50 Fun Facts About 8th Grade Math

Slope is defined as rise over what?

It is usually written m; a line that goes up from left to right has positive slope.

What is the slope of the line through the points (1, 2) and (3, 8)?

Rise is 8 − 2 = 6 and run is 3 − 1 = 2, so the slope is 6/2 = 3.

Which letter is traditionally used for slope in the equation y = mx + b?

Nobody is sure why; it first appears in English in O'Brien's 1844 textbook.

What does a line with positive slope do as you read it from left to right?

A larger absolute value of slope means a steeper line.

Where does the graph of y = 2x − 5 cross the y-axis?

Set x = 0 and you get y = −5; the y-intercept is the constant term in slope-intercept form.

A function with slope 0 has a graph that is what kind of line?

It is a constant function, f(x) = b, which some authors do not even count as linear.

What is the solution to the system x + y = 10 and x − y = 2?

Add the equations: 2x = 12, so x = 6, and then y = 4. Both equations must hold at once.

What is a solution to a system of linear equations?

Graphically, it is where the lines cross; parallel lines give no solution.

A function assigns to each input how many outputs?

That is why a vertical line can only cross the graph of a function once.

Which set of ordered pairs is NOT a function?

The input 1 is paired with two different outputs, 2 and 3, which a function cannot do.

In a directly proportional relationship, what stays constant?

If the product is constant instead, the quantities are inversely proportional.

What is 2 to the power of 5?

It is 2 × 2 × 2 × 2 × 2, which you can read as "2 to the 5th".

What is 3 to the power of −2 equal to?

A negative exponent means the reciprocal of the positive power: 1 divided by 3 squared.

In scientific notation, the coefficient m is usually kept in what range?

The integer power of ten is called the exponent and m is the significand or mantissa.

How is 4,500,000 written in scientific notation?

Move the decimal six places to get a coefficient between 1 and 10.

In the number 6.022 × 10^23, what is the 23 called?

Calculators often display it as 6.022E23.

What is the principal square root of 9?

Both 3 and −3 square to 9, but the radical sign means the nonnegative one.

Which two numbers are square roots of 16?

Squaring either one gives 16; the principal root is the positive one.

The square root of a positive integer that is not a perfect square is always what kind of number?

The ancient Greeks knew this; the square root of 2 is the classic example.

What is the cube root of 64?

A cube root is the number whose third power gives the original: 4 × 4 × 4 = 64.

How many real cube roots does every real number have?

Unlike square roots, negative numbers have a real cube root: the cube root of −8 is −2.

A number that can be written as a fraction of two integers is called what?

The denominator must be nonzero; 0.75 qualifies because it equals 3/4.

A decimal that repeats or terminates represents what kind of number?

0.333... is 1/3, and 0.25 is 1/4; an irrational number's decimal never repeats.

Which of these numbers cannot be written as a ratio of two integers?

Pi, e and the golden ratio are also irrational; 22/7 is only an approximation of pi.

What is (3 + 5)²?

The parentheses force the addition first: 8 squared is 64, whereas 3 + 5² would be 28.

Pi is approximately equal to what?

It is the ratio of a circle's circumference to its diameter; 2.71828 is e and 1.61803 is the golden ratio.

Who is credited with the first proof that some lengths cannot be written as a ratio of integers?

He probably found them while studying the sides of a pentagram.

The Pythagorean theorem relates the square on the hypotenuse to what?

In symbols, a² + b² = c², where c is the side opposite the right angle.

A right triangle has legs of 3 and 4. How long is the hypotenuse?

9 + 16 = 25, and the square root of 25 is 5; (3, 4, 5) is the most famous Pythagorean triple.

Which of these is a Pythagorean triple but NOT a primitive one?

It is just (3, 4, 5) doubled; primitive triples share no common factor.

The hypotenuse is always the longest side of which kind of triangle?

It is the side opposite the right angle, the largest angle in the triangle.

What do the interior angles of any triangle add up to?

That is a straight angle, or π radians.

An exterior angle of a triangle is equal to the sum of what?

Euclid's original version only says it is greater than either remote interior angle.

What is the sum of the interior angles of a hexagon?

Use (n − 2) × 180 with n = 6: 4 × 180 = 720.

When a transversal crosses two parallel lines, what is true of the alternate interior angles?

Corresponding and vertical angles are equal too, while consecutive interior angles are supplementary.

Consecutive interior angles formed by a transversal on parallel lines are what?

Supplementary means they add up to 180 degrees.

Two figures with the same shape and size are called what?

One can be moved onto the other by translations, rotations and reflections.

Two figures with the same shape but possibly different sizes are called what?

One can be obtained from the other by uniform scaling plus moves and flips.

Which transformation moves every point the same distance in the same direction?

It is a slide; the figure's size and orientation do not change.

Which transformation produces a mirror image across a line?

The points on the line of reflection itself stay fixed.

Which transformation is determined by a center point and a scale ratio?

Mathematicians call it a homothety; a ratio bigger than 1 enlarges, smaller than 1 shrinks.

Which transformations are rigid motions that preserve size and shape?

These rigid motions are called isometries.

By convention, a clockwise rotation has what sign?

A counterclockwise turn is treated as a positive angle.

The volume of a cone is what fraction of a cylinder with the same base and height?

So a cone's volume is (1/3)πr²h.

Archimedes showed the volume of a sphere is twice the volume of what?

That leads to the familiar formula V = (4/3)πr³.

A sphere with diameter 1 m has about what percentage of the volume of a 1 m cube?

That is π/6 of the cube, about 0.524 cubic metres.

In a scatter plot, dots running from lower left to upper right show what?

Dots running from upper left to lower right indicate a negative correlation.

What is the absolute value of −7?

Absolute value can be thought of as distance from zero, so it is never negative.

Which of these is NOT an integer?

Integers are zero, the positive natural numbers and their negatives.

Which fraction is commonly used to approximate pi?

It equals about 3.142857, close to pi's 3.14159 but not exact, since pi is irrational.