← ISEE® Upper Level

Mathematics Achievement (47 questions, 40 minutes)

Forty-seven achievement items aligned to NCTM grades 8 to 11, reaching second-year algebra, where every wrong choice is a process error.

56 topics · 221 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.

Integers and Operations With Signed Numbers

  1. How it worksHow does a number line turn every signed addition into a movement left or right?
  2. Worked exampleFind the value of -7 - (-12) + (-4) one step at a time on a number line.
  3. Worked exampleFind the value of (-3)(-4)(-2) and show where each sign change happens.
  4. Common mistakeWhy isn't -32 equal to 9 when (-3)2 is?
Your turneasy1 of 3
Find the value of -7 - (-12) + (-7).

Order of Operations

  1. How it worksHow does the order of operations decide which piece of an expression collapses first?
  2. Worked exampleFind the value of 8 + 12 divided by 4 times 3 minus 2 squared, in the correct order.
  3. Common mistakeWhy doesn't multiplication always come before division inside the same expression?
  4. Going furtherWhat changes when a minus sign sits outside a set of parentheses that is being squared?
Your turneasy1 of 3
Find the value of 8 + 31 divided by 4, times 3, minus 2 squared.

Factors, Multiples, Primes, GCF and LCM

  1. How it worksHow does a prime factor tree build both the greatest common factor and the least common multiple?
  2. Worked exampleFind the greatest common factor and the least common multiple of 84 and 126.
  3. Worked exampleFind how many positive factors 360 has without listing a single one of them.
  4. Common mistakeWhy isn't 1 counted as a prime number?
  5. Going furtherWhat happens to the least common multiple when two numbers share no factor above 1?
Your turneasy1 of 3
A number's prime factorisation is 24 times 33 times 5. How many positive factors does it have?

Fractions, Decimals, Percents and Converting Among Them

  1. How it worksHow does dividing the top by the bottom turn any fraction into its decimal?
  2. Worked exampleFind 5/8 as a decimal and a percent, then order it against 0.6 and 63 percent.
  3. Worked exampleFind the sum of 2/3, 3/4 and 5/6 using a common denominator.
  4. Common mistakeWhy doesn't a decimal with more digits automatically stand for a bigger number?
Your turneasy1 of 3
Write 5/8 as a percent.

Percent Change and Percent Word Problems

  1. How it worksHow does the word of in a percent problem tell you what the denominator has to be?
  2. Worked exampleFind the percent increase when tomatoes go from 1.20 dollars a pound to 1.50 dollars a pound.
  3. Worked exampleFind the original price of a jacket that costs 68 dollars after a 15 percent discount.
  4. Common mistakeWhy isn't the percent change from 40 up to 50 the same size as the change from 50 down to 40?
  5. Going furtherWhat changes about a percent problem when the base itself is the unknown you want?
Your turneasy1 of 3
Tomatoes rise from 1.448 dollars a pound to 2.319 dollars a pound. What is the percent increase?

Ratio, Proportion and Rates

  1. How it worksHow does cross multiplying turn a proportion into one ordinary linear equation?
  2. Worked exampleFind x when 7 is to 12 as x is to 96.
  3. Worked exampleFind how long 5 painters need for a job that 3 painters finish in 10 hours.
  4. Going furtherWhat happens to the ratio 3 to 5 when 4 is added to both of its parts?
Your turneasy1 of 3
Find x when 7 is to 12 as x is to 102.

Powers, Square Roots and Radicals as Numbers

  1. How it worksHow do you simplify a radical by pulling perfect squares out from underneath it?
  2. Worked exampleFind the simplified form of the square root of 72 added to the square root of 50.
  3. Common mistakeWhy isn't the square root of a sum equal to the sum of the square roots?
Your turneasy1 of 3
The sum of the square root of (72 times 2) and the square root of (72 times 2) equals k times the square root of 2. Find k.

Scientific Notation

  1. How it worksHow does the exponent in scientific notation count the places a decimal point travels?
  2. Worked exampleFind 3.2 times 105 divided by 4.0 times 10−2, written in scientific notation.
  3. Worked exampleFind which is larger, 6.1 times 10−4 or 5.9 times 10−3.
  4. Common mistakeWhy doesn't a negative exponent make the number it describes negative?
Your turneasy1 of 3
Find the value of 7 times 105 divided by 7 times 10−2.

Absolute Value

  1. How it worksHow does one absolute value equation split into two separate equations?
  2. Worked exampleFind both solutions when the absolute value of 2x - 5 equals 9.
  3. Worked exampleFind the solution set when the absolute value of x + 1 is less than 4.
  4. Going furtherWhat happens when an absolute value expression is set less than a negative number?
Your turneasy1 of 3
The absolute value of 2x - 5 equals 16. What is the larger of the two solutions for x?

The Real Number System and Irrational Numbers

  1. How it worksHow do whole numbers, integers, rationals and reals nest inside one another?
  2. Worked exampleFind which of 0.375, the square root of 16, π divided by 2, and 22/7 are irrational.
  3. Common mistakeWhy doesn't a decimal that never ends have to be irrational?
  4. Going furtherWhat happens when one irrational number is multiplied by another irrational number?
Your turneasy1 of 3
Of the numbers 0.375, the square root of 22, π divided by 2, and 22/7, which are irrational?

Simplifying Expressions and the Distributive Property

  1. How it worksHow does the distributive property spread one multiplier across every term in a bracket?
  2. Worked exampleFind the simplified form of 4(2x - 3) - 2(x + 5).
  3. Common mistakeWhy doesn't a minus sign in front of a bracket apply to only the first term inside it?
Your turneasy1 of 3
Write 4(2x - 7) - 2(x + 7) in its simplest form, in terms of x.

Linear Equations in One Variable

  1. How it worksHow do you peel operations off a variable in the reverse of the order they were applied?
  2. Worked exampleWalk me through solving 3(x - 2) = 5x + 8 for x.
  3. Worked exampleFind x when (x + 4)/3 minus (x - 1)/2 equals 1.
  4. Common mistakeWhy isn't an equation with no solution the same as one whose solution is zero?
Your turneasy1 of 3
Solve 3(x - 2) = 5x + 15 for x.

Linear Inequalities and Compound Inequalities

  1. How it worksHow does the answer to an inequality become a whole stretch of the number line?
  2. Worked exampleFind the solution set when -2x + 1 lies between 5 and 9, endpoints included.
  3. Worked exampleFind the largest integer x for which 7 - 3x is still greater than 1.
  4. Going furtherWhat changes about the picture when a compound inequality uses or in place of and?
Your turneasy1 of 3
The expression -2x + 1 lies between 5 and 16, endpoints included. What is the smallest value x can take?

Systems of Linear Equations

  1. How it worksHow does elimination kill one variable by adding two equations together?
  2. How it worksHow does substitution turn a two-variable system into one equation in one unknown?
  3. Worked exampleFind x and y when 3x + 2y = 16 and x - 2y = 0.
  4. Common mistakeWhy doesn't every pair of linear equations have exactly one solution?
Your turneasy1 of 3
Solve for x when 3x + 2y = 31 and x - 2y = 0.

Slope and the Equations of Lines

  1. How it worksHow does rise over run give the same slope no matter which two points you pick?
  2. Worked exampleFind the slope of the line through (2, 5) and (6, 13), then write its equation.
  3. Worked exampleFind the equation of the line through (0, -3) running parallel to y = 4x + 1.
  4. Common mistakeWhy isn't a line of slope -5 gentler than a line of slope 2?
  5. Going furtherWhat happens to slope when a line stands vertical instead of lying horizontal?
Your turneasy1 of 3
Find the slope of the line through the points (2, 5) and (6, 20).

Functions and Function Notation

  1. How it worksHow does the vertical line test decide whether a curve is a function at all?
  2. Worked exampleFind f(-2) and f(3) when f(x) = 2x2 - 5x + 1.
  3. Worked exampleFind f(g(2)) when f(x) = 3x - 1 and g(x) = x2 + 1.
  4. Common mistakeWhy doesn't a function have to hit each output value exactly once?
Your turneasy1 of 3
For f(x) = 2x2 - 5x + 1, find f(3).

Domain and Range

  1. How it worksHow do a zero denominator and a negative under a radical carve holes out of a domain?
  2. Worked exampleFind the domain of 1/(x - 4) and then the domain of the square root of x - 4.
  3. Going furtherWhat happens to the range of y = x2 + 3 compared with the range of y = x2?
Your turneasy1 of 3
Compare the domain of 1/(x - 9) with the domain of the square root of x - 9. How do they differ?

Exponent Rules, Including Negative and Fractional Exponents

  1. How it worksHow does adding exponents fall straight out of counting repeated factors?
  2. Worked exampleFind the simplified form of (2x3 y−2)3 divided by 4x2 y.
  3. Worked exampleFind the value of 16 raised to the three quarters power without a calculator.
  4. Common mistakeWhy isn't x2 times x3 equal to x6?
Your turneasy1 of 3
Write (2x3)3 divided by x2 in its simplest form, in terms of x.

Polynomial Operations

  1. How it worksHow does distribution generate every single term when two polynomials multiply?
  2. Worked exampleFind the product of 2x - 3 and x2 + 4x - 5.
  3. Worked exampleFind the result of subtracting 3x2 - x + 7 from 5x2 + 2x - 1.
  4. Going furtherWhat happens to the degree of the answer when two cubic polynomials are multiplied?
Your turneasy1 of 3
Write the product of 2x - 7 and x + 7, in terms of x.

Factoring Quadratics and the Difference of Squares

  1. How it worksHow do a sum and a product together pin down the factors of x2 + bx + c?
  2. Worked exampleFind the factors of x2 - 7x + 12 and then the roots of that quadratic.
  3. Worked exampleFind the factors of 9x2 - 49 by spotting a difference of two squares.
  4. Common mistakeWhy isn't x2 + 16 factorable the way x2 - 16 is?
Your turneasy1 of 3
Write x2 - 92 in factored form, in terms of x.

The Quadratic Formula and the Discriminant

  1. How it worksHow does the discriminant under the radical count where a parabola meets the x axis?
  2. Worked exampleFind both solutions of 2x2 - 5x - 3 = 0 using the quadratic formula.
  3. Worked exampleFind the discriminant of x2 - 6x + 9 = 0 and say what it predicts about the graph.
  4. Common mistakeWhy doesn't a negative discriminant mean you have made an arithmetic slip?
Your turneasy1 of 3
Find the larger solution of 2x2 - 5x - 7 = 0.

Completing the Square

  1. How it worksHow does halving the coefficient of x and squaring it supply the missing corner?
  2. Worked exampleWalk me through completing the square on x2 + 8x + 5 to reach vertex form.
  3. Worked exampleFind the center and radius of x2 + y2 - 6x + 4y = 12 by completing the square twice.
  4. Going furtherWhat happens to the completing-the-square steps when the leading coefficient is not 1?
  5. Ties togetherHow does completing the square on ax2 + bx + c produce the quadratic formula itself?
Your turneasy1 of 3
Writing x2 + 8x + 14 as (x + 4)2 + k, find k.

Rational Expressions

  1. How it worksHow does factoring the top and the bottom let a rational expression cancel down?
  2. Worked exampleFind the simplified form of x2 - 9 over x2 + x - 12.
  3. Worked exampleFind the sum of 3/(x + 2) and 5/(x - 1) as one single fraction.
  4. Common mistakeWhy doesn't the x cancel out of the top and bottom of (x + 3)/(x + 5)?
Your turneasy1 of 3
Write (x2 - 72)/(x + 7) in its simplest form, in terms of x.

Radical Expressions

  1. How it worksHow does multiplying by a conjugate clear a radical out of a denominator?
  2. Worked exampleFind the simplified form of 6 divided by the quantity square root of 3 minus 1.
  3. Worked exampleFind the product of 2 plus the square root of 5 and 2 minus the square root of 5.
  4. Going furtherWhat changes when a cube root rather than a square root sits under the sign?
Your turneasy1 of 3
Find the value of 10 divided by the quantity square root of 3 minus 1.

Sequences and Patterns

  1. How it worksHow does the common difference become the coefficient of n in the nth term rule?
  2. Worked exampleFind the expression for the nth term of the sequence -3, 2, 7, 12, 17.
  3. Worked exampleFind the 20th term of a geometric sequence that starts at 3 and doubles each time.
  4. Common mistakeWhy isn't a sequence that grows faster and faster automatically geometric?
Your turneasy1 of 3
A sequence begins at -3 and rises by 7 at every step. Write the value of term number n, in terms of n.

Translating Word Problems Into Algebra

  1. How it worksHow do you turn a sentence about ages or coins into an equation term by term?
  2. Worked exampleFind Maria's age when she is 4 years older than twice her brother and their ages total 34.
  3. Worked exampleFind how many dimes sit in a jar of 30 dimes and quarters worth 4.20 dollars.
  4. Going furtherWhat changes about the setup when a problem hands you two unknowns instead of one?
Your turneasy1 of 3
Maria is 4 years older than twice her brother's age, and their ages total 40. How old is Maria?

Angle Relationships and Parallel Lines

  1. How it worksHow do corresponding, alternate interior and same-side angles all follow from one slide along the transversal?
  2. Worked exampleFind every angle at both crossings when a transversal meets two parallel lines at 68 degrees.
  3. Common mistakeWhy doesn't a pair of angles that looks equal have to be equal without the parallel marks?
Your turneasy1 of 3
A transversal crosses two parallel lines, making an angle of 68 degrees. What is the measure of the angle next to it on the same line?

Triangle Properties and the Triangle Inequality

  1. How it worksHow does the biggest angle in a triangle always end up opposite the longest side?
  2. Worked exampleFind the range of possible third sides for a triangle whose other two sides are 7 and 11.
  3. Worked exampleFind the three angles of a triangle when they measure 3x, 4x and 5x degrees.
  4. Going furtherWhat happens to an exterior angle when the two far interior angles change size?
Your turneasy1 of 3
A triangle has sides of 7 and 17. What is the largest whole number the third side can be?

The Pythagorean Theorem

  1. How it worksHow does rearranging four copies of one right triangle prove a squared plus b squared equals c squared?
  2. Worked exampleFind the hypotenuse of a right triangle whose legs measure 9 and 40.
  3. Worked exampleFind the long diagonal of a rectangular box measuring 3 by 4 by 12.
  4. Common mistakeWhy doesn't the Pythagorean relationship hold in a triangle with no right angle?
  5. Ties togetherHow does the distance formula on a coordinate grid turn out to be the Pythagorean theorem in disguise?
Your turneasy1 of 3
A right triangle has legs of 9 and 31. Find the length of its hypotenuse.

Special Right Triangles

  1. How it worksHow does slicing an equilateral triangle in half generate the 30-60-90 side ratios?
  2. Worked exampleFind both missing sides of a 30-60-90 triangle whose shortest leg measures 5.
  3. Going furtherWhat changes when a 45-45-90 triangle hands you the hypotenuse instead of a leg?
Your turneasy1 of 3
A 30-60-90 triangle has a shortest leg of 9. Find the length of the longer leg.

Congruence and Similarity

  1. How it worksHow does a scale factor grow one triangle into another without bending a single corner?
  2. Worked exampleFind YZ when triangle RST is similar to triangle XYZ, RS is 3 cm, XY is 2 cm and ST is 9 cm.
  3. Common mistakeWhy doesn't matching angles plus matching area make two triangles congruent?
  4. Going furtherWhat happens to the ratio of areas when the ratio of sides between similar figures is 2 to 3?
Your turneasy1 of 3
Triangle RST is similar to triangle XYZ, with RS = 5 cm, XY = 2 cm and ST = 16 cm. Find YZ, in centimetres.

Quadrilaterals and Polygons

  1. How it worksHow does chopping a polygon into triangles produce the 180 times n minus 2 rule?
  2. Worked exampleFind the measure of one interior angle of a regular decagon.
  3. Worked exampleFind the fourth angle of a quadrilateral whose other three measure 85, 110 and 47 degrees.
  4. Common mistakeWhy isn't every rhombus a square when every square is a rhombus?
Your turneasy1 of 3
Find the measure of one interior angle of a regular polygon with 10 sides, in degrees.

Circles: Circumference, Area, Arcs and Sectors

  1. How it worksHow does a central angle slice the arc and the sector in exactly the same proportion?
  2. Worked exampleFind the area and the circumference of a circle with a radius of 7.
  3. Worked exampleFind the arc length and sector area cut by a 45 degree central angle in a circle of radius 12.
  4. Going furtherWhat happens to a circle's area when its radius grows by 10 percent?
Your turneasy1 of 3
Find the area of a circle with radius 9.

Coordinate Geometry: Distance, Midpoint and Slope

  1. How it worksHow do the distance and midpoint formulas both come out of the legs of a right triangle?
  2. Worked exampleFind the distance and the midpoint between the points (-3, 4) and (5, -2).
  3. Worked exampleFind the area of the triangle with corners at (0, 0), (6, 0) and (6, 8).
  4. Common mistakeWhy doesn't a segment's midpoint sit at the average of only its x coordinates?
Your turneasy1 of 3
Find the distance between the points (-3, 4) and (5, 0).

Transformations in the Plane

  1. How it worksHow does reflecting a point across the line y = x simply swap its two coordinates?
  2. Worked exampleFind where (4, -3) lands after a reflection in the y axis followed by a 90 degree turn about the origin.
  3. Going furtherWhat happens to a figure's area under a dilation with a scale factor of 3?
Your turneasy1 of 3
A shape with 6 labelled corners is reflected and then rotated. Which move reverses its handedness?

Three-Dimensional Figures

  1. How it worksHow do the faces, edges and vertices of a prism follow from the shape of its base?
  2. Worked exampleFind the number of faces, edges and vertices of a hexagonal prism.
  3. Common mistakeWhy isn't a cylinder a prism when both have two identical parallel ends?
  4. Going furtherWhat changes about a cross section when a cube is sliced at an angle instead of straight across?
Your turneasy1 of 3
A prism has a base with 10 sides. How many edges does it have?

Units, Conversion and Precision

  1. How it worksHow does a conversion factor written as a fraction equal to 1 cancel the units you do not want?
  2. Worked exampleFind the expression giving 8 miles per hour in feet per second, using 5,280 feet to a mile.
  3. Common mistakeWhy isn't a reading of 4.0 centimeters the same claim as a reading of 4 centimeters?
  4. Going furtherWhat changes about a conversion when the unit is squared, as in square feet to square yards?
Your turneasy1 of 3
A rectangle measures 22 feet by x feet. Write its area in square yards, in terms of x.

Perimeter and Area of Plane Figures

  1. How it worksHow does a triangle's area come from cutting a rectangle exactly in half?
  2. Worked exampleFind the area of a trapezoid whose parallel sides are 8 and 14 with a height of 6.
  3. Worked exampleFind the perimeter of a rectangle whose area is 96 and whose length is 12.
  4. Going furtherWhat happens to a parallelogram's area when it is sheared over without moving its base?
Your turneasy1 of 3
A trapezoid has parallel sides of 8 and 24 with a height of 6. Find its area.

Surface Area of Solids

  1. How it worksHow does unrolling a cylinder turn its curved side into a plain rectangle?
  2. Worked exampleFind the total surface area of a cylinder with a radius of 3 and a height of 10.
  3. Worked exampleFind the surface area of a rectangular box measuring 5 by 4 by 7.
  4. Ties togetherHow does surface area fit the scale factor pattern that governs length and volume?
Your turneasy1 of 3
Find the total surface area of a cylinder with radius 3 and height 15.

Volume of Prisms, Cylinders, Cones, Pyramids and Spheres

  1. How it worksHow does base area times height cover every prism and every cylinder at once?
  2. Worked exampleFind the volume of a cone with a radius of 6 and a height of 15.
  3. Worked exampleFind the volume of a sphere of radius 3 and set it against a cube of edge 6.
  4. Going furtherWhat happens to a cylinder's volume when the radius is halved and the height quadrupled?
Your turneasy1 of 3
Find the volume of a cone with radius 6 and height 16.

Composite Figures

  1. How it worksHow do you split an awkward shape into rectangles and circle parts you already know?
  2. Worked exampleFind the area of a 10 by 6 rectangle with a semicircle of diameter 6 stuck on one end.
  3. Common mistakeWhy doesn't the outline of a composite figure include the seams where the pieces join?
Your turneasy1 of 3
A rectangle measures 10 by 10, with a semicircle of diameter 10 stuck on one end. Find the total area.

Scale Drawings and Indirect Measurement

  1. How it worksHow do similar triangles turn a map scale into a real distance on the ground?
  2. Worked exampleFind a flagpole's height when it throws a 24 foot shadow while a 6 foot person throws 4 feet.
  3. Worked exampleFind the real distance when a map drawn at 1 inch to 25 miles shows a gap of 3.5 inches.
  4. Going furtherWhat changes about a scale drawing when the scale is given as a bare ratio with no units?
Your turneasy1 of 3
A flagpole casts a shadow of 33 feet at the same moment a 6 foot person casts one of 4 feet. How tall is the flagpole?

Mean, Median, Mode and Range

  1. How it worksHow does the mean act like a balance point while the median just holds the middle seat?
  2. Worked exampleFind the mean, median, mode and range of 4, 7, 7, 9, 13 and 20.
  3. Worked exampleFind the sixth score when five scores average 82 and all six together average 85.
  4. Common mistakeWhy doesn't the median have to be one of the numbers actually in the data set?
  5. Going furtherWhat happens to all four measures when every value in a data set is doubled?
Your turneasy1 of 3
Find the mean of 4, 7, 7, 9, 13 and 25.

Weighted Average

  1. How it worksHow does weighting by group size drag a combined mean toward the larger group?
  2. Worked exampleFind the combined average when 20 students average 78 and 30 students average 88.
  3. Common mistakeWhy doesn't a 90 on a test worth 40 percent count the same as a 90 on one worth 10 percent?
Your turneasy1 of 3
One class of 31 students averages 78 and another of 29 students averages 88. What is the combined average?

Reading Tables and Bar Graphs

  1. How it worksHow do you pull a single comparison out of a table with four columns and six rows?
  2. Worked exampleFind the total and then the percent choosing baseball from bars of 20 football, 18 baseball, 10 basketball and 12 other.
  3. Going furtherWhat changes about a stacked bar chart compared with bars set side by side?
Your turneasy1 of 3
Bars read 20 football, 25 baseball, 10 basketball and 12 other. What percent of the students chose baseball?

Line Graphs and Circle Graphs

  1. How it worksHow does a category's fraction of the total become its central angle out of 360 degrees?
  2. Worked exampleFind the central angle for basketball when 10 of 60 surveyed students name it their favorite sport.
  3. Worked exampleFind the percent drop when a line graph falls from 80 in March to 56 in April.
  4. Common mistakeWhy doesn't a bigger slice on one circle graph have to stand for a bigger number than a smaller slice on another?
Your turneasy1 of 3
Of 60 students surveyed, 16 name basketball as their favourite sport. What central angle does basketball get on a circle graph?

Histograms, Stem-and-Leaf Plots and Box Plots

  1. How it worksHow does a box plot pin down a whole data set with only five numbers?
  2. Common mistakeWhy isn't a histogram just a bar graph with the bars pushed together?
  3. Worked exampleFind the median and the interquartile range of 3, 7, 8, 12, 14, 15, 21 and 28.
  4. Worked exampleFind the median of a stem-and-leaf plot where stem 3 carries leaves 1, 4, 8 and stem 4 carries 0, 2, 5, 9.
Your turneasy1 of 3
Find the median of the ordered data 3, 7, 8, 12, 14, 15, 21, 28.

Scatterplots and the Line of Best Fit

  1. How it worksHow does a line of best fit balance the dots above it against the dots below it?
  2. Worked exampleFind the value predicted at x = 9 by a best fit line running through (1, 4) and (5, 16).
  3. Common mistakeWhy doesn't a strong correlation between two measurements prove that one causes the other?
  4. Going furtherWhat happens to a line of best fit when one far-off outlier is dropped onto the plot?
Your turneasy1 of 3
A line of best fit passes through (1, 4) and (5, 16). What value does it predict at x = 16?

Probability of a Simple Event

  1. How it worksHow does writing out a sample space turn a chance question into a plain fraction?
  2. Worked exampleFind the probability of rolling a sum of 7 with two fair six-sided dice.
  3. Worked exampleFind the probability of drawing a face card from a standard 52 card deck.
  4. Going furtherWhat happens to a probability when the event is certain or completely impossible?
Your turneasy1 of 3
Two fair six-sided dice are rolled. What is the probability that the two faces total 8?

Compound and Independent Events

  1. How it worksHow does a tree diagram multiply along each branch and then add across the ends?
  2. Worked exampleFind the chance of rolling red then blue on a cube with 2 red faces, 3 blue faces and 1 green face.
  3. Common mistakeWhy doesn't the word or in a probability question always mean you simply add the two chances?
  4. Going furtherWhat changes about the arithmetic when two events are able to happen at the same time?
Your turneasy1 of 3
A cube has 2 red faces, 3 blue faces and the rest green. What is the probability of rolling red and then blue?

Dependent Events and Drawing Without Replacement

  1. How it worksHow does the denominator shrink at every step when nothing gets put back?
  2. Worked exampleFind the chance that both cards are hearts when two are dealt from a 52 card deck without replacement.
  3. Worked exampleFind the chance of drawing two greens in a row from a bag of 5 green and 7 white marbles.
  4. Common mistakeWhy isn't grabbing two at once any different from drawing one after the other without replacing?
Your turneasy1 of 3
A deck of 48 cards contains 13 hearts. Two cards are dealt without replacement. What is the probability that both are hearts?

Counting Principles, Permutations and Combinations

  1. How it worksHow does the multiplication principle count a set of choices slot by slot?
  2. How it worksHow does dividing by the arrangements within a chosen group turn a permutation into a combination?
  3. Worked exampleFind how many 4 digit codes use the digits 1 through 9 with no digit repeated.
  4. Worked exampleFind how many 3 person committees can be chosen out of a group of 8 people.
  5. Common mistakeWhy isn't picking 3 of 8 people the same count as lining 3 of 8 people up in order?
Your turneasy1 of 3
How many 4 digit codes can be made from the digits 1 through 8 if no digit repeats?

Drawing Inferences From a Sample

  1. How it worksHow does a biased sampling method tilt a conclusion before a single number is collected?
  2. Worked exampleFind the estimated number of left handers in a school of 900 when 12 of a random 150 are left handed.
  3. Common mistakeWhy doesn't a larger sample repair a sample that was gathered in a biased way?
  4. Going furtherWhat happens to the reliability of an estimate when the sample size is quadrupled?
Your turneasy1 of 3
In a random sample of 150 students, 17 are left handed. Estimate the number of left handed students in a school of 900.

Pacing 47 Items in 40 Minutes

  1. How it worksHow do you mark an item and move past it so you can return with time still on the clock?
  2. Worked exampleWalk me through spending 30 seconds each on the first 15 items to bank time for the last 10.
  3. Common mistakeWhy isn't three minutes on one hard item ever worth it when every item scores the same?
Your turneasy1 of 3
The Mathematics Achievement section runs 40 minutes for 47 items. A student spends 42 seconds on each of the first 15 items. How many seconds remain?

No Calculator, No Scratch Paper, and Which Conversions Are Given

  1. How it worksHow does mental arithmetic stand in for a calculator across 47 items with no scratch paper?
  2. Worked exampleFind 15 percent of 240 in your head by taking 10 percent and adding half of that.
  3. Worked exampleFind 48 times 25 mentally by turning it into 48 divided by 4, times 100.
  4. Going furtherWhat changes about your working when the only place to write is the test booklet itself?
Your turneasy1 of 3
Find 26 percent of 240 without a calculator.

The Terminology the Section Assumes

  1. How it worksHow do integer, factor, multiple, mean, median and perimeter each pin down a different object?
  2. Worked exampleFind the product of the smallest prime number and the greatest common factor of 18 and 24.
  3. Common mistakeWhy isn't a factor the same thing as a multiple when both describe one division?
  4. Going furtherWhat changes when a question asks for consecutive even integers instead of consecutive integers?
Your turneasy1 of 3
Find the product of the smallest prime number and 22.

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