50 Fun Facts About Middle School Math

Which property lets you compute 7 × 98 as 7 × 100 minus 7 × 2, the shortcut behind much mental arithmetic?

Distributive multiplication over subtraction, a(b − c) = ab − ac, is the trick behind mental arithmetic like 7 × 98 = 700 − 14 = 686.

What is (−8) × (−6), a product that trips up anyone who expects two negatives to stay negative?

48 is the answer because the product of two negative numbers is positive, which is why squaring any negative number gives a positive result.

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: 5² + 12² = 169, and the square root of 169 is 13. Adding the legs gives 17, a common slip.

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, the shortcut cashiers use instead of subtracting.

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; forgetting the division gives 15, the most common wrong answer.

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 (1-6, 2-5, 3-4, 4-3, 5-2, 6-1) add to 7, which is why craps players dread a seven.

What is (−3)², a value that differs from −3² because the parentheses square the minus sign too?

9 is (−3)² since (−3) × (−3) is positive, while −3² means −(3²) = −9 by the usual convention that exponents bind before a 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 variable?

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 any triangle is exactly half a parallelogram with the same base and height, as Euclid proved.

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, and the same box has a surface area of 162 in², a figure students mix up with volume on tests.

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

x = 35 because x/5 must equal 7 after adding 3, and 7 × 5 = 35; multiplying before adding gives the tempting 20.

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

√10 is irrational, about 3.1623 with digits that never repeat, whereas √25 is simply 5 and 22/7 is only a rational approximation of pi.

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

Translation is the slide, the one rigid motion that keeps every segment parallel to its original; 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 inequality solves −2x > 8, remembering that dividing by a negative number reverses the direction of the sign?

x < −4 is the solution because dividing both sides by −2 flips the inequality; x > −4 is the answer of anyone who forgot the flip.

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, since circumference equals pi times the diameter; using the 12 as a radius doubles it to 75.4.

By what percent did a class grow when it went from 60 pupils to 75, measured against the starting number?

25% is the increase: the class gained 15 pupils, and 15 ÷ 60 = 0.25; dividing by the new size of 75 instead gives a misleading 20%.

What is the mode of 3, 3, 8, 12, 16, 20 and 22, a set whose mean and median both equal 12?

3 is the mode because it is the only value listed twice; a data set can have no mode at all, or several, which is why it is the least used average.

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

Perpendicular lines form four right angles where they cross, which is why a carpenter's square checks a corner with a single touch.

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 lower still.

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 × 3², one face times six; a cube of side 6 is the one whose surface area and volume share a number, 216.

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; 1/3 is the error of leaving the red ones out of the total.

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; compound interest would instead add each year's $12 to the balance and pay slightly more, $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 87 and 93 have digit sums divisible by 3 and 91 is seven times thirteen, the factor pair that fools almost everyone.

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 24π trap is the lateral surface area, 2πrh, which measures the curved wall instead.

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; adding instead of subtracting gives 31, 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 usual 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, a power whose negative exponent signals a reciprocal rather than a negative result?

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; dividing by 2 instead of 2.5 gives 90, and multiplying gives the absurd 450.

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 the average base is 7 and 7 × 6 = 42; forgetting to halve the sum of the bases gives 84, twice the true figure.

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.