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48 + 27 = 75. Regroup 15 ones as 1 ten and 5 ones; 4 + 2 + 1 = 7 tens.
48 + 27 = 75. Regroup 15 ones as 1 ten and 5 ones; 4 + 2 + 1 = 7 tens.
302 − 200 + 2 = 104. Subtracting 200 removes two too many, so add two back.
302 − 200 + 2 = 104. Subtracting 200 removes two too many, so add two back.
50 × 6 = 300
50 × 6 = 300
500
500
Multiplication can represent six equal groups of four, totaling 24; adding 6 and 4 gives 10.
Multiplication can represent six equal groups of four, totaling 24; adding 6 and 4 gives 10.
56
8 × 7 = 56
Start with the known part and count on to ten.
Grades K–1Count both groups without counting any item twice.
Grades K–1Start with the whole and take away the second number.
Grades K–1Make ten first when it helps.
Grades 1–2Check subtraction by adding the result back to the amount removed.
Grades 1–2A group of ten can be exchanged for ten ones.
Grades 1–2Find the repeated difference between neighbouring terms.
Grades 1–3For positive whole numbers, inspect the ones digit.
Grades 2–3Regroup ten ones as one ten when needed.
Grades 2–3Exchange one ten for ten ones when needed.
Grades 2–3Multiplication represents equal groups.
Grades 2–4Use the related multiplication fact; the divisor cannot be zero.
Grades 3–4The remainder is at least zero and smaller than the positive divisor.
Grades 3–5Multiply hours by sixty, then add the extra minutes.
Grades 2–4A square has four equal sides. Perimeter uses linear units.
Grades 3–5Split into equal groups using the denominator, then take the numerator groups.
Grades 3–6Keep the common denominator while finding the missing numerator.
Grades 3–5A factor divides a whole number without a remainder.
Grades 4–7Use positive whole numbers. A common multiple is divisible by both.
Grades 4–7The tenths place is one position after the decimal separator.
Grades 4–5Scale both numbers by the same power of ten.
Grades 5–7Multiply kilograms by one thousand to express the same mass in grams.
Grades 4–6Percent means per hundred. Simplify the resulting fraction.
Grades 5–7Distance is never negative.
Grades 6–8Subtracting a negative is adding its opposite.
Grades 6–8The exponent counts factors, not additions.
Grades 6–9The principal square root is non-negative.
Grades 6–9Negative numbers have negative real cube roots.
Grades 8–10Add exponents only when multiplying the same base.
Grades 8–10Subtract exponents; the base must not be zero.
Grades 8–10Substitute with brackets, then follow the order of operations.
Grades 6–9Only terms with the same variable part can be combined this way.
Grades 6–9Perform the same operation on both sides. Here a is different from c.
Grades 8–10Keep the point order the same in numerator and denominator.
Grades 8–10Average the x-coordinates separately from the y-coordinates.
Grades 8–10The hypotenuse is opposite the right angle and is the longest side.
Grades 8–10Use the radius, not the diameter. These exercises request the coefficient of π.
Grades 7–10The diameter is twice the radius.
Grades 7–10a and b are the parallel sides; h is the perpendicular distance between them.
Grades 6–9Multiply circular base area by perpendicular height.
Grades 8–10For a simple polygon, n is the number of sides and n ≥ 3.
Grades 7–9Square the input before multiplying by its coefficient.
Grades 9–11For real coefficients with a ≠ 0: positive Δ gives two real roots; zero gives one repeated root; negative gives no real roots.
Grades 10–12Set each factor equal to zero and check the original equation.
Grades 9–11The vertex lies on this vertical axis for y = ax² + bx + c.
Grades 10–12Add the equations to eliminate y, then solve for x.
Grades 9–11The first term is n = 1. Use one fewer step than the term number.
Grades 9–12Pair the first and last terms; each pair has the same total.
Grades 10–12Multiply repeatedly by the common ratio; do not add it.
Grades 10–12A logarithm asks which exponent produces the argument.
Grades 10–12Name the angle first; opposite is the side across from that angle.
Grades 9–12The adjacent leg is next to the angle; it is not the hypotenuse.
Grades 9–12A half turn is 180° or π radians.
Grades 10–12Multiply by the exponent and reduce that exponent by one. These tasks use positive integer powers.
Grades 11–12Use an antiderivative at the upper bound minus its value at the lower bound.
Grades 12–12Polynomials are continuous at every real input, so direct substitution applies.
Grades 11–12Use combinations when order does not matter and items are chosen without replacement.
Grades 10–12Use permutations when order matters and no item is repeated.
Grades 10–12Sort the data or find its two extreme values.
Grades 4–8Multiply each value by its frequency; divide by total frequency, not by the number of values.
Grades 9–12An event and its complement cover all outcomes without overlap.
Grades 7–10Work out one side, then subtract the known part.
Grades 2–4Add the counts for all categories.
Grades 1–3Use consistent length and area units.
Grades 3–6Convert to improper fractions; then subtract with a common denominator.
Grades 4–7Divide by the percentage as a decimal, not by the whole-number percent.
Grades 6–9Apply each percentage to the value at that stage.
Grades 7–10One kilometre is 1000 metres; one hour is 3600 seconds.
Grades 6–9Add the volumes only when the boxes do not overlap.
Grades 5–8Use the perpendicular height and radius, not the diameter.
Grades 8–12A length scale factor k changes area by k squared.
Grades 7–10The image is the same distance on the opposite side of the mirror line.
Grades 6–9This rule rotates around the origin in the coordinate plane.
Grades 7–10Undo addition first, then undo multiplication.
Grades 8–11Apply the inside function first.
Grades 9–12The middle term is twice the product, not zero.
Grades 8–11Halve the coefficient of x, then square it.
Grades 9–12Use n terms, starting with the first term a.
Grades 10–12Use the decimal growth rate; assume the same rate each period.
Grades 9–12For real logarithms: b > 0, b ≠ 1, u > 0 and v > 0.
Grades 10–12Differentiate term by term before substituting x.
Grades 11–12Evaluate the antiderivative at the upper limit minus the lower limit.
Grades 11–12Replacing the first item keeps the second probability unchanged.
Grades 7–10Restrict the denominator to the group named in the condition.
Grades 9–12This is population variance: divide by N, not N − 1.
Grades 10–12Use medians of the lower and upper halves for this exercise.
Grades 8–12Rules, conditions and small examples in one printable reference.
Grades K–2Rules, conditions and small examples in one printable reference.
Grades 3–5Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades K–2Rules, conditions and small examples in one printable reference.
Grades 3–5Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 1–2Rules, conditions and small examples in one printable reference.
Grades 3–3Rules, conditions and small examples in one printable reference.
Grades 2–2Rules, conditions and small examples in one printable reference.
Grades 3–5Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–9Rules, conditions and small examples in one printable reference.
Grades 3–5Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 3–5Rules, conditions and small examples in one printable reference.
Grades 6–7Rules, conditions and small examples in one printable reference.
Grades 4–5Rules, conditions and small examples in one printable reference.
Grades 6–7Rules, conditions and small examples in one printable reference.
Grades 5–5Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 8–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 10–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 11–12Rules, conditions and small examples in one printable reference.
Grades 1–2Rules, conditions and small examples in one printable reference.
Grades 3–5Rules, conditions and small examples in one printable reference.
Grades 6–8Rules, conditions and small examples in one printable reference.
Grades 9–10Rules, conditions and small examples in one printable reference.
Grades 11–12A focused reference to use alongside your practice.
Grades 4–8A focused reference to use alongside your practice.
Grades 5–9A focused reference to use alongside your practice.
Grades 10–12A focused reference to use alongside your practice.
Grades 10–12A correction is stronger when it names the mistaken rule or place value.
Grades 2–5A correction is stronger when it names the mistaken rule or place value.
Grades 2–5Equivalent fractions require multiplying or dividing numerator and denominator by the same nonzero number.
Grades 5–8Equivalent fractions require multiplying or dividing numerator and denominator by the same nonzero number.
Grades 5–8Apply the same valid operation to both sides and respect restrictions such as division by zero.
Grades 8–11Apply the same valid operation to both sides and respect restrictions such as division by zero.
Grades 8–11Over real numbers, denominators cannot be zero and even-root inputs must be nonnegative.
Grades 10–12Over real numbers, denominators cannot be zero and even-root inputs must be nonnegative.
Grades 10–12A derivative describes local rate of change where it exists.
Grades 11–12A derivative describes local rate of change where it exists.
Grades 11–12Equal counting works only when the individual outcomes are equally likely.
Grades 6–10Equal counting works only when the individual outcomes are equally likely.
Grades 6–10Perimeter uses length units, area uses square units and volume uses cubic units.
Grades 4–8Perimeter uses length units, area uses square units and volume uses cubic units.
Grades 4–8Identify what is counted and make sure categories are not accidentally counted twice.
Grades 3–7Identify what is counted and make sure categories are not accidentally counted twice.
Grades 3–7Comparisons require the same whole. Equal-sized parts give the denominator meaning.
Grades 4–7Comparisons require the same whole. Equal-sized parts give the denominator meaning.
Grades 4–7Tenths, hundredths and thousandths are different-sized units.
Grades 3–6Tenths, hundredths and thousandths are different-sized units.
Grades 3–6A part-to-part ratio is not the same as a part-to-whole fraction.
Grades 6–9A part-to-part ratio is not the same as a part-to-whole fraction.
Grades 6–9Simplifying an expression and solving an equation are different tasks.
Grades 6–9Simplifying an expression and solving an equation are different tasks.
Grades 6–9A drawing may not be to scale. Apply the stated properties.
Grades 5–9A drawing may not be to scale. Apply the stated properties.
Grades 5–9Check the axis, units, scale, population and sample size.
Grades 6–10Check the axis, units, scale, population and sample size.
Grades 6–10A function assigns exactly one output to each allowed input.
Grades 9–12A function assigns exactly one output to each allowed input.
Grades 9–12A derivative describes local rate; a definite integral describes signed accumulation.
Grades 11–12A derivative describes local rate; a definite integral describes signed accumulation.
Grades 11–12Compare common representations and check the whole before using a percentage.
Grades 4–8Use scale factors correctly for lengths, areas and volumes.
Grades 5–9Read the domain restrictions, especially around zero and fractional powers.
Grades 8–12Do not assume independence or mutual exclusion unless justified.
Grades 9–12Connect an equation, its graph and a table without guessing from appearance.
Grades 7–10A compact check card for elementary derivatives and definite integrals.
Grades 11–12Explain what a digit represents, not just which digit is largest.
Grades 1–3Explain what a digit represents, not just which digit is largest.
Grades 1–3Use the same whole and equal-size parts when interpreting fractions.
Grades 3–6Use the same whole and equal-size parts when interpreting fractions.
Grades 3–6Preserve equality and check a solution in the original equation.
Grades 6–9Preserve equality and check a solution in the original equation.
Grades 6–9Choose a numerical summary that matches the question and the data.
Grades 6–10Choose a numerical summary that matches the question and the data.
Grades 6–10Connect calculations to their meanings and domains.
Grades 11–12Connect calculations to their meanings and domains.
Grades 11–12Substitution reverses a chain rule; integration by parts reverses a product rule.
UniversitySubstitution reverses a chain rule; integration by parts reverses a product rule.
UniversityThe coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.
UniversityThe coefficient of (x−a)^n is f⁽ⁿ⁾(a)/n!, not just the derivative.
UniversityThe gradient collects first partial derivatives and determines directional derivatives through a dot product.
UniversityThe gradient collects first partial derivatives and determines directional derivatives through a dot product.
UniversityDraw or describe the region before changing integration order.
UniversityDraw or describe the region before changing integration order.
UniversityA basis both spans a vector space and is linearly independent.
UniversityA basis both spans a vector space and is linearly independent.
UniversityAv=λv requires a nonzero vector v.
UniversityAv=λv requires a nonzero vector v.
UniversityThe closest point in a subspace is found by an orthogonal residual.
UniversityThe closest point in a subspace is found by an orthogonal residual.
UniversitySolving a differential equation produces constants; initial conditions select a particular solution.
UniversitySolving a differential equation produces constants; initial conditions select a particular solution.
UniversityFor constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.
UniversityFor constant coefficients, an exponential trial function reduces the homogeneous ODE to a polynomial.
UniversityP(A∣B)=P(A and B)/P(B) requires P(B)>0.
UniversityP(A∣B)=P(A and B)/P(B) requires P(B)>0.
UniversityA density is integrated over a set; its value at a point is not the point probability.
UniversityA density is integrated over a set; its value at a point is not the point probability.
UniversityTruncation error comes from approximation; round-off comes from finite-precision arithmetic.
UniversityTruncation error comes from approximation; round-off comes from finite-precision arithmetic.
UniversityFor every ε>0 there exists N such that every n≥N satisfies the error bound.
Advanced universityFor every ε>0 there exists N such that every n≥N satisfies the error bound.
Advanced universityThe cutoff N may depend on both ε and the point x.
Advanced universityThe cutoff N may depend on both ε and the point x.
Advanced universityIn R^n, compactness is equivalent to being closed and bounded.
Advanced universityIn R^n, compactness is equivalent to being closed and bounded.
Advanced universityA derivative must have the same limit along every complex direction.
Advanced universityA derivative must have the same limit along every complex direction.
Advanced universityThe coefficient of (z−a)^(−1) in the Laurent expansion is the residue.
Advanced universityThe coefficient of (z−a)^(−1) in the Laurent expansion is the residue.
Advanced universityAn associative operation has an identity and an inverse for every element; commutativity is optional.
Advanced universityAn associative operation has an identity and an inverse for every element; commutativity is optional.
Advanced universityA structure-preserving map respects the relevant operation.
Advanced universityA structure-preserving map respects the relevant operation.
Advanced universityOn [−π,π], distinct integer sine or cosine modes are orthogonal under integration.
Advanced universityOn [−π,π], distinct integer sine or cosine modes are orthogonal under integration.
Advanced universityA sine mode for u_t=κu_xx decays as exp(−κk²t).
Advanced universityA sine mode for u_t=κu_xx decays as exp(−κk²t).
Advanced universityValues on a line segment lie below the corresponding chord of function values.
Advanced universityValues on a line segment lie below the corresponding chord of function values.
Advanced universityProve a base case and a step from n to n+1; neither alone suffices.
UniversityProve a base case and a step from n to n+1; neither alone suffices.
UniversityThe arithmetic average of numerical observations.
UniversityThe arithmetic average of numerical observations.
UniversityA possible result of a random process.
UniversityA possible result of a random process.
UniversityThe full group a study aims to describe.
UniversityThe full group a study aims to describe.
UniversityA value computed from sample data to approximate a population quantity.
UniversityA value computed from sample data to approximate a population quantity.
UniversityDifferentiate before substituting. The value at zero can hide a nonzero product-rule term.
UniversityDifferentiate before substituting. The value at zero can hide a nonzero product-rule term.
UniversityConvergence requires k > 0. The upper limit is a limit, not a substitution of a finite number.
UniversityConvergence requires k > 0. The upper limit is a limit, not a substitution of a finite number.
UniversityHere n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.
UniversityHere n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.
UniversityThe requested coefficient is not the nth derivative; divide by n factorial.
UniversityThe requested coefficient is not the nth derivative; divide by n factorial.
UniversityHold the other independent variable fixed while differentiating.
UniversityHold the other independent variable fixed while differentiating.
UniversityNormalize a direction vector before the dot product. Here (3/5,4/5) is already a unit vector.
UniversityNormalize a direction vector before the dot product. Here (3/5,4/5) is already a unit vector.
UniversityA positive determinant alone does not distinguish a local minimum from a maximum; inspect fₓₓ as well.
UniversityA positive determinant alone does not distinguish a local minimum from a maximum; inspect fₓₓ as well.
UniversityThe rectangular region allows two independent single-variable integrations.
UniversityThe rectangular region allows two independent single-variable integrations.
UniversityAn integral change of variables uses the absolute determinant; this question asks for its signed value.
UniversityAn integral change of variables uses the absolute determinant; this question asks for its signed value.
UniversityMatch each component to its own coordinate derivative.
UniversityMatch each component to its own coordinate derivative.
UniversityThe order of subtraction sets the orientation.
UniversityThe order of subtraction sets the orientation.
UniversityThis applies when the vector field is the gradient of a single-valued potential on the path.
UniversityThis applies when the vector field is the gradient of a single-valued potential on the path.
UniversityOff-diagonal entries do not enter this special-case determinant; this is not true for every matrix.
UniversityOff-diagonal entries do not enter this special-case determinant; this is not true for every matrix.
UniversityVerify the vector is nonzero. Distinct eigenvectors can have distinct eigenvalues.
UniversityVerify the vector is nonzero. Distinct eigenvectors can have distinct eigenvalues.
UniversityThe coefficient multiplying u differs from the scalar projection onto the unit vector u/||u||.
UniversityThe coefficient multiplying u differs from the scalar projection onto the unit vector u/||u||.
UniversityThe dimension is that of the domain, not the codomain. Rank cannot exceed the domain dimension.
UniversityThe dimension is that of the domain, not the codomain. Rank cannot exceed the domain dimension.
UniversityThe coefficient determinant is nonzero by construction.
UniversityThe coefficient determinant is nonzero by construction.
UniversityUse the initial condition after solving the homogeneous differential equation.
UniversityUse the initial condition after solving the homogeneous differential equation.
UniversityFor y′ + ay = b with constant a > 0, b/a is the equilibrium value.
UniversityFor y′ + ay = b with constant a > 0, b/a is the equilibrium value.
UniversityThis transform integrates over t ≥ 0. Substitute a positive s after transforming.
UniversityThis transform integrates over t ≥ 0. Substitute a positive s after transforming.
UniversityA single iteration is an approximation, not a guaranteed exact root. The derivative at x₀ must be nonzero.
UniversityA single iteration is an approximation, not a guaranteed exact root. The derivative at x₀ must be nonzero.
UniversityThis is an approximation to the integral. For a convex quadratic it overestimates the exact area.
UniversityThis is an approximation to the integral. For a convex quadratic it overestimates the exact area.
UniversityAssume a fixed number of independent trials with the same success probability.
UniversityAssume a fixed number of independent trials with the same success probability.
UniversityUse the base rate and both conditional likelihoods. This is an abstract classification model, not a medical test.
UniversityUse the base rate and both conditional likelihoods. This is an abstract classification model, not a medical test.
Universityλ is the mean count for the entire specified interval, not a rate without a time unit.
Universityλ is the mean count for the entire specified interval, not a rate without a time unit.
UniversitySxx must be positive. These are centered sums, so no extra division by sample size is required.
UniversitySxx must be positive. These are centered sums, so no extra division by sample size is required.
UniversityThis uses the real Fourier convention on [−π,π]. Distinct sine modes are orthogonal.
Advanced universityThis uses the real Fourier convention on [−π,π]. Distinct sine modes are orthogonal.
Advanced universityThe numerator is analytic near a and the pole is simple.
Advanced universityThe numerator is analytic near a and the pole is simple.
Advanced universityDifferentiate the analytic numerator once; its value p(a) is not the residue for a double pole.
Advanced universityDifferentiate the analytic numerator once; its value p(a) is not the residue for a double pole.
Advanced universityA bound uniform in x proves more than pointwise convergence at a chosen x.
Advanced universityA bound uniform in x proves more than pointwise convergence at a chosen x.
Advanced universityThe operation is addition modulo n. Element order is the least positive multiple giving zero.
Advanced universityThe operation is addition modulo n. Element order is the least positive multiple giving zero.
Advanced universityW must be a linear subspace of a finite-dimensional vector space V.
Advanced universityW must be a linear subspace of a finite-dimensional vector space V.
Advanced universityFor u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.
Advanced universityFor u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.
Advanced universityFor a string with fixed endpoints, length L, wave speed c, and positive integer mode n.
Advanced universityFor a string with fixed endpoints, length L, wave speed c, and positive integer mode n.
Advanced universityThis separable quadratic has positive curvature in each coordinate.
Advanced universityThis separable quadratic has positive curvature in each coordinate.
Advanced universityState the logarithm base; base 2 gives bits. The probabilities sum to one.
Advanced universityState the logarithm base; base 2 gives bits. The probabilities sum to one.
Advanced universityPractise improper integrals, series, gradients and multiple integrals, with explicit conditions and worked examples.
Develop matrix, vector-space and projection reasoning before using least squares.
Connect initial conditions, linear differential equations and numerical approximation with error and conditioning.
Work with conditional probability, normalization, densities and expectation.
Build epsilon proofs and distinguish pointwise convergence, uniform convergence and compactness.
Study groups, vector-space quotients and proof techniques with carefully scoped definitions.
Study complex analysis, residues, Fourier series, PDE boundary conditions and convexity through focused lessons.
Learn descriptive statistics, probability, sampling and inference with worked reasoning.