311 core topics
+ 14 prerequisite topics taught
as needed · approximately 96 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. |
| Integer Exponents & Laws
[M] |
Add exponents to multiply, subtract to divide, multiply to raise a power. |
| Scientific Notation
[E] |
a × 10ⁿ with 1 ≤ a < 10. |
| Square & Cube Roots
[E] |
√ undoes squaring; ∛ undoes cubing. |
| Rational vs Irrational Numbers
[E] |
Irrational numbers never end or repeat. |
| Functions: Inputs & Outputs
[E] |
One input, exactly one output. |
| Rate of Change & Initial Value
[M] |
Slope is the rate of change; the y-intercept is the initial value. |
| Linear vs Nonlinear Functions
[E] |
Linear graphs are straight lines with constant slope. |
| Systems of Linear Equations
[M] |
The solution is where the two lines intersect. |
| The Pythagorean Theorem
[M] |
a² + b² = c² for right triangles. |
| Distance on the Coordinate Plane
[M] |
Horizontal and vertical gaps are the legs of a right triangle. |
| Transformations
[M] |
Translations slide, reflections flip, rotations turn. |
| Dilations & Similarity
[M] |
Dilations scale from a center; similar figures share shape. |
| Volume of Cylinders, Cones & Spheres
[M] |
V = πr²h, ⅓πr²h, and 4⁄3·πr³. |
| Scatter Plots & Association
[E] |
Positive, negative, or no association between two variables. |
| 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. |
| Building y = mx + b from a Story
[M] |
The per-unit amount is m; the one-time starting amount is b. |
| Using a Linear Model
[M] |
Substitute into y = mx + b - or solve backward for x. |
| Interpreting the Slope
[E] |
Slope answers: how much does y change for each one unit of x? |
| Interpreting the y-Intercept
[E] |
The intercept is the value of y when x = 0 - the starting amount. |
| A Model from Two Data Points
[M] |
Rate first (Δy over Δx), then walk back to x = 0. |
| Comparing Rates Across Forms
[M] |
Put both functions' rates in the same form, then compare numbers. |
| Comparing Linear Functions' Values
[M] |
Rebuild the table's equation, then evaluate both functions. |
| Variables on Both Sides in Context
[M] |
Model each side, set them equal, and collect the variable on one side. |
| One, None, or Infinitely Many Solutions
[M] |
Compare the x-coefficients first, then the constants. |
| Two-Way Tables
[E] |
Rows are one category, columns the other; totals come from adding. |
| Relative Frequency in a Table
[M] |
Divide a cell by its row total to see the pattern, not just the count. |
| Predicting with a Trend Line
[M] |
A fitted line turns a cloud of points into predictions. |
| Adding Integers
[M] |
Same signs add and keep the sign; different signs subtract. |
| Subtracting Integers
[M] |
Subtracting is adding the opposite: a − b = a + (−b). |
| Multiplying Integers
[M] |
Like signs give a positive product; unlike signs give a negative one. |
| Dividing Integers
[M] |
The sign rule for division is the same as for multiplication. |
| Order of Operations
[M] |
PEMDAS: parentheses, exponents, ×/÷, then +/−, left to right. |
| Greatest Common Factor
[M] |
The GCF is the largest number dividing both values evenly. |
| Least Common Multiple
[M] |
The LCM is the smallest number both values divide into. |
| Simplifying Fractions with the GCF
[M] |
Divide numerator and denominator by their GCF to reach lowest terms. |
| Fractions to Decimals
[M] |
A fraction is a division: n/d = n ÷ d. |
| Decimals to Fractions
[M] |
Put the decimal over its place value, then simplify. |
| Fractions to Percents
[M] |
Percent means per hundred: multiply the fraction by 100. |
| Percent of a Number
[M] |
p% of n = (p/100)·n. |
| Percent Increase and Decrease
[M] |
Increase multiplies by (1 + p/100); decrease by (1 − p/100). |
| Simplifying Ratios
[M] |
Divide both parts of a ratio by their GCF. |
| Solving Proportions
[M] |
Cross-multiply: a/b = c/x means a·x = b·c. |
| Unit Rates
[M] |
A unit rate is the amount per single unit - divide total by quantity. |
| Evaluating Powers
[M] |
aⁿ means a multiplied by itself n times. |
| The Product Rule for Exponents
[M] |
Multiplying powers of the same base adds the exponents. |
| Negative Exponents
[M] |
A negative exponent means reciprocal: a⁻ⁿ = 1/aⁿ. |
| Square Roots
[M] |
√x asks which non-negative number squared gives x. |
| Cube Roots
[M] |
∛x asks which number cubed gives x. |
| Estimating Square Roots
[M] |
Trap a non-perfect square between the nearest perfect squares. |
| One-Step Equations
[M] |
Undo the single operation with its inverse on both sides. |
| Two-Step Equations
[M] |
Undo addition/subtraction first, then the multiplication. |
| Two-Step Equations with Negatives
[M] |
Dividing by a negative coefficient flips the sign of the result. |
| Solving Inequalities
[M] |
Solve like an equation to find the boundary value. |
| Flipping the Inequality Sign
[M] |
Multiplying or dividing by a negative reverses the inequality. |
| Absolute Value
[M] |
|x| is the distance from 0 - always non-negative. |
| Absolute Value Equations
[M] |
|x − a| = k means x − a = k or x − a = −k, giving two solutions. |
| Scientific to Standard Form
[M] |
A positive exponent moves the point right; a negative one moves it left. |
| Standard to Scientific Form
[M] |
Place the decimal after the first nonzero digit; count the shift. |
| Multiplying in Scientific Notation
[M] |
Multiply the coefficients and add the exponents. |
| Quadrants of the Coordinate Plane
[M] |
The signs of (x, y) place a point in one of four quadrants. |
| Distance on the Coordinate Plane
[M] |
Points on a horizontal or vertical line are |difference| apart. |
| Perimeter of a Rectangle
[M] |
Perimeter is the distance around: P = 2(l + w). |
| Area of a Triangle
[M] |
A = ½ · base · height. |
| Volume of a Rectangular Prism
[M] |
V = length · width · height. |
| The Mean (Average)
[M] |
Add the values and divide by how many there are. |
| The Median
[M] |
Sort the data; the median is the middle value (or the mean of the two). |
| The Mode
[M] |
The mode is the value that appears most often. |
| The Quotient Rule for Exponents
[H] |
Dividing powers of the same base subtracts the exponents. |
| Power of a Power
[H] |
Raising a power to a power multiplies the exponents. |
| Zero and Negative Exponents Together
[H] |
Anything nonzero to the 0 power is 1; a negative power is a reciprocal. |
| Combining the Exponent Laws
[H] |
Chain the rules: powers multiply, products add, quotients subtract. |
| Estimating Cube Roots
[H] |
Trap the number between the two nearest perfect cubes. |
| Square Roots to the Nearest Whole Number
[H] |
Compare the number to the perfect squares on each side, then pick the closer root. |
| Finding the Whole from a Percent
[H] |
If p% of the whole is known, divide by p and scale to 100%. |
| What Percent Is It?
[M] |
Divide part by whole, then multiply by 100 - percents can pass 100. |
| Ratio Shares of a Total
[H] |
Add the ratio parts, divide the total by that sum, then scale each part. |
| Discount and Tax Pipelines
[H] |
Apply each percent change as its own multiplier, one after the other. |
| Scaling a Recipe Both Ways
[H] |
Set up cups per serving and scale in whichever direction is asked. |
| Comparing Unit Prices
[H] |
Divide each price by its count, then compare the per-item rates. |
| Equations with Distribution
[H] |
Distribute first (or divide both sides), then solve the two-step leftover. |
| Variables on Both Sides
[H] |
Collect the x-terms on one side, the numbers on the other, then divide. |
| Equations with Fractions
[H] |
Clear the fraction by multiplying both sides by the denominator. |
| Distribute, Combine, Solve
[H] |
Expand every set of parentheses, merge the like terms, then solve. |
| Solving a Formula for a Variable
[H] |
Treat the other letters as numbers and undo operations step by step. |
| Two-Variable Equations at a Point
[H] |
Substitute the known value, then solve the equation that remains. |
| Slope from Two Points
[H] |
Slope is rise over run: the change in y divided by the change in x. |
| Midpoint of a Segment
[H] |
Average the x-coordinates and average the y-coordinates. |
| Reflecting Points Across the Axes
[M] |
x-axis reflections flip the y sign; y-axis reflections flip the x sign. |
| Pythagorean Theorem: the Hypotenuse
[H] |
Square the legs, add, and square-root: c = sqrt(a^2 + b^2). |
| Pythagorean Theorem: a Missing Leg
[H] |
A leg is the square root of the hypotenuse squared minus the other leg squared. |
| Pythagorean Theorem in Word Problems
[H] |
Draw the right triangle hiding in the story, then apply a^2 + b^2 = c^2. |
| Mean Absolute Deviation from Scratch
[H] |
Find the mean, then average the distances of the data from it. |
| Quartiles and the IQR
[H] |
Order the data, split it at the median, and take the medians of the halves. |
| How an Outlier Moves the Mean
[H] |
Compare the mean with and without the extreme value. |
| Missing-Digit Divisibility Puzzles
[H] |
Use the digit-sum tests: divisible by 3 or 9 when the digit sum is. |
| Spotting Multiples of 6 and 4
[H] |
Divisible by 6 means even AND digit-sum divisible by 3; by 4, check the last two digits. |
| Reading a Prime Factorization
[M] |
Factor down to primes, then count how many times each prime appears. |
| Counting Factors from Prime Powers
[H] |
Add 1 to each exponent in the prime factorization and multiply. |
| LCM Scheduling Problems
[H] |
Events that repeat every a and b units line up again at LCM(a, b). |
| Modeling Integer Addition with Arrows
[H] |
Adding a positive number slides right on the number line; adding a negative number slides left. |
| Subtraction as a Directed Move
[H] |
Subtracting a positive number moves left; subtracting a negative number moves right. |
| Distance Between Two Rational Points
[H] |
The distance between two numbers is the absolute value of their difference. |
| Finding an Endpoint from a Distance
[H] |
A known point and a distance give two possible endpoints, one on each side. |
| Net Change from a Sequence of Signed Moves
[H] |
Add the signed changes to a starting value to get the final position. |
| Absolute Value Bars Inside a Chain
[H] |
Absolute value bars group like parentheses: finish the inside, then take the size. |
| Tolerance Ranges with Absolute Value
[H] |
The statement |x - c| at most t describes every value within t units of the center c. |
| Absolute Deviation from a Target
[H] |
The absolute deviation of a measurement is the distance between it and the target, ignoring direction. |
| Greatest Value Versus Greatest Absolute Value
[H] |
Order compares position on the number line, while absolute value compares distance from zero. |
| Negative Bases Versus Negated Powers
[H] |
Parentheses decide whether the minus sign is part of the base or applied after the power. |
| Nested Grouping Symbols with Negatives
[H] |
Work outward from the innermost grouping symbol, keeping every sign attached. |
| The Fraction Bar as a Grouping Symbol
[H] |
A fraction bar acts as parentheses around its numerator and around its denominator. |
| Placing Parentheses to Force a Value
[H] |
Parentheses override the default order of operations, so different placements give different values. |
| Adding and Subtracting Across Forms
[H] |
Convert both numbers to one form, either all fractions or all decimals, before adding or subtracting. |
| Multiplying and Dividing Across Forms
[H] |
Rewrite each decimal as a fraction over a power of ten, then multiply or divide fractions. |
| Signed Mixed Numbers and Decimals
[H] |
Convert signed mixed numbers to improper fractions before combining them with decimals. |
| Repeating Decimals as Fractions
[H] |
A repeating block of n digits becomes that block over n nines, then reduce. |
| Which Fractions Terminate
[H] |
A reduced fraction terminates exactly when its denominator has no prime factors besides 2 and 5. |
| Completing a Fraction, Decimal and Percent Table
[H] |
Every rational number has one fraction form, one decimal form and one percent form; each converts to the others. |
| Percents Above 100 and Below 1
[H] |
A percent is hundredths, so dividing by 100 converts any percent to a decimal, however large or small. |
| Ordering Values Given in Different Forms
[H] |
Rewrite every value in one common form before comparing them. |
| Comparing Negative Fractions
[H] |
Among negatives the larger the absolute value, the smaller the number. |
| Locating a Number Between Two Others
[H] |
The midpoint of two numbers is their average; a fractional part of the way is the start plus that fraction of the gap. |
| Naming the Property That Justifies a Step
[H] |
Commutative changes order, associative changes grouping, and distributive spreads a factor across a sum. |
| Regrouping to Compute Mentally
[H] |
The commutative and associative properties allow terms or factors to be paired into friendly numbers. |
| Distributing Over a Friendly Split
[H] |
Split an awkward factor into a round number plus or minus a small number, then distribute. |
| Factoring a Common Factor from a Sum
[H] |
The greatest common factor of the terms can be pulled outside the sum, leaving the quotients inside. |
| Translating a Sentence into a Numeric Expression
[H] |
Phrases such as sum, difference, product, quotient and square name the operation and its grouping. |
| Long Chains with Signs, Powers and Roots
[H] |
Evaluate roots and powers first, then multiply and divide left to right, then add and subtract. |
| The Largest Prime Factor
[H] |
Strip out small prime factors one at a time; the last factor left is the largest prime factor. |
| Smallest Multiplier for a Perfect Square
[H] |
A perfect square has every exponent even in its prime factorization, so supply exactly the primes with odd exponents. |
| GCF and LCM from Prime Factorizations
[H] |
Take the lower exponent of each shared prime for the GCF and the higher exponent of every prime for the LCM. |
| GCF Packaging and Bundling Problems
[H] |
Splitting several totals into the largest number of identical groups uses the greatest common factor. |
| LCM in Tiling and Gear Problems
[H] |
The first length or count that both cycles reach together is their least common multiple. |
| Deciding Between the GCF and the LCM
[H] |
Splitting things into equal groups calls for the GCF, while events meeting again calls for the LCM. |
| Divisibility by 9 and by 11
[H] |
The remainder on division by 9 matches the digit sum, and the remainder on division by 11 matches the alternating digit sum. |
| Remainder Reasoning Without Dividing
[H] |
Remainders add and multiply just as the numbers do, then reduce below the divisor. |
| What Must Divide a Sum or a Product
[H] |
A common factor of both terms divides their sum, and the factors of both numbers together divide their product. |
| Square Roots of Perfect-Square Fractions and Decimals
[H] |
The square root of a quotient is the quotient of the square roots, and a decimal square root has half as many decimal places. |
| Solving x Squared Equals a Perfect Square
[H] |
Squaring destroys sign information, so x squared equal to a positive number has one positive and one negative solution. |
| Recognizing Perfect Squares
[H] |
A perfect square is the square of a whole number, so test candidates against the nearby squares. |
| Rounding to a Named Place
[H] |
Look at the digit one place to the right of the rounding place and round up when it is 5 or more. |
| Estimating with Rounded and Compatible Numbers
[H] |
Round each quantity to a friendly place first, then compute exactly with the rounded values. |
| Deciding Whether an Estimate Lands High or Low
[H] |
Rounding every factor up overestimates a product and rounding every factor down underestimates it. |
| Unit Rates from Complex Fractions
[H] |
A rate with fractional quantities is still amount divided by time, so divide the two fractions. |
| The Two Unit Rates of One Relationship
[H] |
Every rate has two unit rates that are reciprocals, and the question decides which one to use. |
| Best Buy Among Three Packages
[H] |
Compare packages by dividing price by size, converting every size to the same unit first. |
| Buying a Partial Amount at a Unit Price
[H] |
Cost equals unit price times amount, and amount equals money divided by unit price. |
| Fuel Economy and Trip Cost
[H] |
Divide miles by gallons for fuel economy, then divide miles by that rate to find the gallons a trip needs. |
| Setting Up a Proportion from a Story
[H] |
A proportion is correct when matching quantities occupy matching positions in both ratios. |
| Proportions from Context: Exact Answers
[H] |
Cross multiply to solve a proportion and keep the exact fraction, even when the unknown sits in the denominator. |
| Missing Side in Similar Figures
[H] |
Corresponding sides of similar figures share one scale factor, and the perimeter scales by that same factor. |
| Indirect Measurement with Shadows
[H] |
At one moment every upright object and its shadow form similar triangles, so height over shadow is constant. |
| Estimating a Population by Proportion
[H] |
The tagged fraction of a fair sample estimates the tagged fraction of the whole population. |
| Finding the Scale Factor
[H] |
The scale factor is actual length divided by drawing length, computed after both are in the same unit. |
| Map Scale in Mixed Units
[H] |
A map scale is a rate: multiply map distance by it to get real distance, and divide to go back. |
| Actual Dimensions from a Scale Drawing
[H] |
Multiply each drawing length by the scale to get the real length, and divide to go from real back to drawing. |
| How Area Changes Under a Scale Factor
[H] |
Scaling every length by k multiplies area by k squared, because both dimensions are scaled. |
| Fractional and Decimal Percents
[H] |
A percent is hundredths, so a fractional percent such as 0.4% means 0.004 of the whole. |
| Percents Greater Than 100
[H] |
A percent above 100 gives a result larger than the whole, since the multiplier exceeds 1. |
| A Percent of a Percent
[H] |
Taking a percent of a percent multiplies the two decimal multipliers, giving a percent of the original whole. |
| Recovering the Original Amount
[H] |
A percent change multiplies the original, so dividing the new amount by that multiplier recovers the original. |
| Percent Changes as Multipliers
[H] |
Every percent change is one multiplication: increases multiply by 1 plus the rate, decreases by 1 minus it. |
| Undoing a Percent Change
[H] |
The percent that reverses a change is measured against the new amount, so it differs from the original percent. |
| Markup from Cost to Selling Price
[H] |
Markup is a percent of the cost, so selling price equals cost times one plus the markup rate. |
| Commission and Base-Plus-Commission Pay
[H] |
Commission is a percent of sales, and base-plus-commission pay is a fixed amount plus that percent. |
| Tip, Total, and Splitting the Bill
[H] |
The tip is a percent of the bill, and an evenly split total divides the after-tip amount by the number of people. |
| Finding the Tax Rate from a Total
[H] |
The tax rate is the tax divided by the pre-tax price, so subtract to find the tax before dividing. |
| Finding the Discount Percent
[H] |
The discount percent is the amount saved divided by the original price, times 100. |
| Simple Interest over Months and Years
[H] |
Simple interest is I = Prt with the time expressed in years, so a month count is divided by twelve. |
| Solving I = Prt for Any Variable
[H] |
Any one of interest, principal, rate, or time can be found by dividing I by the product of the other two. |
| Comparing Two Simple-Interest Offers
[H] |
Comparing accounts or loans requires computing each total separately, since a higher rate need not mean a higher cost. |
| Percent Error
[H] |
Percent error is the size of the difference divided by the accepted value, expressed as a percent. |
| Tolerance Ranges from a Percent
[H] |
A percent tolerance sets an acceptable interval reaching that percent of the target above and below it. |
| Net Change from Successive Percent Changes
[H] |
Successive percent changes multiply their multipliers, so the net change is not the sum of the changes. |
| Does the Order of Percent Changes Matter?
[H] |
Stacked percent changes multiply, so their order never matters, but stacking is not the same as adding the percents. |
| Finding the Missing Second Change
[H] |
Divide the overall multiplier by the known multiplier to recover the missing percent change. |
| Metric Conversions Across Several Steps
[H] |
Each metric prefix step is a factor of ten, so converting multiplies or divides by a power of ten. |
| Customary Unit Conversions
[H] |
Customary conversions use memorized factors such as 12 inches per foot and 16 ounces per pound. |
| Arithmetic in Mixed Units
[H] |
A measurement written in two units becomes a single number once the larger unit is converted to the smaller one. |
| Converting Between Customary and Metric
[H] |
A conversion between systems multiplies by the given factor, or divides by it to go the other way. |
| Square and Cubic Unit Conversions
[H] |
Converting square units squares the length factor and converting cubic units cubes it. |
| Converting a Rate with Dimensional Analysis
[H] |
Multiply a rate by conversion fractions arranged so the unwanted units cancel. |
| Converting Both Units of a Rate
[H] |
When both the top and bottom units change, apply one conversion factor to each part of the rate. |
| Choosing the Conversion Factor
[H] |
The correct conversion factor is the one whose units cancel the unit being replaced. |
| Flooring Cost with a Unit Change
[H] |
Find the area, convert it to the unit the material is priced in, then multiply by the price. |
| Fencing a Garden in Mixed Units
[H] |
Convert every side to one unit, add for the perimeter, then round the count of whole materials upward. |
| Filling a Tank with Rate Conversions
[H] |
Time equals volume divided by rate, using one consistent unit and the net rate when something drains. |
| Writing an Expression from a Phrase
[H] |
Each phrase names an operation and an order; the order decides the expression. |
| Reading an Expression in Words
[H] |
Read the last operation performed, then describe the pieces it joins. |
| A Total Written as One Expression
[H] |
Model each quantity in the same variable, then add and combine like terms. |
| Consecutive Integers as Expressions
[H] |
Name the smallest n; the others are n + 1, n + 2, or n + 2, n + 4 when stepping by two. |
| Evaluating at Negative Values
[H] |
Substitute the value inside parentheses, then follow the order of operations. |
| Two-Variable Expressions at Signed Values
[H] |
Replace every occurrence of each variable, then simplify one operation at a time. |
| Powers of a Negative Value
[H] |
An even power of a negative value is positive; -x squared negates after squaring. |
| Evaluating a Formula at Given Values
[H] |
Substitute every named value into the formula, then finish with the order of operations. |
| Like Terms in Two Variables
[H] |
Only terms with an identical variable part combine; xy is its own kind of term. |
| Distributing Over Negatives
[H] |
A negative factor multiplies every term inside, so every sign inside changes. |
| Subtracting an Entire Expression
[H] |
Subtracting a grouped expression subtracts each of its terms. |
| Expand, Then Combine
[H] |
Expand every set of parentheses first, then gather x-terms and constants separately. |
| A Perimeter as a Simplified Expression
[H] |
Add the side expressions, then combine like terms into one linear expression. |
| One-Step Equations with Rational Coefficients
[H] |
Undo a fractional coefficient by multiplying both sides by its reciprocal. |
| Two-Step Equations with Fractional Coefficients
[H] |
Undo the constant first, then multiply by the reciprocal of the coefficient. |
| Equations with Decimal Coefficients
[H] |
Multiply both sides by a power of ten to clear decimals, or work with them directly. |
| Clearing Denominators with the LCD
[H] |
Multiply every term by the least common denominator to remove all fractions at once. |
| Fractions with the Variable on Both Sides
[H] |
Clear the denominators first, then collect the variable terms on one side. |
| Distributing on Both Sides
[H] |
Expand each side completely, combine like terms, then collect the variable on one side. |
| The Coefficient That Changes Everything
[H] |
Matching variable coefficients give either no solution or every solution, decided by the constants. |
| Building and Solving a Multi-Step Equation
[H] |
Name the unknown, express every other quantity from it, then set the total equal to the given value. |
| Rearranging a Formula, Then Using It
[H] |
Isolate the named variable with inverse operations, then substitute the given values. |
| Choosing the Correct Rearrangement
[H] |
Undo operations in reverse order, and divide by a whole factor rather than one piece of it. |
| Isolating a Variable That Appears Twice
[H] |
Collect every term containing the variable, factor it out, then divide by the bracket. |
| Graphing a One-Variable Inequality
[H] |
The boundary uses an open circle for a strict inequality and a closed circle otherwise; shading follows the solved direction. |
| Multi-Step Inequalities
[H] |
Distribute and combine like terms first; reverse the symbol only when dividing by a negative. |
| Inequalities with the Variable on Both Sides
[H] |
Collect the variable on one side; if the surviving coefficient is negative, dividing reverses the symbol. |
| Counting the Integers in a Solution Set
[H] |
Solve for the endpoints, then count the integers strictly or weakly inside them. |
| Constraint Word Problems
[H] |
Translate at most or at least into an inequality, solve, then round toward the values the context allows. |
| Inequalities with Fractional Coefficients
[H] |
Multiply by the reciprocal; a negative reciprocal reverses the symbol. |
| A Missing Side from the Perimeter
[H] |
Subtract the known sides from the perimeter, then divide by the number of equal sides left. |
| A Missing Dimension from the Area
[H] |
Substitute the known values into the area formula and solve the resulting equation. |
| Working Backward from a Circumference
[H] |
Recover the radius from C = 2 pi r or A = pi r squared, then use it in the other formula. |
| Perimeter of a Composite Figure
[H] |
Trace the actual boundary; a corner cut leaves the perimeter unchanged, while a notch or a hole adds edge. |
| Area in a Cost Application
[H] |
Find the area that is actually covered, then multiply by the price per unit area. |
| Volume of a Triangular Prism
[H] |
Volume of any prism is the area of its cross-section times its length. |
| Surface Area of a Prism
[H] |
Total surface area is twice the end-face area plus the end-face perimeter times the length. |
| Cylinder Volume with pi as 22/7
[H] |
Volume is the circular base area times the height; 22/7 keeps the arithmetic exact. |
| Surface Area of a Cylinder
[H] |
The curved surface unrolls to a rectangle of width 2 pi r and height h; each end adds pi r squared. |
| A Missing Dimension from a Volume
[H] |
Divide the volume by the dimensions you know; recovering a radius needs a square root. |
| Splitting an Angle Sum by a Ratio
[H] |
Add the ratio parts, divide the angle total by that sum, then scale each part. |
| An Angle Described by Its Complement
[H] |
Write the complement as 90 minus x and the supplement as 180 minus x, then solve the sentence as an equation. |
| Triangle Angle Sums with Expressions
[H] |
The three angles of a triangle sum to 180 degrees, so their expressions form one equation. |
| The Exterior Angle of a Triangle
[H] |
An exterior angle equals the sum of the two remote interior angles. |
| Vertical Angles and Linear Pairs in a Chain
[H] |
Vertical angles are equal and a linear pair sums to 180, so one solved value unlocks the rest. |
| Completing a Parallelogram on the Grid
[H] |
In parallelogram ABCD the step from A to B repeats from D to C, so D = A + C - B. |
| Area of a Figure from Its Vertices
[H] |
Coordinate differences give the base and the height, then the usual area formula applies. |
| Distance Along Grid Streets
[H] |
A path along grid lines has length equal to the sum of the horizontal and vertical changes. |