Place Value and Unitising

The big idea

The value of a digit depends on its position, and any number can be treated as a collection of units of different sizes. This flexibility, seeing 34 as 3 tens and 4 ones, or as 34 ones, or as 2 tens and 14 ones, is what makes calculation, comparison, and later algebraic reasoning possible. Place value is not a primary-school topic left behind; it is the foundation on which every other concept in this framework rests.

Why this matters for secondary maths

Place value is the single concept that every other section in this framework assumes. If it is not secure, nothing built on top of it will be stable.

The deeper idea beneath place value is unitising: treating a group of objects as a single countable unit. When a pupil says “3 tens”, they are treating “ten” as a unit that can be counted, just as they might count 3 apples. This same reasoning reappears throughout mathematics:

3 tenths + 4 tenths = 7 tenths (decimal arithmetic)
$3 x + 4 x = 7 x$ (algebra: $x$ is the unit being counted)
3 fifths + 4 fifths = 7 fifths (fraction arithmetic)
3 metres + 4 metres = 7 metres (measurement)

In each case, the underlying reasoning is identical: when the unit is the same, we count the number of those units. Pupils who understand this principle from place value are equipped to collect like terms, add fractions with common denominators, and combine measurements because they recognise the same structure in a new context.

What goes wrong without it: pupils who lack flexible place value understanding struggle with column methods (especially when zeros or exchanges are involved), cannot estimate sensibly, misread decimal place value (treating 0.35 as “thirty-five” rather than “3 tenths and 5 hundredths”), and find the transition to algebraic terms (“what do you mean, $3 x$ ?”) bewildering rather than natural.

Key representations

Place value counters and columns

Physical or drawn counters placed in labelled columns (thousands, hundreds, tens, ones, tenths, hundredths) allow pupils to see the value of each digit and to exchange between columns. For example, 304 shown as 3 counters in the hundreds column, 0 in the tens, and 4 in the ones. When subtracting, a counter can be exchanged from the hundreds column for 10 counters in the tens column, making the exchange process visible and meaningful.

Strengths: Makes regrouping/exchange concrete. Extends naturally to decimals. Supports understanding of zero as a placeholder.

Limitation: Can become cumbersome for very large numbers or for calculations where mental methods are more efficient.

Dienes blocks (base-ten blocks)

Physical blocks in which a unit cube represents 1, a rod represents 10, a flat represents 100, and a large cube represents 1000. These provide a proportional model, the rod is physically ten times the size of the cube, which reinforces the multiplicative relationship between place value columns.

Strengths: The proportional sizing makes the $\times 10$ relationship between adjacent columns physically tangible. Effective for exchange and regrouping.

Limitation: Difficult to extend to decimals (what is one tenth of a unit cube?). Can become unwieldy beyond thousands.

Number lines

A number line can show where a number sits in relation to its neighbours and to multiples of 10, 100, 1000. Zooming in on a section of the number line, for example, the interval from 3.4 to 3.5, divided into ten equal parts, develops decimal place value. This also prepares pupils for the idea that between any two numbers there are infinitely many others.

Strengths: Connects place value to magnitude and ordering. Supports estimation. Extends seamlessly into decimals and negative numbers.

Limitation: Does not make exchange or regrouping as visible as counters or blocks.

Expanded form and partitioning diagrams

Writing a number in expanded form, e.g. $4507 = 4000 + 500 + 0 + 7$ , makes the contribution of each digit explicit. Part–whole diagrams showing a number decomposed into different combinations of units ( $34 = 30 + 4 = 20 + 14 = 10 + 24$ ) develop flexible partitioning.

Strengths: Directly supports structural calculation. Essential for understanding why written methods work.

Worked examples

Example 1: Flexible partitioning

Partition 63 in three different ways.

$63 = 60 + 3$ (standard place value partition)
$63 = 50 + 13$ (useful if, for example, subtracting 7 from 63, because $13 - 7$ is easy)
$63 = 40 + 23$ (useful in other contexts)

The point is not that one partition is correct and the others are wrong. All three represent the same number. Choosing which partition to use is a strategic decision. The same kind of strategic decision pupils will later make when choosing how to rearrange an algebraic expression.

Example 2: Unitising across the decimal point

What is $0.3 + 0.4$ ?

If we think in tenths: 3 tenths + 4 tenths = 7 tenths = 0.7. The calculation is as simple as 3 + 4, because the unit (tenths) is the same. This is the same reasoning as: if I have 3 apples and gain 4 apples, I have 7 apples. The unit is what matters.

Example 3: Understanding exchange

Calculate $403 - 178$ using place value understanding.

We need to subtract 8 ones from 3 ones.
There are no tens to exchange, so we exchange 1 hundred for 10 tens, giving 3 hundreds, 10 tens, 3 ones.
Then exchange 1 ten for 10 ones, giving 3 hundreds, 9 tens, 13 ones.
Now subtract: 3 hundreds − 1 hundred = 2 hundreds; 9 tens − 7 tens = 2 tens; 13 ones − 8 ones = 5 ones.
Result: 225.

Place value counters make every step of this process visible. The pupil sees why the exchange works, not just how to perform it.

Example 4: Place value and multiplication

Calculate $6 \times 34$ .

Partition 34 as $30 + 4$ . Then $6 \times 34 = 6 \times 30 + 6 \times 4 = 180 + 24 = 204$ . This uses the distributive law, but it depends on place value: the pupil must see 34 as made of 30 and 4 before the law can be applied.

The bridge to algebra

The unitising principle is the direct conceptual bridge from place value to algebra.

Place value: 3 tens + 4 tens = 7 tens

Algebra: $3 x + 4 x = 7 x$

The reasoning is identical. In both cases, we are counting units of the same kind. The tens are the unit in the first case; $x$ is the unit in the second. A pupil who deeply understands that “3 tens + 4 tens = 7 tens” is already doing the cognitive work of collecting like terms, they just don’t know it yet.

This extends further:

Flexible partitioning ( $63 = 50 + 13$ ) parallels algebraic decomposition: $5 x + 13$ might be rewritten as $3 x + 2 x + 13$ if useful.
Expanded form ( $4507 = 4000 + 500 + 7$ ) parallels expanding expressions: $4 x^{3} + 5 x^{2} + 7$ has the same additive structure across powers.
The idea that $12 x = 10 x + 2 x$ is a place-value-style decomposition of a term.

The key message: collecting like terms is unitising. If pupils arrive in algebra with a strong sense that you can count anything as a unit, tens, tenths, apples, $x$ ’s, then $3 x + 4 x = 7 x$ is not a new rule. It is the same principle in a new notation.

Key vocabulary

Term	Definition
Place value	The value of a digit determined by its position in a number. In 352, the 5 has a value of 50 (5 tens).
Unitising	Treating a group or quantity as a single countable unit. ‘Ten’ is a unit; ‘one fifth’ is a unit; ‘ $x$ ’ is a unit.
Partition / decompose	To break a number into parts, typically by place value. $47 = 40 + 7$ , or $47 = 30 + 17$ .
Regroup / exchange	To rearrange the units in a number without changing its value. Exchange 1 ten for 10 ones.
Placeholder	A zero used to indicate that a particular place value column is empty. In 305, the 0 shows there are no tens.
Expanded form	A number written as the sum of the values of each digit: $4507 = 4000 + 500 + 0 + 7$ .
Like terms	Terms that have the same unit and can therefore be combined: $3 x$ and $5 x$ are like terms; $3 x$ and $5 y$ are not.

What we don’t say

Avoid	Why	Say instead
‘You can’t take 8 from 3’	This is mathematically false (the answer is −5). It also discourages flexible thinking. In directed number work, pupils learn that subtraction can produce negative results.	‘In this column, $3 - 8$ would give a negative result, so we need to regroup’, or use a structural method that avoids the issue.
‘Borrow from the tens column’ (without explanation)	Borrowing is a shorthand for exchanging, which is a place value operation. If pupils do not understand why borrowing works, it becomes a mysterious ritual.	‘Exchange one ten for ten ones.’ Better still, show the exchange with counters so the pupil sees the value being preserved.
‘The decimal point separates big numbers from small numbers’	The decimal point marks the boundary between whole number units and fractional units. ‘Big’ and ‘small’ are vague and misleading.	‘The decimal point separates whole number place values from fractional place values. Ones from tenths.’
‘Just add a zero’ (when multiplying by 10)	This rule breaks immediately with decimals: $3.5 \times 10 = 35$ , not $3.50$ . It hides the multiplicative structure: multiplying by 10 shifts every digit one place to the left.	‘Multiplying by 10 makes every digit ten times as large. Each digit shifts one place to the left.’ Use place value columns to show the shift.

Common misconceptions and how to surface them

Misconception 1: The digit tells you the value.

Pupils who say “the 5 in 352 is 5” have not internalised place value. They see the digit but not its position. Ask: ‘What is the value of the 3 in 4,302?’ and ‘What is the value of the 3 in 4,032?’ A pupil who gives the same answer for both has not connected digit to position.

Misconception 2: Treating decimals as whole numbers after the point.

Pupils who think $0.35 > 0.4$ (because “35 is more than 4”) are reading the decimal part as a separate whole number rather than as fractional place value. Ask: ‘Which is larger: $0.8$ or $0.35$ ?’ A pupil who says $0.35$ needs work on decimal place value. Show both on a number line zoomed into 0 to 1.

Misconception 3: Rigid partitioning.

Some pupils can partition 47 as $40 + 7$ but cannot see it as $30 + 17$ . They have learned one decomposition as ‘the answer’ rather than understanding that infinitely many decompositions exist. Ask: ‘Can you partition 63 in a way that makes it easy to subtract 8?’ A pupil who can only offer $60 + 3$ needs exposure to flexible partitioning.

Misconception 4: ‘Adding a zero’ when multiplying by 10.

This procedural shortcut works for whole numbers but fails for decimals ( $3.5 \times 10 \neq = 3.50$ ). More importantly, it obscures the structural idea that multiplication by 10 shifts digits one place to the left. Use place value columns to show what happens to each digit when we multiply by 10, 100, or 1000. Ask: ‘If $3.5 \times 10$ means adding a zero, what do we get?’ The absurd result (3.50, the same number) reveals the error.

Diagnostic questions

Question 1: Write 4.07 as a sum of its parts using place value. Then explain why 4.07 is less than 4.7.

What this reveals: A pupil who writes $4 + 0.07$ and can explain that 0.07 means 7 hundredths while 4.7 means 4 + 7 tenths, and that 7 tenths is larger than 7 hundredths, understands decimal place value. A pupil who cannot explain the difference, or who thinks 4.07 is larger because “7 comes after 0”, has a misconception about fractional place values.

Question 2: A pupil says: ‘ $3 x + 4 x = 7 x$ because you just add the numbers at the front.’ Is this explanation good enough? What would a better explanation sound like?

What this reveals: The ‘add the numbers at the front’ explanation is procedurally correct but structurally empty. A better explanation: ‘Three x’s plus four x’s is seven x’s, just like three tens plus four tens is seven tens, we’re counting the same unit.’ A pupil who can give the unitising explanation demonstrates the conceptual understanding the framework aims for. This question also tests whether the pupil sees the connection between place value and algebra.

Progression spine

Stage	Key ideas	Notes
Primary (Y1–Y6)	Partitioning into tens and ones, hundreds/tens/ones. Reading and writing numbers to 1,000,000. Decimal place value to hundredths. Exchange in column methods.	Pupils arrive in Year 7 with varying degrees of flexibility. Many can partition by place value but only in one way. The concept of unitising is rarely made explicit.
Year 7	Flexible partitioning. Unitising as a named principle. Decimal place value extended to thousandths. Connection between place value and the ×10 multiplicative structure. Expanded form as preparation for structural calculation and algebra.	The emphasis should be on flexibility and on naming the unitising principle explicitly, so pupils can transfer it to new contexts.
Years 8–11	Standard form (a × 10ⁿ). Metric unit conversions as place value shifts. Collecting like terms as unitising. Index notation and powers of 10. Decimal arithmetic in context.	By KS4, place value reasoning should be automatic. Standard form is the most prominent application: writing 4,500,000 as 4.5 × 10⁶ is a place value operation.