are irrational

• Rational because all square roots are rational

• Rational because it is a perfect square that can be simplified to 12, which is rational

Fractions, decimals, and percentages

• Move the decimal to the right two places to make a decimal into a percent

  Ex. 1: 0.2 =

• Ex. 2: 4.5 =

• Ex.4: 0.425 =

Multiple Choice

What is 1.86 as a percent?

• 1.86%

• 186%

• 18.6%

• .086%

Multiple Choice

What is 0.04 as a percent?

• 40%

• 400%

• 4%

• 0.04%

Fractions, decimals, and percentages

• To go from a fraction to a decimal or percent: Multiply to make the denominator 100, use the new numerator as your percent or decimal

• Ex. 1: 3/4 to a percent and a decimal

• Ex. 2: 47/50 to a percent and a decimal

Multiple Choice

Convert 1/20 to a percent

• 5%

• 50%

• 120%

• 1%

• 4%

Multiple Choice

A basketball player made 7 out of 10 baskets. What percent of the baskets did he make?

• 7%

• 14%

• 710%

• 70%

Fractions, decimals, and percentages

• To go from a percent or a decimal to a fraction: put the number over 100 and simplify

• Ex. 1: 14% as a fraction

• Ex. 2: 0.50 as a fraction

Multiple Choice

What is 0.22 as a simplified fraction?

• 22/10

• 11/50

• 2/10

• 1/5

Multiple Choice

I received a 90% on a quiz. What fraction of the questions did I get correct?

• 90/1

• 99/100

• 3/5

• 9/10

Laws of exponents

• The product rule says to multiply exponents with the same base, we add the exponents

• Ex. 10⁸x 10³

• Ex. y⁵ x y⁴

Multiple Choice

Simplify the following: 77⁴ x 77⁹

• 7,983

• 77³⁶

• 77⁵

• 77¹³

Multiple Choice

Simplify the following: y⁴ x y³ x y⁵

• y⁶⁰

• y¹⁴

• y¹²

• y⁴³⁵

Laws of Exponents

• The quotient rule says to divide exponents with the same base, we subtract the exponents

•  $\frac{10^8}{10^5}\$  

•  $\frac{y^5}{y^4}$  

Multiple Choice

Simplify the following:

 $\frac{10^{10}}{10^8}\$  

•  $10^4$   = 10,000

•  $10^2$   = 100

•  $10^3$  = 1,000

•  $10^{18}$   = 1,000,000,000,000,000,000

Laws of Exponents

• The power rule says when you raise a power to a power, you multiply the powers

• Ex. (100⁴)⁵

• Ex. (y¹⁰)⁷

Multiple Choice

Simplify the following: (x³)¹¹

• x³³

• x¹⁴

• x⁸

• x⁴

Laws of exponents

• The negative exponent rule says a number raised to a negative power is the same as its reciprocal raised to a positive power

• Ex. (2/3)^-6

• Ex. 8^-2

• Ex. y^-10

Multiple Choice

Which of the following is equivalent to (2/5)^-4

• 25⁴

• 2^-4

• (2/5)⁴

• (5/2)⁴

Multiple Choice

Which of the following is equivalent to $x^{-10}$  

•  $\frac{1}{x^{10}}$  

•  $x^{10}$  

•  $\frac{1}{x^{-10}}$  

Scientific Notation

• To go from Scientific Notation to Standard Notation: Use the exponent to tell you how many places to move the decimal. If it is a positive exponent, move the decimal to the right. If it is a negative exponent, move the decimal to the left.

• Ex. 3.45 x 10⁸

• Ex. 1.2 x 10^-3

Fill in the Blank

What is 2.19 x 10⁴ in standard notation? (make sure to put commas in the correct place if they are needed)

Fill in the Blank

What is 1.2 x 10^-5 in standard notation?

Scientific Notation

• To go from standard notation to scientific notation: Write the non-zero numbers, put a decimal behind the first number, count how many places the decimal needs to move

• Ex. 675,000,000

• Ex. 0.000675

Multiple Choice

What is 321,000 in scientific notation?

• 3.21 x 10^-5

• 321 x 10⁷

• 3.21 x 10⁵

• 32.1 x 10⁴

Multiple Choice

What is 0.000023 in scientific notation?

• 23 x 10^-4

• 2.3 x 10^-5

• 2.3 x 10⁵

• 2.3 x 10⁴

Part Two

Calculator

Solving Equations

• Complete distributive property or combine like terms first, them complete inverse operations to solve

• Ex. 3x + 5 + 6x =41

• Ex. 7(x + 4 ) = 7x + 21

Multiple Choice

Solve the following equation: 4(x + 3) = 2x + 12 + 2x

• x = 12

• no solution

• infinite solutions

• x = 4

Multiple Choice

Solve the following equation: 2y + 6 + 5y = 7y + 14

• y = 12

• y = 14

• no solution

• infinite solutions

Multiple Choice

Solve the following equation: 5(x + 3) = 2x + 30

• x = 5

• x = 12

• no solution

• infinite solutions

Missing angles of triangles

• The interior angles of triangles add up to 180 degrees

• The two non-adjacent interior angles add up to be the same as the exterior angles

• The adjacent interior angle and exterior angle add up to be 180 degrees

• Examples (draw in)

Fill in the Blank

_ $\degree$  

Find the missing angle:

Fill in the Blank

_ $\degree_{ }$  

Find the missing angle: 

Types of Pairs of Angles

• Corresponding angles are angles on the same "corner" of parallel lines and are congruent

• Alternate Interior Angles are angles on opposite sides of the transversal and within the parallel lines. These are congruent.

• Alternate Exterior Angles are angles on opposite sides of the tranversal and outside the parallel lines. These are congruent.

• Same-side (consecutive) Interior Angles are angles on the same side of the transversal and within the parallel lines. These are supplementary.

• Same-side (consecutive) Exterior Angles are angles on the same side of the transversal and outside the parallel lines. These are supplementary.

Multiple Choice

Angles b and h are...

• Same-side interior

• Corresponding

• Vertical

• Alternate exterior

Multiple Choice

Angles c and h are...

• Same-side interior

• Alternate interior

• Same-side exterior

• Corresponding

Fill in the Blank

If <c is 45 degrees, <d is _ degrees

Fill in the Blank

If <d is 50 degrees, <e is _ degrees

Pythagorean Theorem

• a²+b²=c²

Multiple Choice

What is 0.04 as a percent?

• 40%

• 400%

• 4%

• 0.04%