Understand the math
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38 guides
Grade 12 · Calculus and continuous change
Build a quantity from rate-times-duration pieces, interpret the limiting integral, and use a derivative to find a stationary member of a family.
Skills: Understand a limit as local or end behavior of a function · Interpret definite integrals as accumulation and signed area · Use derivatives for motion, approximation, related rates, and optimization
Grade 10 · Grade 11 · Grade 12 · Number and quantity
Interpret a + bi as both an algebraic number and a point, use i squared equals negative 1, and measure its distance from zero.
Skills: Perform arithmetic with complex numbers
Grade 1 · Grade 2 · Grade 3 · Operations and computation · Number and quantity
Doubling makes two equal copies; halving reverses that action when a quantity can be split into two equal whole-number groups.
Skills: Double and halve whole-number quantities · Determine whether a group up to 20 is even or odd
Grade 2 · Grade 3 · Grade 4 · Measurement and units
Treat time as a distance between two moments, using friendly jumps in hours and minutes instead of fragile clock arithmetic.
Skills: Tell time to the minute and solve elapsed-time problems
Grade 2 · Grade 3 · Grade 4 · Operations and computation
Use multiplication to combine equal groups and division to reverse the relationship by finding a group size or a number of groups.
Skills: Interpret a product as equal groups or a rectangular array · Interpret whole-number quotients as sharing or grouping
Grade 8 · Grade 9 · Algebraic structure and equations
Treat equations as balanced statements, transform complete sides together, and eliminate a variable without losing the original system.
Skills: Solve linear equations in one variable · Solve systems of two linear equations algebraically or graphically
Grade 5 · Grade 6 · Grade 7 · Grade 8 · Algebraic structure and equations
Read what each symbol means, substitute known values carefully, follow the operation order, and check whether the result fits the situation.
Skills: Evaluate expressions for specified variable values
Grade 4 · Data, statistics, and probability
Expected value describes the average outcome over many repetitions, so it can be meaningful even when it is not a possible result of one trial.
Skills: Interpret a non-whole expected value
Grade 12 · Algebraic structure and equations · Number and quantity · Functions and change · Calculus and continuous change · Discrete mathematics and logic · Measurement and units · Data, statistics, and probability
Translate each formula before calculating, then use repeated totals as checksums that can expose and locate a corrupted entry.
Skills: Perform matrix operations and interpret matrices as transformations or data structures · Represent and operate on vectors geometrically and by components · Understand logarithms as inverses of exponential functions · Perform arithmetic with complex numbers · Compute derivatives using standard rules · Count permutations and combinations with constraints · Find basic antiderivatives and indefinite integrals · Convert measurement units using ratio reasoning · Use expected value to compare decisions and games · Understand a limit as local or end behavior of a function · Use overlapping checks to locate an error
Grade 2 · Grade 3 · Grade 4 · Number and quantity
A fraction names equal-sized parts of one whole; the denominator names the partition and the numerator counts selected parts.
Skills: Understand a/b as a copies of unit fraction 1/b · Find one tenth of a quantity
Grade 10 · Grade 11 · Grade 12 · Functions and change
Track functions as an ordered input-output pipeline, then undo the pipeline by applying inverse functions in reverse order.
Skills: Compose functions and interpret composition · Find and verify inverse functions on appropriate domains
Grade 6 · Algebraic structure and equations · Ratios, rates, and proportional reasoning · Number and quantity
Follow a repeated halving rule, compare each amount with a percentage of the original whole, and explain why a tiny positive amount is still different from zero.
Skills: Identify and explain arithmetic patterns · Interpret percent as a rate per 100 and solve basic percent problems · Reason about whether multiplication by a fraction enlarges or shrinks a quantity
Grade 1 · Grade 2 · Grade 3 · Number and quantity
Use the chart's fixed row and column structure: horizontal moves change by 1, while vertical moves change by 10.
Skills: Use hundred-chart structure
Grade 11 · Grade 12 · Algebraic structure and equations
Use the equation T(v)=0 to find what a linear transformation erases, then connect the kernel to output collisions and one-to-one behavior.
Skills: Perform matrix operations and interpret matrices as transformations or data structures · Connect matrix operations with systems of linear equations · Interpret a matrix as a linear transformation · Solve systems of two linear equations algebraically or graphically
Grade 9 · Grade 10 · Grade 11 · Grade 12 · Number and quantity
Compute with congruences, combine compatible remainder clues, and recognize when familiar zero-product reasoning fails.
Skills: Compute and reason with congruences modulo n
Grade 2 · Grade 3 · Grade 4 · Operations and computation
Turn a longer situation into a short chain of questions, keep track of what each number means, and check the final result against the story.
Skills: Solve one- and two-step addition/subtraction stories within 100
Grade 12 · Algebraic structure and equations
Construct left and right unit isomorphisms for a chosen product, verify naturality, and distinguish a monoidal unit from a terminal object.
Skills: Verify natural transformations · Classify monoidal unit objects
Kindergarten · Grade 1 · Grade 2 · Grade 3 · Number and quantity · Geometry and spatial reasoning
Track an ordered position by fixing a direction, using equal steps, and distinguishing a location from the distance traveled to reach it.
Skills: Represent whole numbers, sums, and differences within 100 on a number line · Use relative-position words to describe shapes and objects
Grade 9 · Grade 10 · Grade 11 · Grade 12 · Algebraic structure and equations · Number and quantity
Test an operation for closure, commutativity, identity, and inverses, then see how complex rotations form a small cyclic group.
Skills: Analyze a binary operation and its properties · Reason with group axioms and symmetry operations · Represent complex numbers in polar form and connect multiplication to rotation
Grade 5 · Grade 6 · Grade 7 · Grade 8 · Grade 9 · Grade 10 · Grade 11 · Grade 12 · Discrete mathematics and logic
When more objects must fit into fewer categories, prove that a collision is unavoidable—and identify exactly what the collision gives you.
Skills: Use an intuitive pigeonhole argument
Grade 1 · Grade 2 · Grade 3 · Number and quantity
Understand a three-digit number as equivalent collections of hundreds, tens, and ones that can be regrouped without changing the total.
Skills: Compose and decompose whole numbers to 1,000
Grade 12 · Discrete mathematics and logic
Enumerate compatible subobjects of a terminal presheaf, order them as truth values, and compute pseudocomplementary negation.
Skills: Enumerate global truth values in finite presheaf models · Compute Heyting negation and test classical laws
Grade 2 · Grade 3 · Grade 4 · Measurement and units
Keep two measurements distinct: area counts square units covering the inside, while perimeter measures length around the boundary.
Skills: Distinguish and compare area and perimeter · Understand area as the number of unit squares covering a region · Find rectangle area by multiplying side lengths · Find perimeters and unknown side lengths of polygons
Grade 9 · Grade 10 · Grade 11 · Algebraic structure and equations · Discrete mathematics and logic
Read a next-step rule, generate a sequence, and use a safe-zone argument to control every future value without calculating forever.
Skills: Define and analyze simple recursive processes and recurrence relations · Rewrite expressions to reveal zeros, extrema, growth, or other features
Grade 6 · Grade 7 · Grade 8 · Functions and change
Use F(F(x))=F(x) to show that every later repeat stabilizes, then use reversibility to decide whether the map must be the identity.
Skills: Reason about repeat-stable iteration
Grade 1 · Grade 2 · Grade 3 · Grade 4 · Discrete mathematics and logic
Find the smallest repeating unit, locate positions within the cycle, and count complete cycles before handling any leftover positions.
Skills: Identify a position in a repeating cycle · Count categories in a repeating pattern
Grade 5 · Grade 6 · Grade 7 · Discrete mathematics and logic
Read a one-way network, trace every possible route from a start, and distinguish a local incoming arrow from a complete path back to the start.
Skills: Reason about networks, routes, vertices, and edges
Grade 3 · Grade 4 · Grade 5 · Discrete mathematics and logic
Turn a finite list of possibilities into a complete proof by showing why each case fails or why one remaining case must work.
Skills: Use systematic elimination of cases
Grade 2 · Grade 3 · Data, statistics, and probability
Use the graph key as a multiplication rule: each picture may represent several objects rather than just one.
Skills: Draw and interpret scaled picture and bar graphs
Kindergarten · Grade 1 · Number and quantity · Operations and computation
Build early number sense by matching one count word to each object, recognizing the total, comparing collections, and composing a number from smaller parts.
Skills: Count objects with one number word per object · Use the last count word as the total · Count up to 20 arranged objects and up to 10 scattered objects · Compare groups as greater, less, or equal · Compose and decompose within 5
Grade 2 · Grade 3 · Grade 4 · Grade 5 · Operations and computation · Algebraic structure and equations
Translate what quantities mean before choosing operations, then use grouping symbols to preserve which quantities belong together.
Skills: Represent joining and separating with objects, drawings, or equations · Understand the equal sign as a relation, not an instruction · Use parentheses, brackets, and braces in numerical expressions
Grade 4 · Grade 5 · Discrete mathematics and logic
Organize a finite set of possibilities so every case is counted once and the finished list supports a complete conclusion.
Skills: Count outcomes systematically without omission or duplication
Grade 11 · Grade 12 · Algebraic structure and equations
Track the direction and uniqueness of arrows into a terminal object, then use those forced arrows to compare any two terminal objects.
Skills: Recognize and prove terminal objects · Terminal objects are uniquely isomorphic
Grade 2 · Grade 3 · Number and quantity · Operations and computation
Help a child add three-digit numbers accurately, compare the results, and keep a stated limit active while choosing a best total.
Skills: Compare three-digit numbers with place-value reasoning and symbols · Add and subtract fluently within 1,000
Grade 8 · Grade 9 · Grade 10 · Number and quantity
Identify the integer units 1 and -1, recognize factors that differ by a unit, and understand the exact qualifiers in uniqueness of factorization.
Skills: Explain factorization uniqueness up to units and associates
Grade 4 · Measurement and units
Connect the volume formula to counting unit cubes: find cubes in one rectangular layer, then multiply by the number of equal layers.
Skills: Understand volume as an attribute measured in unit cubes · Find rectangular-prism volume using multiplication formulas
Grade 4 · Grade 5 · Operations and computation
Keep place value organized while multiplying whole numbers, dividing by a two-digit divisor, and interpreting a quotient with a remainder in context.
Skills: Fluently multiply multidigit whole numbers with the standard algorithm · Divide up to four-digit dividends by two-digit divisors · Solve multistep whole-number problems and interpret remainders
Grade 8 · Grade 9 · Number and quantity
Extend exponent and product rules to their zero boundary by preserving the multiplicative identity.
Skills: Use and explain properties of integer exponents
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