49 Fun Facts About 8th Grade Math

Slope is defined as rise over what?

Run. Dividing the change up or down by the change across turns steepness into one number, so a slope of 2 means two units up for every unit along.

What is the slope of the line through the points (1, 2) and (3, 14)?

6. The line climbs 12 while moving only 2 across, and 12 ÷ 2 = 6; dividing the wrong way round, the classic slip, gives 1/6.

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

m. Nobody knows why that letter was chosen: the popular link to the French monter, 'to climb', has no evidence behind it, and Descartes, who was French, never used it.

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

It goes up. A positive slope means y grows as x grows, and the bigger the number the steeper the climb; a slope below zero runs downhill instead.

Where does the graph of y = 2x + 6 cross the y-axis?

At (0, 6). On the y-axis x is zero, so the 2x term vanishes and only the constant is left, which is why that constant is called the y-intercept.

A line with a slope of 0 is what kind of line?

Horizontal. With zero slope y never changes, so the line runs flat; a line straight up and down is the opposite case, with a slope that is undefined.

What is a solution to a system of linear equations?

Values that satisfy every equation at once. On a graph that is the point where the lines cross, which is why parallel lines, which never meet, give no solution.

Which test tells whether a graph shows a function?

Vertical line test. A function gives each input only one output, so no upright line may cross its graph twice: a U-shaped parabola passes and a circle fails.

Which of these sets of ordered pairs is a function?

{(1, 2), (2, 2), (3, 2)}. Outputs may repeat as often as they like; what a function forbids is one input paired with two outputs, which each of the other sets has.

In a directly proportional relationship, what stays constant?

The ratio of the two quantities. Double one and the other doubles, so y divided by x never changes; when it is the product that stays fixed instead, the two are inversely proportional.

What is 2 to the power of 5?

32. Each extra power of 2 doubles the last (2, 4, 8, 16, 32), so the 5 counts doublings rather than multiplying 2 by 5.

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

1/9. A negative power means divide rather than make negative: stepping down the powers of 3 (9, 3, 1, 1/3, 1/9) divides by 3 each time.

Which of these numbers, written in scientific notation, is the largest?

2 × 10^6. The power of ten outweighs the digit in front: two million beats nine hundred thousand, so compare the powers of ten first.

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

4.5 × 10^6. The decimal point hops six places to the left to leave a single digit in front, and the 6 simply counts the hops.

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

The exponent. Calculators that can't show a raised number print it after an E instead, so the same value appears as 6.022E23.

Which root does √9 mean?

The positive root. The radical sign always means the principal root, here 3, which is why formulas that need both roots have to write ± in front of it.

Which two numbers are square roots of 16?

4 and −4. A negative times a negative is positive, so every positive number has two square roots, mirror images on either side of zero.

√50 lies between which two whole numbers?

7 and 8. Since 7 × 7 is 49 and 8 × 8 is already well past 50, the root sits just above 7, at about 7.07.

What is the cube root of 64?

4. Three 4s multiplied together make 64, so a cube built from 64 small blocks is exactly 4 blocks along every edge; 8 is the square root instead.

Which of these numbers has a real cube root but no real square root?

−27. Three negatives multiply back to a negative, so −3 cubed is −27, but no real number squared can ever come out below zero.

Which repeating decimal is exactly equal to 1?

0.999… with the nines going on forever. A third is 0.333…, and three of those make 0.999…, yet three thirds are exactly 1.

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

Rational. Every such decimal can be turned back into a fraction of two whole numbers: 0.25 is 1/4, and 0.272727… is 3/11.

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

√2. No fraction squares to exactly 2, and in one telling Hippasus, the follower of Pythagoras sometimes credited with the proof, was drowned at sea for revealing it.

What is (3 + 5)²?

64. The parentheses make the sum happen first, so it is 8 squared; without them only the 5 would be squared and the answer would drop to 28.

Pi is a circle's circumference divided by its what?

Diameter. Every circle, from a coin to the equator, is about 3.14159 times as far around as it is across, which is why one number serves them all.

Which of these equations has no solution?

x + 1 = x + 2. Take x from both sides and it claims that 1 equals 2, which no value of x can repair; x + 1 = 1 + x is the opposite case, true for every x.

The square on the hypotenuse equals what?

The sum of the squares on the other two sides. Though named for Pythagoras, the rule was in everyday use in Babylon more than a thousand years before he was born.

What is the hypotenuse of a right triangle whose legs are 3 and 4?

5. The squares on the legs, 9 and 16, add to 25, and it works backwards too: mark out any triangle with sides 3, 4 and 5 and you have made a perfect right angle.

Which of these Pythagorean triples is just a smaller triple scaled up?

(6, 8, 10). Every side is even, so halving gives (3, 4, 5); the other three have no common factor, which is what makes a triple primitive.

Which of these equations graphs as a curve rather than a straight line?

y = x². Squaring makes y grow faster and faster as x grows, so the graph bends into a parabola; the other three change by the same amount for every step along.

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

180 degrees. Tear the three corners off any paper triangle and they fit together along a straight line, whatever shape the triangle was.

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?

720 degrees. A hexagon cuts into four triangles from one corner, so its angles total four times a triangle's; in a regular hexagon each corner gets an equal sixth of that.

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

They are equal. It works in reverse too: if a line crossing two others makes this pair of angles equal, the two lines must be parallel.

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

Supplementary. Together the pair makes a half-turn, which is why neighbouring corners of any parallelogram always add up to a straight angle.

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

Congruent. For triangles, three matching sides are enough to guarantee it, but three matching angles are not, since one triangle can simply be bigger.

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

Similar. One is a scaled copy of the other, so every angle matches while every length is multiplied by the same number, as on a map or a model.

Which geometric move shifts every point of a figure the same distance in the same direction?

Translation. It is a pure slide: every point shifts by the same arrow, so the figure keeps its size and still faces the same way.

Reflecting a point across the x-axis flips which coordinate's sign?

y only. The mirror line is the x-axis itself, so a point keeps its place along it and only swaps above for below: (2, 5) lands on (2, −5).

Enlarging a shape by a scale factor of 3 multiplies its area by what?

9. Both the width and the height triple, so the enlargement covers nine times the area, which is why a pizza twice as wide is four times the pizza.

Which transformations are rigid motions that preserve size and shape?

Translation, rotation and reflection. None of the three changes a single length or angle, so the image would fit exactly over the original; such moves are called isometries.

By convention, which direction of rotation is taken as positive?

Counterclockwise. Angles are measured from the positive x-axis round towards the positive y-axis, which means the hands of a clock turn, oddly, in the minus direction.

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

One third. Three cones of water exactly fill a cylinder of the same base and height, which is where the 1/3 in the cone's volume formula comes from.

Archimedes showed that a sphere fills what fraction of the smallest cylinder that can hold it?

Two thirds. Archimedes was so proud of the result that he asked for a sphere inside a cylinder to be carved on his tomb, and Cicero later found the grave by it.

A ball that just fits a cubic box fills roughly what part of it?

About half. The exact share is π/6, a little over 52 percent, so nearly half of the box is empty space in the corners.

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

A positive correlation. As one quantity rises so does the other, like ice cream sales and temperature, though a pattern in the dots can never prove that one causes the other.

Which of these numbers has the greatest absolute value?

−30. Absolute value throws away the sign and keeps only the distance from zero, so −30 sits thirty steps out, five farther than 25.

Which of these numbers is an integer?

−8. Integers are the whole numbers, their negatives and zero; the word is Latin for 'whole' or, literally, 'untouched', a number with nothing broken off.

Which fraction is commonly used to approximate pi?

22/7. It comes to 3.1428…, right to two decimal places, and Archimedes proved more than 2,200 years ago that pi lies just below it.