3rd Grade Math Checklist: What Your Child Should Know

Measuring length (age 7+)

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, metre sticks, and measuring tapes

• Choose the most appropriate tool for measuring a given object
• Align the zero mark of a ruler with the end of the object and read the measurement
• Measure lengths in inches, feet, centimetres, and metres using the correct tool

Area (age 8+)

Measure areas by counting unit squares (square cm, square m, square in, square ft)

• Count unit squares to find the area of an L-shaped figure
• Measure the area of a book cover using square-centimetre tiles
• Compare areas of two shapes by counting their unit squares

Understanding Area

Understand that a unit square has one square unit of area and that the area of a plane figure is the number of unit squares that cover it without gaps or overlaps

• Identify a unit square and state its area is 1 square unit
• Explain why a figure covered by 12 unit squares has area 12 square units
• Distinguish between area and perimeter as different measurements

Area by Tiling

Find the area of a rectangle by tiling it with unit squares and show that the result equals the product of the side lengths

• Tile a 4×6 rectangle and count 24 squares, then verify 4×6=24
• Explain why the number of rows times the number in each row gives the area
• Draw a rectangle on squared paper, tile it, and write the multiplication

Understanding angles (age 8+)

Multiply side lengths to find areas of rectangles and represent whole-number products as rectangular areas

• Calculate the area of a 7 cm × 9 cm rectangle as 63 cm²
• Draw a rectangle with area 36 square units and label its side lengths
• Solve: A garden is 8 m by 5 m — what is its area?

Area and the distributive property

Use tiling to demonstrate the distributive property: the area of a rectangle with sides a and (b+c) equals a×b + a×c; use area models to represent the distributive property

• Tile a 3×(4+2) rectangle and show it decomposes into 3×4 and 3×2
• Use an area model to compute 6×13 as 6×10 + 6×3
• Draw an area model showing 5×(7+3) = 5×7 + 5×3

Calculating with measurements

Measure, compare, add, and subtract lengths (m/cm/mm), mass (kg/g), and volume/capacity (l/ml) using standard units

• Measure a length in centimetres and millimetres
• Weigh an object using grams and kilograms
• Add two measurements in the same unit (e.g. 250 ml + 400 ml = 650 ml)

Converting measurement units

Convert between different units of measure (e.g. kilometre to metre, hour to minute, minute to second, year to month, week to day)

• Convert 3 km to 3000 m
• State 2 hours = 120 minutes
• Convert 5 weeks to 35 days

Perimeters of polygons

Solve problems involving perimeters of polygons: find perimeter from side lengths, find an unknown side length, and explore rectangles with same perimeter but different areas (or vice versa)

• Calculate the perimeter of a rectangle with sides 8 cm and 5 cm
• Find the missing side of a pentagon with perimeter 30 cm and four known sides
• Draw two rectangles both with perimeter 24 cm but different areas

Measuring Perimeters

Measure the perimeter of simple 2-D shapes

• Measure each side of a rectangle and add the lengths to find the perimeter
• Calculate the perimeter of a regular shape given the side length
• Explain that perimeter is the total distance around a shape

Time Units and Calendar Facts

Know the number of seconds in a minute and the number of days in each month, year, and leap year

• State that there are 60 seconds in a minute
• Name the months and state the number of days in each
• Explain that a leap year has 366 days and occurs every 4 years

Numbers on a number line

Solve word problems involving elapsed time by adding and subtracting time intervals in minutes, including using a number line

• A film starts at 2:15 and lasts 47 minutes — when does it end?
• Calculate how many minutes between 9:20 and 10:05
• Use a number line to show the elapsed time between two events

Comparing Time Durations

Compare durations of events and calculate the time taken by particular events or tasks

• Calculate how long an activity lasted given start and end times
• Compare the duration of two events and identify which was longer
• Solve a problem such as 'The lesson starts at 10:15 and ends at 11:00. How long is it?'

Telling time to the minute (age 7+)

Tell and write time from analogue and digital clocks to the nearest five minutes, using a.m., p.m., and 12-hour and 24-hour notation

• Read the time to five minutes on an analogue clock face
• Write the time using digital notation (e.g. 3:25)
• Distinguish between a.m. and p.m. and relate to daily events

Estimating answers (age 7+)

Estimate and read time with increasing accuracy to the nearest minute; record and compare time in terms of seconds, minutes, and hours

• Read an analogue clock to the nearest minute
• Estimate how long an activity takes in minutes or seconds
• Compare two durations and determine which is longer

Halves and quarters (age 8+)

Generate measurement data by measuring lengths to the nearest half and quarter inch; display the data on a line plot with a scale marked in whole numbers, halves, and quarters

• Measure five objects to the nearest 1/4 inch
• Create a line plot showing the lengths of classmates' pencils in half-inches
• Read a line plot and answer questions about the data

Telling time to the minute (age 8+)

Tell and write time to the nearest minute using analogue and digital clocks

• Read 7:43 from an analogue clock face
• Write 11:06 on a digital display given a clock with hands
• Match analogue and digital times to the nearest minute

Addition and subtraction word problems

Solve word problems involving lengths within 100, using addition and subtraction with drawings and equations

• Solve 'The rope is 45 cm long. I cut off 18 cm. How long is it now?'
• Draw a diagram (e.g. a ruler drawing) to represent a length word problem
• Write an equation with a symbol for the unknown to represent the problem

Measuring & Plotting Lengths

Generate measurement data by measuring lengths to the nearest whole unit and display the data on a line plot

• Measure the lengths of several objects and record the data
• Create a line plot with a horizontal scale marked in whole-number units
• Interpret a line plot to answer questions about the data

Estimating and comparing money

Estimate, compare, and calculate different measures including money in pounds and pence

• Estimate the length of the classroom in metres
• Compare 1.5 kg and 1200 g and identify which is heavier
• Calculate the total cost of 3 items at £2.45, £1.30, and £0.75

Measuring Liquids & Masses

Measure and estimate liquid volumes and masses of objects using grams, kilograms, and litres; solve one-step word problems involving mass or volume

• Estimate the mass of a textbook in grams or kilograms
• Read a scale to measure liquid volume in litres
• Solve: 3 bags weigh 250 g each — what is the total mass?

Halves and quarters (age 7+)

Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately

• Determine the total value of a collection of coins
• Solve a word problem about making change with US currency
• Use $ and ¢ symbols correctly in answers

Comparing lengths by measuring

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit

• Measure two objects and calculate the difference in length
• Express the difference using the correct unit (e.g. '7 cm longer')
• Solve a comparison problem: 'How much taller is the bookshelf than the desk?'

Area of compound shapes

Recognise area as additive; find areas of rectilinear figures by decomposing into non-overlapping rectangles and summing their areas

• Decompose an L-shape into two rectangles, find each area, and add them
• Find the area of a floor plan shaped like a T by splitting into rectangles
• Solve: A room is L-shaped (3m×5m plus 2m×4m) — what is the total area?

12-hour and 24-hour time

Read, write, and convert time between analogue and digital 12-hour and 24-hour clocks

• Convert 3:45 pm to 15:45 in 24-hour time
• Read 19:30 and state the 12-hour equivalent as 7:30 pm
• Match a set of 12-hour and 24-hour times

Giving Change

Add and subtract amounts of money to give change, using both £ and p in practical contexts

• Calculate the total cost of two or three items priced in pounds and pence
• Work out the change from £5 or £10
• Record money calculations using £ and p notation correctly (e.g. £3.47)

Estimating Lengths

Estimate lengths using units of inches, feet, centimetres, and metres

• Estimate the length of a classroom object before measuring it
• Use a known reference (e.g. width of a finger ≈ 1 cm) to make reasonable estimates
• Check estimates by measuring and evaluate how close they were

Measuring with different units

Measure the length of an object using two different length units and describe how the measurements relate to the size of the unit chosen

• Measure a desk in both centimetres and inches and compare the two numbers
• Explain that measuring with a smaller unit gives a larger number
• Predict whether a measurement in centimetres will be greater or less than in inches

Fraction Notation

Read, write, and use fraction notation correctly — fraction, numerator, denominator, unit fraction, non-unit fraction, proper fraction, improper fraction, mixed number, equivalent fraction, simplest form — and understand what each term describes, including the roles of the numerator and denominator in expressing parts of a whole

• Point to and name the numerator and denominator in any given fraction and explain what each tells you
• Correctly classify fractions as unit, proper, improper, or mixed number with an example of each
• Explain in own words why 2/4 and 1/2 are equivalent fractions

Tenths

Count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and from dividing one-digit numbers or quantities by 10

• Count: one tenth, two tenths, three tenths … up to ten tenths (one whole)
• Show that dividing a shape into 10 equal parts gives tenths
• Explain that 3 ÷ 10 = 3/10

Fractions on a number line

Recognise and use fractions as numbers: place unit fractions and non-unit fractions with small denominators on a number line

• Place 1/2, 1/4, 3/4 on a number line from 0 to 1
• Identify that 1/3 lies between 0 and 1/2 on the number line
• Understand that a fraction is a single number, not just 'part of a shape'

Fractions of a whole

Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a/b as a parts of size 1/b

• Given a shape divided into 5 equal parts, identify one shaded part as 1/5
• Explain that 3/4 means 3 parts each of size 1/4
• Draw a model showing 2/6 as 2 pieces of a whole cut into 6

Splitting shapes into equal parts (age 7+)

Partition circles and rectangles into two, three, or four equal shares; describe shares as halves, thirds, and fourths; recognise that equal shares of identical wholes need not have the same shape

• Partition a circle into 3 equal parts and label each 'a third'
• Partition a rectangle into 4 equal shares in more than one way
• Explain that two different-looking shares can still be equal in size

Equivalent fractions on a number line

Understand two fractions as equivalent if they are the same size or the same point on a number line; recognise and show families of common equivalent fractions using diagrams

• Use fraction strips to show 1/2 = 2/4 = 3/6
• Verify on a number line that 2/3 and 4/6 land on the same point
• Identify at least three fractions equivalent to 1/2 using diagrams

Equivalent fractions (age 8+)

Generate simple equivalent fractions and explain why they are equivalent using visual fraction models

• Given 1/3, generate 2/6 as equivalent and show with area model
• Simplify 4/8 to 1/2 and justify with a fraction strip
• Complete equivalence chains: 1/4 = ?/8 = ?/12

Equivalent fractions

Recognise and show, using diagrams, equivalent fractions with small denominators

• Show that 1/2 = 2/4 using a diagram of equal parts
• Use a fraction wall or bar model to find equivalent fractions
• Explain why two fractions are equivalent by comparing the shaded areas

Fractions on a number line (age 8+)

Represent fractions on a number line: partition the interval 0 to 1 into b equal parts to locate 1/b, then mark off a lengths of 1/b from 0 to locate a/b

• Partition a 0-to-1 number line into 4 equal parts and mark 1/4
• Explain that each part on the line has size 1/b
• Locate 1/3 and 1/6 on separate number lines

Decimal equivalents of tenths and hundredths

Recognise and write decimal equivalents of any number of tenths or hundredths (e.g. 3/10 = 0.3, 27/100 = 0.27)

• Write 7/10 as 0.7 and vice versa
• Convert 45/100 to 0.45
• Place 0.3 and 3/10 at the same point on a number line

Tenths (age 8+)

Count up and down in hundredths; recognise that hundredths arise when dividing an object by 100 or dividing tenths by 10

• Count from 3/100 to 12/100 in hundredths
• Explain that 1/10 ÷ 10 = 1/100
• Place several hundredths on a number line between 0 and 1/10

Decimal & Percent Notation

Read, write, and use decimal and percentage notation correctly — decimal, decimal point, tenths, hundredths, thousandths, percentage, per cent, % symbol, convert, terminating decimal — and understand the relationships between fractions, decimals, and percentages as three ways of expressing the same value

• Read and write decimal numbers correctly, identifying the value of each digit (ones, tenths, hundredths)
• Use the % symbol correctly and explain that per cent means 'out of 100'
• Convert between simple fractions, decimals, and percentages (e.g. 1/2 = 0.5 = 50%) and explain why they are equal

Fraction-Decimal Equivalents

Recognise and write decimal equivalents of 1/4, 1/2, and 3/4

• State that 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75
• Match fractions to decimals in a sorting activity
• Explain why 1/4 = 25/100 = 0.25 using a hundredths grid

Decimal place value (age 8+)

Compare numbers with the same number of decimal places up to two decimal places

• Order 0.45, 0.54, 0.39 from smallest to largest
• Compare 3.72 and 3.27 using place-value reasoning
• Place three two-decimal-place numbers on a number line in order

Simple Fraction Sums

Add and subtract fractions with the same denominator within one whole (e.g. 5/7 + 1/7 = 6/7)

• Calculate 2/5 + 2/5 = 4/5
• Calculate 6/8 − 3/8 = 3/8
• Explain that when denominators are the same, you add/subtract the numerators

Dividing by 10 and 100

Find the effect of dividing a one- or two-digit number by 10 and 100, identifying the value of the digits as ones, tenths, and hundredths

• Calculate 37 ÷ 10 = 3.7 and identify 3 as ones and 7 as tenths
• Calculate 4 ÷ 100 = 0.04 and identify 4 as hundredths
• Explain that dividing by 10 shifts each digit one place to the right

Adding Fractions (Same Denominator)

Add and subtract fractions with the same denominator, including results greater than one whole (e.g. 5/8 + 6/8 = 11/8)

• Calculate 3/5 + 4/5 = 7/5 and explain it equals 1 2/5
• Subtract 2/6 from 5/6
• Solve addition problems where the sum exceeds the whole: 7/8 + 3/8

Comparing fractions

Compare and order unit fractions, and fractions with the same denominator

• Order 1/2, 1/3, 1/4, 1/5 from largest to smallest
• Explain that a larger denominator means smaller unit fractions
• Compare 2/5 and 4/5 and explain that 4/5 is larger because it has more fifths

Comparing fractions (age 8+)

Compare two fractions with the same numerator or the same denominator by reasoning about size; record comparisons with >, =, or < symbols

• Compare 3/8 and 3/4: same numerator, larger denominator means smaller pieces so 3/8 < 3/4
• Compare 5/6 and 2/6: same denominator so 5/6 > 2/6
• Justify a comparison using a visual model and explain why both fractions must refer to the same whole

Fractions of a whole (age 8+)

Express whole numbers as fractions (e.g. 3 = 3/1) and recognise fractions equivalent to whole numbers (e.g. 4/4 = 1, 6/1 = 6)

• Write 5 as 5/1 and explain why
• Locate 4/4 and 1 at the same point on a number line
• Identify which fractions from a list equal a whole number: 6/3, 8/4, 5/2

Unit fractions

Recognise, find, and write fractions of a discrete set of objects: unit fractions and non-unit fractions with small denominators

• Find 1/4 of 12 objects by dividing into 4 equal groups
• Find 3/4 of 12 objects
• Write the fraction of a set that is shaded or selected

Fractions of amounts (harder)

Solve problems involving increasingly harder fractions to calculate quantities, including non-unit fractions where the answer is a whole number

• Find 3/5 of 20
• Calculate 2/3 of 18 and explain the two-step process (divide then multiply)
• Solve: A bag has 24 sweets, 3/8 are red — how many red sweets?

Decimals and fractions

Solve simple measure and money problems involving fractions and decimals to two decimal places

• Calculate 1/4 of £3.20
• A rope is 2.5 m long; how much is left after cutting 0.75 m?
• Find 3/10 of 1 kg and express the answer in grams and as a decimal of a kg

Comparing fractions (age 7+)

Solve problems involving counting in tenths, fractions of quantities, equivalence, fraction addition/subtraction, and fraction comparison

• Solve a word problem requiring finding a fraction of a quantity
• Solve a problem that requires comparing or ordering fractions
• Choose and apply appropriate fraction knowledge to a multi-step problem

Decimal place value

Round decimals with one decimal place to the nearest whole number

• Round 3.7 to 4 and 3.2 to 3
• Place 6.5 on a number line between 6 and 7 and decide it rounds to 7
• Round a set of one-decimal-place numbers and explain the rounding rule

Fractions as parts of shapes

Partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole

• Partition a rectangle into 6 equal-area parts and label each 1/6
• Show two different ways to partition a square into 4 equal parts
• Given a pre-partitioned shape, write the unit fraction for one part

Times tables (age 7+)

Recall and use multiplication and division facts for the 3, 4, and 8 multiplication tables

• Recall 3 × 1 through 3 × 12 and corresponding division facts
• Recall 4 × 1 through 4 × 12 and corresponding division facts
• Recall 8 × 1 through 8 × 12 and corresponding division facts

Arrays for multiplication (age 7+)

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and 5 columns; write an equation to express the total as a sum of equal addends

• Count objects in a 3×4 array and write 4 + 4 + 4 = 12
• Explain that each row has the same number of objects so the total can be found by repeated addition
• Draw a rectangular array to model a given repeated-addition equation

What Multiplication Means

Interpret products of whole numbers (e.g. 5 × 7 as the total number of objects in 5 groups of 7)

• Explain that 4 × 6 means 4 groups of 6 objects
• Draw a picture or array to represent a multiplication expression
• Match a multiplication expression to a word problem involving equal groups

All times tables to 12×12

Recall multiplication and division facts for multiplication tables up to 12 × 12

• Recall any fact from the 1–12 times tables rapidly
• Recall the corresponding division fact for any multiplication fact
• Use known facts to check or derive answers in calculations

Written Multiplication

Multiply two-digit and three-digit numbers by a one-digit number using formal written layout

• Set out and solve 47 × 6 using short multiplication
• Set out and solve 234 × 5 using short multiplication with carrying
• Check the answer using estimation (e.g. 234 × 5 ≈ 200 × 5 = 1000)

Written Multiplication & Division

Write and calculate mathematical statements for multiplication and division using known tables, including two-digit × one-digit, using mental and progressing to formal written methods

• Calculate 23 × 4 using partitioning (20 × 4 + 3 × 4)
• Calculate 96 ÷ 8 using known table facts
• Begin to use a formal written layout for short multiplication

Properties of Operations

Apply properties of operations (commutative, associative, distributive) as strategies to multiply and divide

• Use commutativity: if 6 × 4 = 24 then 4 × 6 = 24
• Use the distributive property: 8 × 7 = 8 × 5 + 8 × 2 = 40 + 16 = 56
• Use associativity to multiply three numbers: 2 × 3 × 5 = 6 × 5 = 30

Fluent multiplication and division facts

Fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division

• Answer any single-digit multiplication fact within 3 seconds
• Answer any related division fact within 3 seconds
• Use known facts to derive unknown facts (e.g. 9 × 7 from 10 × 7 − 7)

Division as Unknown Factor

Understand division as an unknown-factor problem (e.g. find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8)

• Explain that 32 ÷ 8 = ? is the same as 8 × ? = 32
• Use a known multiplication fact to find a division answer
• Describe the relationship between multiplication and division as inverse operations

What Division Means

Interpret whole-number quotients (e.g. 56 ÷ 8 as the number of objects in each share or the number of equal groups)

• Explain that 56 ÷ 8 can mean sharing 56 into 8 equal groups or making groups of 8
• Draw a picture to represent a division expression
• Match a division expression to a word problem involving equal sharing or grouping

Multiply & Add Problems

Solve problems involving multiplying and adding, including using the distributive law, integer scaling problems, and harder correspondence problems

• Use the distributive law to solve 14 × 6 as 10 × 6 + 4 × 6
• Solve a scaling problem (e.g. 'A tower is 3 times as tall as a 15 m building')
• Solve a correspondence problem (e.g. '3 types of bread, 4 types of filling — how many different sandwiches?')

Multi-Step Multiply & Divide

Solve problems involving multiplication and division, including scaling problems and correspondence problems where n objects are connected to m objects

• Solve a scaling problem (e.g. 'the ribbon is 3 times as long')
• Solve a correspondence problem (e.g. '4 shirts and 3 trousers — how many outfits?')
• Solve a missing-number multiplication or division problem

Factor Pairs & Commutativity

Recognise and use factor pairs and commutativity in mental calculations

• List all factor pairs of a given number (e.g. 24: 1×24, 2×12, 3×8, 4×6)
• Use commutativity to reorder a multiplication for easier mental calculation
• Explain what a factor pair is and how commutativity helps

Patterns in Times Tables

Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations

• Notice that all products of 5 end in 0 or 5 and explain why
• Observe that the sum of two even numbers is always even
• Identify a pattern in the multiplication table and explain it using commutativity or the distributive property

Multiplying by Tens

Multiply one-digit whole numbers by multiples of 10 in the range 10–90 using strategies based on place value and properties of operations

• Calculate 7 × 30 = 210 by reasoning 7 × 3 tens = 21 tens = 210
• Explain why multiplying by a multiple of 10 adds a zero to the product
• Use this skill to estimate products of larger numbers

Mental multiplication and division

Use place value, known and derived facts to multiply and divide mentally, including multiplying by 0 and 1, dividing by 1, and multiplying together three numbers

• Calculate 40 × 6 = 240 mentally using place value
• Explain why any number × 0 = 0 and any number × 1 = the number itself
• Multiply three numbers mentally by choosing a useful pair first (e.g. 2 × 7 × 5 = 2 × 5 × 7 = 70)

Multiplication and Division Word Problems

Use multiplication and division within 100 to solve word problems involving equal groups, arrays, and measurement quantities

• Solve an equal-groups word problem using multiplication
• Solve a measurement division problem (e.g. 'How many 4-cm pieces from a 28-cm ribbon?')
• Solve an array/area word problem using multiplication

Unknown in Multiplication & Division

Determine the unknown whole number in a multiplication or division equation relating three whole numbers (e.g. 8 × ? = 48, ? × 6 = 42)

• Find the missing factor in 7 × ? = 63
• Find the missing dividend in ? ÷ 5 = 9
• Explain the strategy used (e.g. using the related multiplication fact)

Rows & Columns in Rectangles

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them

• Divide a rectangle into equal rows and columns of unit squares
• Count the total number of squares using repeated addition or skip counting
• Relate the rows-and-columns structure to a rectangular array

Fluent adding and subtracting within 100

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and the relationship between addition and subtraction

• Add two-digit numbers within 100 efficiently (e.g. 46 + 37 using place-value partitioning)
• Subtract two-digit numbers within 100 fluently (e.g. 83 − 25)
• Choose between strategies (decomposing, compensating, using known facts) based on the numbers involved

Fluent adding and subtracting within 20

Fluently add and subtract within 20 using mental strategies; know from memory all sums of two one-digit numbers

• Answer any single-digit addition fact within 3 seconds
• Recall subtraction facts within 20 from memory (e.g. 15 − 8 = 7)
• Use known addition facts to derive related subtraction facts rapidly

Addition and subtraction within 1000

Add and subtract within 1000 using concrete models, drawings, and strategies based on place value; understand composing and decomposing tens and hundreds

• Add two three-digit numbers using base-ten blocks or drawings, composing a ten or hundred when necessary
• Subtract three-digit numbers, decomposing a ten or hundred when necessary
• Relate the concrete/drawn strategy to a written method and explain why it works

Adding and subtracting (age 7+)

Add and subtract numbers with up to three digits using formal written methods of columnar addition and subtraction

• Set out a columnar addition correctly with digits aligned by place value
• Carry out columnar subtraction with exchange (borrowing) when needed
• Check the answer using the inverse operation or estimation

Estimating by rounding

Estimate the answer to a calculation and use inverse operations to check answers; apply to increasingly large numbers using rounding and inverse reasoning

• Round numbers to the nearest 10 or 100 to estimate a sum or difference before calculating
• Use addition to check a subtraction answer, or vice versa
• Identify when a calculated answer is unreasonable based on the estimate

Missing number problems (age 7+)

Solve addition and subtraction problems including missing-number problems, using number facts, place value, and more complex methods

• Solve a missing-number problem such as 245 + ? = 380
• Choose an appropriate method (mental, written, or combination) based on the numbers
• Solve multi-step problems combining addition and subtraction of three-digit numbers

Two-Step Word Problems

Solve one- and two-step word problems within 100 using addition and subtraction, with unknowns in all positions

• Solve a two-step word problem involving adding to and taking from
• Represent a word problem with an equation using a symbol for the unknown
• Solve comparison problems (how many more/fewer) within 100

Addition and subtraction strategies (age 7+)

Explain why addition and subtraction strategies work, using place value and the properties of operations

• Explain why adding tens and ones separately gives the correct total
• Describe why a compensation strategy works (e.g. 'I added 1 too many, so I subtract 1')
• Use place-value language to justify a written method step by step

Two-Step Equations

Solve two-step word problems using the four operations; represent problems using equations with a letter standing for the unknown quantity

• Solve a two-step problem that combines addition/subtraction with multiplication/division
• Write an equation using a letter for the unknown (e.g. 3 × n + 5 = 26)
• Assess the reasonableness of the answer using estimation and mental computation

Adding and subtracting (age 8+)

Add and subtract numbers with up to four digits using formal written methods of columnar addition and subtraction

• Set out and solve a columnar addition with up to four-digit numbers
• Set out and solve a columnar subtraction with exchange across multiple columns
• Check the answer using estimation or inverse operations

Two-step addition and subtraction problems

Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and why

• Identify the two steps needed to solve a contextual problem
• Choose between mental and written methods for each step based on the numbers
• Explain why the chosen operations and methods are appropriate

Numbers on a number line

Represent whole numbers as lengths on a number line and represent sums and differences within 100 on a number line diagram

• Place whole numbers on a number line with equally spaced points
• Show an addition as a jump forward on the number line (e.g. 38 + 27 shown as jumps)
• Show a subtraction as a jump backward on the number line

Mental addition and subtraction (age 7+)

Mentally add and subtract a three-digit number and ones

• Calculate 345 + 7 mentally
• Calculate 462 − 5 mentally
• Explain that only the ones digit changes (unless bridging through a ten)

Fluent adding and subtracting within 1000

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and the relationship between addition and subtraction

• Add two three-digit numbers fluently using an efficient method
• Subtract three-digit numbers fluently, including with regrouping
• Choose the most efficient strategy based on the numbers involved

Adding numbers

Add up to four two-digit numbers using strategies based on place value and properties of operations

• Add three or four two-digit numbers by grouping tens and ones
• Look for pairs that make multiples of 10 to simplify addition
• Explain the strategy used to combine multiple addends

Mentally adding hundreds to 3-digit numbers

Mentally add and subtract a three-digit number and hundreds

• Calculate 345 + 200 mentally
• Calculate 762 − 400 mentally
• Explain that only the hundreds digit changes

Mentally adding tens to 3-digit numbers

Mentally add and subtract a three-digit number and tens

• Calculate 345 + 40 mentally
• Calculate 462 − 30 mentally
• Explain that only the tens digit changes (unless bridging through a hundred)

A Hundred Is Ten Tens

Understand that 100 can be thought of as a bundle of ten tens — called a 'hundred'

• Explain that 10 groups of 10 ones make 100
• Bundle ten tens sticks into one hundred and describe what happened
• Represent 100 using base-ten blocks showing 10 tens

The three digits of a three-digit number

Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones

• State the value of each digit in a three-digit number (e.g. in 362, the 3 represents 3 hundreds)
• Partition a three-digit number into hundreds, tens, and ones (e.g. 485 = 400 + 80 + 5)
• Explain why 706 has 7 hundreds, 0 tens, and 6 ones

Place value of each digit

Recognise the place value of each digit in a four-digit number (thousands, hundreds, tens, and ones)

• State the value of each digit in a four-digit number (e.g. in 7,345 the 7 represents 7 thousands)
• Partition a four-digit number into thousands, hundreds, tens, and ones
• Compose a four-digit number from given place-value parts (e.g. 3000 + 400 + 50 + 2 = 3,452)

Ordering Numbers to 1000

Compare and order numbers up to 1000 using >, =, and < symbols, based on place-value understanding

• Compare two three-digit numbers by examining hundreds first, then tens, then ones
• Use >, =, and < correctly to record comparisons of three-digit numbers
• Order a set of numbers up to 1000 from smallest to largest

Comparing Large Numbers

Order and compare numbers beyond 1000

• Compare two four-digit numbers using >, <, and = by examining digits from the highest place value
• Order a set of numbers up to 10,000 from smallest to largest
• Justify the ordering using place-value reasoning

Negative Numbers

Count backwards through zero to include negative numbers

• Count backwards from 5 through zero: 5, 4, 3, 2, 1, 0, −1, −2 …
• Place negative numbers on a number line
• Understand that negative numbers are less than zero and use them in context (e.g. temperature)

Representing Numbers

Identify, represent, and estimate numbers using different representations including the number line

• Place a two-digit number on a 0–100 number line in approximately the right position
• Estimate 'about how many' objects are in a set of 30–50 without counting
• Represent the same number using base-ten blocks, a number line, and a drawing

The multiples of 100

Understand that the multiples of 100 (100–900) each represent a number of hundreds with 0 tens and 0 ones

• Identify that 300 means 3 hundreds, 0 tens, 0 ones
• Place multiples of 100 on a number line to 1000
• Read and write multiples of 100 and explain their place-value structure

Rounding to 10, 100, 1000

Round any number to the nearest 10, 100, or 1000

• Round 4,367 to the nearest 10 (4,370), 100 (4,400), and 1000 (4,000)
• Explain the rounding rule using a number line (which multiple is closer)
• Apply rounding to estimate calculations

Numbers to 10,000

Identify, represent, and estimate numbers up to 10,000 using different representations

• Represent a four-digit number on a place-value chart or with base-ten materials
• Estimate where a number falls on a 0–10,000 number line
• Match different representations of the same number (e.g. expanded form, place-value counters, numeral)

Place Value Problem-Solving

Solve number and practical problems involving place value with increasingly large positive numbers

• Solve a problem requiring rounding, comparing, or ordering numbers beyond 1000
• Apply place-value knowledge in a practical context (e.g. population figures, distances)
• Explain the strategy used, referencing place-value understanding

Reading and writing numbers to 1000

Read and write numbers to 1000 in numerals, number names, and expanded form

• Read a three-digit numeral aloud correctly
• Write a three-digit number in words (e.g. 'three hundred and forty-two')
• Write a number in expanded form (e.g. 600 + 30 + 7)

10 or 100 More or Less

Find 10 or 100 more or less than a given number up to 1000

• Given a three-digit number, state what is 10 more and 10 less
• Given a three-digit number, state what is 100 more and 100 less
• Explain the strategy using place-value understanding (e.g. only the tens/hundreds digit changes)

1000 More or Less

Find 1000 more or less than a given number

• Given 4,562, state that 1000 more is 5,562 and 1000 less is 3,562
• Explain using place value that only the thousands digit changes
• Apply this skill to numbers beyond 10,000

Place Value to 1000

Solve number problems and practical problems involving place value of numbers up to 1000

• Solve a problem that requires identifying how many hundreds, tens, or ones are in a number
• Apply place-value knowledge to a practical context (e.g. counting money in pounds)
• Explain the strategy used to solve a place-value problem

Roman numerals to 100

Read Roman numerals to 100 (I to C) and understand that the numeral system changed over time to include zero and place value

• Read and write Roman numerals I, V, X, L, C and combinations up to 100
• Convert between Roman numerals and Hindu-Arabic numerals (e.g. XLIV = 44)
• Explain that Roman numerals have no zero and no place-value system

Odd or Even

Determine whether a group of objects (up to 20) has an odd or even number of members

• Pair objects and determine whether there is one left over (odd) or not (even)
• Count a group by 2s to determine if the total is even
• Write an equation to express an even number as a sum of two equal addends (e.g. 8 = 4 + 4)

Mathematical Precision

Communicate with mathematical precision: use correct fraction/decimal vocabulary, name angle types accurately, specify units in measurement and money, and use notation (=, <, >, ÷, ×) correctly

• Distinguish between 'three fourths' and 'three quarters' and use both correctly
• State an answer in the correct unit: '63 square centimetres' not just '63'
• Write 15:45 in 24-hour notation and explain the distinction from 3:45 pm

Multi-Step Problem Solving

Make sense of multi-step problems involving four operations, fractions, and area/volume by identifying sub-steps, choosing a strategy, and monitoring progress

• Break a two-step word problem into parts and explain a plan before calculating
• Choose between drawing a diagram or writing equations for a perimeter problem
• Check a fraction-of-quantity answer by estimating: 3/5 of 20 must be more than half of 20

Justifying mathematical reasoning (age 8+)

Construct and present multi-step mathematical arguments; critique the reasoning of others and explain clearly why a method works or fails

• Explain why 1/3 > 1/5 using the idea that more parts means smaller pieces
• Find and explain an error in a peer's column subtraction with exchanges
• Present a chain of reasoning: since 6×8=48 and 6×2=12, then 6×10=60 so 6×8=60−12=48

Multi-Step Problem Solving

With teacher support, make sense of multi-step problems involving larger numbers or mixed operations by breaking them into parts, choosing strategies, and checking answers for reasonableness — children at this stage are developing the habit with guidance; independent strategy evaluation comes later

• Break a two-step word problem within 1000 into sub-problems and solve each part
• Estimate an answer before calculating to set a reasonableness benchmark
• Check an answer using a different method or inverse operation and revise if needed

Understanding fractions (age 7+)

Communicate with mathematical precision: use correct place-value and fraction vocabulary, specify units in measurement answers, and use notation accurately

• Consistently specify units in measurement answers (e.g. '35 cm' not '35')
• Use fraction vocabulary precisely (numerator, denominator, equivalent)
• Write equations with correct notation including £/p, >, <, = and fraction symbols

Choosing mathematical tools

Select and use appropriate tools and representations strategically: choose between mental, written, and diagrammatic methods; use calculators for checking; select fraction models suited to the task

• Decide to use mental multiplication for 25×4 but a written method for 167×3
• Choose fraction strips rather than a number line to compare 3/8 and 1/4
• Use a ruler and squared paper to verify area by counting squares after calculating l×w

Modelling with multiplication and fractions

Model real-world problems involving multiplication, area, fractions, and unit conversion by choosing appropriate representations and interpreting mathematical results in context

• Model a tiling/area problem with an array and write the corresponding multiplication
• Represent a recipe-scaling problem as a fraction calculation and interpret the answer in grams
• Use a bar model to set up a unit conversion problem (metres to centimetres)

Times tables (age 8+)

Recognise and use repeated reasoning to generalise: extend patterns in times tables and equivalent fractions, derive unknown facts from known facts efficiently, describe general rules

• Notice that all fractions equivalent to 1/2 have a numerator that is half the denominator
• Use the pattern 3×4=12, 3×40=120, 3×400=1200 and explain the generalisation
• Derive 8×7 from 8×5=40 plus 8×2=16 and describe the strategy as a general approach

Fractions on a number line

Move fluently between real-world situations, diagrams, number lines, and symbolic equations involving multiplication, fractions, and decimals, explaining what each representation shows

• Represent a sharing problem as both a fraction diagram and a division equation
• Explain how a bar model for 4 × 23 connects to the area model and the written method
• Translate a decimal on a number line into a fraction and explain the equivalence

Justifying mathematical reasoning

Construct and follow multi-step mathematical arguments; identify errors in reasoning and explain why a method works or does not work

• Explain step by step why a columnar addition method gives the correct answer
• Find and explain an error in a worked example (e.g. incorrect regrouping)
• Construct a simple argument for why a general statement is true (e.g. 'adding two even numbers always gives an even number')

Working with money

Model real-world problems involving measurement, money, and time by choosing appropriate representations and interpreting results in context

• Choose whether to use a bar model, number line, or equation for a measurement problem
• Model a multi-step money problem with equations and interpret the final answer as change or total cost
• Create a line plot from measurement data and use it to answer questions about the real-world situation

Using Mathematical Structure

Look for and use mathematical structure: exploit place-value patterns for ×10/×100, use the distributive property to break apart multiplications, apply fraction equivalence to compare and compute, use shape properties to classify quadrilaterals

• Decompose 7×13 into 7×10 + 7×3 using the distributive property
• Explain why multiplying by 10 shifts digits one place left using place-value structure
• Use the fact that a square is a special rectangle to reason about quadrilateral properties

Understanding fractions

Move fluently between real-world situations, diagrams, and symbolic equations involving three-digit numbers and fractions, explaining what each part represents

• Write an equation with three-digit numbers to match a measurement or money word problem
• Draw a bar model to represent a fraction problem and use it to solve
• Explain how a number line diagram relates to the quantities in a word problem

Choosing the right strategy

Select and use appropriate tools and representations strategically, including choosing between mental methods, jottings, formal algorithms, and calculators for arithmetic with multi-digit numbers, decimals, and fractions

• Choose a mental method for 345 + 200 but a written method for 345 + 278
• Select a ruler vs a metre stick based on the object being measured
• Decide when a number line is more useful than base-ten blocks for a given problem

Shape patterns (age 7+)

Look for and use mathematical structure: apply place-value patterns to three-digit operations, use multiplication/division relationships, and exploit shape properties to classify

• Use the structure of place value to explain why adding hundreds only changes the hundreds digit
• Use commutativity and the relationship between multiplication and division to derive unknown facts
• Classify shapes by their structural properties (number of sides, right angles, parallel lines)

Extending Table Patterns

Recognise and use repeated reasoning to generalise: extend multiplication table patterns, derive unknown facts from known ones, and describe rules for sequences

• Notice that all multiples of 4 are even and use this to check answers
• Derive 8 × 7 from 8 × 5 + 8 × 2 by spotting the pattern
• Describe a rule for a growing pattern (e.g. 'add 50 each time') and use it to predict the next terms

Right Angles & Turns

Identify right angles; recognise that two right angles make a half-turn, three make three-quarters, and four make a complete turn

• Use a right-angle checker to identify right angles in shapes and the environment
• Classify angles as right angles, less than a right angle, or greater than a right angle
• Explain that 4 right angles make a full turn (360°)

Understanding angles

Recognise angles as a property of shape or a description of a turn

• Identify angles at the corners of 2-D shapes
• Describe a turn (e.g. quarter turn, half turn) in terms of the angle made
• Explain that an angle measures the amount of turn between two lines meeting at a point

Types of angles

Identify acute and obtuse angles; compare and order angles up to two right angles by size

• Classify given angles as acute, right, or obtuse
• Order four angles from smallest to largest by visual comparison
• Identify all acute and obtuse angles in a given triangle or quadrilateral

Types of angles (age 8+)

Use and interpret standard geometric diagram conventions: mark right angles with a small square, equal lengths with single or double tick marks, and equal angles with arc marks; label angles in three-letter notation (∠ABC) and individual angles with a single letter or number; draw diagrams showing angles at a point, angles on a straight line, and angles inside polygons with these conventions; read diagrams with these marks to identify given information and find unknown values

• Mark a right angle with a small square symbol in a diagram and explain what it means
• Use tick marks to show equal lengths in a shape and double tick marks for a second pair of equal sides
• Read and interpret angle notation (e.g. angle ABC) and identify the angle being referred to in a diagram

Parallel and perpendicular lines

Identify horizontal and vertical lines and pairs of perpendicular and parallel lines

• Point out horizontal and vertical lines in shapes and real-world contexts
• Identify a pair of parallel lines (lines that never meet and are always the same distance apart)
• Identify perpendicular lines (lines that meet at a right angle)

Angles in triangles (age 7+)

Recognise and draw shapes having specified attributes (e.g. a given number of angles or equal faces); identify triangles, quadrilaterals, pentagons, hexagons, and cubes

• Draw a shape with exactly 5 sides (pentagon)
• Identify all quadrilaterals in a set of mixed shapes
• Name and draw a hexagon, explaining it has 6 sides and 6 angles

Understanding angles (age 8+)

Understand that shapes in different categories may share attributes defining a larger category; classify quadrilaterals (rhombuses, rectangles, squares) and draw examples of quadrilaterals not in those subcategories

• Explain that a square is a special rectangle and also a special rhombus
• Sort shapes into a Venn diagram: quadrilaterals vs rectangles vs squares
• Draw a quadrilateral that is not a rectangle, rhombus, or square

First Quadrant Coordinates

Describe positions on a 2-D grid as coordinates in the first quadrant

• Read the coordinates of a point on a grid as (3, 5)
• Explain that the first number is the horizontal distance and the second is the vertical distance
• Identify the coordinates of all vertices of a shape plotted on a grid

Coordinates (age 8+)

Plot specified points on a coordinate grid and draw sides to complete a given polygon

• Plot points (1,1), (1,4), (5,4), (5,1) and join to make a rectangle
• Given three vertices of a square, plot the fourth vertex
• Complete a triangle by plotting the third vertex at given coordinates and drawing sides

2-D shapes (age 7+)

Draw 2-D shapes and make 3-D shapes using modelling materials; recognise 3-D shapes in different orientations and describe them

• Draw a triangle, rectangle, pentagon, or hexagon accurately
• Construct a cube or cuboid from modelling materials (e.g. straws and connectors)
• Recognise a 3-D shape (e.g. a pyramid) when it is rotated or seen from a different angle

Transformations on a grid

Represent and carry out geometric transformations on squared paper or a coordinate grid: reflections (in horizontal, vertical, and diagonal mirror lines, including the axes), translations (described as a vector or as left/right/up/down moves), and rotations (90° or 180° about a stated centre point); describe each transformation precisely using the correct language; identify which transformation maps one shape onto its image by comparing position, orientation, and size

• Reflect a shape in a given mirror line on a grid and label the new coordinates
• Translate a shape by a given number of squares horizontally and vertically and describe the movement
• Rotate a shape 90° or 180° about a given centre on a grid and check the image is congruent to the original

Nets of 3-D Shapes

Identify, draw, and interpret nets of common 3-D shapes — cubes, cuboids, triangular prisms, and square-based pyramids — by predicting which 3-D shape a given flat arrangement of faces will fold into, checking whether a net will close completely, and sketching a net from a description or 3-D model; understand the relationship between the number of faces and the structure of the net

• Draw the net of a cube, cuboid, or triangular prism and fold it mentally to identify which faces connect
• Build a 3-D shape from its net and check that all faces, edges, and vertices match
• Identify which of several given nets will fold into a specific 3-D shape and explain why the others won't

Describing Movements

Describe movements between positions as translations of a given unit to the left/right and up/down

• Describe moving from (2,3) to (5,3) as 3 units to the right
• Translate a shape 4 units right and 2 units up and state the new coordinates
• Predict where a point will be after a given translation

2-D shapes (age 8+)

Identify lines of symmetry in 2-D shapes presented in different orientations; recognise line-symmetric figures and draw lines of symmetry

• Find all lines of symmetry in a rectangle, square, and equilateral triangle
• Determine whether a given shape has a line of symmetry when rotated
• Identify which shapes in a set have exactly one line of symmetry

Lines of symmetry

Complete a simple symmetric figure with respect to a specific line of symmetry

• Given half a butterfly shape and a mirror line, complete the other half on a grid
• Complete a symmetric pattern on squared paper with a vertical line of symmetry
• Check a completed figure by folding along the mirror line

Sorting Data into Categories

Organise and represent data with up to three categories by counting objects in each category and sorting categories by quantity

• Sort a set of objects into 2-3 given categories and count each group
• Create a simple table or list showing category names and counts
• Order categories from most to fewest or fewest to most

Pictograms and tally charts (age 6+)

Read, write, and use the vocabulary of data collection and display — data, tally, tally chart, frequency, frequency table, survey, pictogram, bar chart, axis/axes, scale, label, category, discrete data, continuous data, line graph, pie chart — and apply these terms when collecting, organising, and presenting data

• Correctly label the axes of a bar chart including a title, axis labels, and scale
• Distinguish between discrete data (counted) and continuous data (measured) with an example of each
• Use 'tally', 'frequency', and 'pictogram' correctly when describing how to record and display data

Pictograms and tally charts

Interpret and construct simple pictograms, tally charts, block diagrams, and simple tables

• Read a pictogram where each symbol represents one item
• Construct a tally chart from collected data
• Draw a block diagram to represent data from a survey

Representing numbers with objects (age 8+)

Draw a scaled picture graph and a scaled bar graph to represent a data set; solve one- and two-step comparison, sum, and difference problems using bar charts, pictograms, and tables

• Draw a bar graph where each square represents 5 pets
• From a scaled pictogram, answer: how many more children chose football than tennis?
• Solve a two-step problem: how many votes in total for the top two choices?

Picture & Bar Graphs

Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories; solve put-together, take-apart, and compare problems using information presented in a bar graph

• Draw a picture graph with a symbol representing one unit for each data point
• Draw a bar graph with labelled axes and a single-unit scale
• Use a bar graph to answer comparison questions (e.g. 'How many more votes did cats get than dogs?')

Bar graphs

Interpret and present discrete and continuous data using appropriate graphical methods, including bar charts and time graphs

• Read a time graph showing temperature changes over a day
• Present data about plant growth over weeks as a time graph
• Explain the difference between a bar chart (discrete) and a time graph (continuous)

Sorting into categories (age 6+)

Interpret categorical data by asking and answering questions about totals, how many in each category, and how many more or less one category has than another

• Answer 'how many?' for each category in a data set
• Calculate the total number of data points across all categories
• Compare two categories using 'how many more/fewer' language

Skip Counting (4s, 8s, 50s, 100s)

Count from 0 in multiples of 4, 8, 50, and 100

• Recite the multiples of 4 from 0 to at least 48
• Recite the multiples of 8 from 0 to at least 96
• Count in steps of 50 and 100 from 0 to 1000

Counting in 6s

Count in multiples of 6, 7, 9, 25, and 1000

• Recite the multiples of 6 from 0 to at least 72
• Recite the multiples of 7 from 0 to at least 84
• Count in steps of 25 from 0 to 1000 and in steps of 1000 up to 10,000

Counting Within 1,000

Count within 1000, including skip-counting by 5s, 10s, and 100s

• Count forwards and backwards within 1000 from any starting number
• Skip-count by 5s from any multiple of 5 to 1000
• Skip-count by 100s from any number (e.g. 150, 250, 350 …)