All problem types
556 problem types · 6,016 generated practice variants across 62 archetypes
Numbers & Place Value87
2 Count numbers built from digit cards 2 Compare numbers with an unknown digit 2 Find a number from digit clues 2 Carry ten units to the next place 2 Ten hundreds make one thousand 2 Different factor pairs, same product 2 Bound an inequality at equality 2 Count groups several equivalent ways 2 Find a number satisfying product bounds 3 Bigger addend, bigger sum 3 Recover hidden digits using regrouping 3 One number, many addition expressions 3 Big digits in high places 3 Ten in a place carries forward 3 Place digit cards for largest or smallest product 3 Estimate a product to solve an inequality 3 Recover missing digits from column carries 3 Ten tenths make one whole 3 Place big digits high to maximize a decimal 3 Place big digits high for largest product 3 Estimate a product to bound an unknown 3 Deduce missing digits in a multiplication 3 Fill long-division blanks from the known 4 Build a number from digit conditions in order 4 Compare place by place to bound an unknown digit 4 Ten of a unit carries to the next place 4 Skip-count by the changing place value 4 Times ten adds one place 4 Smaller divisor makes a bigger quotient 4 Deduce hidden digits from the ones place up 4 A number divides by its factor pieces 4 Build largest and smallest mixed numbers from cards 5 Combine part expressions into one expression 5 Make result largest or smallest 5 Build the largest proper fraction 5 Decompose into sums of unit fractions 5 Inclusive vs exclusive range boundaries 5 Largest difference uses max and min bounds 5 Overlap of two ranges on a number line 5 Round by looking at lower-place digits 5 Find original number range from rounded result 5 List two-digit numbers from digit conditions 5 Decimal point position predicts the product 5 Factor out a common factor with parentheses 5 Place big digits in high places to maximize product 6 Multiplying both sides keeps the inequality direction 6 Find the number before it was rounded 6 Simplify the expression before comparing 6 Dividing a smaller number by a bigger one shrinks the quotient 6 Dividing a bigger number by a smaller one grows the quotient 6 Rounding is an up or a down 7 Fix primes by sum, exponents by divisor count 7 Smallest square that is a common multiple 7 Split by common divisors, count |b| values 7 Opposite signs turn |b|=2|a| into b=-2a 7 ac=1 makes a and c same-sign reciprocals 7 Product and sum signs fix how many are negative 7 Combine sign cases to get extreme distances 7 Place numbers by sign to maximize and minimize 7 Fix the signs, then compare reciprocals 7 Turn directions and distances into signed coordinates 7 Recover each letter's sign from a quadrant 7 Three reflections make a rectangle sized by absolute values 7 A square on a given side has two sides to grow 8 Turn leading digits into an inequality 8 Make every prime exponent even 8 Count numerators that cancel the denominator's 3s 8 Chain the given inequality twice 8 Match the exponents, then compare bases 8 Pair 2s with 5s to make powers of ten 8 Same exponent, so match the bases 8 Rewrite in base 2 and factor the sum 8 Multiply top and bottom to use the given power 8 Solve for the building blocks, then rebuild 8 Let the sign decide the exponent's parity 8 Match interval endpoints, splitting on the sign 8 Turn decimal-digit clues into inequalities 8 Rounding to n means a half-open interval 8 Bound the smallest variable, then list cases 8 Triangle inequality bounds isosceles legs 8 Moving-point triangle area by subtraction 8 One win count sets both players' scores 8 Average cost under a flat fee plus surcharge 8 Two thresholds on one mixture give a range 8 Spend downstream time first, bound upstream speed 8 Fix one unknown and count the rest 8 Too few equations: let digit limits decide
Operations & Word Problems127
2 Skip-count backward to the start 2 Recover the unknown, then recompute 2 Multiplying by one and by zero 2 Solve shape-coded products in order 2 Count sticks needed via multiplication 2 Undo more/less to find original length 2 Find two lengths from sum and difference 3 Choose digits for sum nearest a target 3 Reduce to one unknown to solve 3 Dividend equals divisor times quotient 3 Division as repeated subtraction 3 Find the total first, then divide 3 Larger dividend gives larger quotient 3 Evaluate a custom-defined operation symbol 3 Reduce relations to a single-unknown equation 3 Reverse the operations to find the start 3 Distance grows in proportion to time 3 Speed is distance per unit time 3 Set up an expression and compute in order 3 Recover the original number from a wrong product 3 Find a rectangle side via one unknown 3 Bouncing ball rises a fraction of its fall 3 Fraction of a number: divide then multiply 3 Smaller error means a better weight estimate 3 Find container weight after adding or removing 3 Net fill rate is inflow minus outflow 3 Set up one-unknown equation for capacities 3 Add poured water to find total capacity 3 Find the per-unit-weight rate first 3 Combine sum and difference of two weights 4 Set up an equation with one unknown angle 4 A new symbol can define a custom operation 4 How many times the unit quantity 4 Find the value of one unit 4 Reverse the moves to recover original 4 Add per-day work fractions to find combined output 4 Apply a defined symbol to compute with fractions 4 Use submerged and exposed parts to find bar length 4 Undo a wrong fraction operation to recover the start 4 Build the decimal closest to a target from cards 4 Place consecutive whole numbers to sum decimals 4 Use weight differences to find the empty container 4 Reverse decimal operations to recover the start 4 Eliminate to find two decimals from sum and difference 4 Undo a wrong decimal operation to recover the start 4 Slope of a time-distance line graph is speed 5 A balanced scale means both sides are equal 5 Add or subtract equations to eliminate unknowns 5 Evaluate a newly defined operation symbol 5 Reverse the operation order to find the unknown 5 Symbolize the unknown, then build an expression 5 Define a custom operator and evaluate it 5 Reverse the steps to recover the original fraction 5 Same numerator: smaller denominator is larger 5 Find empty container weight by subtracting fractions 5 Recover the original number from a wrong calculation 5 Closest number may be above or below target 5 Express smaller units as larger using fractions 5 Decimals convert small units to larger ones 6 Find the daily gap between a clock and true time 6 Dividing by a whole number is multiplying by its reciprocal 6 Both sides of an equation can be multiplied or divided by the same number 6 Unit distance is total distance divided by time 6 Set the whole job to 1 and write a day's work as a fraction 6 Write it as an equation with a single unknown 6 Use how multiplication and division undo each other 6 Name the unknown with a symbol and build an equation 6 Find the unit amount first 6 Use what was poured out to weigh the container 6 Change the unit using a decimal 6 Find the unit amount 6 Write the daily gain or loss as a proportion 6 Equal heights make the area ratio the base ratio 6 Reduce the investment ratio to whole numbers 6 Write the rate as a fraction to read off both terms 6 Scale a simple ratio up to bigger numbers 6 Scaling both terms keeps the ratio 6 Moving some across leaves the total unchanged 6 Name both unknowns and write two equations 7 Equal side products cancel the shared corner 7 A birthday trick is just 100x+y in disguise 7 Measure a border path strip by strip 7 Linear means the square term's coefficient is zero 7 Split off a multiple of 5 to read the quotient 7 Turn the condition into x+y in terms of xy 7 Track grams of salt through two pourings 7 Sum pyramid: symmetric terms cancel at the top 7 True for every k means an identity in k 7 Peel a nested defined operation from the outside 7 Rewrite numerators using the given total 7 No solution means 0 times x equals a nonzero number 7 Each wrong solve still reveals one true constant 7 A relation fixes a ratio, so the expression is constant 7 Write the solution in a, then filter by wholeness 7 Solve for the opposite-hands moment, then step back 7 Split buyers into only-coffee, only-cake, both 7 Find the time first, not the zigzags 7 Express years married two ways, then equate 7 Subtract the full-time pump's share first 7 Count the children two ways and equate 7 Compare two drain-outs to isolate the inflow 7 Swapping mixtures shrinks the concentration gap 7 Salt balance turns a mixture into proportion 8 Recover each correct part from a wrong decimal 8 Chain two ratios, then evaluate by ratios 8 Match coefficients of like terms 8 Pair each monomial with a term by degree 8 Clear denominators to reshape the condition 8 Sum of reciprocals is sum over product 8 Undo the wrong step to recover the polynomial 8 Pin the boundary so exactly three naturals fit 8 No solution forces a zero coefficient 8 Solve in terms of a, then demand positivity 8 Equation fixes the form, inequality bounds it 8 A misread answer still fits the untouched equation 8 Write the solution in terms of the constant 8 Rename repeated reciprocals to get a linear system 8 Same way: lap gap; opposite way: lap sum 8 Find the mixing ratio, then the bottleneck 8 Subtract team rates to isolate one worker 8 Times give equations in reciprocal speeds 8 Count the gaps, then add and subtract 8 Track the salt through each pour 8 Plug special inputs into the identity 8 Find the x that makes the input right 8 Build values up from the sum rule 8 Each wrong graph still keeps one right coefficient
Patterns & Reasoning94
2 Count ways to make an amount with coins 2 Find the constant skip-count step 2 Guess-and-check a legs problem 2 Gaps are one fewer than the objects 2 Subtract the overlap from total length 2 Weekdays repeat every seven days 2 Solve graph values step by step from clues 2 Find the rule of changing clock times 2 Complete addition and multiplication tables 2 Find the rule of a growing block stack 2 Multiplication table folds to equal cells 2 Sum a repeating number sequence 2 Find shape and color repeats separately 2 Use the seven-day weekday cycle 3 Sum consecutive numbers via the middle 3 Overlap shrinks the total length 3 Perimeter of the nth figure by rule 3 Numbers divisible by several are common multiples 3 Narrow candidates by divisibility conditions 3 Objects versus gaps when spacing along a line 3 Overlaps are one fewer than the strips 3 Write a growing pattern as an expression 3 Find right and wrong counts from total score 3 Consecutive integers around a middle value 3 Posts versus gaps on lines and loops 3 Sum of evenly spaced numbers equals middle times count 3 Adjust the total to leave no remainder 3 Exact division means remainder zero 3 Count posts via length divided by spacing 3 Recover the dividend from quotient and remainder 3 Remainder must be less than the divisor 3 Find the rule in a fraction sequence 3 Round the quotient up to carry the remainder 3 Solve missing data from ratio or sum clues 3 Fill the table by solving knowns first 4 Find one tick interval, then read the value 4 Locate and compare numbers on a number line 4 The remainder is always less than the divisor 4 Intervals differ on open paths versus closed loops 4 Use total and clues to fill missing bars 4 Sum-and-difference rules in a number grid 4 Pair consecutive numbers with constant sum 4 Use neighbor differences to find rule 4 Pattern in equations predicts the answer 4 Find shape and number rule together 4 Find patterns in mixed-number sequences and sum 4 Subtract the overlap when joining fraction lengths 4 Fill a table and complete a line graph from clues 5 Fixed sum: one side falls, other rises 5 Events coincide at common multiples of intervals 5 Count equal-share ways using divisors 5 Meshing gears realign at LCM of teeth 5 Blinking lights coincide at the LCM 5 A multiple is divisible by its factor 5 Knowing the multiple aids factoring into products 5 Product of two numbers equals GCF times LCM 5 Subtract the remainder to find a divisor 5 Constant increase yields a correspondence equation 5 Cuts-to-pieces correspondence rule 5 Separate fixed and changing parts of a pattern 5 Lines all crossing make the most points 5 Check multiple relation to find correspondence rule 5 Find the rule linking position to term 5 Model two correspondences with separate rules 5 Common factor over 1 means not in lowest terms 5 Both A and B is part of A or B 5 Choose round up or round down in context 5 Solve multi-step decimal problems in order 5 Overlaps equal tape count minus one 6 A denominator that cancels to 1 leaves a whole number 6 Use the overlap as the unit to find a figure's area 6 Find the rule and write it as an expression 6 The path's shape decides how many gaps there are 6 A repeating block gives any decimal place 6 Subtract the remainder and the division comes out exact 6 A denominator that cancels to 1 leaves a whole number 6 Rebuild the dividend with multiplication and addition 6 Some decimals repeat forever 6 To divide exactly, drop the leftover or top it up 6 Equal after removing the shared part means equal to begin with 7 Same shortfall turns remainders into one LCM 7 Divisor of dividend minus remainder, within bounds 7 Equal remainders: divisor divides every difference 7 Repeated digit pair always divisible by 3 7 Smallest fraction: LCM of denominators over GCF 7 Group by numerator plus denominator, then count 7 Peel off whole parts and flip the remainder 7 Split the total gap into equal steps 7 Run the sum rule backwards on unit fractions 7 Regroup a long sum of differences 8 Skip the lead-in, then add whole blocks 8 Reversed repeating blocks add to the digit sum 8 Over 11 means the block is a multiple of 9 8 Let the LCM shrink the candidate solutions
Fractions & Decimals36
3 Compare decimals from the highest place down 3 Compare fractions sharing numerator or denominator 3 A fraction is a count of unit fractions 3 Fractions require equal partitioning of the whole 3 Find fractions between two on the number line 3 The whole is one; find the remaining fraction 3 Match fraction forms to compare sizes 3 Recover the whole from a fractional part 3 Split a whole into proper-fraction portions 3 Same numerator: smaller denominator is larger 4 Compare the next digit to fill a blank in decimals 4 Find the extreme decimal satisfying an inequality 5 Common denominators reveal fraction order 5 Match denominators or numerators to compare 5 Scaling numerator and denominator keeps value 5 Smaller difference means fractions are closer 5 Apply conditions step by step, then count 5 Sum fractional parts of a whole 5 Bouncing ball: repeated fraction-ratio heights 5 Fraction of a quantity by multiplying 5 A fraction of a quantity equals whole times fraction 5 Map length equals real distance times scale 5 Find the whole from a fractional part 5 Smaller denominator means larger unit fraction 5 Whole as 1: find remaining fraction 6 Concentration is salt over solution 6 Discount rate is the discount over the original price 6 Interest rate is interest over principal 6 A percent is a rate out of 100 6 Population density is people over area 6 A rate gives the area of a part 6 A rate is the ratio written as a number 6 Speed is distance over time 6 Get the whole from a fractional part 7 Match numerators to bound the denominator 7 Rewrite both ends over one denominator
Measurement & Time18
2 Find one handspan from measured counts 2 Smaller unit, more measured counts 2 Convert to one unit, then compare lengths 2 Compare hours first, then minutes 2 Find a future date using days per month 2 Elapsed time between two clock times 2 Reason about a fast or slow clock 2 Reach the next hour, then add the rest 2 Read a mirrored clock, then find elapsed time 2 Read the hour when the hand is between 3 Sum distances and times across legs 3 AM and PM make a 24-hour day 3 Match units before comparing path lengths 3 A drifting clock accumulates daily error 3 Total length is the sum of its parts 3 Time regroups in base sixty 4 Bigger units express numbers more simply 5 Express minutes as fractions of an hour
Geometry & Figures104
3 Count all shapes hidden in a grid 3 Count segments between points on a line 3 Segment through chained centers as radius multiples 3 Diameter is twice the radius 3 Rectangle sides are multiples of the diameter 3 Center to edge distance is the radius 3 Perimeter splits into radius arcs and straight parts 3 Perimeter of shape joining circle centers 3 Perimeter of shapes from tangent circles 3 Find each radius and diameter from segments 3 All radii in one circle are equal 3 Polygon side equals sum of two radii 3 Triangle side equals sum of two radii 4 A clock face is twelve equal 30-degree parts 4 Small angles combine into larger angles 4 Use the 180-degree line to find figure angles 4 Split a polygon into triangles to sum angles 4 Quadrilateral angles sum to 360 degrees 4 A straight line is 180 degrees 4 Triangle angles sum to 180 degrees 4 Even number of flips returns original 4 Flip across a line creates symmetry 4 Flipping or turning digits forms new digits 4 Four 90-degree turns return original 4 Center of rotation stays fixed 4 Express counts of dots, lines, faces 4 Equal point distances make isosceles triangles on a peg board 4 Systematically count all triangles in a composite 4 Use equilateral 60-degree angles and equal sides 4 Folded angles are equal; chain to the unknown 4 Chain isosceles base angles to find unknown angles 4 Two radii form an isosceles triangle; find angles 4 Spot equal sides to build isosceles triangles 4 Rotation preserves side lengths and angle measures 4 Draw an auxiliary parallel line to find a bent angle 4 Corresponding and alternate angles are equal 4 Count all the rhombuses in a figure of small triangles 4 Folding makes congruent corresponding angles 4 Adjacent angles of a parallelogram sum to 180 4 Perpendicular is half of a straight angle 4 Exterior angle sum of a regular polygon is 360 4 Isosceles triangle from a rectangle diagonal 4 Diagonals of a parallelogram bisect each other 4 Polygon angle sum via triangulation from a vertex 4 Square diagonals are equal and perpendicular bisectors 5 Symmetry axis splits figure into congruent halves 5 Congruent figures have equal corresponding sides 5 Overlapping congruent shapes leave congruent leftovers 5 Use congruence to find lengths for area 5 Fold or half-turn overlap reveals symmetry 5 Draw axis to split into congruent halves 5 Symmetry axis or center bisects corresponding-point segment 5 Dashed edges in a sketch are hidden 5 Adjacent faces in a net meet perpendicularly 5 Faces are bounded by edges joining vertices 5 Each prism face: 4 perpendicular, 1 parallel 5 Opposite faces of a prism are parallel 5 Exposed faces depend on a cube's position 5 Ribbon length matches parallel box edges 6 In a prism net the two congruent faces are the bases 6 A segment in the net is an edge of the solid 6 Cut an edge to unroll the sides into one rectangle 6 Face, edge and vertex counts identify a prism 6 Face, edge and vertex counts identify a pyramid 6 Ribbon parallel to an edge is as long as that edge 6 Find every point the line passes through 6 More faces means more vertices and edges 6 Count how many fit along each edge 6 Cutting a piece away removes that much volume 6 A submerged stone displaces its own volume 6 Work back from volume or surface area to a missing edge 6 Build the prism from its top, front and side views 6 Volume needs every length in the same unit 6 Scaling every edge by ■ scales volume by ■ cubed 6 Same boxes, different stacking, different surface area 6 Surface area is the sum of every face 6 Volume is the base stacked to the height 6 Per-cell counts give the front and side views 6 Find the cubes removed twice over 6 Layer and position fix how many faces show 6 Spot the piece added onto a shape 6 Height on a plan makes it a solid 6 The longest side sets the cube's edge 6 Start where the front and side views pin the count down 6 See the shape after cubes are removed 6 Surface area is the sum of every face 6 Get a cylinder's height or radius back from its surface area 6 A cylinder seen from the front is its vertical cross-section 6 Find the matching lengths in a cylinder net 6 Same height, radius times ●, volume times ● squared 7 Decide which side the halving line crosses 7 A line through the origin sweeps between endpoints 8 Find the common point from the known lines 8 A second solution forces the two lines to coincide 8 Find which side the bisector hits first 8 Rotate a line about its fixed point 8 Swapped-coefficient lines always meet at x = 1 8 When three lines fail to make a triangle 8 Area bisectors of a square pass through its center 8 A zero product graphs as two lines 8 No triangle: a parallel pair or one common point 8 Two lines through the origin meeting twice coincide 8 Bisect a pentagon by a line from one vertex 8 Pivot a line past a triangle's extreme vertices
Perimeter & Area41
2 Cut a rectangle into a grid of cards 2 Ribbon around a box: four equal parts 3 Base perimeter from multiple views 3 Find a side from equal perimeters 3 Side lengths from overlapping rectangles 3 Perimeter plus tiles needed to build it 3 Perimeter by tracing every side 3 Recover side length from number of cut pieces 3 Find perimeter path length by multiplication 3 Perimeter as side length times side count 4 Find a missing side, then sum sides as fractions 4 Perimeter as decimal side sums minus overlaps 4 Perimeter of overlapped squares and rectangles 4 Distance between parallel sides equals the side length 4 Count distinct shapes built by adding congruent tiles 4 Identify tiling pieces and total area 5 Perimeter is the sum of all side lengths 5 Composite perimeter sums boundary edges 5 Equal overlap area yields a segment length 5 Overlapping rectangles' edges form the outline 5 Perimeter as a multiple of a unit length 5 Slide edges to rectangle; perimeter unchanged 5 Use shared base to find composite area 5 Solve known areas first, step by step 5 Triangle height depends on chosen base 5 Complete point-symmetric figure to find perimeter 5 Decompose composite figures into rectangles 5 Find cube edge from net perimeter 6 Count how many unit shapes make the whole 6 One ■-th of the whole is the whole divided by ■ 6 Angles adding past 360° cover more than one circle 6 A part's share of a circle is its central angle's share 6 Draw a line to split it into straight and curved pieces 6 For equal thickness the material ratio is the area ratio 6 One full roll moves it its own circumference 6 Slide pieces so the shaded part becomes one lump 6 A circle's snug square has the diameter as its side 6 Measure straight parts as sides, curved parts as arcs 6 Match the traced circle's circumference to the cone 7 Box the triangle, subtract the corner triangles 7 Trapezoid minus two right triangles, in letters
Data & Graphs49
2 Read totals and differences from a graph 2 Find the value of one grid interval 2 Set the scale, then complete the graph 2 Complete a table from its graph and back 2 Total equals the sum of the rows 2 Compare two data sets in one graph 3 Translate between table and pictograph 3 Determine the value each picture symbol stands for 3 Convert a pictograph into a frequency table 3 Total all categories from a pictograph 4 Read and compare multiple bars 4 Combine two graphs to find totals 4 Choose a scale that fits the largest value 4 Convert a table into a bar graph 4 Read a double bar graph 4 Find the value of one grid square 4 Compare line graphs that use different gridline scales 4 Largest vertical gap between two line graphs 4 Estimate an unmeasured midpoint value on a line graph 4 Read max, min, and gridline scale from a line graph 4 Sum all data values read from a line graph 5 Average distance per hour from total time 5 Model sum as rectangle area for means 5 Average times count gives the total 5 Recompute mean after adding or removing data 5 Set up one-unknown equation from mean 5 Use a partial average to find an unknown value 5 Probability is favorable over total outcomes 6 An item's angle out of 360° gives its percent 6 Knowing what 1 cm stands for gives the whole 6 Percent gives the weight of a part 6 Use a ratio to split a percent between items 6 Redraw with one item as the new whole 6 Change the total and every share changes 6 One item's count and share give the total 6 A percent gap plus a count gap gives the total 7 Run a counting rule backward to every input 7 Constant xy fixes the triangle's area 7 Equal xy at two points fixes the ratio 7 Count one quadrant, then use point symmetry 7 Read each runner's rate from one graph point 8 Write the overall average as a weighted average 8 Meshed gears share teeth times turns 8 Count functions by multiplying output choices 8 Exactly one prime sum makes a function 8 Two equal inputs break a function 8 Every difference quotient is the slope 8 Trade a vertical shift for a horizontal one 8 Graph a moving point's area piece by piece
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