1. Manipulate
Drag, slice, shade
Students change parameters and watch the model respond. The math is in the response, not in a symbol.
41 free tools · No install · Embeddable
Free Virtual Math
Manipulatives — K-8
41 browser-based interactive math tools — base-ten blocks, fraction bars, arrays, protractors, balance scales, coordinate planes, formula derivations. Free to use, free to embed in Google Sites, Canvas, or Schoology. No install, no plugin, no account.
Why visual-first
See it before you symbolize it.
The Concrete-Pictorial-Abstract sequence (often credited to Singapore Math but rooted in Bruner's earlier work) consistently beats symbol-first instruction in K-8 math. Virtual manipulatives are the digital "Pictorial" middle layer — students can drag, slice, partition, and rearrange a math idea before they write any equation about it.
2. Notice
Watch what stays the same
Equivalent fractions look identical when stacked. Equal ratios scale together. Pattern is visible before it's spoken.
3. Symbolize
Now write the formula
Once the structure is visible, the equation is shorthand for what was just shown — not a magic incantation.
Catalog
7 categories · 41 tools.
Every tool is a standalone Vue manipulative — opens at a dedicated URL, runs in any browser, embeds via iframe.
8 tools
Number & Operations
Place value, multiplication, division, and the structures behind multi-digit arithmetic. The bedrock of K-5 math.
- Free
Base-Ten Block Builder
Place value you can build
Build 347 from hundreds, tens, and ones. This is a standalone Fun Math game for seeing why each digit has a different weight.
Open tool →
- Free
Number Line Hopper
Addition as motion
Hop along a number line and leave a visible trail. This standalone playground makes direction, distance, and operation signs visible.
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- Free
Array Builder
Rows, columns, multiplication
Change rows and columns and watch equal groups turn into a multiplication structure.
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- Free
Groups Sharing Lab
Division as equal sharing
Change the total and the number of groups. The model separates the number of groups from the size of each group.
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- Free
Partial Products Box
Why multi-digit multiplication works
Change the factors and let the area model reveal the partial products. The four boxes are the visible source of the standard algorithm.
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- Free
Long Division Stepper
Quotient and remainder in context
Step through division as a sequence: quotient, product, subtraction, and remainder.
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- Free
Area Model Multiplication
Split and conquer big numbers
Watch 12×13 fracture along the tens place into four colored regions: 10×10, 2×10, 10×3, and 2×3. Each region's area is a partial product — add them up for the total.
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- Free
Sieve of Eratosthenes
Hunt every prime by elimination
Tap the smallest active number — its multiples chain-collapse to gray. Whatever stays alight is prime. Four levels escalate from 1–30 up to a 1–100 speedrun where the optimal play is exactly four taps: 2, 3, 5, 7.
Open tool →
3 tools
Fractions & Decimals
Equal parts, equivalence, and the same idea written in three notations — fractions, decimals, percents.
- Free
Fraction Partition Lab
Equal parts of one whole
Partition one whole into equal pieces, then change how many pieces are shaded. The bar makes numerator and denominator roles visible.
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- Free
Equivalent Fraction Bars
Same amount, different denominators
Stacked bars all show the same proportional amount. Change the denominator and slide the numerator — green dots mark exact equivalents, pink bars show close-but-not-equal approximations.
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- Free
Percent Grid Lab
Percent means per 100
Slide across a hundred-grid and watch percent, fraction, and decimal names stay connected.
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2 tools
Ratios & Proportions
The multiplicative cousin of addition. Equivalent ratios, unit rates, and the bar-model bridge to algebra.
- Free
Ratio Reactor Relay
Equivalent ratios in motion
A richer animation game for proportional reasoning: tune two streams, launch the reactor, and watch the tokens form repeated ratio batches. The reactor only locks when the ratio and the total both match.
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- Free
Ratio Tape Diagram
Scale both bars together
Scale a tape diagram and watch both bars grow together. The bars compare by multiplication, not subtraction.
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13 tools
Geometry & Measurement
Area, perimeter, volume, angles, coordinates, time, and the formula derivations behind πr² and the Pythagorean theorem.
- Free
Grid Tiling & Boundary
Area versus perimeter
Change a rectangle freely and watch area and perimeter update separately.
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- Free
Angle Protractor Lab
Degrees as rotation
Rotate a ray freely. The baseline and vertex alignment matter as much as the number you read.
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- Free
Coordinate Plane Navigator
x first, then y
Move across a four-quadrant grid. Ordered pairs become movement instructions.
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- Free
Cube Stacker Volume
Length × width × height
Stack a 4 by 3 by 2 prism from unit cubes and see why each layer has 12 cubes.
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- Free
Surface Net Unfolder
Surface area as a wrapper
Unfold a rectangular prism into six faces and add the visible rectangles.
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- Free
Where Does πr² Come From?
Slice, rearrange, discover
The animation plays automatically: a circle is sliced into 4 → 8 → 16 → 32 wedges and staggered into a row. See it? That row is a near-perfect parallelogram with base πr and height r.
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- Free
Drag the Radius, Feel r²
See area grow quadratically
Drag the radius and watch the circle area update live. When the radius grows by 1, the area grows by a new square layer, not by a straight line.
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- Free
Unroll a Circle into a Rectangle
From 4 slices to 32
Tap + to add slices: 4 → 8 → 16 → 32. The right-hand row of wedges quietly straightens — the more slices, the more it resembles a real parallelogram.
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- Free
Four Quarter-Circles in a Square
Compose the leftover region
A square has four quarter-circles cut from the corners. Change the side length and watch the leftover area follow square minus one full circle.
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- Free
Square Minus Inscribed Circle
Compute the four corner gaps
A square with the largest possible circle carved out. Change the side length and watch the four corner gaps scale together.
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- Free
Stadium-Shape Area
Decompose an athletic track
Change a stadium shape and watch its area decompose into one rectangle plus one full circle.
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- Free
Pythagorean Theorem — a² + b² = c²
Watch the squares fill and flow
Build squares on all three sides of a right triangle, then watch the area of the two smaller squares flow into the largest one. Change the triangle shape with the triple picker and replay the proof.
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- Free
Clock Trainer
Time as rotating hands
Turn the clock in five-minute steps. The minute hand moves directly while the hour hand keeps drifting between hours.
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4 tools
Algebra & Expressions
Variables as objects, equations as balance, and the order-of-operations as a tree to collapse from the inside out.
- Free
Expression Tiles Lab
Variables become objects
Change x-tiles and unit tiles to see coefficients and constants as objects.
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- Free
Equation Balance Lab
Do the same to both sides
Remove the same amount from both pans and watch x stay balanced.
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- Free
Order of Ops Tree
Collapse the expression in order
Collapse 4 + 6 × 3 by reducing the multiplication node before the final addition.
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- Free
GCF / LCM Venn Lab
Sort factors before calculating
Let the factors of 24 and 36 fall into a Venn diagram and discover their greatest shared factor.
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1 tools
Data & Statistics
Build, raise, lower, and compare bar charts before computing means or medians.
10 tools
Olympiad & Problem-Solving
Visual models for the classic word-problem archetypes — chicken-rabbit cages, sum-difference-multiple, pursuit, intervals, inclusion-exclusion.
- Free
Chicken & Rabbit Cage
Heads fixed, legs reveal the mix
A classic Olympiad model game: keep the head count fixed, swap one chicken into one rabbit, and watch the leg count jump by two. The all-chickens assumption makes the hidden mix visible.
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- Free
Honor Guard Formation
Square numbers and the 4(n−1) ring
Drag a slider to grow an n × n honor guard, then hollow out the inside to discover why the outer ring holds 4(n − 1) people. Four Olympiad-style levels — build, predict, decide, and a three-layer parade-square story — turn the square-formation problem (方阵问题) into a hands-on visualizer.
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- Free
Cows on the Meadow
Newton's grazing problem, animated
Drag cows onto a meadow that regrows daily. Each day, +g tiles bloom, then N cows crop the grass — so the net change Δ = g − N becomes a meter you can watch breathe. Four levels build the cow grazing problem (牛吃草问题) from free play to a leaky water tank, the same model wearing a different skin.
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- Free
Gem Vault Balance
Sum-difference-multiple bar model
Pull the gold bonus gap out of a larger vault, then split the remaining blue energy into equal units. This animated 和差倍 game turns bar modeling into a vault puzzle.
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- Free
Catch-Up Track
Distance gap divided by speed gap
Run two racers on the same track and watch the distance gap shrink by the speed difference. The chase ends when the faster runner has eaten the whole starting gap.
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- Free
Tree Spacing Lab
Intervals and endpoints
Plant seedlings on a line or loop and watch gaps light up before the trees count. Endpoint rules become visible instead of memorized.
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- Free
Venn Overlap Lab
Why the overlap is subtracted once
Drag badges into two club circles and run scanners that count each circle. Overlap badges flash when they are counted twice, making A + B - both visible.
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- Free
Pattern Growth Lab
Find rules by watching new growth
Run a growth factory where the conveyor sends only the new layer. Fresh tiles reveal first differences, making sequence rules easier to see.
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- Free
Choice Tree Builder
Counting principle through splitting paths
Open choice gates and watch every existing path split through all new options. The multiplication principle becomes a moving path tree.
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- Free
Pythagorean Tile Forge
Guess c, then watch area move
A standalone Pythagorean theorem H5 game: choose the missing hypotenuse, then launch the moving tile proof that shows why a² + b² equals c².
Open tool →
For teachers
Embed any manipulative in your site.
Every tool ships with a one-click "Embed on your site" button that copies a ready-made iframe snippet. Drop it into Google Sites, Canvas, Schoology, Seesaw, WordPress, Notion, or any HTML page. No plugin, no LMS integration, no API key.
<!-- Drop this into any HTML page --> <iframe src="https://inquiryai.zogmath.com/embed/{slug}/" width="100%" height="600" loading="lazy" ></iframe>
Virtual manipulatives catalog
Who this page helps, and where to go next.
Free virtual math manipulatives for teachers and parents who need concrete visual models before equations and algorithms.
Best for
- Teachers projecting a manipulative before independent practice.
- Parents who need to show the model behind a homework method.
- Students who learn better by dragging, partitioning, balancing, or measuring.
Problems solved
- Symbol-first explanations leave students memorizing procedures without a model.
- Physical manipulatives are not available at home or on a Chromebook.
- Interactive tools often lack a next step into guided grade-level practice.