Noetic Learning Math Contest (Grades 2-8)

270 core topics + 19 prerequisite topics taught as needed · approximately 92 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

Noetic Contest - Grades 2-4 · 50 topics

Skip-Counting Puzzles [H]	Spot the constant jump and continue the count.
Find the Missing Number [H]	Work backwards from the total to find the missing part.
What Number Am I? (Place Value) [H]	Build a number from its hundreds, tens, and ones.
Doubling Patterns (Grades 2-4) [H]	Each number is twice the one before it.
Counting Coins (Grades 2-4) [H]	Add up coin values: quarter 25¢, dime 10¢, nickel 5¢, penny 1¢.
Making Change (Grades 2-4) [H]	Change is what is left after you pay.
Fair Sharing with Leftovers [H]	Share equally, then count what cannot be split.
Elapsed Time (Grades 2-4) [H]	Count the minutes from a start time to an end time.
Perimeter of a Rectangle (Grades 2-4) [H]	Perimeter is the distance all the way around.
Counting Odd Numbers (Grades 2-4) [H]	An odd number ends in 1, 3, 5, 7, or 9.
Adding Consecutive Numbers [H]	Add a run of numbers in a row.
Working Backwards (Grades 2-4) [H]	Undo each step in reverse to find the start.
Digit Sums (Grades 2-4) [H]	Add up the separate digits of a number.
Comparing Numbers [H]	Line up place values to see which number is biggest.
Repeating Shape Patterns [H]	Use the length of the repeat to find any position.
Counting Legs (Grades 2-4) [H]	Multiply legs per animal, then add the groups.
Rows and Columns (Arrays) [H]	Equal rows make a multiplication.
How Many Can You Buy? [H]	Divide your money by the price and ignore the leftover.
Age Problems (Grades 2-4) [H]	Everyone ages by the same amount each year.
Making Equal Groups [H]	Divide the total by the group size.
Ten and Hundred More [H]	Only the tens (or hundreds) digit changes.
Rounding to the Nearest Ten [H]	Look at the ones digit to decide which way to round.
Days and Weeks [H]	There are 7 days in each week.
Position in a Line (Grades 2-4) [H]	Front position and back position add up to one more than the total.
Missing Operation [H]	Test each operation to see which gives the answer.
Halving Numbers [H]	Half means splitting into two equal parts.
Dozens [H]	A dozen means 12.
Find the Pattern Rule [H]	The rule is the gap between neighbouring terms.
Triangular Dot Stacks [H]	Stacking rows 1, 2, 3, … makes a triangle of dots.
Counting Grid Paths (Grades 2-4) [H]	Every shortest path uses the same number of right and down moves.
Overlapping Groups (Grades 2-4) [H]	Counting both groups counts the overlap twice.
Sum-and-Difference Puzzles [H]	Guess and check, or split the sum evenly and adjust.
Counting Coin Combinations (Grades 2-4) [H]	List the possibilities in an orderly way so none are missed.
Reversing Digits [H]	Swap the tens and ones digits.
Biggest Number from Digits [H]	Put the biggest digit in the biggest place.
Smallest Number from Digits [H]	Put the smallest digit in the biggest place.
Sum of Odd Numbers (Grades 2-4) [H]	Adding the first n odd numbers always makes a perfect square.
How Many More? (Grades 2-4) [H]	Subtract what you have from what you want.
Repeated Halving [H]	Each halving divides by 2 again.
Logic: Who Is Heaviest? [H]	Chain the comparisons together.
Counting Squares of All Sizes [H]	Do not forget the bigger squares hiding inside.
Patterns with Growing Jumps [H]	The gap itself gets bigger by a steady amount.
Reading a Data Chart [H]	Add every category to find the grand total.
The Number Just Before [H]	Step backwards by the counting amount.
Average of Two Numbers [H]	The average sits halfway between the two numbers.
Digit Riddles [H]	Turn the clues into the tens and ones digits.
Counting Within a Pattern [H]	Count whole blocks first, then the leftover beads.
Sharing Money Equally [H]	Divide the money by the number of people.
Totals from Comparisons [H]	Find each amount, then add them.
Multiply-then-Add Patterns [H]	Apply the same two-step rule to the last term.

Noetic Contest - Grades 5-6 · 50 topics

nth Term of a Sequence [H]	Jump straight to any term instead of listing them all.
Geometric Sequences (Grades 5-6) [H]	Each term multiplies the one before by a fixed factor.
When Do Cycles Line Up? (LCM) [H]	Events that repeat coincide at their least common multiple.
Largest Equal Groups (GCD) [H]	The biggest equal split uses the greatest common factor.
Counting Handshakes (Grades 5-6) [H]	Each pair shakes once - count the pairs.
Adding 1 to n Quickly [H]	Pair the ends to add a long run instantly.
Consecutive Integer Sums [H]	The middle number is the average of the run.
Find the Missing Value from an Average [H]	The total is the average times the count.
Percent of a Number (Grades 5-6) [H]	Percent means per hundred.
Fraction of a Group [H]	Split into equal parts, then take the parts you want.
Sharing in a Ratio (Grades 5-6) [H]	Split the total into equal shares, then group them.
Distance = Rate × Time (Grades 5-6) [H]	Distance is speed multiplied by time.
Elapsed Time Across the Hour (Grades 5-6) [H]	Convert both times to minutes, then subtract.
Find a Side from the Area [H]	Area = length × width, so divide to undo.
Counting Rectangles in a Grid (Grades 5-6) [H]	A rectangle is fixed by choosing two vertical and two horizontal lines.
Diagonals of a Polygon (Grades 5-6) [H]	Each vertex connects to all but itself and its two neighbours.
Reversed-Digit Differences [H]	The difference is always a multiple of 9.
Counting Multiples (Grades 5-6) [H]	Divide and drop the remainder.
Units Digit of a Power (Grades 5-6) [H]	Last digits repeat in a short cycle.
Spotting Multiples of 3 [H]	A number is a multiple of 3 when its digit sum is.
Multi-Step Working Backwards [H]	Reverse the steps in the opposite order.
Chickens and Rabbits [H]	If all were chickens, every extra pair of legs is a rabbit.
How Many in Neither Group? [H]	Subtract the union from the total.
At Least One (Inclusion-Exclusion) [H]	Add the groups but subtract the overlap once.
Systematic Counting of Numbers [H]	Count choices for each place and multiply.
Choosing a Team (Combinations) [H]	Order does not matter when you choose a group.
Ordered Arrangements (Permutations) [H]	When order matters, multiply the shrinking choices.
Perfect Squares in a Range [H]	Perfect squares are 1, 4, 9, 16, 25, …
Counting Factors (Grades 5-6) [H]	Factors come in pairs that multiply to the number.
Adding Unlike Fractions (Grades 5-6) [H]	Rewrite over a common denominator first.
Leftovers When Sharing [H]	The leftover is the remainder of the division.
Angle Between Clock Hands [H]	The minute hand moves 6° per minute; the hour hand 0.5°.
Sum of an Arithmetic Series [H]	Average the first and last term, then multiply by how many.
Insert the Operation [H]	Test each operation until one hits the target.
The Value of a Digit [H]	A digit's worth depends on its place.
Rounding to the Nearest Hundred [H]	Look at the tens digit to decide the direction.
Scaling a Recipe (Grades 5-6) [H]	Find the per-person amount, then multiply.
Best Buy (Unit Price) [H]	Compare price per single item.
Arranging Letters with a Repeat [H]	A repeated letter halves the count of arrangements.
Reversed-Digit Sums [H]	A number plus its reverse is 11 times the digit sum.
The Next Perfect Square [H]	Find the square root, round up, and square again.
Finding the Median (Grades 5-6) [H]	The median is the middle value once sorted.
Finding the Mode (Grades 5-6) [H]	The mode is the value that appears most often.
Finding the Range (Grades 5-6) [H]	The range is the gap from smallest to largest.
Percent Increase (Grades 5-6) [H]	Add the extra percent onto the original.
Least Common Multiple (Grades 5-6) [H]	The smallest number both divide into.
Greatest Common Factor (Grades 5-6) [H]	The largest number that divides both.
Simple Probability (Grades 5-6) [H]	Probability is favourable outcomes over total outcomes.
Counting Divisible Numbers [H]	Check each number for an even split.
Average Speed (Grades 5-6) [H]	Average speed is total distance over total time.

Noetic Contest - Grades 7-8 · 50 topics

Summing an Arithmetic Series [H]	Number of terms times the average of the ends.
Summing a Geometric Series [H]	Use the closed form a(rⁿ − 1)/(r − 1).
Square-Number Patterns [H]	When first differences grow steadily, the rule is quadratic.
Sum of the First n Squares [H]	There is a formula: n(n + 1)(2n + 1) ÷ 6.
Choosing Three (Combinations) [H]	Divide the ordered count by the arrangements of the chosen items.
Ordered Selections [H]	Multiply the shrinking list of choices, one per position.
Sum of Even Numbers (Grades 7-8) [H]	Factor out the 2 and use the triangle formula.
Polygon Diagonals (Grades 7-8) [H]	Each vertex joins all but itself and its two neighbours.
Interior Angle Sum [H]	Split the polygon into triangles from one vertex.
Interior Angle of a Regular Polygon (Grades 7-8) [H]	Share the angle sum equally among the vertices.
Pythagorean Triples [H]	In a right triangle, leg² + leg² = hypotenuse².
Working Together (Grades 7-8) [H]	Add the rates, not the times.
Mixture Percentages [H]	Percent juice is juice ÷ total, times 100.
Percent of a Percent [H]	Apply the two percents one after another.
Three-Part Ratios (Grades 7-8) [H]	Total parts set the size of one share.
Age Puzzles with Algebra [H]	Set up an equation for the future ages.
Coin Value Puzzles [H]	Assume all one coin, then account for the extra value.
Remainders of Powers (Grades 7-8) [H]	Remainders of powers fall into a repeating cycle.
Sum of Divisors (Grades 7-8) [H]	Add every factor, pairing them for speed.
GCF × LCM = Product [H]	The product of two numbers equals GCF times LCM.
Sums of Consecutive Integers (Grades 7-8) [H]	Subtract the fixed 'staircase' extra, then divide.
Guaranteeing a Match (Pigeonhole) [H]	One more than the number of colors forces a repeat.
Distributing Identical Items [H]	Place dividers in the gaps between items.
Binomial Coefficients (Grades 7-8) [H]	C(n, k) counts unordered selections of k from n.
Shortest Lattice Paths [H]	Every path is an arrangement of right and up moves.
Counting Ways to Make Change [H]	Build up the count coin type by coin type.
Weighted Averages (Grades 7-8) [H]	Weight each average by its group size.
Average After Adding a Value [H]	Rebuild the total, then divide by the new count.
Fibonacci-Style Sequences (Grades 7-8) [H]	Each term is the sum of the previous two.
Constant Second Differences [H]	Extend the differences to extend the sequence.
Fraction-of-Remainder Problems [H]	Each fraction acts on what is left, not the original.
Meeting-in-the-Middle [H]	They close the gap at the sum of the speeds.
Up Then Down by the Same Percent [H]	The result is below the start - the percents apply to different bases.
Area of an L-Shape [H]	Subtract the missing rectangle from the whole.
Perimeter with Algebra [H]	Turn the perimeter into an equation for the width.
Similar Figures [H]	Corresponding sides share one scale factor.
Multiplying Powers (Grades 7-8) [H]	Same base: add the exponents.
Powers of Ten [H]	Multiplying powers of ten adds the exponents (and the zeros).
Checkerboard Coloring [H]	On an odd board the corner color has one extra square.
Triangulating from a Vertex [H]	Diagonals from one vertex cut an n-gon into n − 2 triangles.
The Next Multiple [H]	Divide, drop the remainder, then step up one.
Complement Probability [H]	The chance of 'not' is one minus the chance of 'yes'.
Average of 1 to n [H]	For an evenly spaced list, the mean is the middle value.
Counting Coprime Numbers [H]	Count values sharing only the factor 1 with n.
Remainder of a Triangular Sum [H]	Add with the formula, then take the remainder.
Distance Between Points (Grades 7-8) [H]	The distance is the hypotenuse of a right triangle.
How Many Terms? [H]	Count the steps, then add one for the start.
Ordered Pairs with a Fixed Sum [H]	The first number can be anything from 1 up to N − 1.
Median of an Even List [H]	With no single middle, average the two central values.
Trailing Zeros of a Factorial (Grades 7-8) [H]	Count the factors of 5 in the product.

Noetic (Grades 2-4) - Number Sense, Patterns & Logic · 40 topics

The Row That Is Not Shown [H]	Subtract the rows you can see from the grand total to find the row that is missing.
Boxes of Ten and Loose Ones [H]	Ten loose ones trade for one box of ten, so a count can be repacked either way.
A Number from Digit Clues [H]	Test the clues one at a time against each candidate until only one number survives.
How Many Numbers in Between? [H]	The whole numbers strictly between two numbers are counted without either end.
The Missing Part of a Total [H]	Add every part you can see and take that away from the total to find the part left out.
Balancing Two Sides [H]	Total the side you can work out, then find what the other side needs to match it.
A Two-Step Number Machine [H]	Run the steps forwards when the start is known and undo them in reverse when it is not.
Equal Jumps on a Number Line [H]	Equal jumps multiply, so the distance traveled is the jump size times the number of jumps.
The Missing Digit in a Sum [H]	Work out the whole two-digit number first, then read off the digit that the box hides.
Gaps in a Skip Count [H]	Find the constant jump from any two neighbouring terms, then fill the gap or extend the count.
Numbers Two Counts Share [H]	A number said by both counts must fit both jumps, so step along one count and test it against the other.
Growing a Dot Array [H]	Equal rows multiply, so adding or completing rows changes only the number of rows.
Matching a Sentence to a Picture [H]	A dot array can be written as a product or as a repeated addition of the row size.
Which Pile Is Biggest? [H]	Turn every description into a single number first, then compare the numbers.
How Many More for an Equal Share [H]	Count up from the total to the next multiple of the group size to see how many more are needed.
Cartons Needed and the Last Carton [H]	A part-filled carton still counts as a carton, so any leftover forces one extra box.
Counting Round a Circle [H]	Handing things round a circle repeats with the number of seats, so a remainder names the seat.
Moving on a Number Chart [H]	One step down a number chart adds a whole row and one step right adds one.
Reading a Dot on a Number Line [H]	Find the size of one space first, then count spaces from a label you can read.
Odd or Even Without Adding [H]	Parity follows a rule of its own, so a sum can be judged odd or even before it is worked out.
Which One Needs Regrouping? [H]	Only the ones digits decide whether a two-digit sum or difference has to regroup.
Two Cards from a Row of Five [H]	Search a small set in order so every pair is tested once and none is tested twice.
A Missing Cell in a Times Table [H]	Each cell of a times table is its row label multiplied by its column label.
Same Total, Different Rows [H]	Doubling the number of rows and halving the row size leaves the total untouched.
A Fraction of a Group [H]	A unit fraction of a set is one of the equal parts the set is split into.
Filling and Doubling a Ten Frame [H]	A ten frame shows at a glance how far a count is from ten and from a teen number.
Counting Colors in a Repeating Strip [H]	A repeating strip is fixed by its unit, so any place is found from the remainder after dividing by the unit length.
A Pattern with Two Different Jumps [H]	When a pattern alternates two jumps the rule only repeats every second step.
Swapping One Thing for Another [H]	A balance gives an exchange rate, and every further swap multiplies or divides by that rate.
Who Owns Which Thing [H]	With one thing each, naming another child's thing removes it from everybody else.
Placing Four Children in a Line [H]	Fix the places a clue names outright, then fit the joined pairs into the gaps that are left.
Numbers Made from Digit Cards [H]	Count the choices place by place, remembering that a used card is gone and that a leading zero is not allowed.
Counting Outfits [H]	Each choice in one list can meet every choice in the other, so the counts multiply.
Handshakes Around a Table [H]	Every pair meets once, so count the pairs and halve only when the two directions are the same event.
The Fewest Coins [H]	Take as many of the largest coin as fit, then move down the list, and the count cannot be beaten.
Paying from a Price List [H]	Total the purchase first and subtract once, rather than making change item by item.
Later and Earlier on a Clock Face [H]	Move the minute hand in whole quarters and carry an hour whenever it passes the twelve.
Reading a Timetable [H]	A row gives a length by subtraction and two rows give a gap the same way.
A Picture Graph with a Key [H]	When one picture stands for several things, every row must be multiplied by the key before it means anything.
Comparing Rows of a Tally Chart [H]	Count each gate of five as five, write the four totals down, and only then compare them.

Noetic (Grades 5-6) - Reasoning & Problem Solving · 40 topics

Mixed Numbers That Regroup [H]	Add and subtract mixed numbers by rewriting one whole as a fraction whenever the fraction parts will not fit.
Comparing Fractions Against a Benchmark [H]	Rewrite every fraction with one common denominator, or measure each against a benchmark, before deciding which is bigger.
Multiplying and Dividing Decimals by Powers of Ten [H]	Multiplying by ten moves every digit one place to the left, and dividing moves it one place to the right.
A Fraction of a Measurement [H]	Convert the measurement into the unit the answer wants, then take the fraction of it - and take the second fraction of what is left, not of the original.
Money in Several Steps [H]	Work in cents so nothing rounds, find every subtotal before the total, and reverse the same chain when a single price is the unknown.
Sharing Into Fractional Servings [H]	Dividing by a fraction asks how many of those pieces fit, so the count is larger than the amount you started with.
Ratios Before and After a Change [H]	Write each quantity as so many equal parts, then let the one extra fact pin down the size of a part.
Reading a Map Scale [H]	A scale is a rate: multiply to go from the drawing to the real world and divide to come back.
Rates With a Unit to Convert [H]	Put both quantities into the same unit of time before you multiply, and never add two rates that are measured per different unit.
From a Ratio to a Fraction or a Percent [H]	A ratio compares part with part, so the denominator of the matching fraction is the SUM of the parts.
Discount, Then Tax [H]	Each percent change acts on the price in front of it, so the changes are applied one after another and never simply added.
Finding the Whole From a Percent [H]	When the part is known and the whole is wanted, divide by the percent instead of multiplying by it.
Percents Across Two Groups [H]	A percent is always a percent OF something, so two groups of different sizes cannot have their percents averaged.
Percents From a Data Table [H]	Read the count from the table, divide by the total that belongs with it, and rebuild the total whenever the data changes.
The Area of a Border [H]	A border is the difference of two rectangles, and its width is added twice to each dimension - once at each end.
Area and Perimeter of an L-Shape [H]	Cutting a corner out of a rectangle takes area away but leaves the perimeter unchanged.
Tiling a Floor and Paying For It [H]	Match the units before counting tiles, then round the number of BOXES upward because part of a box cannot be bought.
Same Perimeter, Different Areas [H]	List every whole-number rectangle that fits the condition and compare, because the extreme is always at the most square shape.
Volume of a Box, Forwards and Backwards [H]	Volume is the area of the base times the height, so a missing dimension is found by dividing the volume by the other two.
Water Levels and Displacement [H]	The volume of water is base area times depth, so pouring or sinking something changes the depth by that volume divided by the base area.
Painted Cubes [H]	Sort the small cubes by where they sit: corners have three painted faces, edges two, face centers one, and the inside none.
Working Out a Timetable [H]	Add the hours and the minutes separately, then trade sixty minutes for an hour, and count the gaps rather than the events.
Buses, Waits and Deadlines [H]	Departures every k minutes are a list of multiples, so a wait is what is left over and a count of departures is one more than the number of gaps.
The Score You Still Need [H]	Turn the target average into a required TOTAL, then subtract the total already scored.
Putting Two Averages Together [H]	Rebuild each group's total before combining, because a bigger group pulls the joint average towards its own.
When One Value Changes [H]	Work with the total: swapping a value changes the total by the difference, and removing values changes the count as well.
The Digit That Makes It Divide [H]	Use the digit tests: threes and nines look at the digit sum, fours look at the last two digits, and two tests together must both pass.
Rectangles With a Given Area [H]	Each rectangle with whole-number sides is one factor PAIR of the area, so counting rectangles means pairing the factors up.
Numbers With a Given Remainder [H]	Numbers leaving a fixed remainder form their own ladder: start at the remainder and step up by the divisor.
Hunting for Primes [H]	Test a candidate by dividing only by primes up to its square root, and remember the composites that look prime.
Growing Matchstick Patterns [H]	A pattern that grows by the same amount each time is the first figure plus that amount one time fewer than the figure number.
Position in a Repeating Pattern [H]	Divide by the length of the cycle: the remainder names the position inside the block and the quotient counts the whole blocks.
Sequences With Two Rules Taking Turns [H]	Find the two rules by looking at the gaps in turn, then continue the pattern by alternating them strictly.
Counting Numbers With a Digit Rule [H]	Count place by place: decide how many digits may sit in each position, then multiply, and treat a leading zero as forbidden.
Counting Choices With a Restriction [H]	Multiply the number of choices at each stage, then subtract exactly the combinations that the restriction forbids.
Counting the Games in a Tournament [H]	A knockout needs one match per team eliminated, while a round robin needs one game per PAIR of teams.
Routes Across a Street Grid [H]	Write on each crossing the number of ways to reach it, adding the crossing above and the crossing to the left.
Who Owns Which Pet [H]	Cross off every pairing a clue forbids and the survivors fall into place one at a time.
Putting a Race in Order [H]	Build the finishing order from the strongest clue outwards, and use the clues that rule a position out to close the gaps.
Balancing Scales by Swapping [H]	A balance says two collections weigh the same, so either side may be swapped for the other wherever it appears.

Noetic (Grades 7-8) - Advanced Reasoning · 40 topics

Direct and Inverse Variation [H]	Decide whether the product or the quotient of the two quantities is the constant, then rebuild the missing value.
Scale Factors for Area and Volume [H]	A length ratio of k makes areas grow by k squared and volumes by k cubed, and every one of those steps is reversible.
Rates Whose Units Must Be Reconciled [H]	Put every rate in the same unit of time before it is scaled, reversed, or added to another rate.
Ratios That Shift [H]	Name the size of one ratio part, then write the change as an equation in that one unknown.
Two Percent Changes in a Row [H]	Multiply the change factors instead of adding the percents, and reverse the same product to recover a hidden change.
Working Back to the Original Price [H]	Divide by the change factor to undo a percent change, and strip away any flat amount before you divide.
Percent More and Percent Less [H]	The two directions of a comparison measure against different bases, so they never give the same percent.
Percent Change in Area [H]	Area is a product, so each dimension contributes its own factor and the factors multiply.
Variables on Both Sides [H]	Clear the brackets or the denominators first, then gather the variable on one side and the constants on the other.
Counting the Integer Solutions [H]	Solve for x first, then count the integers inside the interval by subtracting the endpoints and adding one.
The Sign Flip in an Inequality [H]	Dividing or multiplying an inequality by a negative number reverses the direction of the sign.
Turning a Sentence into an Equation [H]	Name the unknown, write each phrase as it is spoken, and only then solve.
Adding and Subtracting Two Equations [H]	Combine the two equations so that one variable disappears, and stop as soon as the quantity asked for appears.
Two Purchases, Two Unknown Prices [H]	Two shopping baskets are two equations, and the difference or the sum of the baskets is often the fastest route.
Power and Quotient Laws for Exponents [H]	Rewrite every term over one common base, then add exponents for a product and subtract them for a quotient.
Zero and Negative Exponents [H]	A negative exponent is a reciprocal, not a negative number, and anything to the zero power is one.
Simplifying a Square Root [H]	Pull the largest perfect square out of the radicand, and only add radicals that already share the same radicand.
Arithmetic in Scientific Notation [H]	Handle the number parts and the powers of ten separately, then renormalize so the leading number sits between 1 and 10.
The Pythagorean Theorem in Context [H]	Name the hypotenuse first, because the theorem treats it completely differently from the two legs.
The Diagonal of a Box [H]	Apply the theorem twice: once across the base, once up to the far corner.
Area and Perimeter of a Right Triangle [H]	The two legs are the base and the height, so the area needs the legs while the perimeter needs the hypotenuse as well.
Measuring by Shadows [H]	Objects and their shadows at the same moment form similar right triangles, so height over shadow is a constant.
A Line Drawn Parallel to a Side [H]	A line parallel to one side cuts off a triangle similar to the whole one, so every length is in the same ratio and every area in its square.
Parallel Lines with Algebra [H]	Decide whether the two marked angles are equal or supplementary, then write that as an equation.
Exterior Angles of a Triangle [H]	An exterior angle equals the sum of the two interior angles it is not next to, which turns three-angle problems into two-angle ones.
Chasing Angles Through Isosceles Triangles [H]	Equal sides force equal base angles, and each new triangle in the figure gets its own angle sum of 180.
Surface Area of a Prism [H]	Add the areas of the faces the solid actually shows, and remember that gluing two solids hides two faces.
Volume of a Cylinder [H]	The volume of a cylinder is the area of the circular base times the height, so the radius counts twice and the height once.
Volume of a Composite Solid [H]	Break the solid into boxes, and treat poured or displaced water as a volume that keeps its size while its shape changes.
Two Draws Without Replacement [H]	The second draw happens in a smaller bag, so both the favourable count and the total count drop by one.
Independent Events in Stages [H]	Independent stages multiply, an OR needs the overlap subtracted once, and at least one is easiest through its complement.
Totals on Two Dice [H]	Count ORDERED pairs of faces, because the two dice are distinguishable even when they look alike.
Counting Numbers with a Restriction [H]	Fill the most restricted position first, then count the choices left for the others.
Arrangements With Two Items Tied or Split [H]	Glue two items into one block to force them together, and subtract that count from the total to force them apart.
Committees With a Condition [H]	A committee is an unordered choice, and at least one is counted as everything minus none.
Sequences Built by a Repeated Rule [H]	Apply the rule one step at a time going forwards, and undo it in the opposite order going backwards.
Which Term Is It? [H]	Invert the rule for the nth term: subtract the start, divide by the step, and add one back on.
Numbers With Given Remainders [H]	A remainder condition repeats every modulus, so two conditions repeat together every lcm of the two moduli.
Units Digits of Powers [H]	Units digits of powers repeat in a short cycle, so the exponent only matters through its remainder.
GCF and LCM from Prime Factors [H]	Compare the two numbers prime by prime: the gcf takes the smaller exponent and the lcm takes the larger.

Prerequisite material - taught automatically when the diagnostic finds gaps

Arithmetic Foundations · 3 topics

Adding & Subtracting Whole Numbers	Multi-digit addition and subtraction.
Multiplication	Multiplying whole numbers.
Division	Dividing whole numbers.

Elementary Math Skills · 7 topics

Place Value & Comparing	Each digit's position multiplies its value by ten.
Multi-Digit Multiplication	Split by place value, multiply each part, add.
Long Division	Divide, multiply, subtract, bring down - repeat.
Time & Elapsed Time	Count up to the next hour, then keep going.
Units & Measurement	Bigger unit → multiply; smaller unit → divide.
Perimeter & Area of Rectangles	Perimeter walks around; area covers.
Multi-Step Word Problems	One sentence at a time: what do I know, what changes, what's asked?

Grade 5: Decimals, Volume & the Coordinate Plane · 3 topics

Powers of 10 & Patterns	Each power of ten shifts every digit one place.
Volume by Unit Cubes	Volume counts the unit cubes that fill a solid.
Converting Measurement Units	Multiply or divide by the conversion factor.

Grade 4: Angles, Lines & Measurement · 1 topics

Hours, Minutes & Seconds

Time converts by sixties: hours to minutes to seconds.

Grade 4: Multi-Step Word Problems · 1 topics

Elapsed Time Across Hours

Count up to the next hour, jump the whole hours, then finish the minutes.

Grade 5: Volume & Measurement Applications · 3 topics

The Volume Formula V = l × w × h	Multiply the three edge lengths - no more counting cubes one by one.
Volume Word Problems	Spot the three dimensions hiding in a story, then multiply them.
Liquid Volume: Liters & Milliliters	Convert between liters and milliliters to solve pouring and filling stories.

Grade 4: Measurement, Data & Geometry - Deeper Practice · 1 topics

Elapsed Time Across the Hour

Total the minutes across each o'clock, then adjust for breaks.

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