Foundations

Count each object once, explain the final count, and distinguish quantity, position and identifying labels. Make zero’s role explicit.

Use number-line positions to compare whole numbers. Read the scale, recognise equality, and distinguish numerical distance from drawing size.

Combine separate counts, remove a part, and find differences or missing parts. Connect objects, equations and number-line moves, then use tens and ones to regroup without changing value.

Count equal groups, distinguish sharing from making groups, and check with multiplication. Split calculations into parts; interpret zero and remainders in context.

Choose a whole, count equal fractional units, and connect sharing with division. Place fractions on the number line, including values beyond one, and explain what their numerator and denominator mean.

Distinguish part-to-part ratios, fractions of a whole and rates with units. Scale quantities together, compare unit rates, and explain when a relationship is proportional. Check zero cases, starting amounts and whether a result fits its context.

Use symbols to name numbers or quantities, describe their allowed values, and evaluate expressions by replacing every occurrence consistently. Preserve grouping, signs, fractions and units, and distinguish a calculated value from solving an equation or proving two rules equivalent.

Read equality as a relationship, find the permitted values that make an equation true, and justify one-step solutions by reversible operations on both sides. Work exactly with negative numbers and fractions, recognize all-solution and no-solution cases, and check the original equation, units and context.

Repeated division computes a greatest common divisor.

Positive integers greater than one have unique prime factors.

Congruence identifies integers differing by a multiple of n.

Coprime moduli specify one combined residue.

A group has associative composition, identity and inverses.

Homomorphisms preserve the group operation.

A ring combines addition with distributive multiplication.

A field allows division by every nonzero element.

Nonconstant complex polynomials factor into linear terms.

A subgroup H⊆G is a group under G's restricted operation.

Left cosets gH={gh:h∈H} partition G.

A subgroup N is normal when gNg⁻¹=N for every g∈G.

A two-sided ideal I of R is an additive subgroup absorbing multiplication by R on either side.

For finite G and subgroup H, |G|=[G:H]|H|.

For a group homomorphism φ:G→H, ker φ is normal and G/ker φ is isomorphic to im φ by g ker φ↦φ(g).

The subgroup generated by a is ⟨a⟩={aᵏ:k∈Z}; its order is the smallest positive n with aⁿ=e, or infinite if none exists.

Over a field F, f=gq+r has unique q,r for nonzero g, with r=0 or deg r<deg g.

A nonconstant polynomial over F is irreducible if it cannot factor into two positive-degree polynomials over F.

A unit has a multiplicative inverse.

For integral domain R, fractions a/b with b≠0 are equivalence classes under a/b=c/d iff ad=bc.

For a commutative ring with identity and proper ideal I, I is prime iff ab∈I implies a∈I or b∈I, equivalently R/I is an integral domain.

A field has characteristic zero if no positive n satisfies n·1=0; otherwise its characteristic is a prime p.