Archetypes
62 reasoning patterns — the same idea, deepening across grades. 51 of them span 2+ grades (the progression spine).
Grade
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Types Numbers & Place Value11
Place-Value Regrouping Ten of any unit bundles into one of the next-higher place, and the reverse on the way down. Children first carry ones into tens and tens into hundreds, then extend the same rule to thousands, to multiplying by ten (which shifts every digit up one place), and finally to tenths regrouping into a whole. The single idea — units convert at a fixed exchange rate — is what every later number topic rests on.
Build the Largest or Smallest Value from Digit Cards To maximize (or minimize) a number, product, quotient, or mixed number, place the biggest available digits in the places that carry the most weight. The reasoning starts with whole numbers, then governs products and quotients (where a small divisor enlarges the result), decimals, and mixed numbers. It is place-value comparison turned into an optimization strategy.
Pin Down a Number from Digit and Range Conditions A hidden number is found by applying clues in turn — digit constraints, place-by-place comparison, and inequality bounds that fence the unknown into a finite range. Early versions narrow a four-digit number from digit clues; later versions bound an unknown digit or an unknown box value under a sum, product, or comparison constraint. The shared move is converting each condition into a restriction until one survivor remains.
Decompose a Number into Parts and Factors One number can be rebuilt many ways — as sums of two addends or as different factor pairs giving the same product. Learners first see that the same quantity has many additive and multiplicative names, then use factor pieces to test divisibility. Flexible decomposition is the engine behind mental arithmetic and later divisibility work.
Recover Hidden Digits from Carries In a partly blanked addition, subtraction, multiplication, or long division, the missing digits are deduced column by column from the lowest place up, using how carries and borrows must propagate. The same cryptarithm logic deepens from three-digit sums to multi-digit products and division. It trains airtight place-value bookkeeping.
Inequality Range Membership Decide or count the whole numbers that satisfy 이상/이하/초과/미만 (greater/less than or equal, strictly greater/less) boundary conditions, often as the overlap of two ranges on a number line.
Read a Number by Its Prime Exponents Write each number as a product of prime powers and the question becomes a question about exponents. A perfect square has every exponent even, the divisor count multiplies (exponent + 1) over the primes, and a fraction terminates exactly when its reduced denominator has no prime but 2 and 5. Grade-school divisibility tested one divisor at a time; this reads the whole factor structure at once.
Exponent Laws: Same Base or Same Exponent Powers multiply by adding exponents, nest by multiplying them, and distribute over a product -- each law read straight off the meaning of a power as repeated multiplication. The problems here rewrite to a common base or a common exponent, or factor a shared power out of a sum, and then compare what is left; pairing 2s with 5s to make powers of ten is the same rewrite, and some use a given inequality to bound the result.
Settle the Signs First With negative numbers in play, a product or quotient of nonzero numbers is negative exactly when an odd number of its factors are negative, and its size depends only on the absolute values. So a problem about unknown signed numbers starts by narrowing the sign candidates from a product or quotient, then brings in the other conditions -- a sum, an order, a quadrant, a reciprocal pair -- to decide which case holds, and only then compares sizes. Elementary ordering never needed this first step.
Signed Position and Distance Every point on the number line (and each coordinate in the plane) is a signed position. On the line the distance between two points is the absolute value of their difference; in the plane the horizontal and vertical gaps are each the absolute difference of one coordinate. Directions become signs: reflecting in an axis flips the sign of one coordinate, reflecting in the origin flips both. |x - a| <= b is just 'within b of a'. Each absolute value hides a two-case split -- this side or that side -- and the extremes come from choosing the cases.
Model with an Inequality, Then Read the Range A situation with a threshold -- at least, cheaper than, before the deadline -- becomes a linear inequality in the unknown. The modelling is the same as for an equation; what changes is that the answer is an interval, so the boundary (included or not, rounded which way) decides it, and two thresholds on one situation give a range squeezed from both sides.
Operations & Word Problems15
Multiplication as Equal Groups Repeated equal groups are counted by multiplication, including the special behaviors of multiplying by one and by zero. This foundational archetype anchors the leap from skip-counting to the multiplication facts that every later rate, area, and scaling problem depends on.
Division as the Inverse of Multiplication Dividing is undoing equal grouping: the dividend equals divisor times quotient, and repeated subtraction reaches the same answer. Learners meet equal sharing, the multiply-then-divide two-step, and the relation that a larger dividend yields a larger quotient. This unknown-factor view is the basis for all later division reasoning.
Multiplicative Comparison and Unit Rate How many times as much, and how much per single unit, are two faces of the same relationship. Learners find a per-unit value (per kilogram, per hour as speed, fraction of a number) and scale it up, including reading speed as the slope of a distance-time graph. Unit-rate reasoning underlies proportions and all later rate work.
Proportion and Proportional Division Two ratios set equal, and a total split in a given ratio. Scaling both terms of a ratio leaves it unchanged, which is what lets an unknown be carried as a multiplier and pinned down by one extra condition -- a sum, a difference, or a product of extremes. Sharing a profit or a length in a ratio is the same idea run forwards.
Work Backwards to Recover a Start Value Given the end state of a chain of operations (or a calculation done wrongly), invert each step in reverse order to recover the original number or position. It begins with undoing simple add/subtract or skip-counts, then reversing a wrong product, undoing geometric moves, and unwinding multi-step decimal and fraction operations. Inversion is the universal 'the result was...' strategy.
Apply a Newly Defined Operation A new symbol or rule is defined, and the task is to evaluate or solve by substituting carefully into that definition and respecting order of operations. Learners meet symbol-coded equations they solve in order, then defined operators over whole numbers and fractions. The skill is reading a rule precisely and executing it. A function rule such as f(a + b) = f(a) + f(b) is the grade-8 form: plug in the inputs the rule makes easy and build outward.
Find Two Unknowns from Sum and Difference When two unknowns are tied by their sum and their difference (or one relation expressing one in terms of the other), rewrite everything in a single unknown and solve. The structure recurs across lengths, weights, capacities, decimals, rectangle sides, and angles. Reducing to one unknown is the gateway to algebraic thinking.
Track a Quantity Through Changes A measured quantity — container weight, water level, bar length, work done — changes as things are added, removed, or accumulate at a net rate; the unknown is recovered by following the running total. Learners handle before/after weight, net fill-with-leak, bouncing-ball fractions, submerged bars, and combined work rates. The thread is conservation: every change is accounted for. In middle school the same ledger runs on grams of salt carried from cup to cup and on whole-job fractions, with a letter standing for the amount that is not yet known.
Get Closest to a Target Value Choose or arrange digits so a sum, weight estimate, or decimal lands as near as possible to a target — the smaller the gap, the better. Learners minimize the difference for sums, estimated weights, and card-built decimals, including placing consecutive numbers to hit a decimal sum. It blends estimation with optimization.
Say It in Letters Name the unknown with a letter and build the quantity forwards as an expression, then let like terms collect. A two-digit number becomes 10a + b, a border path becomes strips of x, a sum pyramid becomes weighted sums of its base. Once written, the expression explains why a trick always works or why a total never depends on the letter -- by simplifying it, not by undoing the steps.
Rewrite the Condition into the Block You Need Some questions ask for an expression rather than the letters -- a ratio x/y, a sum over a product, a reciprocal sum -- and some systems hide the letters inside a repeated block such as 1/x. Instead of solving for each letter, reshape the given condition until it produces exactly that block and substitute it whole, or rename the repeated block as a new unknown and solve the simpler system it makes. Sometimes the value comes out though the letters are never pinned down; when the blocks themselves are solved, the original letters can be recovered from them.
Conditions on Coefficients: Identity, Degree, No Solution Some conditions are about the shape of an equation or inequality, not its solution. 'True for every x' means the two sides are the same polynomial, so matching coefficients gives the unknowns; 'linear' means the square term's coefficient is zero while the first-degree coefficient is not; 'no solution' means the x terms cancel to 0 * x and what is left is false -- 0 * x equal to a nonzero number in an equation, or a constant inequality that fails in an inequality. The move is to collect like terms and read the condition off the coefficients.
Count One Quantity Two Ways and Equate A word problem becomes an equation when the same quantity -- children, years, buyers -- can be expressed twice in terms of the unknown. Choose the unknown, write both expressions, set them equal. The elementary sum-and-difference problems are an earlier, narrower strategy worth recalling; here the two expressions can be anything the story gives.
Solve in Terms of the Parameter, Then Impose the Condition When an equation, inequality or system carries a letter constant, solve it as if that letter were a number: the solution comes out as an expression in it. The extra demand -- a natural-number solution, both solutions positive, exactly three whole numbers in range, a fraction in lowest terms -- then becomes a condition on the parameter, and that is what you solve. Usually the solving is ordinary; only the filter is new.
Relative Speed: Meeting and Chasing Two movers on one path close their gap at the sum of their speeds when they head toward each other and at the difference when one chases the other; on a loop, meeting again means one has gained a whole lap. Find the common time first and everything else is rate times time. Clock hands are two runners on a circular track.
Patterns & Reasoning9
Generalize a Growing Pattern into a Rule A figure or sequence that grows by a fixed amount each step is captured by a rule (often multiplication plus a constant) that predicts any term, including table symmetries and neighbor-difference rules. Learners move from skip-count steps and block stacks to perimeter-of-nth-figure and fraction-sequence rules. Generalizing from a few cases to a formula is core algebraic reasoning.
Repeating Cycle Patterns A pattern that repeats with a fixed period lets you find a far-off term using its position within the cycle (remainder on dividing by the period). Learners use weekday and shape/color cycles, repeating clock times, and number-line intervals. Periodicity plus remainder reasoning answers 'what is the Nth item' without listing all.
Repeating Decimal Period A division that never terminates still has a rule: the remainders cycle, so the digits after the point repeat in a fixed block. Once a child finds the block, any far-off decimal place is a remainder question rather than a long division -- the 18th or the 20th digit costs the same as the 3rd. This is where 'the division does not end' turns from a dead end into a pattern. In grade 8 the same block runs the other way: a k-digit repeating block is that block over k nines, so sums, reversals and digit conditions on the block become fraction arithmetic.
Divisibility and Remainder Reasoning A remainder is what is left after exact grouping, and it must stay smaller than the divisor. Problems range from finding common multiples and numbers meeting divisibility clues, to recovering a dividend from quotient-and-remainder, adjusting a total to leave no remainder, rounding bundles up, and maximizing a remainder. The controlling fact is the division algorithm: dividend = divisor x quotient + remainder.
Sum of Evenly Spaced Numbers via the Middle A run of consecutive or evenly spaced numbers adds up to the middle value times the count (or pairs with a constant sum). Children first add short runs, then generalize to long sequences and grid relations where pairing collapses a tedious sum into one product. The insight reappears as the arithmetic-series shortcut.
Solve a Table or Graph Step by Step from Clues When a table or graph has several blanks linked by clues (a total, a ratio, a difference), solve the entries you can pin down first, then use them to unlock the rest in sequence. Learners fill missing data from conditions, complete graphs from clues, and chain knowns to unknowns. It is constraint-propagation applied to data displays.
Two-Category Counts from a Total Two interacting categories (right vs wrong answers, animals with different leg counts, coins of different values) combine into one total; the counts are recovered by assuming one case and adjusting, or by systematic guess-and-check. This is the elementary precursor to systems of equations.
Overlap Reduces the Total When pieces are joined with overlap, the combined length is the sum of the parts minus the overlapping portion — and with several strips the overlaps number one fewer than the strips. The idea runs from taped lengths to multiplied strips to fraction distances. It is the first taste of inclusion-exclusion.
Objects versus Gaps (Fencepost Counting) Spacing objects along a line or loop ties the object count to the gap count: on an open path posts are one more than gaps, on a closed loop they are equal. Problems compute trees, posts, or intervals from length and spacing. The off-by-one fencepost relation is a classic trap the archetype trains learners to see.
Fractions & Decimals4
Part-Whole Fraction Reasoning A fraction names equal parts of a whole that equals one; from a part you can find the whole, the remaining fraction, or a fraction of a number. Learners partition equally, see a fraction as a count of unit fractions, and recover the whole from a given part. This part-whole model is the foundation of all fraction operations.
Compare Fractions and Decimals by Structure Sizes are compared by structure: same denominator favors the larger numerator, same numerator favors the smaller denominator, decimals compare highest place first, and the number line locates values between. Learners order fractions, mixed and improper forms, and decimals, often to fill an inequality blank. It builds magnitude sense across number systems.
Equivalent Fractions by Scaling Generate equivalent fractions by multiplying numerator and denominator by a common factor, then count or pin them down under digit, range, or difference constraints.
Ratio, Rate and Percent One number compared against a chosen base, written as a fraction, a rate or a percent. Speed, density, discount, interest and concentration are all the same move -- decide what the base is, divide by it, and read the answer as a comparison. The work in these problems is almost never the arithmetic; it is identifying which quantity is the base.
Measurement & Time2
Length as Sum of Parts with Unit Matching A total length or distance is the sum of its segments, but only after every measurement is expressed in the same unit. Learners unify handspans and rulers, convert km/m/cm, and accumulate multi-leg distances. Matching units before combining is the habit that prevents the most common measurement errors.
Elapsed Time and Base-Sixty Regrouping Time arithmetic regroups at sixty (seconds, minutes) and at twelve/twenty-four (hours, AM/PM), with days and weekdays cycling on top. Problems find elapsed time between clock readings, future dates, and the error of a fast or slow clock. It is place-value regrouping in a mixed-base system.
Geometry & Figures11
Angle Facts in a Figure Unknown angles fall out of a small set of facts: a straight line is 180 degrees, a triangle sums to 180, a quadrilateral to 360, angles combine and split, and parallel lines give equal corresponding/alternate angles. Learners apply these from clock faces and single figures up to polygon angle sums via triangulation and auxiliary lines. The method is choosing the right fact and chaining equations.
Isosceles and Equilateral Angle Chaining Equal sides force equal base angles, so spotting isosceles or equilateral triangles (often from equal radii or a diagonal) lets you chain known angles to an unknown one. Learners identify equal-side triangles inside circles and polygons and step through the angle relations. It fuses side-equality with angle-sum facts.
Transformations Preserve Measures Flips, turns, and folds move a figure without changing its side lengths or angle measures, and repeated transformations can return it to start (two flips, four quarter-turns) or be undone in reverse. Learners reason about flip/turn identities, rotation centers, and folded figures whose matched parts are congruent. The invariance principle underlies all of congruence.
Quadrilateral Diagonal Properties Special quadrilaterals are characterized by their diagonals: a parallelogram's diagonals bisect each other, a square's are equal and perpendicular bisectors. Learners use these defining properties to find lengths and angles. It sharpens classification of quadrilaterals by structural property rather than appearance.
Radius and Diameter Relationships Every radius of a circle is equal and the diameter is twice the radius; chaining these through touching or nested circles turns center-to-center segments and polygon sides into sums of radii. Learners fit circles in rectangles, build polygons joining centers, and compute perimeters mixing arcs and straight radius parts. It is geometric reasoning by equal-length substitution.
Systematically Count Shapes in a Figure Small pieces combine into larger ones, so counting all triangles, quadrilaterals, segments, or vertices in a composite figure requires an organized sweep by size or type to avoid missing or double-counting. Learners count hidden shapes in grids and composites and tally dots, lines, and faces. The discipline is exhaustive, non-overlapping enumeration.
Net and Solid Structure Reason about a rectangular prism, cube, or die through its net, faces, edges, and vertices: which faces are opposite or adjacent, hidden versus visible edges, folding a net, and positional face counts.
Surface Area and Volume of Solids How much skin a solid has and how much space it fills. Surface area is every outward face added once, so faces that touch stop counting; volume is a base stacked to a height, so cutting a piece away removes exactly that piece. Both run backwards too: a known area or volume gives back a missing edge.
Cube Stacks and Their Views A pile of unit cubes described by what it looks like from above, in front and from the side. Each view is a projection that keeps only the tallest cube in its line, so views constrain the pile without fixing it -- some positions are pinned down and others are still free, which is what makes counting all the possible piles a real question.
Equations Are Lines: Solutions Are Crossings Each linear equation in two unknowns is a line, and a solution of the system is where the lines cross. So 'no solution' is parallel lines, 'infinitely many' is the same line, an equation saying a product of two linear expressions is zero is the union of two lines, and three distinct lines fail to make a triangle exactly when two are parallel or all three pass through one point.
Pivot a Line: Meeting a Segment, Halving an Area A line through a fixed point turns like a hand on a dial. When the whole segment lies on one side of the vertical line through the fixed point, the slope to a point of the segment runs from one endpoint's slope to the other's, so 'meets the segment' is the slope range between the endpoints, ends included. Halving an area works the same way: decide which side the line must cross, then fix the crossing point by area. A line through the centre of a centrally symmetric figure always halves it.
Perimeter & Area4
Perimeter by Tracing Every Side The perimeter of a composite or overlapped figure is found by following its boundary and summing each side, using rectangle/square side equalities and overlap relations to recover unknown segments. Learners trace joined rectangles, ribbons around boxes, and overlapped polygons, later with fractional and decimal side lengths. The discipline is accounting for every boundary segment exactly once.
Tile and Cut Figures with Congruent Pieces A region is cut into, or built from, congruent pieces; the piece count, a side length, or the total area follows from how the copies fit. Learners recover a side from the number of cut pieces, count grid cards, identify tiling pieces and area, and read a base shape from solid views. It links multiplication and division to spatial structure.
Area by Decomposition Find a region’s area by cutting it into rectangles and triangles (or relating shared and overlapping sub-regions) and adding or subtracting their areas; choose the base–height pair that makes each piece computable.
Circumference and Area of a Circle Length and area for a shape with no straight sides. A part of a circle is measured by its share of the whole turn, so a central angle out of 360 does for a circle what a fraction does for a bar. Mixed boundaries -- some straight, some curved -- are handled piece by piece, and rolling turns circumference into distance.
Data & Graphs6
Read and Scale a Data Graph Pictographs, bar graphs, and line graphs encode values through a scale — symbol value or grid-square value — that must be read before any total, difference, or comparison is computed. Learners translate between tables and graphs, choose scales, read double bars and two line graphs, and estimate intermediate values. Decoding the scale is the load-bearing step at every grade.
Average from Total Use the relationship sum = average × count to recover a total, a missing data value, or a combined/weighted average, including changes when data is added or removed.
Probability as Outcome Fraction Express the probability of an event as the fraction of equally likely favorable outcomes out of all outcomes, and compare such probabilities.
A Function Sends Each Input to Exactly One Output A function is a correspondence that gives every input one output and only one. That single condition decides whether a rule is a function at all, which constant makes it one, and how many functions exist between two finite sets (multiply the allowed choices for each input). Running the rule backwards recovers every input that lands on a given output.
Slope Is the Rate of Change A linear function changes by the same amount for every unit step, so any two points on it give the same difference quotient -- the slope -- and a graph through the origin is a direct proportion whose slope is the unit rate. Shifting a line or reading a runner's speed off a graph uses that constant rate; a graph made of straight pieces, like an area that grows and then shrinks as a point moves, has its own constant slope on each piece and changes rate only where the pieces join.
Inverse Proportion: The Product Stays Fixed When y = a/x, the product xy is the same at every point, so every rectangle drawn from a point on the curve to the axes has the same area, the curve is symmetric about the origin, and its lattice points are the factor pairs of a -- while points between the curve and the axes satisfy a product inequality instead. Meshed gears are the same law: teeth times turns is shared.