A visual guide to the models children meet in mathematics. Explore each representation, understand why it works, and try a small question together.
Alphabetical within this strand · 15 representations
The same quantity can be organised into different groups, rows or patterns.
Why we use it: It helps children see that a number stays the same even when its arrangement changes.
Show 8 objects in two different arrangements. What changed? What stayed the same?
Objects children can touch, move and count to represent a quantity.
Why we use it: Physical objects make an abstract number visible and help children build a strong sense of quantity.
Make 6 using buttons, blocks or coins. Move them around. Does the total change?
A sketch or picture used to represent a number or mathematical idea.
Why we use it: A drawing records mathematical thinking when real objects are not available.
Draw 5 birds. Then draw 5 stars. How are the pictures different? What mathematics is the same?
A number written as the value of each digit added together.
Why we use it: It reveals the place-value parts hidden inside a numeral.
Write 3,406 in expanded notation. Which place value has no tens?
The written symbol used to name a number, such as 7 or 425.
Why we use it: Numerals let us record, read and compare numbers efficiently—but the visual model helps us understand what the digits mean.
What numeral represents 6 hundreds, 3 tens and 9 ones? Sketch a model to prove it.
A straight line where numbers are placed in order at equal intervals.
Why we use it: It makes number order, distance and change visible.
Start at 3 and move four equal spaces. Where do you land? What calculation did the jump show?
Two connected models showing both the value of each digit and the number’s position.
Why we use it: It connects place-value structure with relative size.
Place 372 in an H T O chart, then estimate where it belongs between 300 and 400.
A number line that marks the lower benchmark, midpoint and upper benchmark used for rounding.
Why we use it: It shows which benchmark a number is closest to rather than relying on a memorised rule alone.
Mark 63 on a line from 60 to 70. Is it before or after 65? Which ten is closer?
A number written using words, such as twenty-four.
Why we use it: It connects spoken number language with written numerals and place value.
Write 132 in words. Which part tells us there is one hundred?
A chart that places each digit into its correct value column.
Why we use it: It shows that a digit’s value depends on its position.
Place 6,205 into a place-value chart. What does the zero tell us?
A place-value chart continues beyond the ones column into tenths, hundredths and smaller decimal places. It is also known as a segmented bar.
Why we use it: It shows that the same place-value system continues to the right of the decimal point.
Place 6.08 in a chart. What does the zero tell us?
A place-value chart paired with an addition statement showing the value of every digit.
Why we use it: It connects the position of each digit with the amount that digit represents.
Show 7,340 in the chart, then write its expanded notation. Which place contributes no value?
A chart that records a starting number and a changed number on different rows.
Why we use it: It makes the place affected by an operation easy to compare.
Start with 425 and add 30. Record both numbers. Which column changes, and which stay the same?
A written record that separates a number into thousands, hundreds, tens and ones.
Why we use it: It exposes the value of each digit and supports flexible calculation.
Partition 8,407 by place value. Why is there no tens part in the expanded notation?
A frame of ten boxes arranged as two rows of five.
Why we use it: It helps children recognise quantities, make ten and see number relationships without counting every object.
Show 7 on a ten frame. How many spaces are empty? What number fact can you see?
A statement using <, > or = to show how two values are related.
Why we use it: It communicates whether one value is less than, greater than or equal to another.
Compare 306 and 360. Which symbol belongs between them? Explain using place value.
Adjoining bars show two or more quantities forming one total.
Why we use it: It makes the parts and the whole visible before children calculate.
Draw bars for 8 and 6. What total should the whole bar show?
A whole bar is matched with known and unknown parts.
Why we use it: It helps children see which quantity is missing and choose an operation.
The whole is 18 and one part is 11. Draw the model and find the missing part.
Aligned bars show two quantities and the difference between them.
Why we use it: It clarifies “how many more” and “how many fewer” situations.
One quantity is 14 and another is 9. Draw aligned bars and mark the difference.
A picture records objects being joined together or taken away.
Why we use it: It connects the action in a story with addition or subtraction.
Draw 9 balloons, then show 3 floating away. How many remain?
Numbers and symbols record a complete mathematical relationship.
Why we use it: It communicates a calculation clearly and efficiently.
Write an equation showing 36 increased by 18.
The opposite operation is used to check whether an answer is correct.
Why we use it: Addition can check subtraction, and subtraction can check addition.
Check 72 − 29 = 43 using addition.
A box, symbol or blank stands for an unknown value in an equation.
Why we use it: It builds relational thinking instead of treating equals as an instruction.
Solve 26 + □ = 41. Which inverse operation helps?
A whole number is connected to the parts that compose it.
Why we use it: It develops flexible understanding of addition and subtraction relationships.
Make a number bond for 15 using 7 as one part. What is the other part?
A blank line records flexible jumps chosen to solve a calculation.
Why we use it: It makes mental strategies and place-value jumps visible.
Solve 48 + 35 using jumps that are easy for you to explain.
Dollar amounts are connected by jumps that show spending, earning or change.
Why we use it: It links number-line strategies to realistic money calculations.
You have $24 and receive $18. Show the change in two helpful jumps.
Jumps connect two numbers, and their combined length shows the difference.
Why we use it: Counting up can make subtraction easier to understand and calculate.
Count up from 58 to 74. What jumps did you use? What is the difference?
Numbers are split into useful place-value parts before calculating.
Why we use it: It makes flexible mental computation clear and recordable.
Partition both numbers to solve 56 + 37.
Two addition and two subtraction facts use the same three numbers.
Why we use it: It shows how addition and subtraction are connected inverse operations.
Write all four facts using 6, 8 and 14.
A model identifies the starting amount, the change and the resulting amount.
Why we use it: It reveals the structure of addition and subtraction stories.
Start with 20, subtract 7 and show the result in the model.
A two-digit factor is partitioned into tens and ones along one side of a rectangle, while the one-digit factor labels the other side.
Why we use it: It makes place-value partitioning and the two partial products visible.
Partition 18 into 10 and 8, then use an area model to solve 6 × 18.
Both two-digit factors are partitioned into tens and ones, making four smaller rectangles and four partial products.
Why we use it: It makes the distributive property and every place-value contribution visible.
Partition 32 and 15, then use a two-digit × two-digit area model to find the product.
Objects or marks are organised into equal rows and columns.
Why we use it: It shows equal groups, multiplication, factors and commutativity.
Draw 5 rows of 7. How many altogether? What related array could you turn it into?
Large multiples are subtracted in chunks, then the quotient is checked by multiplication.
Why we use it: It records division thinking and verifies that the result is reasonable.
Solve 96 ÷ 8 using a 10-group chunk first. Check your answer with multiplication.
One factor is split into useful parts and each partial multiplication is combined.
Why we use it: It turns a difficult fact into easier known facts.
Split 17 into 10 and 7 to solve 6 × 17.
A written division layout places the divisor outside the bar, the dividend inside and the quotient above.
Why we use it: It organises the repeated divide, multiply, subtract and bring-down steps by place value.
Use a division bar to solve 168 ÷ 4. Explain what happens in each place.
A drawing shows the same number of items inside each group.
Why we use it: It connects multiplication language with a visible quantity.
Draw 4 groups of 6 and write the matching multiplication equation.
Circles or regions organise a total into groups of identical size.
Why we use it: It supports both grouping and sharing interpretations of division.
Show 28 divided into 7 equal groups. How many belong in each group?
Counters or other materials are physically organised into equal sets.
Why we use it: Moving objects makes equal grouping and remainders concrete.
Use 18 counters to make 3 equal groups. How many are in each group?
A table lists pairs of whole numbers whose product is the target number.
Why we use it: It provides an organised way to find every factor.
List every factor pair for 36. How do you know the list is complete?
A composite number is repeatedly split into factors until only prime factors remain.
Why we use it: It reveals the prime-factor structure of a number.
Create a factor tree for 48. Can a different first split give the same prime factors?
A two-digit factor is written vertically above a one-digit factor, with place values aligned before each digit is multiplied.
Why we use it: It provides a compact, reliable record and makes regrouping visible.
Use the formal algorithm to solve 47 × 6. Show any regrouping.
A three-digit factor is multiplied by the ones and then the tens of a two-digit factor, creating two aligned partial products.
Why we use it: It shows each place-value contribution before the partial products are combined.
Use the formal algorithm to solve 214 × 32. Explain the place-value shift in the second row.
Two multiplication and two division facts use the same three numbers.
Why we use it: It shows the inverse relationship between multiplication and division.
Write the four related facts for 7, 8 and 56.
A number is written as its prime factors arranged in ascending order.
Why we use it: It gives a consistent final record of prime factorisation.
Write 90 as an ordered product of prime factors.
Separate products are calculated from partitioned factors and then added.
Why we use it: It exposes the place-value thinking inside multidigit multiplication.
Find the partial products for 26 × 13, then combine them.
A multiplication equation is paired with its inverse division equation.
Why we use it: It helps children use known facts to solve unfamiliar division calculations.
Use 9 × 6 = 54 to write and solve a related division equation.
A whole strip is partitioned into equal sections to show division.
Why we use it: It shows the relationship between the whole, number of groups and group size.
Partition a strip representing 30 into 5 equal groups. What is each group worth?
A square divided into 100 equal cells represents hundredths, decimals and percentages.
Why we use it: It makes the connection between a part of 100, a decimal and a percentage visible.
Shade 42 squares. Write the amount as a fraction, decimal and percentage.
The same fraction is shown using two different shapes or models.
Why we use it: It helps children recognise that the value stays the same even when the representation changes.
Show 2/3 using both a circle and a bar. What must stay the same?
A table places fractions beside decimals with the same value.
Why we use it: It organises common equivalences so patterns and relationships are easy to compare.
Add 1/5, 2/5 and 4/5 to a fraction–decimal table.
A table matches fractions, decimals and percentages that represent the same quantity.
Why we use it: It allows three forms of the same value to be read and compared together.
Complete the row for 3/5 as a decimal and percentage.
An equation uses an equals sign to connect fractions with the same value.
Why we use it: It records how the numerator and denominator can change while the quantity remains equal.
Complete 2/3 = □/6 = 8/□ and prove the equivalence with a drawing.
Moveable strips show one whole partitioned into different fractional units.
Why we use it: Children can align, combine and compare pieces to explore fraction size and equivalence.
Use the strips to find two different ways to make one whole.
Equal-length wholes are partitioned differently and aligned for direct comparison.
Why we use it: Keeping the wholes the same size makes equivalent fractions and relative size trustworthy.
Align halves, fourths and eighths. Which pieces cover exactly the same length as 1/2?
A shape is divided into equal parts and labelled to connect the picture with fraction notation.
Why we use it: It clarifies that the denominator counts equal parts in the whole and the numerator counts selected parts.
Divide a rectangle into 6 equal parts, shade 5 and label the fraction.
Decimals are positioned on a line according to their distance from zero.
Why we use it: It shows decimal order, magnitude and the intervals between values.
Mark 0.35 on a number line from 0.3 to 0.4. Explain its position.
Two aligned scales show corresponding values in different forms.
Why we use it: It makes proportional relationships and equivalent fractions, decimals or percentages visible.
Make aligned fraction and percentage scales, then locate 2/5 and 40%.
Fractions are placed at equal intervals to show their value and distance from zero.
Why we use it: It treats fractions as numbers and supports ordering, comparison and equivalence.
Mark 1/4, 2/4, 3/4 and 5/4 on a number line.
A place-value chart continues beyond the ones column into tenths, hundredths and smaller decimal places. It is also known as a segmented bar.
Why we use it: It shows that the same place-value system continues to the right of the decimal point.
Place 6.08 in a chart. What does the zero tell us?
A balance compares an object with known masses until both sides are level.
Why we use it: It makes equal mass visible and connects combining weights with measuring an object.
The object balances 500 g, 200 g and 50 g. What is its mass?
A labelled outline shows the lengths around the outside of a shape.
Why we use it: It emphasises that perimeter is the total distance around a boundary.
Add the labelled sides. What is the perimeter?
A picture of a container and scale shows how much liquid it can hold.
Why we use it: It links millilitres and litres to visible amounts in familiar containers.
Combine 600 mL and 350 mL. How much liquid is there altogether?
A complex shape is split into familiar shapes whose areas can be found separately.
Why we use it: It turns an unfamiliar area problem into smaller, manageable parts.
Find each rectangle’s area, then combine them. What is the total area?
A drawing records an object’s dimensions beside the parts being measured.
Why we use it: It helps children match each number and unit to the correct dimension.
Sketch a table and label its length, width and height.
A prism is drawn as layers of equal cubes to show the space inside it.
Why we use it: It connects length × width × height with counting cubic units.
How many cubes are in one layer? How many in all the layers?
A table records the mass of several objects with an appropriate unit.
Why we use it: It makes measurements easy to compare and encourages sensible unit choices.
Which listed object is heaviest? Which unit suits each object?
A table organises objects, measurements and units in aligned columns.
Why we use it: It supports accurate recording and clear comparison of measurement data.
Measure three classroom objects and record each length and unit.
A linked sequence shows how metric units relate through multiplication and division.
Why we use it: It makes the scale changes between kilometres, metres, centimetres and millimetres explicit.
Convert 3.5 m to centimetres. Which link in the chain helps?
A place-value table aligns metric units so a measurement can be renamed in another unit.
Why we use it: It shows that converting changes the unit and numeral, not the actual length.
Use the table to rename 2.35 m in centimetres and millimetres.
Equal square units are arranged in rows and columns to cover a rectangle.
Why we use it: It connects counting squares with multiplying length by width.
A rectangle is 6 units by 4 units. How many square units cover it?
A clock face uses hour and minute hands to show time.
Why we use it: It shows how hours and minutes move around a circular scale.
Show half past four. Where should each hand point?
A digital display writes the hour and minutes using numerals separated by a colon.
Why we use it: It supports reading exact times and interpreting 12-hour and 24-hour displays.
Read 07:45 in words. How many minutes until 8:00?
The same time is shown on an analogue face and a digital display.
Why we use it: It connects the movement of clock hands with digital notation.
Make 2:15 on an analogue clock. How do both displays show a quarter past two?
A timeline marks a start time, an elapsed duration and a finish time.
Why we use it: It makes the passage of time and duration visible from left to right.
Start at 9:20 am and add 45 minutes. What is the finish time?
A table of times is matched to events positioned along a timeline.
Why we use it: It connects listed information with the order and duration of real events.
How long passes from leaving home to the lesson starting?
Tally marks record durations or frequencies, with the same information placed on a timeline.
Why we use it: It links counted time intervals with their position and length in a schedule.
Which activity lasts longest? How do the tallies and timeline both show this?
A timetable organises times and activities, with the needed entry made easy to locate.
Why we use it: It develops the skill of scanning a schedule and selecting only relevant information.
What happens at 9:45? How long until the next listed activity?
Two drawings show a shape before and after it slides, flips or turns.
Why we use it: It makes the movement clear while showing which properties stay unchanged.
Slide the triangle four squares right. What changed, and what stayed the same?
A numbered grid locates points using an ordered pair: across first, then up.
Why we use it: It gives a precise language for position and helps reveal shapes made by plotted points.
Plot (2, 4). Which coordinate tells you how far across?
A shape is cut into parts and the same parts are rearranged to make another shape.
Why we use it: It shows how shapes and area relationships can be understood through decomposition.
Cut a rectangle diagonally. What two shapes are made?
A three-dimensional object is represented as it appears from one direction.
Why we use it: It develops spatial visualisation by connecting a model with its flat views.
Build the block model. Draw what you see from the top.
A route is traced across a square grid using directions and distances.
Why we use it: It connects position, movement and directional language.
Begin at the school. Follow 3 right and 4 up. Where do you finish?
An angle is drawn with an arc and labelled by its size or type.
Why we use it: It connects the amount of turn with acute, right and obtuse angle language.
Draw a 40° angle. Is it acute, right or obtuse?
A shape is annotated with names, side lengths or other known properties.
Why we use it: Labels make the mathematical information in a diagram precise and easy to discuss.
Draw a pentagon, name its vertices and label two side lengths.
A net shows all the flat faces arranged so they can fold into a three-dimensional object.
Why we use it: It links two-dimensional faces with the object they form.
Which edges of this cube net will meet when it is folded?
A space is drawn as if viewed directly from above.
Why we use it: It supports reading maps, arranging spaces and describing relative position.
Sketch your classroom from above. Where are the door and whiteboard?
The square corner of an object is used as a 90° reference for comparing angles.
Why we use it: It helps estimate and classify angles without measuring every one.
Use a paper corner to find an angle smaller and an angle larger than 90°.
A table records and compares features such as sides, vertices, faces or angles.
Why we use it: It helps children notice patterns and classify shapes by their properties.
Add a hexagon to the table. How many sides and vertices does it have?
A drawing uses visible and hidden edges to represent a solid object.
Why we use it: It helps identify faces, edges and vertices when a physical model is unavailable.
Choose a prism. Draw it and label one face, edge and vertex.
A line joins plotted values to show how a quantity changes across an ordered scale, often time.
Why we use it: It makes trends, rises, falls and periods of change visible.
What happened to the temperature after midday?
Separate horizontal bars show the frequency for each category.
Why we use it: Bar lengths make category comparisons quick and clear.
Which sport has the most votes? How many more than tennis?
Categories and their counted frequencies are organised in aligned columns.
Why we use it: It summarises categorical data in a form that is easy to compare or graph.
Which pet is least popular? How many students chose it?
A table records the outcomes and frequencies from repeated chance trials.
Why we use it: It reveals patterns in experimental results and supports comparisons with predictions.
Which spinner colour occurred most often? Does that prove it will occur next?
Separate vertical columns show the frequency for each category.
Why we use it: Column heights make differences between categories visible.
On which day were 8 books borrowed?
Each data value is shown as a dot above a numbered scale, with repeated values stacked.
Why we use it: It keeps individual values visible while showing clusters, gaps and frequency.
How many students have exactly two pets?
A table shows how many times each value or outcome occurs.
Why we use it: It condenses a data set into an organised count.
How many students read at least two books?
Relevant values, bars or features are marked to support an answer.
Why we use it: It connects a conclusion directly to mathematical evidence in the data display.
Highlight the evidence that proves option B received the most votes.
Every result that could occur in a chance experiment is written systematically.
Why we use it: It helps ensure no possible outcome is missed or repeated.
List all possible results when two coins are tossed.
Repeated pictures represent quantities according to a stated key.
Why we use it: It makes data visually engaging while developing understanding of scale.
If one apple represents two pieces, how many were sold on Tuesday?
The possibilities from two chance events are crossed in a table to show every combination.
Why we use it: It provides a systematic way to build a complete sample space.
Use the table to find all outcomes for tossing two coins.
Marks are recorded as events occur, with every fifth mark crossing the previous four.
Why we use it: Groups of five make live counting and checking totals efficient.
How does the tally show seven votes for apples?
A table organises categories, tally marks and their numerical totals.
Why we use it: It records data as it is collected and prepares it for analysis or graphing.
Check each tally against its total. Which travel method occurred most?
Familiar arrangements show a quantity in a pattern that can be recognised without counting one by one.
Why we use it: It develops subitising and helps children see how numbers are composed.
How do the dice, domino and fingers each show 5? What parts can you see?
Objects are organised in pairs to show whether every object has a partner.
Why we use it: It makes the structure of odd and even numbers visible.
Arrange 7 counters in pairs. What proves that 7 is odd?
Beads arranged in groups of five and ten are moved to represent numbers and calculations.
Why we use it: It supports subitising, making ten and explaining efficient calculation strategies.
Show 7 + 6. How can making ten help you see the total?
A fraction is represented by selecting equal groups from a collection of separate objects.
Why we use it: It shows that fractions can describe sets as well as parts of one continuous whole.
Three of 12 counters are shaded. What fraction of the collection is shaded?
A ruler, jug, thermometer or scale uses equal intervals and labelled benchmarks to show a measurement.
Why we use it: It teaches children to interpret intervals, units and values between labelled marks.
What does each small interval represent? Read the measurement shown.
A triangular dot grid guides a drawing so three-dimensional edges follow consistent directions.
Why we use it: It helps children record and interpret three-dimensional structures accurately.
Build the cube model, then draw it on the grid. Which edges stay parallel?
Straws or sticks form edges and connectors form vertices, leaving faces open.
Why we use it: It makes the edge-and-vertex structure of prisms and pyramids easy to inspect.
Build a triangular prism. How many edges and vertices does it have?
A line of symmetry divides a figure into matching reflected halves.
Why we use it: It helps children recognise reflection and complete missing symmetrical parts.
Complete the other half of the butterfly. How can you check that it matches?
A shape or group of shapes repeats to cover a surface with no gaps or overlaps.
Why we use it: It develops spatial reasoning and reveals which angles and shapes fit together.
Will this shape tessellate? Continue the pattern to justify your answer.
Each symbol or scale interval represents more than one item, according to a key.
Why we use it: It displays larger quantities compactly and develops multiplicative interpretation of scales.
If one symbol represents 5 votes, what do three symbols represent?
Real objects are sorted and aligned into categories to make a concrete graph.
Why we use it: It connects collecting and sorting with the structure of later picture and column graphs.
Which category has the most objects? How many more than the least?
Events are placed from impossible to certain, sometimes using values from 0 to 1.
Why we use it: It gives precise visual meaning to chance language such as unlikely, even chance and likely.
Where would you place “rolling a number less than 7 on a standard die”? Why?
Two related data sets are shown with paired columns for each category.
Why we use it: It allows direct comparison within each category and across the whole display.
In which category is the difference between the two groups greatest?
A sequence of drawings grows according to a consistent numerical rule.
Why we use it: It connects visible growth with number patterns and generalisation.
The figures show 1, 3, 5 and 7 squares. How many will the next figure have?
Inputs are changed by the same rule to produce corresponding outputs.
Why we use it: It makes a rule testable and helps children describe relationships between quantities.
If the rule is × 3, what output belongs with an input of 6?
A short unit of colours, shapes, movements or sounds repeats in the same order.
Why we use it: It helps children identify the repeating unit, continue a pattern and predict later terms.
Circle the repeating unit. What will the next three shapes be?