100 Fun Facts About Middle School Math

Which property lets you compute 7 × 98 as 7 × 100 minus 7 × 2?

Distributive is the property: multiplying spreads across a bracket, a(b − c) = ab − ac, the trick behind mental arithmetic like 7 × 98 = 700 − 14 = 686.

What is (−8) × (−6), the product of two negative numbers?

Positive 48, because each minus sign reverses direction on the number line and two reversals cancel, just as losing eight debts of 6 leaves you 48 better off.

What is the longest side of a right triangle with shorter sides 5 and 12, a classic whole-number triple?

13 completes the 5-12-13 triple, since 5² + 12² = 169 and the square root of 169 is 13; stopping at 169 forgets the root and gives a side longer than both legs laid end to end.

What is 7⁰, the value every nonzero number takes when raised to the power zero?

1 is the value of 7⁰ because each drop of one in the exponent divides by 7: 7² = 49, 7¹ = 7, and 7 ÷ 7 = 1.

What is 30% of 70, the discount on a $70 jacket in a sale that takes 30 percent off?

$21 comes from 0.30 × 70; the sale price is then $49, which is 70% of the original, so multiplying by 0.7 gets there in one step.

Which numbered region of the coordinate plane holds the point (−3, 5), counting anticlockwise from the upper right?

II is the upper-left region, where x is negative and y is positive; the numbering runs counterclockwise from I in the upper right.

What value of x solves 3x + 7 = 22, an equation solved by undoing the addition before the multiplication?

x = 5 because 22 − 7 leaves 3x = 15, and 15 ÷ 3 = 5; stopping at 15 forgets the final division, and multiplying by 3 instead gives 45.

Which name do the equal angles facing each other across the crossing point of two straight lines go by?

Vertical is the name, from the shared vertex rather than any up-and-down direction, and the two angles in each pair are always equal.

What is the probability that two fair dice total 7, the sum that has more ways to come up than any other?

1/6 is the chance because six of the thirty-six equally likely pairs, from 1-6 round to 6-1, add to 7, making it the likeliest total and the roll that wins outright on a craps shooter's first throw.

What is (−3)², where the parentheses mean the minus sign is squared too?

Positive 9, since (−3) × (−3) multiplies two negatives, while −3² without parentheses means −(3²) = −9, because in written maths the power is applied before the minus sign.

What is the mean of the scores 6, 12, 16, 20 and 41, a figure dragged upward by one high score?

19 is the mean: the scores sum to 95 and 95 ÷ 5 = 19, while the median stays at 16, which is why one extreme value can mislead an average.

What is 12 boys to 18 girls as a ratio in simplest form, found by dividing both counts by a shared factor?

2:3 is the ratio once 12 and 18 are both divided by 6; 4:6 and 6:9 are equal to it but not fully reduced, and 3:2 reverses the order.

Which name does the 4 in the term 4x go by, the number that multiplies the letter?

Coefficient is the word, and the unwritten 1 in front of a bare x is still a coefficient, which is why x + x = 2x.

What is 2⁵, the number of different heads-or-tails outcomes when five coins are tossed at once?

32 is 2 × 2 × 2 × 2 × 2, and each extra coin doubles the count of outcomes, which is why ten coins already give 1,024 ways.

What is the area of a triangle with base 8 cm and height 7 cm, half that of the matching rectangle?

28 cm² is half of 8 × 7 = 56, because two copies of any triangle fit together into a parallelogram with the same base and height.

Which word describes two angles adding to 180°, such as the neighbouring pair formed when one line crosses another?

Supplementary is the term, and when the two angles sit side by side their outer arms form a straight line, a pairing geometry calls a linear pair.

What is the volume of a box 3 by 6 by 7 inches, the product of its three dimensions?

126 in³ is 3 × 6 × 7: a 3-by-6 layer holds 18 inch cubes, and seven such layers stack up to fill the box.

What value of x solves x/5 − 3 = 4, where adding 3 to both sides comes before multiplying by 5?

x = 35 because adding 3 to both sides leaves x/5 = 7, and 7 × 5 = 35; stopping at 7 forgets that x is still being divided by 5.

Which of these numbers is irrational, impossible to write exactly as one whole number over another?

√7 is irrational, about 2.6458 with digits that never repeat, whereas √49 is simply 7 and 22/7 is only a rational stand-in for pi.

Which geometric move slides every point of a shape the same distance one way, without turning or flipping it?

Translation is the slide, a rigid motion that keeps every segment parallel to its starting position; a rotation turns and a reflection flips.

What is the greatest common factor of 20 and 70, the largest number that divides both without a remainder?

10 is the greatest common factor, since 20 = 2 × 10 and 70 = 7 × 10 share nothing larger; 140 is their least common multiple instead.

What is the least common multiple of 6 and 8, the first number that appears in both times tables?

24 is the least common multiple: the sixes run 6, 12, 18, 24 and the eights 8, 16, 24, so 24 is where they first meet, well short of the product of the two.

Which of these values of x satisfies the inequality −2x > 8?

Negative 5 works, since −2 × −5 = 10; dividing an inequality by a negative number reverses its sign, so the solutions are the numbers below −4, not above it.

What is the distance around a circular table 12 feet across, found by multiplying the diameter by pi?

About 37.7 ft is π × 12, because any circle is a little over three times as far around as it is across; treating the 12 as a radius doubles the answer to 75.4.

Which percent change is bigger, a class going from 60 pupils to 75 or from 75 to 60?

The increase is 25%, since 15 ÷ 60 = 0.25, while the same 15 pupils are only 20% of 75; a percent change always divides by the starting number.

What is the mode of the five values 3, 3, 8, 26 and 80?

3 is the mode because it is the only value listed twice; the median here is 8 and the mean 24, and unlike them a data set can have several modes or none at all.

Which word describes two lines that meet at exactly 90°, like the hands of a clock at three o'clock?

Perpendicular lines cut the space around their crossing into four equal angles, each a right angle, so checking one corner with a set square checks them all.

What is −15 + 9 − (−4), an expression where subtracting a negative becomes adding a positive?

−2 is the result: −15 + 9 = −6, then −6 − (−4) = −6 + 4 = −2; treating the double minus as a single subtraction drops the total to −10.

What is the slope of the line through (2, 3) and (5, 18), the rise between the points divided by the run?

5 is the slope: y rises 15 while x runs 3, and 15 ÷ 3 = 5, meaning the line climbs five units for every one it moves right.

Which name does the 7 in 63 ÷ 9 = 7 go by, the partner term to dividend and divisor?

Quotient is the name, from the Latin quotiens, 'how many times', since 63 ÷ 9 asks how many times 9 fits into 63.

What is the surface area of a cube with 3 cm edges, the total of its six identical square faces?

54 cm² is 6 × 9, one 3-by-3 face times six; the tempting 27 is the cube's volume, which counts the centimetre blocks inside rather than the squares that wrap it.

What is the probability of drawing a red marble from a bag holding 3 red, 4 blue and 5 green marbles?

1/4 is the chance, since 3 of the 12 marbles are red and 3/12 reduces to 1/4; the remaining 3/4 is the chance of drawing a colour other than red.

Which word names a statement that two ratios are equal, such as 6:8 matching 3:4?

Proportion is the word, and cross-multiplying its two sides, 6 × 4 = 8 × 3, is the standard test for whether the ratios really match.

How many degrees is each interior angle of a regular hexagon, the shape of every cell in a honeycomb?

120° is each corner of a regular hexagon, so three of them meet to fill the full 360° around a point, which is why hexagons tile a floor with no gaps.

How much simple interest does $200 earn in 3 years at 6% a year, with nothing added to the principal?

$36 is 200 × 0.06 × 3, twelve dollars a year; with compound interest each year's payout would itself earn interest, lifting the total to $38.20.

What is the median of the five readings 16, 3, 20, 8 and 5, found only after putting them in order?

8 is the median once the readings are sorted 3, 5, 8, 16, 20; picking the middle of the unsorted list gives 20, and the mean is a different 10.4.

Which of these odd numbers is prime, the only one that cannot be split into two smaller whole-number factors?

59 is prime, while 119 is 7 × 17, 247 is 13 × 19 and 529 is 23 squared, composites that hide well because none of their factors is 2, 3 or 5.

What is the volume of a cylinder with radius 3 and height 4, the area of its circular base times its height?

36π cubic units is π × 3² × 4, about 113; the 9π trap is only the area of the circular base, before it is stacked four units high.

Which word describes two figures of exactly the same shape and size, so one laid over the other matches perfectly?

Congruent figures can always be matched by some mix of sliding, turning and flipping, whereas similar figures share only their shape and may differ in size.

What is the range of the temperatures 7, 16, 2, 29 and 12, the gap between the highest and the lowest?

27 is the range, 29 − 2, the simplest measure of spread; subtracting the first reading listed from the last, 12 − 7, gives a misleading 5, and 66 is just the total.

How long is each side of a square patio with an area of 121 square feet, the square root of that area?

11 ft is the side because 11 × 11 = 121, making 121 a perfect square; halving or quartering the area, 60.5 or 30.25, is the tempting wrong turn.

How is 4,500,000 written in scientific notation, with one nonzero digit before the decimal point?

4.5 × 10⁶ puts the decimal point after the 4 and counts six jumps to the right; 45 × 10⁵ is the same number but is not in normalised form.

What is 2⁻³, the number 2 raised to the power of negative three?

1/8 is 2⁻³ because a negative exponent means one over the positive power, 1/2³; the answer is small and positive, never −8.

Which graph marks each student's height against shoe size as a single dot, to show whether the two are related?

Scatter plot is the graph, and when its dots drift upward together the variables show a positive correlation, the pattern a trend line then summarises.

What is the average speed of a car that covers 180 miles in 2.5 hours, the distance divided by the time?

72 mph is 180 ÷ 2.5, a unit rate: the car covers 72 miles in each full hour, 144 in two, and the last 36 in the final half hour.

Which trio of whole numbers, like 3, 4 and 5, satisfies a² + b² = c² and so forms a right triangle?

8, 15, 17 works because 64 + 225 = 289 = 17²; 9, 12, 16 is the near miss, since 81 + 144 = 225 makes the true third side 15.

What is the name of the point (0, 0), where the x-axis and y-axis of a coordinate grid cross?

Origin is the name, and every point's coordinates are simply its distances from the origin along each axis, which is why it splits each axis into a positive and a negative half.

What is the area of a trapezoid with parallel sides 5 and 9 and height 6, found from the average base?

42 is the area, because a trapezoid covers the same ground as a rectangle whose width is the average of the parallel sides, here 7 by 6; forgetting to halve the sum gives 84.

Which word describes two events, like two coin flips, where the first result has no effect on the odds of the second?

Independent events multiply, so two heads in a row has a probability of one in four, and a coin that has landed heads twenty times is still an even call on the next flip.

What is 5 + 3 × 4², an expression where the exponent is settled before the multiplication and the addition?

53 is the value: 4² = 16, times 3 makes forty-eight, plus 5 is 53; working strictly left to right gives 1,024 instead, and adding before multiplying gives 128.

A bat and a ball cost $1.10 together, and the bat costs $1 more than the ball; how much is the ball?

5 cents is right: the bat then costs $1.05 and the pair totals $1.10. Most people blurt out 10 cents, which would push the total up to $1.20.

A lily pad patch doubles daily and covers a pond on day 30; on which day was half the pond covered?

Day 29 is when half the pond was covered, because one more doubling fills the rest. The tempting answer of day 15 forgets that doubling growth does most of its work at the very end.

One million seconds, counted without a break, lasts roughly how long, given that a day holds 86,400 seconds?

About 12 days is all it takes, since 1,000,000 ÷ 86,400 comes to roughly 11.6. A billion seconds is a different story, lasting nearly 32 years.

Which name, coined by a nine-year-old boy, is given to the number 1 followed by 100 zeros?

Googol was the suggestion of Milton Sirotta, nephew of the American mathematician Edward Kasner. The search engine Google takes its name from a misspelling of the word.

A 16-inch pizza has how many times the area of an 8-inch pizza, since area grows with the radius squared?

4 times is right: doubling the width of a circle doubles it in both directions, so the area quadruples. Two 8-inch pizzas hold only half as much as one 16-inch.

Which polygon gives the stop sign its shape in many countries, with every interior angle measuring 135 degrees?

Octagon was the shape American highway officials chose in the 1920s, wanting the outline alone to carry the meaning because many drivers had trouble reading the words on a sign.

After a share price falls by 50%, what percentage rise is needed to bring it back to the starting price?

100% is needed, because the rise is measured against the new, smaller price, which has to double. A 50% rise would win back only three-quarters of the starting value.

A sheet of paper folded in half seven times ends up how many layers thick, doubling with every fold?

128 layers result, because seven doublings multiply seven 2s together. In 2002 a California student, Britney Gallivan, folded one long strip twelve times, beating a supposed limit of eight.

Which number comes next in the pattern 2, 3, 5, 8, 13, 21, where each term adds the previous two?

34 follows because 13 + 21 makes it. The pattern is the Fibonacci sequence, which reached Europe in a book of 1202 as a puzzle about breeding rabbits.

What is the square root of 144, the number of square inches in one square foot?

12 is the answer, since 12 × 12 = 144, which is why a foot of 12 inches squares to 144 square inches. A set of 144 items is called a gross.

What is 1/2 + 1/3, a sum that needs both fractions rewritten in sixths before adding?

5/6 is the total, because 1/2 becomes 3/6 and 1/3 becomes 2/6. Adding the tops and the bottoms separately gives the classic wrong answer of 2/5, which is smaller than 1/2 on its own.

Which percentage matches the fraction 3/4, the share of a dollar that three quarters make up?

75% is the match, since 3 ÷ 4 = 0.75 and a percentage is simply a number out of 100. Three quarter coins are worth 75 cents of a 100-cent dollar.

Two angles of a triangle measure 40° and 75°; how large is the third, given that all three make a half turn?

65° is left over, because the three angles of any flat triangle total 180°. The rule holds however long or thin the triangle is, so two angles always fix the third.

What is the probability of drawing an ace from a standard 52-card deck, which holds four of them?

1/13 is the chance, because 4 aces out of 52 cards simplifies when both numbers are divided by 4. The same odds apply to any single rank, from twos up to kings.

A weather forecast gives a 30% chance of rain tomorrow; what is the chance that tomorrow stays dry?

70% is the chance of a dry day, because an event and its opposite always share out the full 100% between them. Subtracting from 100% is often the quickest route to a probability.

How many millilitres are in a 2.5-litre bottle of soda, given that milli- means one thousandth?

2,500 mL fills the bottle, since each litre holds 1,000 millilitres. One millilitre of water also takes up one cubic centimetre and weighs about a gram.

A $60 video game goes on sale at 25% off; how much does a shopper pay at the till?

$45 is the price, because a quarter of $60 is $15 and that comes off the tag. A shortcut is to multiply by 0.75, the three-quarters of the price that remain.

Which chart of overlapping circles takes its name from the English logician who introduced it in 1880?

Venn diagram honours John Venn, who set out the idea in a paper of 1880. Each circle stands for a set, and the overlap holds whatever belongs to both.

Which Greek thinker, born on Samos around 570 BC, gives his name to the rule a² + b² = c²?

Pythagoras gets the credit, yet Babylonian scribes were using the same rule more than a thousand years before he was born.

The word algebra comes from the title of a ninth-century mathematics book written in which language?

Arabic supplied the word, from al-jabr, a term for reuniting broken parts that was also used for setting bones. The author's own name, al-Khwarizmi, later gave English the word algorithm.

The x and y coordinate grid takes its name from which French philosopher, who published the idea in 1637?

René Descartes gives his name to Cartesian coordinates, the idea that let problems in geometry be written and solved as algebra. He published it while living in the Netherlands.

Which symbol did the Welsh mathematician Robert Recorde invent in 1557 to avoid writing the same phrase repeatedly?

Equals sign is the one: Recorde drew a pair of parallel lines of the same length, reasoning that no two things could be more alike. It was not used in print again for 61 years.

Which is the smallest perfect number, a number equal to the sum of its own smaller divisors?

6 qualifies because its smaller divisors 1, 2 and 3 add up to 6 again. The next perfect number is 28, and only four were known to the mathematicians of ancient Greece.

What is 4³, the number of small cubes needed to build a solid 4 × 4 × 4 block?

64 is the total, since cubing a number multiplies three copies of it together. A chessboard happens to have the same number of squares, though it gets there as 8 × 8.

What is the sum of the first seven odd numbers, 1 through 13, a total that always forms a perfect square?

49 is the sum, which is 7 × 7. Adding the odd numbers in order always lands on a square, because each new odd number wraps one more L-shaped border around the last square.

Using the rule F = 1.8C + 32, which Fahrenheit temperature matches a mild spring day of 20°C?

68°F is the match: 1.8 × 20 gives 36, and adding 32 completes the sum. The two scales agree at just one reading, −40, which is equally cold in both.

What is the value of 2x² when x = 5, where the power applies only to the x?

50 is correct: squaring comes first, giving 25, and only then is the result doubled. Multiplying first and squaring afterwards produces the common slip of 100.

Which expression is the same as x³ × x⁴, a product of two powers that share the same base?

x⁷ is the result, because three x's multiplied by four more x's makes seven of them in all. Adding the two small raised numbers is a shortcut for that count.

What is the 10th term of the sequence 4, 7, 10, 13, which goes up by 3 each time?

31 is the 10th term, because the rule 3n + 1 gives 3 × 10 + 1. Only nine jumps of 3 separate the first term from the tenth, a detail that catches many people out.

Three consecutive whole numbers add up to 45, so the middle one is their average; which is the largest?

16 is the largest, since 45 ÷ 3 puts 15 in the middle with one number either side of it. The same shortcut finds the centre of any evenly spaced run of numbers.

Between which two whole numbers does the square root of 90 lie, a little under the root of 100?

9 and 10 trap it, because 9 × 9 = 81 and 10 × 10 = 100, with 90 in between. The root is about 9.49, and its decimal digits run on forever without settling into a repeating pattern.

Which point is the mirror image of (3, 4) when it is reflected in the x-axis of a graph?

(3, −4) is the image: a flip in the x-axis leaves the first coordinate alone and reverses the sign of the second. Reflecting in the y-axis would give (−3, 4) instead.

What is the midpoint of the line segment joining (2, 4) and (8, 10), found by averaging the coordinates?

(5, 7) is the midpoint, because 2 and 8 average to 5 while 4 and 10 average to 7. Adding the coordinates without halving them gives (10, 14), a point twice as far out.

How is 0.0003 written in scientific notation, the form scientists use for very small measurements?

3 × 10⁻⁴ is right, because the decimal point moves four places to reach the 3. A negative power of ten marks a number smaller than 1, never a number below nought.

A $200 deposit earns 10% a year, with each year's interest added on; what is the balance after two years?

$242 is the balance: the first year adds $20, and the second adds $22 because interest is now earned on $220. Interest paid only on the first deposit would have stopped at $240.

A 2 m pole casts a 3 m shadow while a tree casts a 21 m shadow; how tall is the tree?

14 m is the height, because the tree's shadow is seven times as long as the pole's. Thales is said to have measured Egypt's pyramids from their shadows more than 2,500 years ago.

The interior angles of a pentagon add up to how many degrees, the total of the three triangles inside?

540° is the sum, since lines drawn from one corner cut a pentagon into three triangles of 180° each. Every extra side adds one more triangle and another 180° to the total.

If 6 friends each shake hands once with every other friend, how many handshakes take place in total?

15 handshakes happen: each of the 6 people shakes 5 hands, making 30, but that counts every handshake twice. The totals for growing groups, 1, 3, 6, 10, 15, are known as the triangular numbers.

What is the sum of the whole numbers from 1 to 100, which young Gauss reputedly found in moments?

5,050 is the total, since the numbers pair off as 1 + 100, 2 + 99 and so on, giving 50 pairs of 101. The tale of Gauss spotting the trick as a schoolboy may be more legend than fact.

On a hiking map drawn at a scale of 1:50,000, a path 2 cm long covers what distance on the ground?

1 km is the distance, because 2 cm × 50,000 makes 100,000 cm, the same as a thousand metres. At this scale each centimetre on the page stands for half a kilometre of ground.

In a pie chart of an election, a slice showing 30% of the votes has which angle at the centre?

108° is the angle, because 30% of the full 360° turn is 0.3 × 360. Each 1% of a pie chart takes up 3.6°, so even a small share shows as a visible wedge.

How many square tiles 20 cm wide are needed to cover a floor measuring 2 m by 3 m?

150 tiles cover it: ten fit along the 2 m side and fifteen along the 3 m side. Dividing 6 by 0.2 gives the tempting wrong answer of 30, which forgets that each tile covers only 0.04 m².

What is the probability of rolling a double six with two fair dice, the only way to total 12?

1/36 is the chance, because each die shows a six one time in six and the two chances multiply. Only one of the 36 equally likely pairs of faces is a double six.

Which name is given to the equal angles on opposite sides of a line that crosses two parallel lines?

Alternate angles lie on different sides of the crossing line, one at each junction, either both between the parallels or both outside them. Their equal size follows from Euclid's parallel postulate.

A class of 20 students averages 80 on a test and a class of 30 averages 90; what do all 50 average?

86 is the overall figure, because the larger class pulls the result towards its own score. Simply averaging 80 and 90 gives 85, which ignores the ten extra students in the second class.

A mother is 34 years older than her son and exactly three times his age; how old is the son?

17 is the son's age: the 34-year gap has to equal twice his age, since three times an age is the age itself plus two more copies. His mother is therefore 51.

A bike lock has four dials, each showing the digits 0 to 9; how many different codes can it show?

10,000 codes are possible, because each of the four dials offers 10 choices and the choices multiply. The codes run from 0000 up to 9999, which is why 9,999 falls one short.

How many quarter-pound burgers can be made from 10 pounds of beef, a division that enlarges the number?

40 burgers come out, because dividing by 1/4 is the same as multiplying by 4. Dividing a positive amount by a fraction between 0 and 1 always gives back more than the starting number.

A train leaves at 9:40 and arrives at 11:15 the same morning; how long does the journey take?

1 h 35 min is the journey: 20 minutes take the clock to 10:00, and another 1 h 15 min reaches 11:15. Subtracting 9.40 from 11.15 on a calculator fails because an hour has 60 minutes, not 100.

Which of these fractions is the largest, as rewriting all four over 120 makes clear?

2/3 is the largest at 80/120, ahead of 5/8 at 75, 3/5 at 72 and 7/12 at 70. A bigger number on the bottom does not make a fraction bigger, which is why a shared base helps.