321 core topics
+ 16 prerequisite topics taught
as needed · approximately 97 hours of instruction
including spaced review
An adaptive diagnostic (up to
40 questions) places the student on the course's knowledge
graph - topics already known are credited, and instruction begins exactly
at the learning frontier. Every topic is taught with a worked-example
lesson and auto-graded practice; a topic is mastered at
75%+ and then maintained through spaced reviews on an
expanding schedule. Mixed checks follow every 6 lessons;
each unit ends with a 12-item quiz, and course-wide assessments appear at
25%, 50%, 75%, and 100% mastery. Prerequisite gaps below the course are detected and taught rather
than skipped, so completion certifies the whole tower, not just the top.
| GCF & LCM
[E] |
Greatest common factor and least common multiple. |
| Prime Factorization
[E] |
Every whole number is a unique product of primes. |
| Integer Operations
[E] |
Fluent four-operation arithmetic with negative numbers. |
| Absolute Value & Distance
[E] |
Absolute value is distance from zero. |
| Unit Rates
[E] |
Per-one comparisons: dollars per item, miles per hour. |
| Ratio Tables & Equivalent Ratios
[E] |
Scaling both parts of a ratio keeps it equivalent. |
| Solving Proportions
[E] |
Cross-multiply to find the missing value. |
| Percent Applications: Tax, Tip & Discount
[E] |
Real-world percents: discounts, tips, and tax. |
| Area: Triangles & Trapezoids
[E] |
Half of base times height - and its trapezoid cousin. |
| Area of Composite Figures
[E] |
Split odd shapes into rectangles and triangles. |
| Volume: Rectangular Prisms
[E] |
Length × width × height, fractional edges included. |
| Surface Area & Nets
[E] |
Unfold the box: surface area is the area of its net. |
| Mean, Median & Range
[E] |
Three ways to summarize a data set with one number. |
| Reading Data Displays
[E] |
Pulling answers out of dot plots, tables, and bar graphs. |
| Probability Basics
[E] |
Favorable outcomes over total outcomes. |
| Compound Probability
[E] |
Independent events multiply. |
| Constant of Proportionality
[M] |
The unit rate k in y = kx. |
| Graphing Proportional Relationships
[E] |
Straight line through the origin; slope = k. |
| Scale Drawings & Maps
[E] |
Scale is a ratio between drawing and reality. |
| Two-Step Inequalities
[M] |
Solve like an equation; flip when you multiply by a negative. |
| Angle Relationships
[E] |
Complementary (90°), supplementary (180°), vertical (equal). |
| Circumference & Area of Circles
[M] |
C = 2πr and A = πr². |
| Simple Interest & Percent Change
[M] |
I = P·r·t; balance = principal + interest. |
| Random Sampling & Inference
[E] |
Use a representative sample to estimate a whole population. |
| Comparing Two Populations
[E] |
Compare centers relative to spread. |
| Probability Models & Simulation
[E] |
Experimental probability from observed frequencies. |
| Opposites in Context
[E] |
Every number has a mirror twin on the other side of zero. |
| Absolute Value in Context
[E] |
Absolute value answers 'how far from zero?' - order answers 'which is greater?' |
| Comparing & Ordering Rational Numbers
[E] |
Smaller means farther left on the number line - even below zero. |
| Rational Numbers on the Number Line
[E] |
Fractions and decimals claim exact spots between the integers. |
| Plotting in All Four Quadrants
[E] |
Negative coordinates open up the other three quadrants. |
| Reflections Across the Axes
[E] |
Reflecting a point just flips the sign of one coordinate. |
| Distances on the Coordinate Plane
[M] |
Same x or same y: subtract and take the absolute value. |
| Rectangles on the Coordinate Plane
[M] |
Coordinates give the side lengths; perimeter and area follow. |
| Dividing Fractions: Word Problems
[M] |
'How many of this size fit?' is a division by a fraction. |
| Decimal Operations in Context
[E] |
Money problems are decimal arithmetic with two places, always. |
| Dividing Decimals in Context
[M] |
Slide both decimal points until the divisor is whole, then divide. |
| Percent of a Quantity: Parts & Wholes
[M] |
Percent problems run in three directions: find the part, the percent, or the whole. |
| Adding & Subtracting Signed Fractions
[M] |
Common denominators first, then the integer sign rules. |
| Adding & Subtracting Signed Decimals
[M] |
Balances, debts, and deposits: decimal arithmetic below zero. |
| Multiplying Signed Rationals
[M] |
Multiply the fractions, then apply the sign rules. |
| Dividing Signed Rationals
[M] |
Flip, multiply, and keep track of the sign. |
| Markup Then Discount
[H] |
Percent changes chain by multiplying, never by adding. |
| Discount, Tax & Tip Chains
[H] |
Work one percent step at a time - each acts on the previous result. |
| Proportional Word Problems
[M] |
Find the rate for one, then scale it to the amount asked. |
| Angle Equations: Complementary & Supplementary
[M] |
Complements sum to 90°, supplements to 180° - write the equation. |
| Angle Equations: Vertical Angles & Linear Pairs
[H] |
Vertical angles are equal; a linear pair sums to 180°. |
| Expanding with Rational Coefficients
[M] |
Distribute fractions and negatives to every term inside. |
| Factoring Linear Expressions
[M] |
Pull the GCF out front; distributing it back must restore the original. |
| Multi-Step Problems with Rationals
[H] |
Chain the operations one sentence at a time - fractions act on what's left. |
| Writing Addition & Subtraction Equations
[E] |
Turn a join-or-separate story into a one-step equation. |
| Writing Multiplication & Division Equations
[E] |
Equal-groups and fair-share stories become px = q or x ÷ p = q. |
| Solving Add & Subtract Equations in Context
[E] |
Undo one addition or subtraction to answer the story's question. |
| Solving Multiply & Divide Equations in Context
[E] |
Undo one multiplication or division to find the missing amount. |
| Is It a Solution?
[E] |
Substitute a value for the variable and see if both sides agree. |
| Money Equations
[M] |
One-step equations where the amounts are dollars and cents. |
| Evaluating Expressions in Context
[E] |
Substitute a number for the variable and follow the order of operations. |
| Dependent & Independent Variables
[E] |
Which quantity drives the relationship, and which one responds? |
| Using Two-Variable Equations
[E] |
Feed the independent variable into a rule to get the dependent one. |
| Writing Inequalities from Constraints
[E] |
Translate 'at least', 'more than', 'at most', 'fewer than' into symbols. |
| Graphing Inequalities on a Number Line
[E] |
Open or closed circle, shaded left or right - read and draw both ways. |
| Boundary Values of an Inequality
[E] |
Find the smallest or greatest whole number an inequality allows. |
| Area of a Circle
[E] |
Area is π times the radius squared. |
| Circumference of a Circle
[E] |
Circumference is π times the diameter. |
| Semicircles
[M] |
Half a circle: halve the area and the circumference. |
| Area of Composite Figures
[M] |
Break the figure into pieces, then add or subtract. |
| Scaling and Area
[M] |
Double the lengths, quadruple the area. |
| Volume of a Triangular Prism
[M] |
Triangle area times the prism's length. |
| Surface Area of a Prism
[M] |
Add the areas of all six rectangular faces. |
| Cross-Sections of Solids
[E] |
The shape revealed when a solid is sliced. |
| Angle Relationships as Equations
[M] |
Complementary sum to 90°, supplementary to 180°. |
| Working Backward from Area
[M] |
Undo πr² to recover the radius. |
| Constant of Proportionality from a Table
[M] |
Divide any y by its x - the ratio is the same every row, even when it is a fraction. |
| Constant of Proportionality from a Graph
[M] |
On a line through the origin, the constant of proportionality is the rise over the run to any point. |
| Constant of Proportionality from an Equation
[M] |
Solve the equation for y = kx; the coefficient of x is the constant, fractions and all. |
| Unit Rates as Complex Fractions
[M] |
A rate can be a fraction over a fraction - divide by multiplying by the reciprocal. |
| Complex-Fraction Rate Problems
[M] |
Real 'per one unit' rates when both quantities are fractions. |
| Redrawing at a New Scale
[M] |
Find the real length once, then convert it into any other drawing's scale. |
| Scale Drawings and Area
[M] |
Lengths scale by the factor; areas scale by the factor squared. |
| Successive Discounts
[M] |
Two discounts in a row multiply - they never simply add. |
| Commission
[M] |
Commission is a percent of sales; total pay may add a base salary. |
| Percent Error
[M] |
How far a measurement is off, as a percent of the true value. |
| Net Percent Change over Several Steps
[M] |
Chain percent changes by multiplying factors, then compare to the start. |
| Proportions with Unit Conversion
[M] |
Convert the units to match the rate before you scale. |
| Comparing Unit Rates (Best Buy)
[M] |
Reduce each deal to a price-per-one and compare those rates. |
| Expand, Then Combine Like Terms
[M] |
Distribute each factor, then gather the x-terms and the constants. |
| Collecting Like Terms
[M] |
Reorder freely, then add coefficients and add constants. |
| Factoring Out the GCF
[M] |
Pull the greatest common factor in front; the numbers left share no factor. |
| Identifying Equivalent Expressions
[M] |
Two expressions are equivalent when they agree for every value of x. |
| Writing Equivalent Expressions
[M] |
Describe a situation two ways, then simplify to a single equivalent form. |
| Two-Step Equation Word Problems
[M] |
Translate the story into ax + b = c, then undo the two operations. |
| Two-Step Equations with Money
[M] |
Subtract the flat fee, then divide by the count - answer to the cent. |
| Two-Step Equations: Geometry & Number Puzzles
[M] |
Perimeter formulas and number riddles both become ax + b = c. |
| Two-Step Inequality Word Problems
[M] |
“At most” and “at least” become ≤ and ≥; solve, then read off the count. |
| Graphing an Inequality's Solution
[M] |
Closed dot for ≤ or ≥, open dot for < or >; shade toward the solutions. |
| Rewriting to Reveal a Relationship
[M] |
A single combined expression can show a relationship the pieces hide. |
| Properties of Operations
[M] |
Apply a property - distribute a negative factor or reorder and combine. |
| Interpreting a Factored Form
[M] |
The factor pulled out front tells you what the value is a multiple of. |
| Comparing Sample Proportions
[E] |
Turn each sample's count into a proportion, then compare across sample sizes. |
| Estimating a Population Count
[M] |
Scale the sample proportion up to a whole population total. |
| A Sample Mean Estimates a Total
[M] |
Population total ≈ sample mean × population size. |
| Mean Absolute Deviation
[M] |
Average distance of the data from its mean - a measure of spread. |
| Comparing Two Means
[M] |
Compute each group's mean, then the gap between them. |
| Comparing Spread with MAD
[M] |
The data set with the larger MAD is the more variable one. |
| Difference as a Multiple of the MAD
[M] |
Measure the gap between two populations in MAD-sized steps. |
| Theoretical Probability of One Event
[M] |
Favorable outcomes over total equally likely outcomes, reduced. |
| Experimental Probability from Data
[M] |
Observed frequency of an outcome over the total number of trials. |
| Theoretical vs. Experimental
[M] |
Expected count = theoretical probability × trials; compare to what happened. |
| Compound Events by Organized List
[M] |
List every equally likely combined outcome, then count the favorable ones. |
| The Counting Principle
[M] |
Multiply the number of choices at each stage; multiply probabilities of independent events. |
| Estimating Probability from a Simulation
[M] |
A simulation's relative frequency estimates the true probability. |
| Is It Proportional? Tables
[H] |
Test every row: proportional means y/x comes out the same each time. |
| Writing y = kx
[H] |
Divide y by x at any point, reduce, and that quotient is the k in y = kx. |
| Solving with y = kx
[H] |
To find x from y, divide by k - undo the multiplication. |
| Points on a Proportional Graph
[M] |
A point (a, b) pairs a amount with b cost; (1, k) shows the unit rate. |
| Comparing Proportional Relationships
[H] |
Reduce each relationship to its unit rate, then compare the two k's. |
| Finding the Original Price
[H] |
The final price is a known percent OF the original - divide to go back. |
| Finding the Percent of Change
[H] |
Percent change = (amount of change) / (original amount) x 100. |
| Tips, Taxes & Splitting the Bill
[H] |
Add each percent of the bill, total it, then divide by the people. |
| Simple Interest: Rate, Time & Principal
[H] |
I = Prt has four quantities - given any three, solve for the fourth. |
| Percents of a Whole Group
[H] |
The percents of all the parts total one hundred - use the leftover part. |
| A Percent of a Percent
[H] |
Take the first percent of the whole, then the second percent of THAT. |
| Temperature Swings with Fractions
[H] |
Rises add, falls subtract - mixed numbers ride the same number line. |
| Average Rate of Change, Signed
[H] |
Average change per day = (final - start) / days, sign included. |
| Rate Times Time, Below Zero
[H] |
A steady change repeats: multiply the signed rate by the time. |
| Terminating or Repeating?
[H] |
In lowest terms, only 2s and 5s in the denominator let a decimal stop. |
| Order of Operations with Signed Fractions
[H] |
PEMDAS survives below zero: multiply and divide before adding. |
| Inequalities with Falling Quantities
[H] |
A quantity dropping r per step crosses a bound after (start - bound)/r steps. |
| The Score You Need
[H] |
An average of n scores at least A means the scores must total at least nA. |
| Choosing the Inequality Model
[M] |
Match each phrase to its symbol: at least is >=, can afford is <=. |
| Finding the Scale
[H] |
Scale = real length per one drawing unit: divide real by drawn. |
| Scale Models (1 : n)
[H] |
A 1:n scale means every real length is n times the model's. |
| Perimeter Through a Scale Drawing
[H] |
Perimeter is a length, so it scales by the scale factor itself. |
| Parallel Lines & a Transversal
[H] |
Corresponding and alternate angles are equal; co-interior sum to 180. |
| Parallel-Line Angle Equations
[H] |
Set equal angles equal, or make co-interior angles sum to 180, then solve. |
| Angles Around a Point
[H] |
A full turn is 360 degrees; the angles at a point account for all of it. |
| Angle Addition
[H] |
A ray inside an angle splits it: part + part = whole. |
| Circles with pi = 22/7
[H] |
Radii that are multiples of 7 make 22/7 cancel to whole numbers. |
| Area of a Ring
[H] |
Ring area = outer circle minus inner circle: (R^2 - r^2) pi. |
| Rolling Wheels & Circumference
[H] |
One rotation covers one circumference: rotations = distance / C. |
| Probability Without Replacement
[H] |
After the first draw, both the bag and the count of favorites shrink. |
| At Least One: Using Complements
[H] |
P(at least one) = 1 - P(none): the complement does the counting. |
| Two Spinners: Products & Parity
[H] |
Lay the outcomes in a grid; a product is odd only when both factors are. |
| Choosing a Fair Sample
[M] |
Random selection from the whole population beats any convenient group. |
| Capture & Recapture
[H] |
Tagged/total in the sample mirrors tagged/total in the population. |
| Predicting a Difference from a Sample
[H] |
Scale both counts by the same factor - or scale their gap directly. |
| Constant of Proportionality from a Description
[H] |
Divide the total by the count: that quotient is k, the cost of one. |
| Interpreting the Constant of Proportionality
[H] |
k is the amount of y for one unit of x - read it with the labels. |
| Is the Graph Proportional?
[H] |
A graph is proportional only if it is a straight line through the origin. |
| Is the Equation Proportional?
[H] |
Only y = kx is proportional: no added constant, x on top, first power. |
| Proportional or Not? Real Situations
[H] |
Proportional means a constant ratio AND zero-goes-to-zero. |
| The Second Unit Rate
[H] |
Every rate has a partner: flip pages-per-minute to get minutes-per-page. |
| Completing a Proportional Table
[H] |
Find k from a full row, then multiply or divide to fill the gap. |
| Matching a Table to Its Equation
[H] |
Compute y/x from one row, reduce, and that is the k in y = kx. |
| Writing y = kx from a Story
[H] |
A per-unit rate in a story is exactly the k that multiplies x. |
| Constant Speed: Distance and Time
[H] |
At a steady speed, distance = speed x time - and rearrange as needed. |
| Density: Mass and Volume
[H] |
Mass is proportional to volume: same density means same mass-to-volume k. |
| Fuel Efficiency
[H] |
Distance is proportional to fuel: find miles per gallon, then scale up. |
| Scaling a Recipe
[H] |
Cups are proportional to servings: multiply by the servings ratio. |
| Shadows and Heights
[H] |
At one moment, height and shadow share a constant ratio for everything. |
| Missing Side of Similar Figures
[H] |
Similar figures scale by one factor: find it, then apply it to any side. |
| Currency Exchange
[H] |
Currency converts by a fixed rate: foreign per dollar times the dollars. |
| Ratios to Counts
[H] |
Scale the ratio: how many groups of the first part make the given count. |
| Markup: Finding the Selling Price
[H] |
A p% markup makes the price (100 + p)% of the cost. |
| Cost, Markup, and Profit
[H] |
Profit is selling price minus cost - build the selling price first. |
| Total Cost with Sales Tax
[H] |
Add the prices first, then tax the subtotal: total is (100 + tax)%. |
| Finding the Sales Tax Rate
[H] |
Tax rate = tax divided by price, times 100. |
| Tip and Total Bill
[H] |
The tip is a percent of the bill; the total is bill plus tip. |
| Coupon or Percent Off?
[H] |
Turn the percent coupon into dollars, then compare the two savings. |
| Base Pay Plus Commission
[H] |
Total pay = fixed base + commission, where commission is a percent of sales. |
| Commission: Finding the Sales
[H] |
Commission is a percent of sales - divide by that percent to recover sales. |
| Simple Interest by the Month
[H] |
Months become years over 12: I = P x r x (months / 12). |
| Comparing Two Interest Offers
[H] |
Same principal and time: the interest gap is P x (rate gap) x time. |
| Tolerance and Acceptable Range
[H] |
Within p% of a target reaches from target minus p% up to target plus p%. |
| Increase Then Decrease
[H] |
Apply each change in turn: the second percent acts on the NEW amount. |
| What Percent More Than
[H] |
Compare to the 'than' quantity: divide the gap by B, not by A. |
| Percent Of vs Percent More Than
[H] |
'Percent of' is a part; 'percent more than' adds that part to the whole. |
| Finding the Whole from a Part
[H] |
If a part is p% of the whole, divide the part by p% to rebuild the whole. |
| Percent of the Remainder
[H] |
The second percent acts on what is LEFT, not on the original amount. |
| Map Scales and Real Distances
[H] |
Real distance = map distance x miles-per-inch; divide to go the other way. |
| Scale Factor Between Two Figures
[H] |
Scale factor = new length divided by matching original length, reduced. |
| Enlarging and Reducing
[H] |
Multiply each dimension by the scale factor - bigger than 1 or smaller. |
| Scale Drawing, Real Area, and Cost
[H] |
Lengths scale by the factor; area scales by its square - then multiply cost. |
| From Real Length to Blueprint
[H] |
To draw a real length, divide it by the meters each drawing centimeter stands for. |
| Scaling a Recipe
[H] |
Multiply the ingredient amount by the scaling factor, fractions included. |
| Fraction of What Is Left
[H] |
Each stage's fraction acts on what remains, not on the original whole. |
| Account Balance with Signed Decimals
[H] |
Deposits add, withdrawals subtract - track the running total to the cent. |
| How Many Servings?
[H] |
Divide the total amount by one serving size - divide by a fraction. |
| All Four Operations with Fractions
[H] |
Do multiply and divide before add and subtract, carrying exact fractions. |
| Combining Decimals and Fractions
[H] |
Rewrite decimals as fractions (or the reverse) so every part matches. |
| Distance from a Fractional Rate
[H] |
Distance = rate times time, even when both are mixed numbers. |
| Two-Step Equations with Fraction Coefficients
[H] |
Undo the addition first, then undo a fractional coefficient by dividing. |
| Consecutive-Integer Puzzles
[H] |
Let the first integer be n; the next ones step up by 1 (or by 2 if even/odd). |
| Age Word Problems
[H] |
Let the unknown age be x; write each person's age from it, then sum. |
| Equations That Need Distribution
[H] |
Distribute the outside factor first, then solve the two-step equation. |
| Perimeter Inequalities
[H] |
Write 2(L + w) <= P, solve for w, then take the greatest whole number. |
| When Do Two Plans Cost the Same?
[H] |
Set the two cost expressions equal and solve for the number of steps. |
| Expand Two Products and Combine
[H] |
Distribute each factor, then join the x-terms and the constants. |
| Distributing a Subtracted Term
[H] |
A minus in front of a parenthesis flips the sign of every term inside. |
| Factoring Out a Negative GCF
[H] |
Pull out the negative of the common factor; the inside signs then flip. |
| Writing a Perimeter Expression
[H] |
Add the side expressions (double them for a rectangle), then combine. |
| Expand, Then Evaluate
[H] |
Simplify the expression first, then substitute the value of x. |
| Complementary Angles with Algebra
[H] |
Complementary angles sum to 90; add the expressions and solve for x. |
| Supplementary Angles with Algebra
[H] |
Supplementary angles sum to 180; set the total to 180 and solve. |
| Vertical Angles with Algebra
[H] |
Vertical angles are equal; set the two expressions equal and solve. |
| Adjacent Angles That Add Up
[H] |
A ray inside an angle splits it: the two parts sum to the whole. |
| Angles Given as a Ratio
[H] |
Split the total (90, 180) into ratio parts to size each angle. |
| Three Angles on a Straight Line
[H] |
Angles that make a straight line add to 180; sum them and solve. |
| From Circumference to Area
[H] |
Circumference 2(pi)r gives the radius; then area is (pi)r squared. |
| Circumference with pi = 3.14
[H] |
Circumference = pi times diameter; use 3.14 for a decimal answer. |
| Area with pi = 3.14
[H] |
Area = pi times the radius squared; square the radius before multiplying. |
| Quarter Circles
[H] |
A quarter circle is one fourth of a full circle's area and arc. |
| A Circle Inside a Square
[H] |
An inscribed circle's diameter equals the square's side; subtract areas. |
| Rectangle Plus a Semicircle
[H] |
Add the rectangle's area to the semicircle's half-circle area. |
| Perimeter of a Composite Figure
[H] |
Trace the outline: add only the edges that lie on the boundary. |
| Area After Cutting a Hole
[H] |
Remaining area = whole shape's area minus the cut-out region's area. |
| Area of an L-Shaped Figure
[H] |
Split the L into two rectangles and add, or take a rectangle minus a corner. |
| Surface Area of a Triangular Prism
[H] |
Add the two triangular ends to the three rectangular side faces. |
| Prism Volume and Capacity
[H] |
Volume is length times width times height; 1000 cm^3 makes 1 liter. |
| Volume of a Trapezoidal Prism
[H] |
Volume = (area of the trapezoid cross-section) times the prism's length. |
| Surface Area for Cost
[H] |
Find the surface area of all six faces, then multiply by the unit cost. |
| Missing Dimension from Volume
[H] |
Divide the volume by the product of the known dimensions. |
| Probability as a Fraction, Decimal, and Percent
[H] |
The same probability can be written as a fraction, a decimal, or a percent. |
| The Complement of an Event
[H] |
The complement's probability is 1 minus the event's probability. |
| Describing How Likely an Event Is
[H] |
A probability near 0 is unlikely, near 1 is likely, and 1/2 is even. |
| Probability of a Number Property
[H] |
Count how many numbers have the property, then divide by the total. |
| Probabilities That Sum to One
[H] |
The probabilities of all outcomes add to exactly one. |
| Probability of One Outcome or Another
[H] |
For a single draw, add the favorable counts of the separate outcomes. |
| Relative Frequency from a Table
[H] |
Relative frequency is a category's count divided by the total count. |
| Trials Needed for an Expected Count
[H] |
Expected count equals probability times trials; solve for the trials. |
| Long-Run Relative Frequency
[M] |
Over many trials the relative frequency settles near the true probability. |
| Deciding Whether a Result Looks Fair
[H] |
Compare the observed count with the expected count; a large gap suggests bias. |
| Listing the Complete Sample Space
[H] |
The sample space lists every equally likely combined outcome exactly once. |
| Counting Outcomes with a Tree Diagram
[H] |
Multiply the number of branches at each stage of the tree. |
| Probability Along a Tree Diagram
[H] |
Multiply the probabilities along the branches of a single path. |
| Three Independent Events
[H] |
For independent events, multiply all their probabilities together. |
| Probability With Replacement
[H] |
Replacing the first draw keeps both draws identical and independent. |
| Probability from a Two-Way Table
[H] |
Divide the count in the chosen cells by the grand total of the table. |
| Probability of a Dice-Sum Event
[H] |
Count the ordered pairs whose sum fits the event out of 36 outcomes. |
| Probability of a Coin-Flip Sequence
[H] |
Each flip has two outcomes, so m flips give two-to-the-m equally likely results. |
| 'And' versus 'Or' in Probability
[H] |
Multiply probabilities for 'both'; add them for 'one or the other'. |
| Designing a Simulation
[H] |
Assign a share of the random numbers equal to the event's probability. |
| Counting Successes in Simulated Trials
[H] |
Count the random digits that fall in the success range, then divide by the trials. |
| Simulating a Compound Event
[H] |
A compound trial succeeds only when every part meets its success rule. |
| Identifying the Population and the Sample
[M] |
The population is the whole group studied; the sample is the part examined. |
| When a Sample Supports a Conclusion
[H] |
A random sample supports an approximate, hedged claim about the population. |
| Which Sample Is More Reliable?
[M] |
Among random samples, a larger sample gives a more reliable estimate. |
| Estimating a Population Percent
[H] |
The sample percent estimates the population percent. |
| The Spread of Estimates from Samples
[H] |
Scale each sample to a population estimate; the estimates spread around the truth. |
| Estimating a Total from a Sample Plot
[H] |
Scale the sample plot's count up by how many plots fill the whole area. |
| Comparing Two Populations by Median
[H] |
Find each data set's median, then the gap between the two medians. |
| Comparing Two Populations by Range
[H] |
The range is largest minus smallest; compare the two ranges. |
| Comparing Spread with the IQR
[H] |
The IQR is Q3 minus Q1; compare the interquartile ranges of the two sets. |
| Comparing Two Populations by MAD
[H] |
Compute each data set's MAD, then the difference between them. |
| Comparing Two Box Plots
[H] |
Read center from the median line and spread from the box width (IQR). |
| Informally Comparing Two Populations
[H] |
A difference in means matters when it is at least about twice the MAD. |
| The Mean from a Dot Plot
[H] |
Each dot is one data value; sum all the values and divide by the count. |
| How an Outlier Affects the Mean
[H] |
Compute the mean with and without the outlier; the difference is its effect. |
| Deriving a Scale, Then Applying It
[H] |
Find how many real units one drawing unit stands for, then scale a new length. |
| Reading an Architect's Floor Plan
[H] |
Multiply the drawing length by the number of real feet each drawn inch represents. |
| Area of a Prism's Cross-Section
[H] |
A plane slices a prism into a rectangle whose sides are two of the prism's edges. |
| Cross-Sections Parallel to a Base
[H] |
A plane parallel to the base of a prism or pyramid reveals that base's polygon. |
| Fencing a Circular Region
[H] |
Find the circumference, then multiply by the cost per unit of length. |
| Covering a Circular Area
[H] |
Find the circle's area, then divide by how much each unit of material covers. |
| Area of a Sector
[H] |
A sector's area is the fraction (angle over 360) of the full circle's area. |
| Arc Length of a Sector
[H] |
An arc is the fraction (angle over 360) of the full circumference. |
| Radius from a Numeric Circumference
[H] |
Undo circumference = 2 times pi times r by dividing by 2 pi. |
| Area of a Stadium Region
[H] |
Add the rectangle to the full circle formed by the two semicircular ends. |
| A Square with Rounded Corners
[H] |
Four equal quarter-circle corners combine into one full circle to subtract. |
| A Plate with Drilled Holes
[H] |
Subtract the area of every circular hole from the whole rectangle. |
| A Rectangle with a Semicircular Notch
[H] |
Subtract half a circle's area from the rectangle for a semicircular cut-out. |
| From x to the Angle Measure
[H] |
Solve the relationship for x, then substitute to report the angle's degree measure. |
| Angle Bisector Relationships
[H] |
A bisector splits an angle into two equal halves; set the halves equal to solve. |
| Complement and Supplement Puzzles
[H] |
Write the complement as 90 minus the angle and the supplement as 180 minus it, then solve. |
| Vertical and Linear-Pair Combined
[H] |
Use the straight-line pair to find x, then vertical angles are equal. |
| Angles Around a Point
[H] |
Angles that fill the space around a point sum to 360 degrees. |
| Angles Split in a Ratio
[H] |
Divide the total by the sum of the ratio parts, then multiply by the larger part. |
| Filling a Tank
[H] |
Divide the tank's volume by the fill rate to get the filling time. |
| Water Depth in a Tank
[H] |
Divide the water's volume by the base area to find its depth. |
| Coating a Box
[H] |
Find the total surface area of all six faces, then divide by the coverage per can. |
| Volume of an L-Shaped Solid
[H] |
Find the L-shaped base area, then multiply by the uniform height. |
| Volume with Unit Conversion
[H] |
Convert all edge lengths to one unit before multiplying, then convert the volume. |
| Missing Height from Surface Area
[H] |
Subtract the two base areas from the total, then divide by the side perimeter. |
| Mass of a Solid Block
[H] |
Multiply the block's volume by its density to get its mass. |
| A Partly Filled Tank
[H] |
The water's height is the given fraction of the tank height; then use base times height. |
| Volume of a Cylinder
[H] |
A cylinder's volume is pi times the radius squared times the height. |
| Volume of a Cylinder in Context
[H] |
Compute pi r squared h with pi = 22/7 for a numeric volume. |
| Cylinder Volume from a Diameter
[H] |
Halve the diameter to get the radius before squaring it in the volume formula. |
| Surface Area of a Cylinder
[H] |
Total surface area is two circular ends plus the curved side: 2 pi r squared plus 2 pi r h. |
| Surface Area of a Cylinder in Context
[H] |
Add both circular ends and the curved side with pi = 22/7 for a numeric total. |
| The Label Around a Can
[H] |
The curved side unrolls into a rectangle of area 2 pi r times the height. |
| Capacity of a Cylindrical Drum
[H] |
Find the cubic-centimeter volume, then divide by 1000 to convert to liters. |
| Height of a Cylinder from Its Volume
[H] |
Divide the volume's pi-coefficient by the radius squared to recover the height. |
| Water Depth in a Cylinder
[H] |
Divide the water's volume by the circular base area to find its depth. |
| Material Cost for a Cylinder
[H] |
Find the total surface area, then multiply by the cost per unit of area. |