← ISEE® Middle Level

Section 4: Mathematics Achievement

Forty-seven items in forty minutes across six NCTM strands for grades 6 to 8, where every wrong choice is a process error somebody actually makes.

53 topics · 205 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Order of Operations

  1. How it worksHow does the order of operations rank brackets, powers, multiplying and adding into one fixed queue?
  2. Worked exampleFind the value of 12 minus 3 times 2 plus 4 squared divided by 8.
  3. Common mistakeWhy isn't multiplication always done before division when both sit in the same expression?
  4. Going furtherWhat changes about the order when a fraction bar acts as an invisible pair of brackets?
Your turneasy1 of 3
Find the value of 12 minus 3 times 3 plus 4 squared divided by 8.

Place Value, Rounding and Estimation

  1. How it worksHow does the digit one place to the right decide whether a number rounds up or stays put?
  2. Worked exampleFind 48,672 rounded to the nearest thousand and then to the nearest ten thousand.
  3. Common mistakeWhy doesn't rounding 4.48 to a whole number give 5 even though 4.48 rounds to 4.5?
Your turneasy1 of 3
Round 66,672 to the nearest thousand.

The Four Operations and Their Properties

  1. How it worksHow do the commutative, associative and identity properties each shuffle numbers in a different way?
  2. Common mistakeWhy isn't subtraction commutative the way addition is?
  3. Worked exampleFind 25 times 16 by regrouping it as 25 times 4 times 4 instead of multiplying straight through.
  4. Going furtherWhat happens to each operation when zero and one step in as its identity element?
Your turneasy1 of 3
Find 25 times 26 by regrouping the multiplication rather than working straight through.

Primes, Factors, Multiples, GCF and LCM

  1. How it worksHow does a factor tree keep splitting a number until only primes are left hanging?
  2. Worked exampleFind the least common multiple of 8, 10 and 12 using prime factorization.
  3. How it worksHow do shared primes build the greatest common factor while all the primes build the least common multiple?
  4. Common mistakeWhy isn't the greatest common factor of two numbers ever bigger than the smaller of them?
  5. Worked exampleFind the greatest common factor of 48 and 60 by lining up their prime factorizations.
Your turneasy1 of 3
A factor tree splits a number all the way down to 2 x 2 x 3 x 13. What was the number?

Divisibility Rules

  1. How it worksHow does adding up a number's digits reveal whether 3 or 9 goes into it?
  2. Worked exampleFind which of 2, 3, 4, 5, 6, 9 and 10 divide 2,340, using rules only.
  3. Common mistakeWhy doesn't a number ending in 4 automatically divide by 4?
Your turneasy1 of 3
What is the sum of the digits of 2,370?

Integers and Operations with Negatives

  1. How it worksHow does the number line turn adding a negative into a step back and subtracting one into a step forward?
  2. Worked exampleFind the value of negative 7 minus negative 12 plus negative 4.
  3. Common mistakeWhy isn't negative 5 bigger than negative 2 just because 5 is bigger than 2?
  4. Going furtherWhat happens to the sign of a product when an odd count of negative factors is multiplied?
Your turneasy1 of 3
The temperature is x degrees, then falls 9 degrees, and later rises 7 degrees. Write an expression for the final temperature, in degrees.

Exponents, Powers and Square Roots

  1. How it worksHow does an exponent count repeated multiplying rather than repeated adding?
  2. Common mistakeWhy isn't negative 3 to the fourth power the same as the negative of 3 to the fourth power?
  3. Worked exampleFind the value of 5 squared plus the square root of 144 minus 2 cubed.
  4. Going furtherWhat happens to a power once the exponent is zero instead of a counting number?
Your turneasy1 of 3
Find the value of 7 squared plus the square root of 144, minus 2 cubed.

Equivalent Fractions and Simplifying

  1. How it worksHow does dividing top and bottom by one shared factor leave a fraction's value untouched?
  2. Worked exampleFind 84/126 in lowest terms by cancelling the greatest common factor once.
  3. Common mistakeWhy doesn't taking 1 off the top and 1 off the bottom leave a fraction alone?
Your turneasy1 of 3
Simplify 84/126 to lowest terms, then multiply the result by 22. Give your answer as a decimal.

Adding and Subtracting Fractions and Mixed Numbers

  1. How it worksHow does a common denominator repartition two bars so their slices finally line up?
  2. Worked exampleFind 3 and 2/3 minus 1 and 5/6 by borrowing one whole into sixths.
  3. Common mistakeWhy isn't 1/3 plus 1/4 equal to 2/7?
  4. Going furtherWhat changes when three fractions with unrelated denominators have to be combined at once?
Your turneasy1 of 3
Find 7 and 2/3 minus 1 and 5/6. Give your answer as a decimal.

Multiplying and Dividing Fractions

  1. How it worksHow does dividing by a fraction turn into multiplying by that fraction flipped over?
  2. Worked exampleFind 2 and 1/4 divided by 3/8 by switching to improper fractions and flipping.
  3. Common mistakeWhy doesn't multiplying two fractions ever call for a common denominator?
  4. Worked exampleFind 3/5 of 2/9, cancelling common factors before any multiplying happens.
  5. Going furtherWhat happens to the answer when a mixed number is multiplied without converting it first?
Your turneasy1 of 3
Find 7 and 1/4 divided by 3/8. Give your answer as a decimal.

Decimal Operations

  1. How it worksHow does counting the decimal places in both factors place the point in the product?
  2. Worked exampleFind 0.45 times 1.2, and then find 3.6 divided by 0.04.
  3. Common mistakeWhy doesn't dividing by a decimal smaller than 1 make the answer smaller?
Your turneasy1 of 3
Find 0.45 times 1.7.

Comparing and Ordering Fractions, Decimals and Percents

  1. How it worksHow does rewriting every value in one single form let one number line rank them all?
  2. Worked exampleCompare 0.58, 5/8, 62 percent and 3/5, then lay them out from smallest to largest.
  3. Common mistakeWhy isn't a longer decimal automatically a larger decimal?
  4. Going furtherWhat changes about the ordering once negative fractions join the list?
Your turneasy1 of 3
How much larger is 0.7 than 0.67? Give your answer as a decimal.

Converting Among Fractions, Decimals and Percents

  1. How it worksHow does dividing the top by the bottom turn any fraction straight into a decimal?
  2. Worked exampleFind 7/8 as a decimal and a percent, then turn 24 percent back into a fraction in lowest terms.
  3. Common mistakeWhy isn't 1/3 exactly 0.33?
Your turneasy1 of 3
Write 5/8 as a percent. Give just the number.

Percent of a Number and Finding the Whole

  1. How it worksHow does the part over whole equals percent over 100 frame settle all three kinds of percent question?
  2. Worked exampleFind the whole amount when 18 of it is 45 percent.
  3. Worked exampleFind 62 percent of 350 with no calculator anywhere in reach.
  4. Common mistakeWhy doesn't the percent sign belong in the answer to a how-many question?
Your turneasy1 of 3
70 is 40 percent of some amount. What is the whole amount?

Percent Increase and Decrease

  1. How it worksHow does a 20 percent discount collapse into one single multiplication by 0.8?
  2. Worked exampleFind the final price of a 50 dollar shirt marked 20 percent off and then a further 10 percent off.
  3. Common mistakeWhy isn't a drop of 20 percent undone by a later rise of 20 percent?
  4. Going furtherWhat happens to a percent change when the starting amount is itself unknown?
  5. Ties togetherHow does stacking two percent changes connect to multiplying two fractions one after the other?
Your turneasy1 of 3
A shirt costs 70 dollars. It is marked 20 percent off, and then a further 10 percent comes off the new price. What is the final price in dollars?

Ratio, Proportion and Unit Rate

  1. How it worksHow does cross multiplying recover the one missing number in a pair of equal ratios?
  2. Worked exampleFind the better buy between 12 pencils for 4.20 dollars and 18 pencils for 6.12 dollars.
  3. Worked exampleFind the missing value when 3 over 7 equals something over 91.
  4. Going furtherWhat changes when a rate must be converted, like dollars per hour into dollars per minute?
Your turneasy1 of 3
A pack of 19 pencils costs c dollars. Write an expression for the price of one pencil, in dollars.

Writing and Evaluating Expressions

  1. How it worksHow does dropping a value into a letter collapse a whole expression to one number?
  2. Worked exampleFind the value of 3a squared minus 2b when a is 4 and b is 5.
  3. Common mistakeWhy doesn't 3a squared mean the whole of 3a gets squared?
Your turneasy1 of 3
A number x is multiplied by 7, and then 15 is subtracted from the result. Write an expression for the result.

Combining Like Terms

  1. How it worksHow does matching both the letter and its power decide which terms are allowed to merge?
  2. Worked exampleFind the simplified form of 7x plus 3y minus 2x plus 5y minus 4.
  3. Common mistakeWhy isn't 3x plus 3x equal to 9x squared?
  4. Going furtherWhat changes about merging when terms carry different powers of the very same letter?
Your turneasy1 of 3
Simplify 7x plus 9y minus 2x plus 5y minus 4.

The Distributive Property

  1. How it worksHow does multiplying across a bracket hand the outside factor to every term inside it?
  2. Worked exampleFind the expanded form of negative 3 times the quantity 2x minus 7.
  3. Common mistakeWhy doesn't the minus sign outside a bracket stay behind when the bracket opens?
  4. Worked exampleFind the value of 4 times the quantity x plus 5, minus 2x, when x is 3.
Your turneasy1 of 3
Expand negative 3 times the quantity 2x minus 12.

One-Step Equations

  1. How it worksHow does the inverse operation peel one number off a letter without upsetting the balance?
  2. Worked exampleFind x when x minus 14 equals 23, and then find x when 6x equals 96.
  3. Common mistakeWhy doesn't a number keep its old sign when it crosses the equals sign?
Your turneasy1 of 3
Find x when x minus 14 equals 45.

Two-Step and Multi-Step Equations

  1. How it worksHow does stripping the addition first and the multiplication second isolate a letter fastest?
  2. Worked exampleWalk me through solving 5x plus 8 equals 43 the way a timed section wants it done.
  3. Worked exampleWalk me through solving x over 4 plus 5 equals 25 and checking the answer in one line.
  4. Common mistakeWhy isn't dividing by 5 the right opening move on 5x plus 8 equals 43?
  5. Going furtherWhat changes once the letter turns up on both sides of the equals sign?
Your turneasy1 of 3
Solve 5x plus 8 equals 73 for x.

Inequalities and Their Number-Line Graphs

  1. How it worksHow does an open circle against a filled circle change which numbers the shading takes in?
  2. Worked exampleFind and graph every solution of 3x minus 4 being less than 11.
  3. Common mistakeWhy doesn't the inequality sign keep its direction when both sides are divided by negative 2?
  4. Going furtherWhat happens to the shaded picture when two inequalities have to hold at once?
Your turneasy1 of 3
For the inequality 3x minus 4 less than 38, at what value of x does 3x minus 4 equal 38? That value is the boundary of the shading.

The Coordinate Plane and Plotting Points

  1. How it worksHow do the signs of the two coordinates sort every point into one of four quadrants?
  2. Worked exampleFind the quadrant holding negative 4 comma 7, then the distance from it to negative 4 comma negative 2.
  3. Common mistakeWhy isn't the point 3 comma 5 in the same place as the point 5 comma 3?
Your turneasy1 of 3
How far is it from the point negative 4 comma 9 to the point negative 4 comma negative 2?

Slope as Rate of Change

  1. How it worksHow does rise over run squeeze a slanted line down to one comparable rate?
  2. Worked exampleFind the slope of the line through 2 comma 3 and 6 comma 11.
  3. Common mistakeWhy isn't a line of slope negative 3 gentler than a line of slope 2?
  4. Going furtherWhat happens to the slope when a line runs perfectly flat or perfectly upright?
  5. Ties togetherHow does slope on a graph connect to the price per item in a shopping problem?
Your turneasy1 of 3
Find the slope of the line through the points 2 comma 3 and 6 comma 18.

Linear Functions and Input-Output Tables

  1. How it worksHow do you read the constant jump down a table and turn it into the multiplier in a rule?
  2. Worked exampleFind the rule that sends 1 to 7, 2 to 11, 3 to 15 and 4 to 19.
  3. Worked exampleFind the cost rule when parking costs 2 dollars for the first hour and 0.60 for every hour after.
  4. Common mistakeWhy doesn't a rule that fits the first row of a table have to fit the rest?
Your turneasy1 of 3
A rule multiplies its input by 7 and then adds 8. Write an expression for the output when the input is n.

Translating a Word Problem into an Equation

  1. How it worksHow do phrases like three times as many and 5 less than land as symbols in a fixed order?
  2. Worked exampleWalk me through turning three times as fast and 1,000 feet apart after 30 minutes into an equation.
  3. Common mistakeWhy isn't 5 less than a number written as 5 minus that number?
  4. Going furtherWhat changes about the setup when a problem asks for a total instead of one part?
Your turneasy1 of 3
A number x is decreased by 11, and the result is then doubled. Write an expression for the final amount.

Number Patterns and Sequences

  1. How it worksHow does the position number multiply the common difference to reach any term directly?
  2. Worked exampleFind the next two terms of 3, 8, 18, 38 and name the rule that produces them.
  3. Common mistakeWhy doesn't every sequence with widening gaps have to be multiplying by a fixed number?
Your turneasy1 of 3
A sequence starts at 7, and each term after that is double the one before it plus 2. What is term number 4?

Angles, Lines and Angle Pairs

  1. How it worksHow do complementary, supplementary and vertical pairs each lock two angles into one relationship?
  2. Worked exampleFind the supplement of an angle whose complement measures 27 degrees.
  3. Common mistakeWhy isn't every pair of angles adding to 180 degrees a linear pair?
  4. Going furtherWhat happens to the vertical-angle relationship when two lines meet at exactly 90 degrees?
Your turneasy1 of 3
An angle has a complement of 59 degrees. How many degrees is its supplement?

Parallel Lines and Transversals

  1. How it worksHow do corresponding and alternate interior angles follow from sliding one crossing onto the other?
  2. Worked exampleFind all eight angles when a transversal cuts two parallel lines at 63 degrees.
  3. Common mistakeWhy doesn't alternate interior equality survive once the two lines stop being parallel?
Your turneasy1 of 3
A transversal cuts two parallel lines, making an acute angle of 65 degrees. How many degrees is each of the obtuse angles?

Triangle Classification and the Angle Sum

  1. How it worksHow does tearing off three corners and fitting them together prove the 180 degree total?
  2. Worked exampleFind the third angle of a triangle holding 47 and 68 degrees, then name its type.
  3. How it worksHow do the labels scalene, isosceles, equilateral, acute, right and obtuse sort every triangle?
  4. Common mistakeWhy isn't a triangle with two right angles possible?
  5. Going furtherWhat happens to the angle total once the shape gains a fourth side?
Your turneasy1 of 3
A triangle holds an angle of 58 degrees and an angle of x degrees. Write an expression for the third angle, in degrees.

The Triangle Inequality

  1. How it worksHow does two sides having to beat the third follow from the shortest path between two points?
  2. Worked exampleFind every whole-number third side a triangle with sides 7 and 10 could have.
  3. Common mistakeWhy isn't checking one pair of sides enough to guarantee a triangle exists?
Your turneasy1 of 3
A triangle has two sides measuring 8 and 17, with 8 the shorter. How many whole-number lengths could the third side take?

Quadrilaterals and Polygons

  1. How it worksHow does splitting a polygon into triangles give its interior angle total from the side count?
  2. Worked exampleFind the interior angle total of a regular octagon and the size of one of its angles.
  3. Common mistakeWhy isn't a rhombus simply another name for a square?
  4. Going furtherWhat changes about the angle formula when the polygon is not regular?
Your turneasy1 of 3
Find the size of one interior angle of a regular polygon with 9 sides, in degrees.

Circles: Radius, Diameter, Circumference and Area

  1. How it worksHow does circumference follow from π being the ratio of the way around to the way across?
  2. Worked exampleFind the circumference and area of a circle of radius 7, leaving π in both answers.
  3. Common mistakeWhy doesn't doubling the radius double the area?
  4. Going furtherWhat happens to the area once the circle is cut down to a single quarter slice?
  5. Ties togetherHow does a circle's area connect to slicing it into wedges and packing them into a rectangle?
Your turneasy1 of 3
A wheel has a radius of r centimetres. Write an expression for the distance it travels in 9 full turns, in centimetres.

Congruence and Similarity

  1. How it worksHow does a single scale factor map every length of a shape onto the matching length of its copy?
  2. Worked exampleFind the missing sides when a triangle of sides 6, 8 and 10 is scaled so the 6 becomes 15.
  3. Common mistakeWhy isn't the area scale factor the same number as the length scale factor?
Your turneasy1 of 3
A triangle with sides 5, 8 and 14 centimetres is enlarged by a scale factor of k. Write an expression for the new length of its longest side, in centimetres.

The Pythagorean Theorem

  1. How it worksHow do the three squares on a right triangle rearrange into a proof that two of them make the third?
  2. Worked exampleFind the diagonal of a rectangle measuring 9 by 12 centimetres.
  3. Common mistakeWhy isn't the theorem usable on a triangle whose angles are 60, 60 and 60?
  4. Worked exampleFind the missing leg when the hypotenuse measures 26 and one leg measures 10.
  5. Going furtherWhat happens to the relationship once the largest angle grows past 90 degrees?
Your turneasy1 of 3
A rectangle measures 10 by 13 centimetres. How long is its diagonal, in centimetres?

Symmetry and Transformations

  1. How it worksHow do a slide, a flip and a turn each move a shape while leaving every length alone?
  2. Worked exampleFind where 3 comma negative 2 lands after a reflection across the x axis and a slide 4 to the left.
  3. Common mistakeWhy doesn't a reflection change the size or the shape of the figure it acts on?
Your turneasy1 of 3
The point x comma negative 2 is reflected across the x axis and then slid 7 units to the left. Write an expression for its new x coordinate.

Units and Conversion

  1. How it worksHow does a conversion factor written as a fraction equal to 1 cancel the unit you want gone?
  2. Worked exampleFind 2,500 centimetres written in metres and 4.2 kilograms written in grams.
  3. Common mistakeWhy doesn't a customary conversion like inches to feet have to be memorised, while a metric one does?
  4. Going furtherWhat changes when a conversion crosses between two systems rather than staying inside one?
Your turneasy1 of 3
Write 7,500 centimetres in metres.

Perimeter and Area of Triangles, Rectangles and Parallelograms

  1. How it worksHow does shearing a rectangle into a parallelogram keep the area while stretching the perimeter?
  2. Worked exampleFind the area of a parallelogram of base 14 and height 6, then of a triangle with the same base and height.
  3. Common mistakeWhy isn't the slanted side of a parallelogram its height?
Your turneasy1 of 3
Find the area of a parallelogram with base 16 and height 9, in square units.

Trapezoids and Composite Figures

  1. How it worksHow does cutting a composite shape into rectangles and triangles turn one hard area into three easy ones?
  2. Worked exampleFind the height of a trapezoid of area 78 square centimetres whose parallel sides are 10 and 16.
  3. Common mistakeWhy doesn't adding up overlapping pieces give the area of a composite figure?
  4. Going furtherWhat changes when a piece has to be taken away, like a round hole punched from a rectangle?
Your turneasy1 of 3
A trapezoid has parallel sides of 10 and 18 centimetres and a height of h centimetres. Write an expression for its area, in square centimetres.

Surface Area of Prisms and Cylinders

  1. How it worksHow does a cylinder unroll into two circles and one rectangle whose width is the circumference?
  2. Worked exampleFind the surface area of a rectangular prism measuring 3 by 4 by 5 centimetres.
  3. Common mistakeWhy isn't surface area measured in cubic units?
Your turneasy1 of 3
Find the surface area of a rectangular prism measuring 6 by 7 by 9 centimetres, in square centimetres.

Volume of Prisms and Cylinders

  1. How it worksHow does base area times height work for every prism and for a cylinder as well?
  2. Worked exampleFind the volume of the rectangular prism folded from a net with edges 3, 4 and 4 centimetres.
  3. Common mistakeWhy doesn't doubling every edge of a cube merely double its volume?
  4. Going furtherWhat happens to the volume when a prism leans so that its sides slant?
  5. Ties togetherHow does volume growing by the cube of a scale factor connect to area growing by its square?
Your turneasy1 of 3
A box has a square base of side x centimetres and a height of 10 centimetres. Write an expression for its volume, in cubic centimetres.

Scale Drawings

  1. How it worksHow does a scale of 1 to 200 turn every drawn length into a real one by a single multiplication?
  2. Worked exampleFind the real length of a wall drawn 7.5 centimetres long on a plan at scale 1 to 40.
  3. Common mistakeWhy doesn't an area on a scale drawing scale by the same number a length does?
Your turneasy1 of 3
A plan is drawn at a scale of 1 to 62. Write an expression for the real length, in centimetres, of a wall drawn d centimetres long on it.

Time, Money and Temperature

  1. How it worksHow does 60 minutes to the hour make time arithmetic run in base sixty rather than base ten?
  2. Worked exampleFind the finishing time of a 2 hour 47 minute film that starts at 6:35 in the evening.
  3. Worked exampleFind the change from a 20 dollar note after buying 3 items at 4.85 dollars each.
  4. Common mistakeWhy isn't 2.5 hours the same length of time as 2 hours and 50 minutes?
Your turneasy1 of 3
A film runs 2 hours and 50 minutes and starts at 6:35 in the evening. How many minutes past 9 o'clock does it finish?

Mean, Median, Mode and Range

  1. How it worksHow does the mean balance a whole set while the median simply counts to the middle of it?
  2. Worked exampleFind the mean, median, mode and range of 12, 15, 15, 18, 20 and 26.
  3. Common mistakeWhy isn't the range a measure of centre at all?
  4. Going furtherWhat changes about finding the median when a set holds an even count of values?
  5. Ties togetherHow does a data set with one long tail connect to a mean sitting far from the median?
Your turneasy1 of 3
Find the mean of 12, 15, 15, 18, 20 and 48.

Reading Tables and Bar Graphs

  1. How it worksHow do you turn gridlines into a value when a bar stops halfway between two of them?
  2. Worked exampleFind the total when four bars read 45, 30, 25 and 60, then what percent the tallest holds.
  3. Common mistakeWhy doesn't a taller bar always mean a larger value when the axis is cut off near the top?
Your turneasy1 of 3
Four bars on a graph read 45, 30, 25 and 75. What do they total?

Line Graphs and Circle Graphs

  1. How it worksHow does a percent become a slice angle once 100 percent is matched against 360 degrees?
  2. Worked exampleFind the slice angle for a category holding 35 percent of a circle graph.
  3. Common mistakeWhy isn't a falling line on a graph the same thing as a negative quantity?
  4. Going furtherWhat changes about a line graph when its horizontal axis measures something other than time?
Your turneasy1 of 3
A category holds 45 percent of a circle graph. How many degrees does its slice measure?

Histograms and Stem-and-Leaf Plots

  1. How it worksHow does a stem-and-leaf plot store every original number and still show the shape of the data?
  2. Worked exampleFind the median and range when stem 2 carries leaves 3, 5, 8 and stem 3 carries leaves 0, 1, 4, 7.
  3. Common mistakeWhy doesn't a histogram let you name the mode of the original values?
Your turneasy1 of 3
A stem-and-leaf plot has stem 2 with leaves 3, 5 and 8, and stem 3 with leaves 0, 1, 4 and 8. What is the range of the data?

Probability of a Simple Event and Its Complement

  1. How it worksHow does every probability land between 0 and 1 with the two complements totalling exactly 1?
  2. Worked exampleFind the chance of not rolling a 5 on a fair six-sided die, as a fraction and as a percent.
  3. Common mistakeWhy isn't a probability of 0.5 a promise of five successes in ten tries?
  4. Going furtherWhat happens to the complement rule once an event has more than two possible outcomes?
Your turneasy1 of 3
What is the probability of not rolling a number above 4 on a fair six-sided die? Give a decimal.

Compound and Independent Events

  1. How it worksHow does multiplying along a run of independent events shrink the combined chance every step?
  2. Worked exampleFind the chance of rolling a 6 and then flipping a head with a fair die and a fair coin.
  3. Common mistakeWhy isn't drawing two cards without replacement a pair of independent events?
  4. Worked exampleFind the chance of pulling two reds in a row from 4 red and 6 blue marbles with no replacing.
  5. Going furtherWhat changes about the multiplying once the first outcome alters what is left behind?
Your turneasy1 of 3
What is the probability of rolling a number above 4 on a fair die and then flipping a head on a fair coin? Give a decimal.

Organized Lists and Tree Diagrams

  1. How it worksHow does a tree multiply choices along each path and then add across the finished paths?
  2. Worked exampleFind how many different meals come from 3 starters, 4 mains and 2 puddings.
  3. Worked exampleFind the chance of exactly two heads in three coin flips by listing all eight outcomes.
  4. Common mistakeWhy doesn't the counting principle hold when one choice removes an option from the next?
Your turneasy1 of 3
A menu offers 3 starters, 7 mains and 2 puddings. How many different three-course meals are possible?

Experimental vs Theoretical Probability

  1. How it worksHow does an experimental fraction wander toward the theoretical one as trials pile up?
  2. Worked exampleFind the experimental probability when a drawing pin lands point up 37 times in 50 drops.
  3. Common mistakeWhy isn't a run of six tails evidence that a coin has been loaded?
  4. Going furtherWhat happens to the gap between the two probabilities once trials climb into the thousands?
Your turneasy1 of 3
A drawing pin is dropped 50 times and lands point up 37 times. What is the experimental probability of landing point up, as a decimal?

Process Errors Hiding in the Answer Choices

  1. How it worksHow is each wrong choice built from one specific slip, a swapped operation, a lost sign or a bad formula?
  2. Worked exampleWalk me through which trap choice catches a student who merges 20 percent and 10 percent off into 30.
  3. Common mistakeWhy doesn't finding your answer among the choices prove your method was sound?
  4. Going furtherWhat changes about checking your work once you know the traps were built around process slips?
Your turnmedium1 of 3
A coat costs 70 dollars and is marked 20 percent off, then 10 percent off the reduced price. A trap choice merges those into a single 30 percent off. By how many dollars is the trap below the true final price?

Terminology and the Formulas You Are Given

  1. How it worksHow do product, quotient, difference, sum, factor, multiple and mean each name one exact move?
  2. Worked exampleFind the difference between the product of 6 and 8 and the quotient of 84 and 7.
  3. Common mistakeWhy isn't a formula printed inside an item a sign that everyone else had memorised it?
  4. Going furtherWhat changes about revision once you know customary conversions arrive inside the question?
Your turneasy1 of 3
Find the difference between the product of 6 and 15 and the quotient of 84 and 7.

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