Home Mathematics Grade 6 Mathematics · Grade 6 Skills Multiplying integers: the sign rules Multiplies two integers using sign rules (−4 × −3 = 12, −4 × 3 = −12). Dividing integers and the inverse link Divides integers using the sign rules and checks by multiplying back (−12 ÷ 3 = −4 because −4 × 3 = −12). Signed fluency and mixed operations Recalls signed products and quotients mentally and chooses the right sign in mixed chains (−2 × 5 + 3 = −7). Negative fractions and decimals on the line Places and orders −¾, −0.5 and 0.2 on one number line (−¾ < −0.5). Adding and subtracting rational numbers Adds and subtracts signed fractions and decimals (−¾ + 0.25 = −½). Number families Sorts numbers into natural, integer, rational and "not a fraction" (π, √2: idea only). Density: always another number between Finds a number between two given fractions or decimals and explains why it never runs out (between 0.3 and 0.31 sits 0.305). Distance between numbers Finds distance as |a − b| across zero (from −7 to 4 is 11). Identify significant figures in a number Identifies the significant digits, ignoring leading zeros (0.004 73 has 3; 48 000 may have 2). Round to 1, 2 or 3 significant figures Rounds to 1, 2 or 3 s.f. (48 215 → 48 000; 0.004 73 → 0.0047). Choose sensible accuracy and estimate to 1 s.f. Chooses a sensible s.f. for a context (a measured length, a population) and estimates 4.8 × 21 as 5 × 20. Write large numbers in standard form Writes large numbers as A × 10ⁿ with 1 ≤ A < 10 and converts back (4 500 000 = 4.5 × 10⁶). Compare numbers in standard form Orders large numbers by exponent first, then by A (3 × 10⁸ > 9 × 10⁷). Zero and negative powers of ten as decimals Reads 10⁰ = 1, 10⁻¹ = 0.1, 10⁻² = 0.01 as place-value columns and moves between powers of ten and decimals. Order of operations with negatives Evaluates with negatives and brackets (−3 × (4 + 5) = −27; 2 − 3 × −4 = 14). Powers and roots inside expressions Evaluates with squares, cubes and roots in the order (2 + 3² = 11; √16 − 2³ = −4). Area model of column multiplication Explains 23 × 14 as four pieces on an area model and links it to the written method. The distributive law Uses a(b + c) = ab + ac to calculate (7 × 98 = 7 × 100 − 7 × 2) and to explain mental tricks. Area model of products of binomials Shows (x + 3)(x + 2) as four area pieces (introduce only). Multiplying powers with the same base Applies aᵐ × aⁿ = aᵐ⁺ⁿ with small whole-number powers (2³ × 2⁴ = 2⁷; introduce only). Instant recall of squares and cubes Recalls squares to 15² and cubes to 5³. Powers of 2 and doubling chains Recalls powers of 2 to 2¹⁰ (2⁸ = 256) and uses them as doubling anchors. Fermi estimates to 1 s.f. Breaks a big question into rough 1 s.f. parts and multiplies (hairs on a head ≈ 100 000). Checking by estimate Judges whether an answer is reasonable by rounding first (48 × 21 is about 1 000). Checking by inverse and by a second method Checks a result by undoing it or by solving another way, and spots which of three solvers is wrong. Square roots of perfect squares Finds √ of perfect squares (√144 = 12) and knows root undoes square. Cube roots of perfect cubes Finds ∛ of perfect cubes (∛64 = 4) and knows root undoes cube. Estimating non-perfect roots Places √20 between 4 and 5, then refines to 4.4 by trial. Prime factorisation with index notation Writes 360 = 2³ × 3² × 5 using a factor tree. HCF from prime factorisations Reads the HCF off shared primes (HCF of 36 and 60 is 2² × 3 = 12). LCM from prime factorisations Reads the LCM off all the primes at their highest power (LCM of 12 and 18 is 36). Roots of perfect squares and cubes via factorisation Finds √ and ∛ of larger perfect powers by halving or thirding the indices (√(2⁴ × 3²) = 12). Counting factors and Euclid's algorithm Counts factors from the exponents (2³ × 3² has 4 × 3 = 12) and finds an HCF by repeated remainders. Remainders, clocks and number patterns Uses remainders (weekday in 100 days, last digit of 7ⁿ), finds patterns in triangular, square, Fibonacci and perfect numbers, and uses the digital root (casting out nines) as a check on sums and products. Adding and subtracting signed fractions Adds and subtracts negative fractions (−½ + ¼ = −¼; ¾ − 1½ = −¾). Multiplying and dividing signed fractions Multiplies and divides negative fractions (−½ ÷ ¼ = −2), and signed decimals or mixed decimal-fraction pairs (−0.5 × 0.2 = −0.1; −1.2 ÷ 0.4 = −3). Fraction of a fraction Finds ¾ of ⅖ = 3/10 by folding or an area model. Complex fractions Simplifies a fraction over a fraction (½ ÷ ¾ shown as (1/2)/(3/4) = 2/3). Unit rates with fractions Finds a rate from fractional quantities (½ mile in ¼ hour is 2 miles per hour). Comparing fractions by cross-multiplication Compares 5/7 and 8/11 by cross-multiplying (55 vs 56). Fractions as terminating or recurring decimals Converts by division (⅐ = 0.142857…) and tests when a fraction terminates. 0.999… = 1 and the five faces of a fraction Proves 0.999… = 1 with ⅓ + ⅓ + ⅓ and switches among part-whole, number, quotient, ratio and operator views of ⅔. Percent as a multiplier Uses ×1.2 for +20% and ×0.85 for −15%, including above 100% (130% of 40 = 52). Repeated percentage change Chains multipliers (two 10% rises make ×1.21; +10% then −10% is not back to start). Reverse percentages Finds the original by dividing by the multiplier (after 20% off it is 48, so 60). Percent error Expresses an error as a percent of the true value (guessed 45, actual 50, error 10%). Pure arithmetic only. Ratios as 1 : n and equivalent ratios Rewrites 4 : 10 as 1 : 2.5 and compares ratios. Sharing in three-part ratios Shares a quantity in a 3-part ratio (£60 as pure number in 2 : 3 : 5 = 12, 18, 30), with mixed units. Direct proportion and the constant k Uses y = kx and finds k from one pair (3 books cost 12, so k = 4). Inverse proportion Recognises that more workers mean fewer days (xy constant: 6 workers take 4 days, so 3 take 8). Additive vs multiplicative reasoning Tells "add the same amount" from "scale by the same factor" (a recipe for 2 scaled to 5 by adding 3 is wrong). Writing expressions from words Turns words into expressions (3 more than twice n is 2n + 3). Terms, coefficients and like terms Names terms and coefficients and spots like terms (3a and 5a are like; 3a and 3b are not). Collecting like terms Simplifies 3a + 2a − a = 4a and 3a + 2b − a + b = 2a + 3b. Substituting positive values Evaluates formulas by substitution (perimeter P = 2l + 2w with l = 5, w = 3). Substituting negative values Substitutes negatives (a = −2 in 3a + 5 gives −1; a² gives 4). Expanding a single bracket Expands 2(x + 3) = 2x + 6 and 3(2x − 1) = 6x − 3. Function machines and working backwards Reads a one- or two-step machine and inverts it (×3 + 2 gives 17, so the input is 5). Solving one-step and two-step equations Solves 3x + 5 = 20 by balancing and checks the answer (x = 5). Forming equations from stories Chooses the equation, then solves ("3 pens and a £2 ruler cost £11" gives 3p + 2 = 11, in pure number). The nth term of an arithmetic sequence Finds the nth term from the common difference (5, 8, 11 … gives 3n + 2) and the 50th term. Taught here only. Parts of a circle and where π comes from Names radius, diameter and circumference, and finds the ratio C ÷ d ≈ 3.14 by rolling. Circumference Uses C = πd and C = 2πr (d = 10 cm gives about 31.4 cm). Area of a circle Uses A = πr² and tells r² from 2r (r = 3 gives about 28.3). Comparing circles Compares circle sizes in context (is one 16 cm pizza bigger than two 10 cm ones?). Volume of cuboids as layers Computes volume as base area × height by layers (3 × 4 × 5 = 60), and recognises a prism as a solid with a constant cross-section, naming that cross-section. Volume of triangular prisms Computes triangular-prism volume as triangle area × length. Volume of circular prisms (cylinder idea) Computes V = πr²h as base area × height. Speed, distance and time Computes speed = distance ÷ time and rearranges (120 km in 2 h is 60 km/h; 60 km/h for 3 h is 180 km). Converting compound units and density-style rates Converts km/h ↔ m/s (72 km/h = 20 m/s), uses g per cm³, keeps units. Angles on a line and at a point Uses 180° and 360° to find missing angles with a reason ("angles on a line sum to 180°"). Vertically opposite angles Knows opposite angles are equal and gives the reason. Corresponding and alternate angles Identifies and uses corresponding and alternate angles on parallel lines (the F and Z shapes). Co-interior angles and multi-step angle chains Uses co-interior angles (sum 180°) and chains two or more rules. Writing angle reasons Writes a short valid reason for each step and spots invalid ones. Perpendicular and angle bisectors Constructs a perpendicular bisector and an angle bisector with compass and ruler. Constructing a triangle from three sides Constructs a triangle from three sides (SSS) with compass arcs. Triangle inequality Knows two sides must sum to more than the third (3, 4, 8 will not close). SSS and congruent triangles (idea) Knows three equal sides force the same triangle (introduce only). Enlarging by a whole-number scale factor Enlarges a shape on a grid from a centre by scale factor 2 or 3. Fractional scale factors and finding the scale factor Enlarges by ½ and finds the scale factor between similar shapes (6 cm to 15 cm is ×2.5). Lengths scale by k, area by k² Knows doubling lengths quadruples the area (k = 3 gives 9 times). Plotting scatter graphs Plots paired data (height vs arm span) with sensible axes. Describing correlation Describes positive, negative or no correlation and spots outliers. Line of best fit by eye Draws a line of best fit by eye and uses it to estimate a value. Correlation is not cause Distinguishes correlation from causation in simple claims. Comparing two data sets with mean and range Defends a claim using mean and range (the team with the higher mean but larger range is less consistent). Weighted average (light intro) Finds a mean when some values count more (2 quizzes at 70 and 1 test at 85 weighted 1 : 1 : 2). Listing outcomes systematically Lists all outcomes of two events in an ordered list or table (coin and spinner). Sample-space tables Builds a 6 × 6 table for two dice and counts favourable cells (P(sum 7) = 6/36). Tree diagrams Draws a tree for two events and counts or multiplies along branches. Probability as a fraction Gives probabilities as simplified fractions and checks they sum to 1. Experiment versus theory Compares experimental frequencies with theory and knows more trials get closer. Fair games Decides whether a game is fair from the sample space and fixes it. Proving with diagrams Shows a number claim is true with a diagram (odd + odd = even with dot pairs). Proving with letters Proves with letters (n + (n+1) + (n+2) = 3(n+1); 2k is even). Counterexamples Refutes a false claim with one counterexample ("squares are bigger than the number": 0.5² = 0.25). Conjecture, test, decide Tests a pattern claim with several cases and decides: true, false or still unproven. Finding and classifying errors Finds an error and tags it as sign, place value, carry or unit. Explaining and correcting a method Writes what went wrong and fixes it (a student adds the unit but forgets to convert).