Home Mathematics Grade 5 Mathematics · Grade 5 Skills Reading and writing numbers into the billions Reads and writes a number in words and digits with the right place groupings (8,431,000 is eight million four hundred thirty-one thousand). Feeling how big a billion is Explains that a billion is a thousand millions by building it from stacked thousands and millions. Comparing and ordering large numbers Orders 4,205,000, 4,250,000 and 420,500 by looking at the highest differing place. Rounding large numbers in a story Rounds a number to a sensible place for the story (8,431,000 people is about 8 million). Negative numbers: order and absolute value Orders −7, 2, −3, 0 and says that −4 is 4 away from zero (absolute value). Adding and subtracting integers on a number line Works out −3 + 5 and 2 − 6 by hopping on a line. One number line for every kind of number Places 0.7, ¾, −1.5 and 3 together on one line in the right spots. Multiply and divide decimals by powers of ten Shifts digits to work out 4.7 × 100 = 470 and 3,500 ÷ 1000 = 3.5 and explains the shift. Multiply and divide by 0.1 and 0.01 Sees that ÷ 0.1 acts like × 10 and × 0.01 like ÷ 100 (6 × 0.1 = 0.6). Read and write powers of ten Writes 1,000 = 10³ and reads 10⁵ as 100,000. Round decimals to a chosen place Rounds 3.476 to 1 d.p. (3.5) and to 2 d.p. (3.48). Choose sensible rounding for a situation Chooses how far to round a length, a crowd or a recipe and says why. Name digit values in decimals to thousandths Says what each digit of 3.472 is worth and writes 3.47 = 3 + 4/10 + 7/100. Order and place decimals to thousandths Orders 0.305, 0.35 and 0.3 and places them on a number line zoomed to thousandths. Multi-digit multiplication by a written method Multiplies 3,482 × 26 with the standard algorithm and checks by estimate. Multiplying and dividing decimals by whole numbers Works out 3.45 × 6 and 7.2 ÷ 4 with a written method. Decimal × decimal Uses scaling to see 0.3 × 0.4 = 0.12 (3 × 4 = 12, then ÷ 100). Long division by a 2-digit number Divides 925 ÷ 25 with a written method. Expressing the remainder to fit the story Gives 50 ÷ 8 as 6 r 2, 6 ¼ or 6.25, whichever the story needs. Decimal ÷ decimal (introduce) Scales both numbers by 10 so 1.2 ÷ 0.3 becomes 12 ÷ 3 = 4. Order of operations Evaluates 2 + 3 × 4 = 14 and 18 − 6 ÷ 3 = 16. Brackets and exponents in expressions Evaluates 2 + 3² × 4 and (2 + 3)² × 4 and sees the brackets change the answer; writes a bracketed expression for a story (3 × (4 + 5)). Choosing a method Picks mental, written or calculator for 4,000 − 1,998 vs 6,284 ÷ 17 and checks against an estimate. Column addition and subtraction of whole and decimal numbers Works out 12.5 + 3.482 and 20 − 7.35 in columns, lining up the decimal point. Mental 2-digit × 1-digit Works out 48 × 6 by splitting into 40 × 6 + 8 × 6. Multiples of 10 and 100 mentally Calculates 70 × 300 = 21,000 from 7 × 3. Simple decimals mentally Calculates 0.6 + 0.7 = 1.3 and 2.5 × 4 = 10 in the head. Deriving new facts from known ones Gets 4 × 0.3 = 1.2 from 4 × 3 = 12 and ⅜ of 24 from ⅛ of 24. Common percentages of an amount Finds 50%, 10% and 1% of 360 in the head (180, 36, 3.6). 25% and 75% by halving and quartering Finds 25% of 80 as half of half, and 75% as 3 quarters. Estimating before calculating Rounds 4.8 × 52 to 5 × 50 and expects about 250 before working it out. Spotting answers that are out by a factor of ten Knows that 4.8 × 52 cannot be 24.96 or 2,496 by checking the magnitude. Common factors Lists the factors of 12 and 18 and picks out the common ones (1, 2, 3, 6). Primes below 100 Lists the primes below 100 and tests a number for primality (is 91 prime? 7 × 13, no). Prime factorisation (introduce) Builds a factor tree for 60 = 2 × 2 × 3 × 5. Highest common factor and co-prime pairs Finds the HCF of 12 and 18 (6) and spots that 8 and 15 are co-prime. Common multiples and lowest common multiple Finds that two buses leaving together meet again after 12 minutes (LCM of 4 and 6). Divisibility tests for 2, 3, 4, 5, 6, 9, 10 Tells without dividing that 1,236 is divisible by 3 and 4 but not by 9. Divisibility by 8 and 11 (introduce) Explores that 1,232 is divisible by 8 (last three digits) and 121 by 11 (alternating sum). Square and cube numbers Knows squares to 12² and cubes to 5³ and reads ² and ³ notation. Triangular, square and cube number patterns Sees 1, 3, 6, 10 grow by 2, 3, 4 and that two neighbouring triangular numbers add to a square. Finding a common denominator Rewrites ⅔ and ¾ as 8/12 and 9/12. Adding and subtracting unlike fractions Works out ⅔ + ¾ = 1 5/12 and ⅚ − ¼ = 7/12. Adding and subtracting mixed numbers Works out 3 ½ − 1 ¾ = 1 ¾ by regrouping. Comparing and ordering fractions above 1 Orders 7/4, 1 ½ and 5/3 and 2 ⅙. Fraction as division Reads ¾ as 3 ÷ 4 and writes 5 ÷ 2 as 2 ½. A fraction times a whole number Works out ⅗ × 20 = 12 and 4 × ⅔ = 2 ⅔. Fraction × fraction with an area model Shades ⅔ of ¾ on a grid to see ⅔ × ¾ = ½ and knows the result is smaller than either. Unit fraction ÷ whole, whole ÷ unit fraction Works out ⅓ ÷ 2 = ⅙ and 4 ÷ ⅓ = 12. Fraction ÷ fraction (introduce) Sees ¾ ÷ ¼ = 3 and 4 ÷ ⅔ = 6 by counting how many fit. Finding the whole from a part Works out ⅗ of ? = 30 is 50. Simplifying fractions Simplifies 12/18 to ⅔ by dividing top and bottom by the common factor 6. Converting fractions, decimals and percent Converts ⅛ = 0.125 = 12.5% by dividing, reads percent as "out of 100" on a 100-square (37% = 37/100 = 0.37) and moves between forms for common values. Ordering a mix of fractions, decimals and percent Orders ⅗, 0.58 and 61% from smallest to largest. Percent of an amount Finds 15% of 80 = 12 as 10% + 5%. Percentage increase and decrease Works out a 25% discount on €40 (€30) and a 10% rise on 60 (66). What percent one quantity is of another Says 12 of 48 is 25%. Finding the whole from a percentage Works out that 45 is 30% of 150. Ratio notation and simplifying Writes 6 red to 4 blue as 6:4 = 3:2 and writes a:b:c for three parts. Sharing a quantity in a ratio Shares 20 in the ratio 3:1 (15 and 5). Proportion by unit value and scaling Solves "3 cost 12, what do 7 cost?" (28). Unit rates and speed Finds that 150 km in 3 h is 50 km/h and uses it to predict a distance. Converting units by ratio reasoning Converts 72 km/h to 20 m/s and 250 cm to km. Writing expressions with a letter Writes n + 5 for "5 more than a number" and 3 × n for "3 times a number". Substituting into a formula Uses P = 2 × (l + w) with l = 7, w = 4 (22). One-step equations and checking Solves n + 7 = 15 and 4 × n = 36 and checks by substituting back. Simple two-step equations Solves 2n + 3 = 13 by undoing the steps (n = 5). Pairs that satisfy an equation Finds pairs with a + b = 10 and a > b (9 and 1, 8 and 2, ...). Stating a sequence rule by position Says "4 × position + 1" for 5, 9, 13, 17. Using a rule to predict far terms Uses the rule to find the 50th term (201). Two patterns from two rules Builds two sequences (add 2 from 0, add 4 from 0) and pairs the terms as (2, 4), (4, 8). Relationships between paired patterns Sees each term in one pattern is twice its partner and says so. Converting metric units with decimals Converts 2.35 km = 2,350 m and 450 ml = 0.45 l. Choosing units and two-step conversions Chooses km, m or cm for a job and converts mm to m through cm. Rough customary equivalents (non-gating) Knows an inch is about 2.5 cm, a foot about 30 cm, a mile about 1.6 km, a pound about 450 g and a pint about 570 ml. Personal benchmarks Uses own hand span and pace to estimate a room (about 6 paces) and own height to estimate a door or a shelf. Area of a parallelogram Finds the area of a parallelogram as base × height (6 × 4 = 24). Area of a triangle Finds the area of a triangle as half of a base × height rectangle. Area and perimeter of compound rectilinear shapes Finds the area and perimeter of an L-shape by splitting it into rectangles. Volume of cuboids and compound cuboids Works out V = l × w × h in cm³ and m³ (4 × 3 × 2 = 24 cm³), including two blocks joined. Volume and capacity: ml and cm³ Knows 1 ml = 1 cm³ and 1 l = 1,000 cm³ (a 10 cm cube holds 1 l). Converting units of time Converts 3 weeks to days and 2 days to hours. Decimal and mixed time units Converts 2.5 h = 150 min and 135 min = 2 h 15 min. Reading timetables and finding durations Finds that a 08:45 bus arriving 10:20 takes 1 h 35 min. Multi-leg journeys across midnight Plans a trip with two legs and a wait and an arrival after midnight. Time in another time zone (introduce) Says that when it is 10:00 in Paris it is 09:00 in London. Scheduling across time zones (introduce) Picks a call time that is a sensible hour in two cities. Comparing offers by unit price Finds 2 for €3 (€1.50 each) is better than €1.75 each. Prices after a percentage discount Works out a 20% discount on €15 (€12). Staying within a budget Chooses items that total at most €20 after a discount. Checking a receipt Finds the wrong line on a till receipt (bad discount, wrong unit price). Shape hierarchy Says every square is a rectangle and every rectangle is a parallelogram. Properties that define quadrilaterals Tells a kite, a rhombus and a trapezium apart by sides, parallels and diagonals. Parts of a circle Names centre, radius, diameter and circumference. Radius and diameter relationship Knows d = 2r, so a circle of radius 4 cm has diameter 8 cm. Angles in a triangle Finds the missing angle in a triangle with 50° and 60° (70°). Angles in a quadrilateral Finds the missing angle in a quadrilateral with 90°, 80° and 100° (90°). Vertically opposite angles Knows that when two lines cross, opposite angles are equal (a 65° angle faces another 65°). Drawing a shape from a specification Draws a triangle with sides 5 cm, 6 cm and 7 cm using ruler, protractor and compasses. Nets of solids Tells which unfolded shapes fold into a cube, cuboid, prism or pyramid. Top, front and side views Matches a solid to its top, front and side views. First-quadrant coordinates in real life Reads and plots a position such as (4, 7) on a map of a park. Plotting in all four quadrants Plots (−3, 2) and (4, −5) and reads a point's coordinates. Completing shapes with coordinates Finds the missing vertex of a rectangle (e.g. (1, 1), (4, 1), (4, 3) gives (1, 3)) and a point's mirror image. Reading a scale on a map Uses 1 cm : 100 m to read that 4.5 cm is 450 m. Using scale on plans Draws or reads a plan at a given scale (a 4 m room at 1 cm : 1 m is 4 cm). Translating a shape Slides a shape 3 right and 2 up and states the move in coordinates. Reflecting a shape Reflects a shape in the y-axis and in a given line and states the new coordinates. Rotating a shape 90° or 180° (introduce) Turns a shape a quarter turn (90°) or a half-turn (180°) about a point and describes it. Mode and range Finds mode 5 and range 7 in 2, 5, 5, 9, 3. Mean Shares out to level the data and calculates the mean (3, 5, 7 gives 5). When the mean misleads Spots that one extreme value drags the mean (1, 2, 2, 3, 40). Median Finds the median of 4, 9, 2, 7, 5 by ordering (5). Reading a pie chart Reads that a quarter-sized sector is 25% and ¼ of the total. Drawing a pie chart (introduce) Turns 50%, 30%, 20% into a pie. Line graphs with two series Reads two lines on one graph and estimates a value between two points. Choosing the right graph Picks a bar chart, line graph or pie chart to suit the data. Spotting a misleading graph Notices a truncated axis, a missing scale or a biased sample. Language of chance and the 0–1 scale Places "unlikely" near 0.2 and "certain" at 1 on a scale. Simple probabilities as fractions Writes the chance of a 5 on a die as 1/6 and a red marble from 3 red in 10 as 3/10. Fair and unfair Says a spinner with unequal sectors is unfair and designs a fair one. Experiments against predictions Predicts 25 heads in 50 tosses, runs it and compares. More trials get closer to the prediction Sees that 1,000 rolls match the prediction better than 10 rolls. Heuristics: work backwards and guess-check-improve Solves "I think of a number, double it, add 3, get 17" by working backwards. Heuristics: simplify and look for a pattern Tries a smaller case to find a pattern before the big one. Missing and surplus information Says which fact is missing or not needed in a story. Finding your own mistake Finds the wrong step in a worked answer. Always, sometimes, never and counterexamples Refutes "multiplying always makes bigger" with ½ × 6. Multi-step fraction, decimal and percent problems Solves a 3-step story such as a shop trip and justifies each step. Bar models Draws a bar model for "⅗ of the sweets are red, 12 are blue". Is the answer reasonable? Rules out an answer by estimate or by the context. Calculator judgment Enters a calculation correctly, checks it by estimate and knows when not to use it.