Projectile
Flight time, range, peak height and the area under the arc of a thrown ball. Drag the angle to find the longest throw.
v₀ = 42 m/s42 m/srange = v₀²*sin(2*θ)/g174.5 mline8 in ft572.4 ftUnitsPlotIntegral
Learn Center
Free math sheets for every grade, from counting coins to calculus, plus everyday math. Each one is a full Dot Decimal sheet: download it, open it, and drag its numbers to see what happens.
136 sheets
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Featured
10 sheetsThe examples built into the app, each showing off one way to explore.
Surfaces
A 3D surface you can spin, with a partial derivative, a gradient, a double integral and a limit.
f(x, y) = sin(x)*cos(y)ƒ(x, y)derivative(f(x,y), x)(cos(x) · cos(y))limit(sin(x)/x, x, 0)13D plotDerivativeLimit
3D surfaces
Four surfaces in one sheet: a wave, a saddle, a ripple and a bump. Spin each one to see where it rises and falls.
plot3d(sin(x)*cos(y), x, -π, π, y, -π, π)3D surfaceplot3d(x² - y², x, -1.5, 1.5, y, -1.5, 1.5)3D surfaceplot3d(cos(√(x²+y²))*exp(-(x²+y²)/40), x, -10, 10, y, -10, 10)3D surface3D plotFunctions
Linear systems
Solve a 3×3 system, find a determinant and eigenvalues, then solve, factor and expand polynomials.
det(A)-1solution = inverse(A)*b2; 3; -1solve(x²-5*x+6, x)x = 2, x = 3MatricesEigenvaluesAlgebra
Waves
Sliders for amplitude, frequency, phase and offset, and an integral whose bounds you drag on the plot.
a = 1.5 in 0 … 31.5f(x) = a*sin(b*x+c)+dƒ(x)integral(f(x), x, lo, hi)2.518SlidersPlotIntegral
Transforms
See a 2×2 matrix as a change to the plane. Drag î and ĵ and watch the determinant become an area.
M = [1.5, 0.5; 0.25, 1]1.5 0.5; 0.25 1det(M)1.375transform(M)2D transformMatricesDeterminant
Geometry
A pentagon on a grid. Drag any corner and the area and perimeter update, and the geometry view shows the area by shoelace, triangles, Pick’s theorem and grid squares.
P = polygon([1, 1; 8, 2; 9, 6; 5, 8; 2, 5])5-gonarea(P)36perimeter(P)24.032PolygonsShoelacePick
Taylor
Build a Taylor polynomial for eˣ and slide its order from 0 to 10 to see where it matches the curve.
n = 4 in 0 … 104T(x) = taylor(f(x), x, a, n)ƒ(x)plot(f(x), x, -4, 4, T(x))2D plotSlidersSeriesPlot
Mortgage
Monthly payment, total paid and total interest on a $320k loan at 6.5% over 30 years.
rate = 6.5%6.5%payment = loan*monthly rate/(1-(1+monthly rate)^(-months))$2,022.6interest = total-loan$408.1kMoneyPercentages
Unit scratch
How much paint a kitchen wall needs, with areas in square metres and square feet.
area = width*height7.68 m²paint = area*coats/(10 m²/L) in L1.536 Larea in ft²82.667 ft²UnitsConversions
Early elementary
27 sheetsKindergarten to grade 2: counting, coins, time and the first facts.
Adding and taking away
Add butterflies, watch some fly away, and split five flowers two ways.
still here = all butterflies - flew away6blue vase = flowers - red vase3red vase + blue vase == flowerstrueFarm animals
Count cows, chickens, and pigs, find the whole farm, and compare.
all animals = cows + chickens + pigs9fewer pigs than cows: pigs < cowstruesame pigs and cows: pigs == cowsfalseMaking ten
Every number under ten has a partner that makes ten.
partner = 10 - first number2check: first number + partner10first number + partner == 10trueTen frames
Two rows of five dots make every number to ten visible.
filled dots = top row + bottom row7empty spots = 10 - filled dots3filled dots + empty spots == 10trueMore, fewer, same
Compare sets with more, fewer, and the same as.
red blocks = 7 in 0 … 107blue blocks = 7 in 0 … 107green blocks = 4 in 0 … 104Shape hunt
Count shapes, then sort the round ones from the ones with corners.
circles = 4 in 0 … 54shapes with corners = triangles + squares5square corners > triangle cornerstrueHopping patterns
A frog starts on a number and hops the same size every time.
after hop one = start at + hop size3after hop two = after hop one + hop size5how far = after hop four - start at8Coin sort
Sort pennies, nickels, and dimes into piles, count each pile, and compare.
pennies = 4 in 0 … 54nickels = 3 in 0 … 53all coins = pennies + nickels + dimes9Days of school
Bundle the days of school into tens and ones on the way to day 100.
bundles of ten = int(day count/10)4check: bundles of ten*10 + extra ones47day count == 100falseMeasuring with blocks
How many blocks tall is each toy? Find the tallest and compare.
robot = 8 in 1 … 158dragon = 12 in 1 … 1512two robots stacked = robot + robot16Addition and subtraction stories
Take-away stories, missing addends, tens and ones, and how many more.
basket apples = 12 in 10 … 2012missing addend = sum goal - first addend7number = tens*10 + ones34Doubles facts
Add a number to itself, then use it for near doubles.
double = my number + my number12double - my number == my numbertruenext number = my number + 17Fact families to 20
Three numbers make two additions and two subtractions.
part two = whole - part one7part one + part two == wholetruepart two + part one == wholetrueAdding three numbers
Combine three piles and look for a pair that makes ten.
red marbles = 77red and blue == 10trueall marbles - green marbles == 10trueComparing two-digit numbers
Tens decide first; ones break the tie.
first tens = 4 in 1 … 94first ones = 5 in 0 … 95third number < fourth numbertrueSkip counting by twos
Count shoes and mittens by twos, forward and back.
two pairs = one pair + 24three pairs = two pairs + 26four pairs = three pairs + 28Time to the half hour
Read hour and half-hour times, then move ahead or back a half hour.
reading = 9:00am9:00 amsnack = reading + 30 min9:30 amrecess = snack + 1 hour10:30 amMeasuring with paper clips
Measure with same-size units, then compare.
storybook > penciltruepencil - crayon1two crayons = crayon + crayon6Place value and arrays
Hundreds, tens, and ones, skip counting, arrays, and a money story.
number = hundreds*100 + tens*10 + ones468second number = start at + count by130third number = second number + count by140Even and odd
Pairs make even numbers; leftovers make odd ones.
team one = int(cookies/2)7team two = cookies - team one7is even = team one == team twotrueTwo-digit addition
Break numbers into tens and ones, then regroup the extras.
total tens = first tens + second tens7total ones = first ones + second ones15answer == first number + second numbertrueAdding money
Add prices in cents, find the change from a dollar, and combine two pockets.
spent = pencil + eraser + sticker85change = paid with - spent15altogether = left pocket + right pocket115Inches and feet
Measure in inches, then convert to feet.
ruler in ft1 ftdesk in ft2 ftdesk == 2 fttrueClass pets bar graph
Turn a survey into a bar graph and compare the bars.
more fish than dogs: fish > dogstruedogs and cats together = dogs + cats10pets in all = dogs + cats + fish + birds21Time to five minutes
Move forward and back on the clock in five-minute jumps.
school ends = 3:05pm3:05 pmsoccer starts = 3:35pm3:35 pmdinner = 6:05pm6:05 pmTwo-step word problems
Solve one part first, then use it for the second part.
has now = left after giving + got more27bottom shelf = top shelf + 823books in all = top shelf + bottom shelf38Chairs in rows
Arrange chairs in equal rows and check there is a seat for everyone.
all chairs = rows*chairs per row20people = students + 119enough chairs = all chairs >= peopletrueUpper elementary
26 sheetsGrades 3 to 5: multiplication, fractions, decimals and measurement.
Multiplication and fractions
Drag factors, share with leftovers, slice a pizza, and time a trip.
product = first factor * second factor42cookies each = int(cookies/friends)4left = slices - eaten5Perimeter and area
Tile a garden, then compare rectangles with the same area.
width = tile side*tiles across3 ftlength = tile side*tiles down5 fttiles needed = tiles across*tiles down15Multiplication properties
Commutative, associative, and distributive with whole numbers.
a*b == b*atruea*(b*c) == (a*b)*ctruea*(b + c) == a*b + a*ctrueDivision facts and remainders
Divide, check with multiplication, and find the leftover.
stickers per page*pages == stickerstruefull bags = int(marbles/bag size)8spare marbles < bag sizetrueEquivalent fractions
Multiply top and bottom by the same number.
new top = top*multiplier4new bottom = bottom*multiplier6new top/new bottom == top/bottomtrueComparing fractions
Same bottom? Count the pieces. Same top? Bigger pieces win.
1/3 > 1/4true1/2 > 1/8true2/3 > 2/6trueLiquid volume
Pour, add, and share water in liters.
tank water = bucket*buckets poured20 Lstill to fill = tank size - tank water8 Leach bottle*bottles == jugtrueWeighing objects
Add and compare masses in grams and kilograms.
both fruits = apple + orange350 gorange > appletruedifference = orange - apple50 gRounding numbers
Round to the nearest ten and hundred.
ten below = int(number/10)*1040ten above = ten below + 1050ones digit = number - ten below7Arrays up close
An array shows a product as equal rows, and splits into smaller arrays.
total = rows*columns24double rows = 2*rows12double rows*columns == 2*totaltrueFractions and decimals
Compare and add fractions, meet tenths and hundredths, and sort angles by size.
3/4 == 9/12true5/6 == 10/12true5/6 > 3/4trueMixed numbers
Whole pizzas plus a half more, then quarters back into wholes.
halves = whole pizzas*2 + 17halves/2 == whole pizzas + 1/2truewhole sandwiches = int(quarters/4)4Comparing decimals
Tenths first, then hundredths.
first > secondtruedifference = first - second0.050.5 == 0.50trueAdding fractions
Add fractions with the same denominator, then tenths and hundredths.
a whole or more = total sevenths >= 7false3/10 == 30/100true3/10 + 45/100 == 75/100trueMultiplying fractions
Multiply a fraction by a whole number, one piece at a time.
thirds poured = friends*210friends*(2/3) == thirds poured/3truewhole cups = int(thirds poured/3)3Angle sums
Angles that share a corner add up: a right angle, a straight line, a full turn.
second part = 90° - first part55°first part + second part == 90°truesmall angle = 180° - big angle60°Factors and multiples
Find factor pairs, list multiples, and test for primes.
1*24 == 24true2*12 == 24true3*8 == 24trueNumber patterns
Find the rule, then predict the tenth term.
second term = start + step9third term = second term + step13tenth term = start + 9*step41Fractions and volume
Multiply fractions, divide by unit fractions, and fill a box.
1/2*1/2 == 1/4true2/3*3/4 == 6/12true6/12 == 1/2trueUnlike denominators
Rename fractions to add and subtract them.
1/3 == 4/12true1/4 == 3/12true4/12 + 3/12 == 7/12trueDividing fractions
How many pieces fit, and how big is each share?
blocks = board feet/(1/3)12blocks == board feet*3true(1/2)/3 == 1/6trueDecimal operations
Add, subtract, multiply, and divide decimals with money.
total cost = price each*quantity$11.25total cost > $10trueeach pays = dinner bill/friends$9.03Additive volume
Two prisms stacked are one shape.
big length = 10 cm10 cmbig width = 5 cm5 cmtotal volume in mL275 mLCoordinate grid
Plot a point, then count across and up to another.
on the x axis = y == 0falseon the y axis = x == 0falsesteps across = abs(library x - x)4Order of operations
Parentheses, then powers of ten, then times and divide, then plus and minus.
2 + 3*414(2 + 3)*4202*3 + 4*526Seedling heights
Seedling heights to the nearest half inch, tallied and shown as a bar graph.
count(heights)10spread = tallest - shortest2.5 intotal(heights)36.5Middle school
25 sheetsGrades 6 to 8: ratios, percents, equations, probability and geometry.
Ratios, rates, and percents
Paint mixes, unit rates, tax and coupons, and temperatures below zero.
speed = distance/time in km/h75 km/hspeed in mph46.603 mphtax amount = tag price*tax rate$1.44Unit conversions
The same length, weight, speed, and volume in many units.
race in m5,000 mrace in mi3.107 mibackpack in g8,000 gDividing fractions
Divide by a fraction by asking how many fit, then multiply by the reciprocal.
ribbon = 3/4 m0.75 mbow = 1/8 m0.125 mbows*bow == ribbontrueInteger operations
Add, subtract, and multiply signed numbers.
end point = start + move5end point > 0truea - b8Which deal is better?
Compare unit prices to find the better buy.
big box price = $6.00$6big box size = 24 oz24 oztwo small boxes = 2*small box price$7.2Mean, median, and spread
One outlier pulls the mean much further than the median.
mean = average(quiz scores)87median = median(quiz scores)88range = highest - lowest13Writing expressions
Turn words into math one phrase at a time.
double it = 2*number14five more = double it + 519five more == 19trueSolving one-step equations
Undo the operation to find the number, then check it.
x = 20 - 812x + 8 == 20truey = 42/67Proportion and probability
Scale a recipe, mark up and discount, and spin a spinner.
profit = sale price - wholesale cost$5.5profit > $0truechance of blue in %37.5%Scale drawings
Every real length is the drawing length times the same factor, and back again.
scale factor = 2 m/(1 cm)200drawing length = 7.5 cm7.5 cmmap scale = 500 m/(1 cm)50,000Percent increase and decrease
Up by 20% and then down by 20% does not cancel.
rise amount = original*rate/100$16risen price = original + rise amount$96fall amount = risen price*rate/100$19.2Simple interest
Simple interest earns the same amount every year.
interest per year = principal*rate$20interest = principal*rate*years$60total = principal + interest$560Probability
Find a theoretical probability, then predict results over many trials.
even chance = even count/sides0.5even chance == 1/2truenot even chance = 1 - even chance0.5Circles: π at work
Circumference and area from one radius.
circumference = pi*2*r43.982 cmarea = pi*r²153.9 cm²double r = 2*r14 cmAngle relationships
Triangle sums, vertical angles, and complementary and supplementary pairs.
top right = 180° - top left70°bottom right = 180° - top right110°bottom right == top lefttrueVolume and surface area
Volume and surface area of prisms, and how many unit cubes fit inside.
length = 20 cm20 cmcubes inside = volume/unit cube3,000triangle area = 1/2*2 m*1.5 m in m²1.5 m²Linear functions
Drag slope and intercept, find a slope from two points, and price a taxi ride.
f(x) = slope*x + y interceptƒ(x)f(0)1f(1)3Systems of equations
Two phone plans cross once; find the crossing point.
plan a = $15 + $0.10*texts$20plan a < plan btruecrossing texts = ($25 - $15)/$0.10100Exponent rules
Multiply and divide powers, meet zero and negative exponents, and use scientific notation.
2^3*2^n1282^3*2^n == 2^(3 + n)true3^5/3^227The Pythagorean theorem
Legs squared add to the hypotenuse squared.
c = sqrt(leg a² + leg b²)5leg a² + leg b² == c²trueshortcut in ft49.213 ftTransformations
Translate, reflect, and rotate a point with coordinate rules.
point = [x, y]3 -2slid = point + [4, 1]7 -1across y axis = [-x, y]-3 -2Cylinders, cones, and spheres
A cone holds one third of the cylinder with the same base and height.
radius = 3 cm3 cmcylinder volume/cone volume3sphere volume in L0.113 LTrend lines
Fit a line to paired data and use it to predict.
hours = [1, 2, 3, 4, 5]1 2 3 4 5slope = (86 - 62)/(5 - 1)6start = 62 - slope*156Function machines
A rule gives every input exactly one output, changing at a constant rate.
f(x) = 3*x - 2ƒ(x)f(0)-2f(1)1Scientific notation
Very big and very small numbers, side by side.
speed of light as sci2.99792 × 10⁸ m/slight year in km as sci9.46073 × 10¹² kmred blood cell = 7e-6 m as sci7 × 10⁻⁶ mAlgebra and geometry
12 sheetsGrades 9 and 10: functions, quadratics, radicals and trigonometry.
Function transformations
Drag h, k, and a, and watch the parabola follow.
h = 0 in -4 … 40g(x) = a*(x - h)² + kƒ(x)plot(g(x), x, -8, 8)2D plotExpanding and factoring
Expand products, factor quadratics, and solve by factoring.
expand((x + 2)²)4 + 4x + x²expand((x + 3)*(x - 1))−3 + 2x + x²expand((x - 4)*(x + 4))−16 + x²Solving linear equations
Whatever you do to one side, do to the other.
x1 = (11 - 3)/242*x1 + 3 == 11truex2 = (v - 3)/57Solving systems two ways
Elimination and substitution solve the same system.
x = (q - p)/(3 - 2)3x by substitution = q - p3x by substitution == xtrueQuadratic functions
Vertex, roots, and the discriminant.
f(x) = a*x² + b*x + cƒ(x)plot(f(x), x, -8, 8)2D plotvertex x = -b/(2*a)1Laws of exponents
Powers of powers, powers of products, and rational exponents.
(2^k)²64(2^k)² == 2^(2*k)true(2*5)^31,000Radicals
Roots, and how they combine.
sqrt(72)8.485sqrt(72) == 6*sqrt(2)truesqrt(50) == 5*sqrt(2)trueTriangle trigonometry
Ladders, kites, and shadows, with sine, cosine, and tangent.
π radians in degrees180°180° in radians3.142 radsin(π/6)0.5The unit circle
Each angle lands on the point (cos θ, sin θ), one unit from the origin.
sind(0)0sind(30)0.5sind(45)0.707Law of sines
Find the missing sides of a triangle from two angles and a side.
angle A = 40°40°side a = 8 cm8 cmangle C = 180° - angle A - angle B75°Law of cosines
Two sides and the angle between them give the third side, and back again.
b = 5 cm5 cmacosd((b² + c² - a²)/(2*b*c))60°hypotenuse == 5 cmtrueTrigonometric identities
Complements, negatives, and Pythagoras.
sind(30) == cosd(60)truesind(35) == cosd(55)truesind(-30) == -sind(30)trueAdvanced math
14 sheetsGrades 11 and 12: logarithms, statistics, calculus, series and vectors.
Sequences and growth
Arithmetic and geometric sequences, their sums, and monthly savings.
last term = start*ratio^(terms - 1)486period rate = annual rate%/120.5%deposited = monthly deposit*months$7,200Logarithms
Logs ask: to what power?
10^31,000log 1000 base 103log2(8)3Exponential growth and decay
Continuous growth, carbon-14 half-life, and how fast money doubles.
start population = 500500P(hours)674.9half lives = elapsed/half life2Statistics: z-scores and polls
Standardize a test score, find its percentile, and a poll's margin of error.
mean = 7575standard deviation = 88score = 91 in 50 … 10091Sine wave explorer
Drag amplitude, period, phase, and offset.
plot(f(x), x, -4, 4)2D ploty max = offset + amplitude1.5y min = offset - amplitude-1.5Polynomial roots
Factor, solve, and read the roots.
solve(x²-5*x+6, x)x = 2, x = 3factor(x²-5*x+6, x)(x − 2)(x − 3)solve(x²+7*x+12, x)x = -4, x = -3Limits, derivatives, integrals
Watch sin(x)/x approach 1, find a tangent slope, and measure area.
f(x) = sin(x)/xƒ(x)f(0.1)0.998f(0.01)1Matrices and 3D surfaces
Transform the plane, solve a system, and spin a surface.
M = [1.5, 0.5; 0.25, 1]1.5 0.5; 0.25 1det(M)1.375transform(M)2D transformMotion and optimization
Position, velocity, and acceleration; related rates; fencing.
position(t) = t³ - 6*t² + 9*tƒ(t)velocity(moment)0acceleration(moment)-6Taylor series
Polynomials that hug a curve.
T(x) = taylor(exp(x), x, 0, 4)ƒ(x)T(1)2.708e2.718Applications of integration
Area between curves, average value, and volume of revolution.
area between == 1/6trueaverage value = integral(x², x, 0, 3)/33average value == 3trueInfinite series
Convergent sums and the harmonic surprise.
sum(1/2^n, n, 1, 10)0.999sum(1/2^n, n, 1, 50)1sum(1/n², n, 1, 1000)1.644Probability distributions
Binomial probabilities, the normal curve, and expected value.
one way = 1/2^flips in %0.391%probability = ways*one way in %21.875%plot(density(x), x, 130, 210)2D plotVectors
Components, magnitude, and the dot product.
magnitude = sqrt(v x² + v y²)5magnitude == 5truedouble magnitude = magnitude*210Everyday math
22 sheetsMoney, cooking, travel, sports, science and more, for any grade.
Grocery run
Unit prices, totals, tax, and the change you get back.
tax = subtotal*tax rate$1.027total = subtotal + tax$18.137total == tax rate on subtotaltrueAllowance and savings goal
Save each week and watch the goal approach.
weeks needed = ceil(goal/weekly saving)30months needed = weeks needed*12/526.923still to save = goal - saved so far$110Tip and split the bill
Tip percents and a fair split.
tip = bill*tip percent%$15.21with tip = bill + tip$99.71with tip == tip percent% on billtruePaycheck and hourly wage
Hours times rate, then withholding comes out.
hourly rate = $18.50$18.5taxes = gross pay*withholding$260.5net pay = gross pay - taxes$923.5Discount stacking
Two discounts are less than they sound.
first off = tag*first coupon%$18after first = tag - first off$72total off = tag - after second$25.2Scaling a recipe
Multiply every ingredient to serve a crowd.
scale factor = guests/recipe servings2.5flour needed = flour*scale factor5 cupmilk needed = milk*scale factor3.75 cupCooking time and temperature
Oven temperatures in both scales, plus the timer.
oven in °C190.6°Crecipe celsius in °F374°Fdifference = oven - recipe celsius1Δ°FNutrition per serving
Calories per serving, then across the day.
breakfast = per serving*bowls220 kcaldifference = total - target-150 kcalover target = total > targetfalseRoad trip
Distance, speed, fuel, and arrival time.
departure = 9:00am9:00 amarrival = departure + driving time4:30 pmgas price = $3.50/gal in USD/gal3.5 $/galTime zones and layovers
What time is it over there, and how long is the trip?
6pm CST in Tokyo9:00 amdeparture = 8:15am8:15 amfirst leg = 2h 45m2 hours 45 minBasketball stats
Points, shooting percentages, and a scoring average.
two pointers = 88three pointers = 33average of 30, 22, 18, 27 and 2524.4Running pace
Pace, speed, and the pace a personal record needs.
race = 10 km10 kmpace = time/race in min/km5 min/kmspeed = race/time in km/h12 km/hMixture lab
Concentrations and dilutions.
stock = 0.5 molar0.5 molarwanted = 0.1 molar0.1 molarmoles = wanted*final volume in mol0.2 molpH of solutions
pH is a negative log of concentration.
hydrogen ions = 3.2e-4 as sci3.2 × 10⁻⁴ph = -log10(hydrogen ions)3.495acid ph = -log10(acid)2Free fall
Drop anything and gravity does the rest.
fall time = sqrt(2*drop height/g)3.029 simpact speed = g*fall time29.714 m/simpact speed in km/h107 km/hEnergy and power
Joules, watts, and the electric bill.
work = mass*g*height in J9,564.8 Jpower = work/time in W318.8 Wprice = $0.23/kWh in USD/kWh0.23 $/kWhBinary and hex
The same number in four bases, hex colors, and bit masks.
value as hex0xADvalue in binary0b10101101value in octal0o255Storage sizes
Bytes to gigabytes, decimal and binary.
photos = floor(storage/photo size)18,2851 GB in MB1,000 MB1 GiB in MiB1,024 MiBThe golden ratio
Phi, the golden rectangle, and Fibonacci ratios.
phi1.618phi == (1 + sqrt(5))/2truelong side = short side*phi16.18 cmFrequencies and intervals
An octave doubles the frequency.
octave up = A4*2880 Hzoctave down = A4/2220 Hzsemitone = 2^(1/12)1.059Temperature conversions
Celsius, Fahrenheit, and how cold the wind makes it feel.
outside = -15°C-15°Coutside in °F5°Finside in °F69.8°FCountdowns
Weeks and days until the big day, and when the doors open.
weeks left = days left/76.429full weeks = int(days left/7)6extra days = days left - full weeks*73No sheets match. Try another word or grade.
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