Sensim Math · Depth
Your kid finished the worksheet in five minutes. Now what?
Depth is where strong students slow down and think — the same topics school is teaching, one level deeper. Multi-step, no formula to copy, built to be reasoned through.
✓ Free to solve · no account neededAligned to the Common Core, standard by standard.
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Stage 3 · Beyond the curriculum Reasoning
Logic, patterns, invariants — thinking no textbook unit teaches.
Explore Reasoning → You are here 2
Stage 2 · Grades 2–6 Depth
The school curriculum, one level deeper — where speed stops being enough.
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One ladder, from fluency to competition. The rungs are connected underneath — every problem here knows what it rests on and what it grows into.
The product, in one problem
What a Depth problem feels like
Try it yourself, right here — no signup, and no need to sit your kid down yet. If it makes you pause and think, it will do the same for them.
Pick your child's grade
Try one · Grade 3
Guess-and-check a legs problem
On Naomi's farm there are 10 animals in all, some cows and some chickens. When she counts all the legs of the cows and chickens, there are 32 legs in total. How many more cows than chickens are there?
Where to start
Start from what you already know
One thing you've noticed about your child is enough to find the right problems.
“She's doing fractions at school right now.”
Match what school is teaching
Pick the topic on this week's homework — get the set that runs one level under the surface of it.
“He knows his facts, but word problems fall apart.”
Start from the struggle
Pick what you're seeing at homework time — each pattern maps to a problem family built to work on exactly that.
“We're aiming for the gifted program next year.”
Start from the goal
“Where do kids usually start here?”
Follow the well-worn path
Our picks for Grade 4 — from the first rung to the deepest. (Real usage rankings will replace these as data accumulates.)
Not sure where to start? There's an optional 10-minute check-up.
Five problems, picked to locate a starting point. No score, no ranking, no signup — it ends with a suggestion, not a grade. And you can skip it forever: everything below works without it.
For parents
The math is free. What you pay for is visibility.
Your child never hits a paywall — problems, hints, and every topic stay open. The paid plan is for you: a clear picture of where they stand and what comes next.
Hints that teach, not tell
When your child gets stuck, Depth doesn't show the answer. It offers a ladder — each rung a smaller nudge toward thinking it through themselves.
1 A question that redirects attention
2 A way to organize the work
3 The method — still not the number
…and the answer itself is never a rung.
Depth for Parents
- Mastery map across every skill — solid, shaky, and untouched
- A weekly email: what they worked on, where they slowed down
- Suggested next path, recalculated as they solve
Kids' side stays 100% free — with or without a subscription.
For the explorers
Prefer to browse? The whole catalog is open.
427 problem types · 4,528 generated practice variants across 46 archetypes — numbers change on every visit, so answers can't be memorized.
Numbers & Place Value
51 problem types
Operations & Word Problems
79 problem types
Patterns & Reasoning
80 problem types
Fractions & Decimals
34 problem types
Measurement & Time
18 problem types
Geometry & Figures
90 problem types
Perimeter & Area
39 problem types
Data & Graphs
36 problem types
Count numbers built from digit cards Base-ten numbers
Compare numbers with an unknown digit Base-ten numbers
Find a number from digit clues Base-ten numbers
Count ways to make an amount with coins Measurement & data
Carry ten units to the next place Base-ten numbers
Skip-count backward to the start Base-ten numbers
Find the constant skip-count step Base-ten numbers
Ten hundreds make one thousand Base-ten numbersMeasurement & data
Different factor pairs, same product Operations
Bound an inequality at equality Operations
Recover the unknown, then recompute Operations
Guess-and-check a legs problem Operations
Count groups several equivalent ways Operations
Multiplying by one and by zero Operations
Find a number satisfying product bounds Operations
Solve shape-coded products in order Operations
Count sticks needed via multiplication Operations
Gaps are one fewer than the objects Measurement & data
Cut a rectangle into a grid of cards GeometryMeasurement & data
Find one handspan from measured counts Measurement & data
Undo more/less to find original length Measurement & data
Subtract the overlap from total length Measurement & data
Ribbon around a box: four equal parts Measurement & data
Smaller unit, more measured counts Measurement & data
Find two lengths from sum and difference Measurement & data
Convert to one unit, then compare lengths Measurement & data
Compare hours first, then minutes Measurement & data
Find a future date using days per month Measurement & data
Elapsed time between two clock times Measurement & data
Reason about a fast or slow clock Measurement & data
Reach the next hour, then add the rest Measurement & data
Read a mirrored clock, then find elapsed time Measurement & data
Read the hour when the hand is between Measurement & data
Weekdays repeat every seven days Base-ten numbers
Read totals and differences from a graph Measurement & data
Find the value of one grid interval Measurement & data
Set the scale, then complete the graph Measurement & data
Solve graph values step by step from clues Measurement & dataOperations
Complete a table from its graph and back Measurement & data
Total equals the sum of the rows Measurement & dataOperations
Compare two data sets in one graph Measurement & data
Find the rule of changing clock times OperationsMeasurement & data
Complete addition and multiplication tables Operations
Find the rule of a growing block stack Operations
Multiplication table folds to equal cells Operations
Sum a repeating number sequence Operations
Find shape and color repeats separately Operations
Use the seven-day weekday cycle Operations
Bigger addend, bigger sum OperationsBase-ten numbers
Choose digits for sum nearest a target Base-ten numbersOperations
Sum consecutive numbers via the middle OperationsBase-ten numbers
Recover hidden digits using regrouping Base-ten numbersOperations
One number, many addition expressions Base-ten numbersOperations
Big digits in high places Base-ten numbers
Overlap shrinks the total length Measurement & dataOperations
Ten in a place carries forward Base-ten numbers
Reduce to one unknown to solve OperationsBase-ten numbers
Count all shapes hidden in a grid GeometryOperations
Count segments between points on a line GeometryOperations
Base perimeter from multiple views Measurement & dataGeometry
Find a side from equal perimeters Measurement & dataOperations
Side lengths from overlapping rectangles Measurement & dataOperations
Perimeter of the nth figure by rule OperationsMeasurement & data
Perimeter plus tiles needed to build it Measurement & dataOperations
Perimeter by tracing every side Measurement & data
Dividend equals divisor times quotient Operations
Division as repeated subtraction Operations
Narrow candidates by divisibility conditions Operations
Find the total first, then divide Operations
Larger dividend gives larger quotient Operations
Recover side length from number of cut pieces OperationsMeasurement & data
Place digit cards for largest or smallest product Base-ten numbersOperations
Estimate a product to solve an inequality Base-ten numbersOperations
Evaluate a custom-defined operation symbol Operations
Recover missing digits from column carries Base-ten numbersOperations
Overlaps are one fewer than the strips Measurement & dataOperations
Write a growing pattern as an expression Operations
Find perimeter path length by multiplication Measurement & dataOperations
Find right and wrong counts from total score Operations
Reverse the operations to find the start Operations
Sum distances and times across legs Measurement & data
AM and PM make a 24-hour day Measurement & data
Match units before comparing path lengths Measurement & data
Distance grows in proportion to time Measurement & dataOperations
A drifting clock accumulates daily error Measurement & dataOperations
Total length is the sum of its parts Measurement & data
Speed is distance per unit time Measurement & dataOperations
Time regroups in base sixty Measurement & data
Ten tenths make one whole Fractions
A fraction is a count of unit fractions Fractions
Fractions require equal partitioning of the whole FractionsGeometry
Set up an expression and compute in order Operations
Consecutive integers around a middle value Operations
Place big digits high for largest product OperationsBase-ten numbers
Estimate a product to bound an unknown Base-ten numbersOperations
Posts versus gaps on lines and loops OperationsMeasurement & data
Deduce missing digits in a multiplication OperationsBase-ten numbers
Adjust the total to leave no remainder Operations
Exact division means remainder zero Operations
Count posts via length divided by spacing OperationsMeasurement & data
Fill long-division blanks from the known Operations
Perimeter as side length times side count Measurement & dataOperations
Find a rectangle side via one unknown Measurement & dataOperations
Remainder must be less than the divisor Operations
Segment through chained centers as radius multiples OperationsGeometry
Diameter is twice the radius OperationsGeometry
Rectangle sides are multiples of the diameter OperationsGeometry
Center to edge distance is the radius Measurement & dataGeometry
Perimeter splits into radius arcs and straight parts Measurement & dataGeometry
Perimeter of shape joining circle centers Measurement & dataOperations
Perimeter of shapes from tangent circles Measurement & dataOperations
Find each radius and diameter from segments OperationsGeometry
All radii in one circle are equal Measurement & dataGeometry
Polygon side equals sum of two radii Measurement & dataGeometry
Triangle side equals sum of two radii Measurement & dataGeometry
Bouncing ball rises a fraction of its fall FractionsOperations
Match fraction forms to compare sizes Fractions
Recover the whole from a fractional part FractionsOperations
Fraction of a number: divide then multiply FractionsOperations
Find the rule in a fraction sequence OperationsFractions
Split a whole into proper-fraction portions FractionsOperations
Smaller error means a better weight estimate Measurement & data
Find container weight after adding or removing Measurement & data
Net fill rate is inflow minus outflow Measurement & dataOperations
Round the quotient up to carry the remainder OperationsMeasurement & data
Set up one-unknown equation for capacities Measurement & dataOperations
Add poured water to find total capacity Measurement & data
Find the per-unit-weight rate first OperationsMeasurement & data
Combine sum and difference of two weights Measurement & dataOperations
Translate between table and pictograph Measurement & data
Solve missing data from ratio or sum clues Measurement & dataFractions
Determine the value each picture symbol stands for Measurement & data
Convert a pictograph into a frequency table Measurement & data
Fill the table by solving knowns first Measurement & dataOperations
Total all categories from a pictograph Measurement & data
Build a number from digit conditions in order Base-ten numbers
Compare place by place to bound an unknown digit Base-ten numbers
Find one tick interval, then read the value Base-ten numbers
Locate and compare numbers on a number line Base-ten numbers
Ten of a unit carries to the next place Base-ten numbers
Skip-count by the changing place value Base-ten numbersOperations
Times ten adds one place Base-ten numbers
A clock face is twelve equal 30-degree parts Measurement & data
Small angles combine into larger angles Measurement & data
Use the 180-degree line to find figure angles Measurement & data
Split a polygon into triangles to sum angles Measurement & data
Quadrilateral angles sum to 360 degrees Measurement & data
Set up an equation with one unknown angle Measurement & data
A straight line is 180 degrees Measurement & data
Triangle angles sum to 180 degrees Measurement & data
Smaller divisor makes a bigger quotient Base-ten numbers
Deduce hidden digits from the ones place up Base-ten numbers
A new symbol can define a custom operation Operations
A number divides by its factor pieces OperationsBase-ten numbers
The remainder is always less than the divisor Base-ten numbers
How many times the unit quantity OperationsBase-ten numbers
Intervals differ on open paths versus closed loops OperationsMeasurement & data
Bigger units express numbers more simply Measurement & data
Find the value of one unit Base-ten numbersOperations
Flip across a line creates symmetry Geometry
Four 90-degree turns return original Measurement & data
Center of rotation stays fixed Measurement & data
Read and compare multiple bars Measurement & data
Combine two graphs to find totals Measurement & dataOperations
Choose a scale that fits the largest value Measurement & data
Use total and clues to fill missing bars Measurement & dataOperations
Convert a table into a bar graph Measurement & data
Read a double bar graph Measurement & data
Find the value of one grid square Measurement & data
Sum-and-difference rules in a number grid Operations
Pair consecutive numbers with constant sum Operations
Express counts of dots, lines, faces Operations
Use neighbor differences to find rule Operations
Pattern in equations predicts the answer Operations
Find shape and number rule together Operations
Find patterns in mixed-number sequences and sum FractionsOperations
Find a missing side, then sum sides as fractions FractionsMeasurement & data
Use equilateral 60-degree angles and equal sides GeometryMeasurement & data
Folded angles are equal; chain to the unknown Measurement & data
Chain isosceles base angles to find unknown angles Measurement & dataGeometry
Two radii form an isosceles triangle; find angles Measurement & dataGeometry
Spot equal sides to build isosceles triangles Measurement & dataGeometry
Rotation preserves side lengths and angle measures Measurement & data
Build the decimal closest to a target from cards Base-ten numbersFractions
Place consecutive whole numbers to sum decimals Base-ten numbers
Use weight differences to find the empty container Base-ten numbersMeasurement & data
Find the extreme decimal satisfying an inequality Base-ten numbersFractions
Reverse decimal operations to recover the start Base-ten numbers
Eliminate to find two decimals from sum and difference Base-ten numbers
Undo a wrong decimal operation to recover the start Base-ten numbers
Perimeter as decimal side sums minus overlaps Base-ten numbersMeasurement & data
Perimeter of overlapped squares and rectangles Measurement & dataGeometry
Perpendicular is half of a straight angle GeometryMeasurement & data
Distance between parallel sides equals the side length GeometryMeasurement & data
Compare line graphs that use different gridline scales Measurement & data
Largest vertical gap between two line graphs Measurement & data
Fill a table and complete a line graph from clues Measurement & data
Estimate an unmeasured midpoint value on a line graph Measurement & data
Read max, min, and gridline scale from a line graph Measurement & data
Sum all data values read from a line graph Measurement & data
Slope of a time-distance line graph is speed Measurement & data
Exterior angle sum of a regular polygon is 360 GeometryMeasurement & data
Isosceles triangle from a rectangle diagonal GeometryMeasurement & data
Square diagonals are equal and perpendicular bisectors GeometryMeasurement & data
Identify tiling pieces and total area GeometryMeasurement & data
A balanced scale means both sides are equal Expressions
Add or subtract equations to eliminate unknowns Expressions
Combine part expressions into one expression Operations
Fixed sum: one side falls, other rises Operations
Evaluate a newly defined operation symbol Operations
Make result largest or smallest Operations
Symbolize the unknown, then build an expression ExpressionsOperations
Events coincide at common multiples of intervals Number systemOperations
Define a custom operator and evaluate it Operations
Count equal-share ways using divisors Operations
Meshing gears realign at LCM of teeth Number system
Blinking lights coincide at the LCM Number system
A multiple is divisible by its factor Operations
Product of two numbers equals GCF times LCM Number system
Subtract the remainder to find a divisor OperationsBase-ten numbers
Cuts-to-pieces correspondence rule Operations
Lines all crossing make the most points Operations
Find the rule linking position to term Operations
Common factor over 1 means not in lowest terms OperationsFractions
Both A and B is part of A or B Fractions
Build the largest proper fraction Fractions
Express minutes as fractions of an hour FractionsMeasurement & data
Perimeter is the sum of all side lengths FractionsMeasurement & data
Decompose into sums of unit fractions Fractions
Sum fractional parts of a whole Fractions
Composite perimeter sums boundary edges Measurement & data
Equal overlap area yields a segment length Measurement & data
Overlapping rectangles' edges form the outline Measurement & data
Perimeter as a multiple of a unit length Measurement & data
Slide edges to rectangle; perimeter unchanged Measurement & data
Use shared base to find composite area GeometryMeasurement & data
Solve known areas first, step by step GeometryMeasurement & data
Closest number may be above or below target Base-ten numbers
Choose round up or round down in context OperationsBase-ten numbers
Inclusive vs exclusive range boundaries ExpressionsNumber system
Largest difference uses max and min bounds ExpressionsBase-ten numbers
Overlap of two ranges on a number line ExpressionsNumber system
Round by looking at lower-place digits Base-ten numbers
Find original number range from rounded result Base-ten numbers
List two-digit numbers from digit conditions Base-ten numbers
Fraction of a quantity by multiplying Fractions
Express smaller units as larger using fractions FractionsMeasurement & data
Map length equals real distance times scale FractionsMeasurement & data
Find the whole from a fractional part Fractions
Whole as 1: find remaining fraction Fractions
Congruent figures have equal corresponding sides GeometryMeasurement & data
Use congruence to find lengths for area GeometryMeasurement & data
Complete point-symmetric figure to find perimeter GeometryMeasurement & data
Draw axis to split into congruent halves GeometryMeasurement & data
Decimal point position predicts the product Base-ten numbers
Decompose composite figures into rectangles Base-ten numbersMeasurement & data
Factor out a common factor with parentheses OperationsBase-ten numbers
Decimals convert small units to larger ones Base-ten numbersMeasurement & data
Place big digits in high places to maximize product Base-ten numbers
Solve multi-step decimal problems in order Base-ten numbers
Overlaps equal tape count minus one Base-ten numbers
Find cube edge from net perimeter GeometryMeasurement & data
Dashed edges in a sketch are hidden Geometry
Exposed faces depend on a cube's position GeometryMeasurement & data
Ribbon length matches parallel box edges GeometryMeasurement & data
Model sum as rectangle area for means Data & probability
Average times count gives the total Data & probability
Recompute mean after adding or removing data Data & probability
Set up one-unknown equation from mean Data & probabilityExpressions
Use a partial average to find an unknown value Data & probability
Probability is favorable over total outcomes Data & probability
A denominator that cancels to 1 leaves a whole number Number system
Find the daily gap between a clock and true time Number systemRatios
Dividing by a whole number is multiplying by its reciprocal FractionsNumber system
Use the overlap as the unit to find a figure's area GeometryExpressionsFractions
Count how many unit shapes make the whole GeometryFractions
Set the whole job to 1 and write a day's work as a fraction RatiosNumber system
Face, edge and vertex counts identify a prism GeometryExpressions
Face, edge and vertex counts identify a pyramid GeometryExpressions
Ribbon parallel to an edge is as long as that edge GeometryMeasurement & data
Find the rule and write it as an expression ExpressionsOperationsNumber system
One ■-th of the whole is the whole divided by ■ GeometryNumber system
The path's shape decides how many gaps there are Number systemOperations
Multiplying both sides keeps the inequality direction ExpressionsNumber system
Find the number before it was rounded ExpressionsNumber systemBase-ten numbers
A repeating block gives any decimal place Number system
Write it as an equation with a single unknown ExpressionsNumber system
Subtract the remainder and the division comes out exact Number system
A rate gives the area of a part RatiosGeometry
A rate is the ratio written as a number RatiosExpressions
Speed is distance over time Ratios
An item's angle out of 360° gives its percent Data & probabilityRatios
Knowing what 1 cm stands for gives the whole Data & probabilityRatios
Percent gives the weight of a part RatiosData & probability
Redraw with one item as the new whole RatiosData & probability
Change the total and every share changes Data & probabilityRatios
One item's count and share give the total RatiosExpressions
A percent gap plus a count gap gives the total RatiosExpressions
Count how many fit along each edge GeometryMeasurement & data
Work back from volume or surface area to a missing edge GeometryExpressions
Volume needs every length in the same unit GeometryMeasurement & dataRatios
A denominator that cancels to 1 leaves a whole number Number system
Use how multiplication and division undo each other Number systemRatios
Name the unknown with a symbol and build an equation ExpressionsGeometry
Simplify the expression before comparing Number systemExpressions
Dividing a smaller number by a bigger one shrinks the quotient Number systemExpressions
Find the unit amount first Number systemRatios
Get the whole from a fractional part Number systemRatiosExpressions
Use what was poured out to weigh the container Number systemExpressionsRatios
Change the unit using a decimal Number systemRatios
Dividing a bigger number by a smaller one grows the quotient Number systemExpressions
Rebuild the dividend with multiplication and addition Number system
Some decimals repeat forever Number system
Rounding is an up or a down Number systemExpressionsBase-ten numbers
To divide exactly, drop the leftover or top it up Number system
Find the unit amount RatiosNumber system
Find the cubes removed twice over Geometry
Spot the piece added onto a shape Geometry
Height on a plan makes it a solid Geometry
The longest side sets the cube's edge GeometryNumber system
Equal heights make the area ratio the base ratio GeometryRatios
Scaling both terms keeps the ratio RatiosExpressions
Moving some across leaves the total unchanged ExpressionsRatios
Name both unknowns and write two equations Expressions
A part's share of a circle is its central angle's share GeometryRatios
Equal after removing the shared part means equal to begin with GeometryExpressions
For equal thickness the material ratio is the area ratio GeometryRatios
Get a cylinder's height or radius back from its surface area GeometryExpressions