8th Grade Math

321 core topics + 4 prerequisite topics taught as needed · approximately 94 hours of instruction including spaced review

How the course runs

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.

Core curriculum

Arithmetic Foundations · 8 topics

Adding & Subtracting Whole Numbers [E]	Multi-digit addition and subtraction.
Multiplication [E]	Multiplying whole numbers.
Division [E]	Dividing whole numbers.
Order of Operations [M]	Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting [M]	Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing [M]	Sign rules for products and quotients.
Exponents [M]	Repeated multiplication in shorthand.
Square Roots [M]	Undoing a square.

Fractions · 7 topics

Equivalent Fractions [M]	Different fractions can name the same amount.
Simplifying Fractions [M]	Reducing a fraction to lowest terms.
Adding Fractions (Like Denominators) [M]	Same-denominator addition.
Adding Fractions (Unlike Denominators) [M]	Rewrite over a common denominator first.
Multiplying Fractions [M]	Multiply straight across.
Dividing Fractions [M]	Multiply by the reciprocal.
Mixed Numbers & Improper Fractions [M]	Converting between forms.

Decimals, Percents & Ratios · 5 topics

Decimal Addition & Subtraction [M]	Line up the decimal points.
Fractions ↔ Decimals [M]	Converting between the two notations.
Percent of a Number [M]	Percent means per hundred.
Percent Increase & Decrease [M]	Applying a percent change to a quantity.
Ratios & Proportions [M]	Two quantities that scale together.

Expressions & Equations · 7 topics

Evaluating Expressions [M]	Substituting a value for a variable.
Combining Like Terms [M]	Adding the coefficients of matching variable parts.
The Distributive Property [M]	Multiplying across a sum.
One-Step Equations [M]	Undoing a single operation.
Two-Step Equations [M]	Undo addition/subtraction first, then multiplication.
Multi-Step Equations [H]	Equations needing distribution or variables on both sides.
Linear Inequalities [M]	Solving with <, >, ≤, ≥.

Linear Functions · 7 topics

The Coordinate Plane [M]	Locating points with (x, y) pairs.
Slope of a Line [M]	Rise over run between two points.
Slope-Intercept Form [M]	y = mx + b describes a whole line.
Finding a Line from Points [H]	Reconstructing y = mx + b from data.
Systems of Equations (Substitution) [H]	Two equations, two unknowns.
Elimination: A First Look [H]	Add or subtract equations so one variable cancels.
Systems: Word Problems [H]	Translating two facts into two equations.

Ratios, Data & Geometry (Middle School) · 16 topics

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.

Grade 7: Proportions, Geometry & Statistics · 10 topics

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.

Grade 8: Functions, Exponents & Geometry · 14 topics

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.

Grade 7: Rational Numbers & Multi-Step Problems · 12 topics

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.

Grade 8: Linear Modeling & Data · 12 topics

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.

Grade 7: Geometry Applications · 10 topics

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.

Grade 8: Functions & Bivariate Data in Depth · 13 topics

Comparing Rates in Different Forms [M]	Pull each function's slope out of its own form before comparing.
Comparing Initial Values in Different Forms [M]	The initial value is the output at x = 0 - recover it from any form.
Comparing Outputs at a Chosen Input [M]	A steeper function is not always the larger one - evaluate both at x.
Building a Function from a Description [M]	The repeated per-unit amount is the rate; the one-time amount is b.
Building a Function from Two Points [M]	Slope from the two points first, then step back to x = 0 for b.
Interpreting a Model's Output [M]	f(input) = output: the answer carries y's units, at that one input.
Linear or Nonlinear from a Table [E]	Equal x-steps with equal y-jumps means linear; changing jumps means not.
Increasing, Decreasing, or Constant [E]	Rising left-to-right is increasing; falling is decreasing; flat is constant.
Matching a Story to a Graph's Shape [M]	Steady rate gives a line; a changing rate gives a curve.
Reading Association in a Scatter Plot [E]	Rising cloud is positive, falling is negative, shapeless is none.
Clusters & Outliers in a Scatter Plot [E]	Clusters are bunched groups; an outlier stands alone, far from the rest.
Predicting with a Line of Best Fit [M]	Substitute into the line to predict; its slope is the per-unit change.
Relative Frequencies in a Two-Way Table [M]	Divide a cell by a total - row, column, or grand - for a relative frequency.

Grade 8: Geometry, Roots & Scientific Notation · 13 topics

The Pythagorean Theorem in Applications [M]	Ladders, screens, wires, and ramps are right triangles in disguise.
The Converse: Testing for a Right Angle [M]	Compare a² + b² to c² to decide if a triangle is right, acute, or obtuse.
Coordinate Distance from Triples [M]	The horizontal and vertical gaps are legs; a triple gives the exact length.
Square Roots of Larger Perfect Squares [E]	Every perfect square has a whole-number square root - even the big ones.
Cube Roots of Perfect Cubes [M]	A cube root undoes cubing: ∛n is the number whose cube is n.
Estimating Irrational Square Roots [M]	Trap an irrational root between the perfect squares on either side.
Classifying Rational vs Irrational [E]	Spot the odd one out: which number's decimal never ends or repeats.
Multiplying in Scientific Notation [M]	Multiply the coefficients, add the exponents, then re-normalize.
Dividing in Scientific Notation [M]	Divide the coefficients, subtract the exponents.
Adding with the Same Power of Ten [M]	Matching powers of ten factor out - just add the coefficients.
Cylinder Volume as a Coefficient of π [M]	V = πr²h - report the coefficient of π and keep the answer exact.
Cone Volume as a Coefficient of π [M]	A cone is a third of its cylinder: V = ⅓πr²h.
Sphere Volume as a Coefficient of π [M]	V = 4⁄3·πr³ for a full sphere; half that for a hemisphere.

8th Grade Math - Deep II · 15 topics

Multi-Step Linear Equations [M]	Collect the variable on one side and the numbers on the other, then divide.
Equations with the Distributive Property [M]	Distribute first to clear the parentheses, then solve the linear equation.
Solving Systems by Substitution [M]	Set the two expressions for y equal, solve for x, then back-substitute.
Solving Systems by Elimination [M]	Scale the equations so one variable cancels when you add or subtract them.
Systems Word Problems [M]	Name two unknowns, write two equations, and solve the system.
Comparing Quantities in Scientific Notation [M]	Divide the leading numbers and subtract the exponents of ten.
Zero and Negative Exponents [M]	A zero exponent is 1; a negative exponent means one over the positive power.
Solving x² = p and x³ = p [M]	Undo a square with a square root and a cube with a cube root.
Angles from Parallel Lines and a Transversal [M]	Corresponding, alternate, and vertical angles are equal; same-side add to 180.
Triangle Angle Sum and Exterior Angle [M]	A triangle's angles total 180°; an exterior angle equals the two remote ones.
Interior Angles of Polygons [M]	The interior angles of an n-gon sum to (n − 2) × 180°.
The Pythagorean Theorem in Three Dimensions [M]	The space diagonal of a box is √(length² + width² + height²).
Finding a Missing Dimension from Volume [M]	Substitute into the volume formula, cancel π, and solve for what's missing.
Repeating Decimals as Fractions [M]	A single repeating digit is that digit over 9; two digits, over 99.
Approximating Irrational Square Roots [M]	Trap √n between the two perfect squares that surround n.

8th Grade Math - Deep III · 23 topics

Checking a Solution of a System [H]	A solution of a system must satisfy every equation, not just one.
How Many Solutions Does a System Have? [H]	Different slopes cross once; equal slopes are parallel or the same line.
Substitution with an Isolated Variable [H]	When one equation already isolates a variable, drop it into the other.
Setting Up a System from a Story [H]	Translate each fact in the story into its own equation before solving.
Making a System Special [H]	Match the slopes for no solution; match the whole equation for infinitely many.
Is the Relation a Function? [M]	A function gives each input exactly one output; repeated outputs are fine.
Rate of Change from a Table [H]	Divide the change in y by the change in x between any two rows.
Initial Value from a Table [H]	Find the rate first, then step the table back to x = 0.
Comparing Functions in Different Forms [H]	Extract slope and intercept from each form, then compare like with like.
Writing y = mx + b from a Table [H]	The rate of change is m; back up to x = 0 for b; then write y = mx + b.
Scientific Notation for Small Numbers [H]	A negative power of ten moves the decimal point to the left.
Fixing an Improper Coefficient [H]	Slide the coefficient into [1, 10) and adjust the exponent to compensate.
Scientific Notation in Context [H]	Very large counts of very small things: multiply coefficients, add exponents.
The Slope of a Line of Fit [H]	Two points on the fit line give its slope: rise over run.
Using a Fit-Line Equation Both Ways [H]	Substitute x to predict y; solve backward when y is what you know.
Spotting an Outlier in a Scatter Plot [H]	The outlier sits far from the line the other points follow.
Completing a Two-Way Table [H]	Every row and column must add up to its total - use that to fill gaps.
Comparing Groups with a Two-Way Table [H]	Compare each group's fraction of its own total, not the raw counts.
Rotations and Rotation Sequences [H]	90° ccw sends (x, y) to (-y, x); 180° sends it to (-x, -y).
Naming the Transformation [H]	Read the coordinate change: which rule turns P into P'?
Exterior-Angle Chases [H]	An exterior angle equals the sum of the two remote interior angles.
Parallel-Line Angle Chases [H]	Chain equal and supplementary pairs to walk an angle across the figure.
Dilations Combined with Rigid Moves [H]	Scale the coordinates first, then apply the slide, flip, or turn.

8th Grade - Linear Relationships & Functions (Deep) · 37 topics

Slope from a Graphed Line [H]	Slope = rise over run: count vertical change over horizontal change.
Comparing the Steepness of Lines [H]	Steepness is the size of the slope, ignoring its sign.
Horizontal and Vertical Lines [H]	A flat line has slope 0; a vertical line has an undefined slope.
The y-Intercept from Slope and a Point [H]	Plug the slope and the point into y = mx + b, then solve for b.
Writing y = mx + b from a Graph [H]	Read b where the line crosses the y-axis; read m as rise over run.
Slope and Intercept from Standard Form [H]	Solve Ax + By = C for y; the coefficient of x is the slope.
The x-Intercept of a Line [H]	The x-intercept is where y = 0; set the equation to 0 and solve for x.
Does the Point Lie on the Line? [H]	A point is on a line only if its coordinates make the equation true.
A Missing Coordinate on a Line [H]	Substitute the coordinate you know, then solve for the one you don't.
Parallel Lines and Equal Slopes [H]	Two lines are parallel exactly when their slopes are equal.
From Slope and a Point to y = mx + b [H]	Find b with b = y - mx, then assemble y = mx + b.
Lines Through the Origin [H]	Direct variation is y = kx: the line passes through (0, 0).
The Vertical Line Test [H]	A graph is a function if no vertical line hits it more than once.
Linear or Nonlinear from an Equation [H]	A linear equation has x only to the first power and no x in a denominator.
Linear or Nonlinear from a Graph [H]	A linear graph is a single straight line; any bend means nonlinear.
Working Backward Through a Function [H]	To find the input for a known output, solve the rule for x.
Rate of Change from a Graph [H]	Rate of change is the slope: change in the vertical over the horizontal.
Initial Value from a Graph [H]	The initial value is the y-intercept - the height where time is 0.
Average Rate of Change in Context [H]	Rate = (change in amount) / (change in time) between the two moments.
Units of the Rate of Change [H]	The slope's units are (vertical unit) per (horizontal unit).
Reading a System's Solution [H]	The solution of a system is where the two lines cross - set them equal.
The Equal-Values Method [H]	When both equations give y, set the two expressions for y equal.
Finding the Break-Even Point [H]	Set the two cost expressions equal and solve for the break-even amount.
Which Plan Costs Less? [H]	Evaluate each plan's linear cost at the given amount, then compare.
Ticket and Admission Mixtures [H]	Let a and c be the counts; use total tickets and total money.
Coin and Bill Systems [H]	One equation counts the coins; the other totals their value.
Two-Number Puzzles [H]	Add the sum and difference equations to find the larger number fast.
Rectangle Systems [H]	Turn 'length is d more than width' and the perimeter into two equations.
Substitution with a Fractional Solution [H]	The method is unchanged when the answer is a fraction - keep it exact.
Comparing a Graph and an Equation [H]	Read each function's slope from its own form, then compare the numbers.
Comparing a Table and a Description [H]	Find the rate from the table's steady jumps and from the words.
Which Function Overtakes the Other? [H]	For large x the function with the greater slope wins, whatever its start.
Writing a Trend Line's Equation [H]	Use two points on the fit line to get the slope, then the intercept.
What a Trend Line's Slope Means [H]	The slope is the predicted change in y for each one-unit rise in x.
What a Trend Line's Intercept Means [H]	The y-intercept is the model's predicted y when x is zero.
Predicting an Input from a Trend Line [H]	To predict x from a target y, substitute y and solve the equation.
Counting Points Above a Trend Line [H]	A point is above the line when its y exceeds the line's y at that x.

8th Grade - Exponents, Roots & Geometry (Deep) · 37 topics

Collapsing an Exponent Chain [H]	Multiplying same-base powers adds exponents; dividing subtracts them.
Evaluating an Exponent Expression [H]	Simplify the exponent first, then work out the single power's value.
Negative and Zero Exponents [H]	A negative exponent flips the base; a zero exponent is always 1.
Solving for an Exponent [H]	Same base on both sides means the exponents must match - solve for x.
Subtracting in Scientific Notation [H]	Line up the powers of ten first, then subtract the aligned amounts.
Raising a Quantity to a Power [H]	Raise the coefficient to the power, multiply the exponent, then normalize.
How Many Times as Large? [H]	Divide the coefficients and subtract the exponents of ten.
Combining Quantities in Context [H]	Expand both quantities to line up place value, then add the totals.
Ordering Numbers in Scientific Notation [H]	Compare the powers of ten first; only break ties with the coefficient.
Arithmetic with Roots [H]	Take each root to a whole number first, then add or subtract.
Side Length from a Square's Area [H]	A square's area is side squared, so the side is the square root of the area.
Edge Length from a Cube's Volume [H]	A cube's volume is edge cubed, so the edge is the cube root of the volume.
The Square Root of a Fraction [H]	Root the numerator and the denominator separately, then simplify.
Finding a Missing Leg [H]	Rearrange a squared + b squared = c squared to leg = sqrt(c squared - other leg squared).
Perimeter of a Right Triangle [H]	Find the missing side with the theorem, then add all three sides.
Perimeter on the Coordinate Plane [H]	Horizontal and vertical sides are easy; the slanted side needs the theorem.
The Space Diagonal of a Box [H]	The longest inside diagonal is sqrt(length squared + width squared + height squared).
Two Right Triangles at Once [H]	Each wire is its own right triangle; find both hypotenuses, then add.
Volume of a Cylinder (Numeric) [H]	V = pi r squared h; square the radius, multiply by height and by 3.14.
Volume of a Cone (Numeric) [H]	A cone holds one third of its cylinder: V = (1/3) pi r squared h.
Volume of a Sphere (Numeric) [H]	V = (4/3) pi r cubed; cube the radius, then take four thirds of pi r cubed.
Volume of a Hemisphere [H]	A hemisphere is half a sphere: V = (2/3) pi r cubed.
Volume of a Composite Solid [H]	Add the pieces: a cylinder plus its cone or hemisphere cap, in terms of pi.
Cone, Cylinder & Sphere Ratios [H]	For the same radius and height: cone = 1/3 cylinder, sphere = 2/3 cylinder.
Water Displacement [H]	The submerged sphere's volume becomes a cylinder of water: rise = V / (pi R squared).
Translating a Point [H]	Add the right/up shift to the coordinates; left and down subtract.
Reflecting a Point over an Axis [H]	Reflecting over the x-axis negates y; over the y-axis negates x.
Rotating a Point about the Origin [H]	90 ccw sends (x, y) to (-y, x); 180 to (-x, -y); 270 ccw to (y, -x).
Dilations and Scale Factor [H]	A dilation multiplies every length by the scale factor k.
Composing Two Transformations [H]	Apply the first move to the point, then feed that image into the second.
Congruent or Similar? [H]	Rigid moves keep congruence; a dilation with k not 1 makes it only similar.
Parallel-Line Angles with Algebra [H]	Set the angle expressions equal (or summing to 180), solve x, then measure.
The Exterior-Angle Relation [H]	An exterior angle equals the sum of the two remote interior angles.
Exterior Angles of a Regular Polygon [H]	The exterior angles of any polygon add to 360, so each one is 360/n.
Interior Angles of a Regular Polygon [H]	Divide the interior-angle sum (n - 2)*180 by n to get each equal angle.
Same-Side Interior Angles [H]	Same-side interior angles between parallel lines are supplementary.
Angles Around a Point [H]	Angles that surround a point add up to a full turn of 360 degrees.

8th Grade - Bivariate Data & Functional Modeling (Deep) · 38 topics

Modeling a Decreasing Linear Relationship [H]	A decreasing model is amount = start minus rate times elapsed input.
When a Model Reaches a Target [H]	Set the model equal to the target value and solve for the input.
Extending a Linear Table Beyond Its Rows [H]	Find the constant rate from the table, then continue it to the new input.
A Missing Entry in a Linear Table [H]	The missing value keeps the table's constant rate of change.
The Starting Amount from a Rate and a Reading [H]	Add back the total amount lost to a later reading to recover the start.
Choosing the Equation for a Falling Quantity [H]	A falling quantity has a negative rate and a positive starting value.
A Linear Model with a Fractional Rate [H]	The rate is amount over interval; multiply that unit rate by the new input.
Independent and Dependent Quantities [H]	The independent variable is the input; the dependent variable responds to it.
What a Negative Slope Means [H]	A negative slope means the output falls by that amount per unit of input.
Which Number Is the Rate, Which the Start [H]	The per-unit charge is the slope; the fixed amount is the intercept.
Total Change Over an Interval [H]	Total change equals the rate of change times the length of the interval.
Average Rate Over an Interval of a Table [H]	Average rate is the change in y divided by the change in x across the interval.
Describing a Graph's Story [H]	Match each rise, fall, or flat stretch of a graph to an event in the story.
Phases of a Journey Graph [H]	On a distance-time graph, standing still shows as a horizontal segment.
Increasing at an Increasing or Decreasing Rate [H]	A curve that steepens grows at an increasing rate; one that flattens slows.
Filling a Container: Height vs Volume [H]	Where a container is narrow the water level rises faster for the same inflow.
Testing Linearity with Unequal x-Steps [H]	A table is linear when the change in y over the change in x stays constant.
Linear or Nonlinear Situations [H]	A relationship is linear only when equal input steps give equal output steps.
Proportional or Just Linear? [H]	A proportional relationship is linear and passes through the origin (b = 0).
Which Table Is Linear? [H]	The linear table adds the same amount for each equal step in x.
Strong or Weak Association [H]	Points hugging a line show a strong association; a loose cloud is weak.
Linear or Nonlinear Scatter [H]	A scatter is linear when its points trend along a straight path, not a curve.
Recognizing No Association [H]	No association means y shows no consistent rise or fall as x increases.
Above or Below the Prediction [H]	A point is above the fit line when its y exceeds the predicted mx + b.
Choosing the Best-Fitting Line [H]	The best fit line follows the point cloud's direction and stays centered in it.
Interpolation vs Extrapolation [H]	Predicting inside the data range is interpolation; outside it is extrapolation.
The Predicted Difference Between Two Inputs [H]	The predicted change equals the slope times the gap between the two inputs.
The Point Farthest from the Fit Line [H]	The farthest point has the greatest gap between its y and the line's prediction.
Row Relative Frequency [H]	A row relative frequency divides a cell by that row's total.
Column Relative Frequency [H]	A column relative frequency divides a cell by that column's total.
Joint Relative Frequency [H]	A joint relative frequency divides one cell by the grand total.
Marginal Frequency and Its Relative Value [H]	A marginal frequency is a row or column total, found by adding its cells.
Association in a Two-Way Table [H]	An association shows when the rows' conditional relative frequencies differ.
A Count from a Relative Frequency [H]	Multiply the total by the relative frequency to recover the actual count.
Solving a Cost Model for the Number of Units [H]	Subtract the fixed fee from the total, then divide by the per-unit charge.
What the x-Intercept Means in Context [H]	The x-intercept is the input at which the modeled quantity reaches zero.
Which Table Shows the Faster Rate [H]	Compare the constant rate of change computed from each table's rows.
Estimating a Value from a Scatter's Trend [H]	Follow the point cloud's straight-line trend to read off a value at a new x.

8th Grade - Real Numbers & Applied Geometry (Deep) · 37 topics

Sums and Products with Irrationals [H]	A rational combined with an irrational stays irrational, unless the roots multiply to a perfect square.
Deciding If a Decimal Is Rational [H]	A decimal is rational when it terminates or eventually repeats, and irrational only when it never repeats.
Counting the Irrational Numbers [H]	Scan a set and count only the roots of non-perfect powers plus pi, ignoring the rational traps.
Terminating or Repeating? [H]	A fraction in lowest terms terminates exactly when its denominator has only 2s and 5s as prime factors.
Delayed Repeating Decimals as Fractions [H]	When a non-repeating digit precedes the block, put the difference over nines followed by zeros.
Repeating Decimals with a Whole Part [H]	Convert the repeating fractional part, then add the whole number to get an improper fraction.
The Square Root of a Decimal [H]	Read the decimal as a fraction over a power of ten, root the top and bottom, then rewrite as a decimal.
Rounding a Square Root to the Nearest Whole Number [H]	Compare the number with the two neighboring perfect squares and pick whichever root it sits closer to.
Approximating a Root to the Nearest Tenth [H]	Test candidate tenths by squaring them: the right one squares to just below and just above the number.
Comparing an Irrational with a Rational [H]	Square both numbers, or approximate the root, then compare the two sizes.
Ordering Numbers in Different Forms [H]	Convert every number to a decimal approximation, then compare the approximations.
Trapping a Root Expression Between Integers [H]	Trap the root between consecutive integers first, then add the whole number to both ends.
Estimating an Irrational Hypotenuse [H]	Add the squares of the legs, then trap that sum's square root between consecutive integers.
The Diagonal Shortcut [H]	The straight diagonal is the hypotenuse; subtract it from the sum of the two legs to find the saving.
Area of an Isosceles Triangle [H]	The height to the base splits it in half, forming a right triangle whose other leg is the height.
Slant Height of a Cone [H]	The radius and height are the legs of a right triangle whose hypotenuse is the slant height.
Bracing a Rectangular Frame [H]	A corner-to-corner brace is the diagonal, the hypotenuse of the right triangle formed by two sides.
Straight-Line Distance in 3-D [H]	Straight-line distance from three perpendicular moves is the square root of the sum of their squares.
How Far the Ladder's Foot Slides [H]	The ladder length stays the hypotenuse in both positions; find each foot distance, then subtract.
Fencing a Right-Triangular Lot [H]	Find the hypotenuse, add all three sides for the perimeter, then multiply by the price per meter.
The Diagonal of a Cube [H]	A cube's space diagonal is the edge times root three; square three edges, add, and trap the root.
Melting One Solid into Another [H]	Recasting keeps the volume, so set the two volume formulas equal and solve for the missing dimension.
Time to Fill a Tank [H]	Divide the tank's volume by the fill rate; when both carry pi, the pi cancels.
Capacity in Liters [H]	Find the volume in cubic centimeters, then divide by 1000 to convert to liters.
How Many Cones Fill the Cylinder? [H]	Divide the cylinder's volume by the cone's; with equal radii it is three times the height ratio.
Empty Space Around an Inscribed Sphere [H]	Subtract the sphere's volume from the enclosing cylinder's; the leftover is one third of the cylinder.
The Cost to Fill a Container [H]	Compute the volume, then multiply by the price per unit of volume.
Which Container Holds More? [H]	Compute each container's volume as a coefficient of pi, then compare the two numbers.
Scaling a Dimension of a Solid [H]	Volume scales by the square of a radius change with fixed height, and by the cube when every length scales.
Water-Level Rise from Added Volume [H]	The added water forms a short cylinder, so the rise is its volume divided by the base area pi r squared.
Melting a Block into Cubes [H]	Divide the block's volume by one small cube's volume, edge cubed, to count the cubes.
Distance, Rate, and Time [H]	Time equals distance divided by rate: divide the coefficients and subtract the powers of ten.
How Many Fit? [H]	Divide the total by the size of one item: divide coefficients and subtract exponents.
A Per-Person Amount [H]	Divide the total by the number of people: divide coefficients and subtract the powers of ten.
Mass, Volume, and Density [H]	Density is mass over volume: divide the coefficients and subtract the exponents of ten.
A Total from a Rate [H]	Total equals rate times amount: multiply the coefficients and add the exponents of ten.
Writing a Measured Quantity in Scientific Notation [H]	Place the decimal point after the first nonzero digit to get the coefficient between 1 and 10.

Prerequisite material - taught automatically when the diagnostic finds gaps

Elementary Math Skills · 4 topics

Place Value & Comparing	Each digit's position multiplies its value by ten.
Multi-Digit Multiplication	Split by place value, multiply each part, add.
Perimeter & Area of Rectangles	Perimeter walks around; area covers.
Reading Graphs & Tables	Read the values first; the math is the easy part.

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