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AMP Mathematics

Practice with a reason for every step.

Placement estimates a starting point. Mixed practice responds to the learner’s work. Explanations make the mathematics visible.

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A starting point,
not a label.

A short placement cannot prove mastery. AMP uses the answers as initial evidence and updates its estimates through practice.

  1. 01

    Find a starting point.

    A short placement adapts to the answers. Separate estimates for different areas keep a strong skill from hiding a gap.

  2. 02

    Work at the edge of understanding.

    Mixed practice selects problems from the learner’s skill estimates and revisits earlier work.

  3. 03

    See why it works.

    Open the explanation or lesson when needed. Use a printable for work on paper.

Inside the product

Look at the mathematics.

These worked examples use AMP’s problem generator, figures, and explanations. They are samples, not a placement assessment.

AMP / Read a fractionSample

What fraction of the whole is shaded?

See the reasoning
  1. 2 of 3 equal parts are shaded.
  2. The fraction is 23\tfrac{2}{3}, equivalent to 23\tfrac{2}{3}.

Answer: 23\tfrac{2}{3}

AMP / Rectangle areaSample

A rectangle is 2 cm wide and 6 cm high. What is its area in square centimeters?

2 cm6
See the reasoning
  1. Each row contains one square centimeter per centimeter of width.
  2. 2×6=122 \times 6 = 12 square centimeters.

Answer: 12

AMP / Solve a linear equationSample

Solve for x: 12x+5=−4312x + 5 = -43.

See the reasoning
  1. Subtract 5: 12x=−4812x = -48.
  2. Divide by 12: x=−4x = -4.

Answer: −4-4

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Coverage, stated precisely

See what your learner can practice.

AMP includes selected skills from early number work through high-school mathematics. It is not a complete K–12 course or a complete SAT preparation program.

Grades K–233 problem types
  • Measure from a shifted start. Read a whole-unit interval from a nonzero ruler position.
  • Count objects. Count up to ten objects.
  • Add within twenty. Add two nonnegative whole numbers with a sum at most twenty.
  • Subtract within twenty. Subtract without a negative result.
  • Value of a digit. Find the value represented by a digit in a three-digit number.
  • Total from a bar chart. Add three category counts from a labeled bar chart.
  • Compare chart categories. Find how many more items one category contains than another.
  • Join two small groups. Model joining within ten with a ten-frame.
  • Take some away. Find a remainder in a removal story within ten.
  • Find the other part. Complete a two-part decomposition of a number through ten.
  • Fill a ten-frame. Find the empty places needed to complete ten.
  • One ten and extra ones. Compose eleven through nineteen from ten ones and extras.
  • Find what was added. Recover an unknown change from a start and total within twenty.
  • Find the starting amount. Recover a starting amount from a removal and remaining amount.
  • Make both sides equal. Find a missing number in an equation with arithmetic on both sides.
  • Combine three groups. Add three quantities with a total at most twenty.
  • Split an addend to make ten. Find the part of an addend left after completing a ten.
  • Split a subtraction at ten. Find the amount still to subtract after reaching ten.
  • Read tens and ones. Compose a two-digit number from ten rods and unit squares.
  • Change one ten. Find ten more or ten less without recounting ones.
  • Make a new ten. Add a one-digit number to a two-digit number by regrouping ones.
  • Subtract bundles of ten. Subtract whole tens with a nonnegative result.
  • Read hundreds tens and ones. Compose a three-digit number including zero-place cases.
  • Complete expanded form. Find a missing place-value amount in an expanded equation.
  • Exchange equal units. Regroup ten ones or ten tens while preserving value.
  • Add several groups. Combine three or four two-digit quantities.
  • Regroup a three-digit sum. Add within one thousand with a required ones-to-tens exchange.
  • Exchange to subtract. Subtract within one thousand including borrowing across a zero tens place.
  • Find the next count. Continue a count by fives tens or hundreds within one thousand.
  • Follow two changes. Solve a join-then-remove story within one hundred.
  • Continue counting. Continue by ones from any supported starting point through 120.
  • Count a unit array. Count unit squares in arrays no larger than five rows by five columns.
  • Pair the counters. Find whether zero or one counter remains after pairing a quantity through twenty.
Grades 3–563 problem types
  • Read the spaces between marks. Read an exact half- or quarter-unit distance.
  • Follow the whole border. Find the complete perimeter of a rectangle.
  • Find a missing side from the border. Recover a rectangle side from perimeter and one perpendicular side.
  • Find the missing factor. Recover a rectangle side from area and one perpendicular side.
  • Choose the right kind of unit. Distinguish boundary length from surface coverage.
  • Same area, different border. Compare perimeters of two rectangles with equal area.
  • Same border, different area. Compare areas of two rectangles with equal perimeter.
  • Rename with smaller units. Convert a whole count of a larger metric length unit to a smaller unit.
  • Rename with larger units. Convert a whole count of smaller metric units to an exact larger-unit value.
  • Bring lengths to one unit. Add lengths expressed in meters and centimeters.
  • One unit around the border. Convert perpendicular rectangle sides to one unit before finding perimeter.
  • Meet both design conditions. Choose a positive whole-number rectangle meeting area and perimeter constraints.
  • Can the design exist?. Decide whether given area and perimeter allow positive whole-number sides.
  • Compare what you must buy. Compare total costs under whole-pack and divisible-length purchase rules.
  • Multiplication facts. Multiply whole numbers from two through twelve.
  • Read a fraction. Express a shaded part of one whole as a fraction.
  • Add fractions. Add two positive fractions with possibly different denominators.
  • Rectangle area. Find area from perpendicular side lengths.
  • Division facts. Divide whole numbers into equal groups.
  • Add decimals. Add two amounts expressed in hundredths.
  • Compare fractions. Choose the larger of two distinct positive fractions.
  • Multiply fractions. Multiply two positive proper fractions.
  • Multiply decimals. Multiply two positive amounts expressed in tenths.
  • Order of operations. Evaluate an expression with parentheses, multiplication and addition.
  • Elapsed minutes. Find a duration up to two hours between times on the same morning.
  • Size of one equal part. Name one part of a whole partitioned into 2, 3, 4, 6, or 8 equal pieces.
  • Count fractional units. Count shaded equal units, including quantities greater than one.
  • Read a fraction location. Read a marked position on a zero-based rational number line through three wholes.
  • Rename a whole in fractional units. Find the numerator when a whole number is expressed using a specified denominator.
  • Rename with smaller pieces. Find a missing numerator when both counts scale by an integer factor.
  • Rename with larger pieces. Find a missing denominator when equal groups are collected.
  • Complete a fractional amount. Find the missing addend with a shared denominator.
  • Remove like fractional units. Subtract positive like-denominator amounts with a nonnegative result.
  • Count the parts in a mixed amount. Find the numerator of a mixed number renamed with its existing denominator.
  • Find full wholes in an amount. Count complete wholes in a positive improper fraction.
  • Find the part beyond full wholes. Find the fractional remainder after removing complete wholes.
  • Subtract through a whole. Subtract mixed numbers with like denominators, requiring one whole to be regrouped.
  • Subtract using a shared unit. Subtract distinct-denominator positive fractions with a nonnegative difference.
  • Measure a fractional gap. Find the distance between two marked rational positions on a common-unit line.
  • Several groups of a fraction. Find the total of a whole number of equal fractional-liter portions.
  • Share whole quantities equally. Share a whole number of liters among a whole-number group, including improper shares.
  • Share one fractional unit. Divide a unit fraction of a liter among a positive whole-number group.
  • Count unit-fraction groups. Count unit-fraction-liter containers in a whole number of liters.
  • Scale a whole amount. Take a proper fraction of a positive whole-liter amount.
  • Different pieces, same amount. Select an equivalent name for a proper fraction among scaling and additive misconceptions.
  • Count complete kits. Count complete fixed-size kits from a whole-item stock, including exact division.
  • Make room for everyone. Find the smallest whole number of vehicles with enough seats, including exact division.
  • Find unused items. Count items left after making as many complete kits as possible.
  • Share equally among trays. Find items per share from a total and a given number of equal shares.
  • Measure equal pieces. Count fixed-length pieces in an exactly divisible whole-centimeter length.
  • Count equal trays. Find a total from equal visible groups of items.
  • Find the group size. Find an unknown factor from a group count and a total.
  • Find the starting total. Find the unknown dividend in a whole-number division equation.
  • Use one part of an array. Find a partial product represented by a disjoint row region of an array.
  • Count groups inside groups. Find the total in crates containing equal bags of equal items.
  • Use zero and one groups. Find products when a group count or group size is zero or one.
  • Find a compared amount. Find a quantity described as a whole-number multiple of another.
  • Find how many times as many. Find the multiplicative comparison factor from two whole-number quantities.
  • Combine deliveries and use some. Find remaining stock after equal deliveries and a known use.
  • Find change after equal purchases. Find whole-dollar change after buying several equally priced items.
  • Find the other factor. Complete one specified factor pair for a whole number through one hundred.
  • Find the tens partial product. Find the tens contribution in a two-digit by one-digit product.
  • Compare neighboring products. Find the change when an equal-row array gains one row.
Grades 6–857 problem types
  • Evaluate a negative power. Evaluate a nonzero rational base raised to a negative integer exponent.
  • Multiply powers with the same base. Combine integer exponents in a product with a shared nonzero base.
  • Divide powers with the same base. Combine integer exponents in a quotient with a shared nonzero base.
  • Raise a power to a power. Combine nested integer powers of a nonzero base.
  • Median of data. Order a finite dataset and find its middle value or the mean of its two middle values.
  • Range of data. Find the distance between the greatest and least values in a finite dataset.
  • Interquartile range. Find quartiles by the stated median-of-halves convention and subtract the lower quartile from the upper.
  • Find a missing leg. Find either leg of an integer right triangle from its hypotenuse and other leg.
  • Recognize a right triangle from its sides. Use the Pythagorean theorem and its converse to decide whether three given triangle sides form a right triangle.
  • Solve a linear equation. Solve an equation with an integer solution.
  • Mean of a dataset. Compute an exact mean of five observations.
  • Pythagorean theorem. Find a hypotenuse from a scaled integer right triangle.
  • Divide fractions. Divide a positive fraction by another positive fraction.
  • Equivalent ratios. Scale a ratio to find a missing quantity.
  • Percent of a quantity. Find an exact percentage of a whole-number quantity.
  • Add signed integers. Combine positive and negative integer changes.
  • Evaluate powers. Evaluate a whole-number base raised to a nonnegative integer exponent.
  • Slope through two points. Find slope as change in y divided by change in x.
  • Probability of one draw. Find a probability from equally likely objects.
  • Arithmetic sequences. Find a term given the first term and a common difference.
  • Proportional cost. Use a constant unit price to find a total cost.
  • Add and subtract signed fractions. Add or subtract two signed fractions with different denominators.
  • Multiply and divide signed fractions. Multiply or divide two signed fractions and reduce the result.
  • Simplify a complex fraction. Simplify a fraction whose numerator and denominator contain fractions.
  • Circle circumference. Find an exact circumference from a radius or diameter.
  • Circle area. Find an exact circle area from a radius or diameter.
  • Find the percent change. Compare an increase or decrease with its original positive quantity, including changes in percentage-valued shares.
  • Apply a percent change. Find the final quantity after one exact percentage increase or decrease.
  • Recover the original quantity. Undo one percentage increase or decrease using its nonzero multiplier.
  • Solve a linear inequality. Find an exact half-line solution set with signed rational coefficients and all four comparison signs.
  • Solve an inequality with grouped terms. Distribute, collect variable terms on both sides, and solve an inequality with an exact rational boundary.
  • Classify an inequality after cancellation. Determine whether an inequality with cancelling variable terms holds for every real number or none.
  • Solve for both coordinates. Find the exact ordered pair satisfying two linear equations, including fractional coordinates and axis-aligned lines.
  • Count a linear system’s solutions. Distinguish one, no, and infinitely many shared ordered pairs in two linear equations.
  • Read a coordinate. Read one coordinate of a labeled point on a unit grid.
  • Slope from a graph. Find exact slope from two labeled lattice points.
  • Distance on a number line. Find the nonnegative distance between two signed integers.
  • Count full and partial groups. Measure a rational total in fractional-liter container units, allowing partial groups.
  • Share a fractional total. Share a positive non-unit fractional-liter amount among a whole-number group.
  • Put a value into a rule. Evaluate a signed linear expression at an integer or half-integer input.
  • Multiply into parentheses. Recognize an equivalent expansion of a signed grouped expression.
  • Collect the x terms. Find only the coefficient of x after combining like terms.
  • Unwrap an equation. Solve a grouped linear equation with an exact rational result.
  • Undo a negative multiplier. Solve an equation with a negative coefficient.
  • Undo a fraction multiplier. Solve an equation whose x coefficient is a noninteger fraction.
  • Bring the unknowns together. Solve a unique-solution equation with variables on both sides.
  • Test a proposed answer. Select the correct substitution check of a proposed integer value.
  • Look past the step sizes. Find a constant rate from linear observations with unequal input steps.
  • Find the output at zero. Recover a zero-input output from a stated rate and another observation.
  • Extend a linear pattern. Predict an output from two points under an explicit linear assumption.
  • One input, one output?. Identify a finite relation with exactly one output for each input.
  • Find a proportional rule. Recognize a real-valued rule y=kx among affine rules with nonzero intercepts.
  • Find when two plans meet. Find the nonnegative input where two affine cost rules agree.
  • Find the first lower price. Find the first whole input at which a fixed-fee plan is strictly cheaper.
  • Read a distance graph. Interpret a level or falling distance-from-home segment without inferring a route or motion speed.
  • One, none, or every number?. Classify the solution set of a linear equation with positive integer coefficients on both sides.
  • Find the graph’s rule. Match an increasing line with positive integer slope and intercept to its equation.
Grades 9–1218 problem types
  • Quadratic minimum. Find the minimum value of a quadratic in vertex form.
  • Geometric sequences. Find a term of a geometric sequence from its first term and ratio.
  • Logarithms. Evaluate a logarithm whose argument is an integer power of its base.
  • Solve a linear system. Find x in a system with a unique integer solution.
  • Evaluate a function. Evaluate a quadratic expression at an integer input.
  • Solve a factored quadratic. Find every real solution of a quadratic written as a product of linear factors.
  • Solve a quadratic by factoring. Factor an expanded quadratic and identify its complete real solution set.
  • Count real quadratic solutions. Use the discriminant to count distinct real solutions without calculating the roots.
  • Add polynomials. Combine like terms in a sum of polynomials.
  • Subtract polynomials. Distribute a subtraction sign and combine like terms.
  • Multiply polynomials. Distribute every term and combine like terms in a polynomial product.
  • Simplify a square root. Extract square factors and give an exact square-root expression.
  • Add and subtract square roots. Simplify radical terms, then combine terms with the same radical part.
  • Multiply square roots. Multiply signed coefficients and nonnegative radicands, then simplify.
  • Rationalize a denominator. Remove a square root from a single-term denominator using an equivalent fraction.
  • Evaluate a fractional power. Evaluate a positive rational base raised to a fractional exponent.
  • Arc length. Find an exact arc length from a radius or diameter and a central angle in degrees.
  • Sector area. Find an exact sector area from a radius or diameter and a central angle in degrees.
Catalog inventory and limitations
171problem types in the catalog
94skills with a problem type
97lessons in the catalog

These are content inventory counts, not evidence of learning outcomes or complete standards coverage. Some practice is taught through book-page bindings. The coverage audit does not certify full coverage of any standard.

The catalog lesson count excludes the per-problem lessons attached by the practice interface.

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