5.1 Whole-Number Place Value, Rounding & Roman Numerals

Mathematics scope and how to use the examples

ISEB specifies National Curriculum mathematics through the end of Year 5, including earlier-year foundations. The official framework lists number/place value, calculations, fractions/decimals/percentages, measurement, geometry and statistics. Check the Department for Education programme of study for the year-by-year detail.

Prioritise core methods before optional extensions. Mean, formal ratio sharing, fraction-by-fraction multiplication, polygon angle formulae, parallel-line angle theorems and cuboid surface-area formulae are later work included for interested readers. Their presence here does not mean ISEB requires them or publishes their frequency. All numerical scenarios in this guide are invented practice examples unless a source says otherwise.

4.1 Place Value, Ordering, Negative Numbers & Rounding

Quick Summary: Place value is the structural foundation of all arithmetic. On selective entrance assessments such as the ISEB Common Pre-Test, mathematics questions test your conceptual mastery of the base-10 system from millions down to thousandths, your fluency in shifting digits when multiplying or dividing by powers of 10, your precision when ordering ragged decimals, your ability to calculate temperature and elevation changes across zero, and your speed in decoding Roman numerals up to 1,000.

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The Base-10 Positional System: From Millions to Thousandths

Our numerical system is a base-10 positional decimal system. In this architecture, the position of each digit determines its actual numerical value. Every step to the left represents a ten-fold increase ($\times 10$), and every step to the right represents a ten-fold decrease ($\div 10$).

A critical distinction tested on selective exams is the difference between a digit's face value and its place value:

• In the integer $4,782,159$, the face value of the digit $8$ is simply eight.
• Its place value is ten thousands ($10,000$).
• Its total value is $8 \times 10,000 = 80,000$ (eighty thousand).

  Millions   H-Th     T-Th    Thousands  Hundreds    Tens     Ones   •  Tenths   Hundredths Thousandths
   (10⁶)    (10⁵)    (10⁴)      (10³)      (10²)     (10¹)    (10⁰)  •  (10⁻¹)     (10⁻²)     (10⁻³)
 ┌────────┬────────┬────────┬───────────┬──────────┬────────┬────────┬─┬────────┬──────────┬───────────┐
 │   4    │   7    │   8    │     2     │    1     │   5    │   9    │.│   3    │    6     │     4     │
 └────────┴────────┴────────┴───────────┴──────────┴────────┴────────┴─┴────────┴──────────┴───────────┘
   4M       700k     80k         2k         100        50       9   •   0.3       0.06       0.004

Master Place Value Reference Table

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Formal Rounding Rules & The 5-and-Above Threshold

Rounding replaces an exact number with an approximate value that is simpler to compute or communicate. On the Pre-Test, rounding items assess whole numbers up to the nearest $100,000$ and decimals to the nearest whole unit, tenth (1 decimal place), and hundredth (2 decimal places).

The Decider Rule

To round any number to a specified target column:

1. Identify the target place value column.
2. Look at the digit immediately to its right—this is the decider digit.
3. Apply the threshold rule:
   • If the decider digit is $0, 1, 2, 3,$ or $4$, round down (the target digit stays the same).
   • If the decider digit is $5, 6, 7, 8,$ or $9$, round up (increase the target digit by 1).
4. All columns to the right of the target column become zeros (for integers) or are dropped entirely (for decimals).

Example: Round 47,825 to the nearest thousand
Target Column: 7 (Thousands)
Decider Digit: 8 (Hundreds, immediately to the right)
Since 8 ≥ 5 → Round UP: 7 becomes 8 → Result: 48,000

Optional extension: rounding to two decimal places

Year 5 requires rounding decimals with two places to a whole number or one decimal place. This example extends the idea to rounding three places to two:

• Problem: Round $4.996$ to 2 decimal places (nearest hundredth).
  • Target column: hundredths ($9$).
  • Decider digit: thousandths ($6$).
  • Since $6 \ge 5$, add $1$ to the hundredths column: $9 + 1 = 10$.
  • The zero remains in hundredths, and $1$ carries into tenths: $9 + 1 = 10$.
  • The zero remains in tenths, and $1$ carries into units: $4 + 1 = 5$.
  • Correct answer: $5.00$.

  Exam Warning: You must write $5.00$, not simply $5$. The question specified two decimal places, which requires explicitly showing the two zero place-holders.

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Roman Numerals: Additive and Subtractive Rules up to 1,000 (M)

The Roman numeral system uses seven capital letters. Unlike our decimal system, Roman numerals do not have a place value system or a symbol for zero.

The Seven Basic Symbols

The Three Essential Syntax Rules

1. Additive Rule: When a symbol is followed by a symbol of equal or smaller value, the values are added together:

   • $\text{VI} = 5 + 1 = 6$
   • $\text{LXX} = 50 + 10 + 10 = 70$
   • $\text{MCC} = 1,000 + 100 + 100 = 1,200$

2. The Repetition Constraint:

   • The symbols I, X, C, and M can be repeated up to a maximum of three times consecutively (e.g. $\text{III} = 3, \text{XXX} = 30, \text{CCC} = 300, \text{MMM} = 3,000$). They cannot be written four times in a row.
   • The symbols V, L, and D represent mid-points ($5, 50, 500$) and are never repeated.

3. Subtractive Rule: To avoid repeating a symbol four times, a smaller symbol placed before a larger symbol indicates subtraction. Only powers of 10 (I, X, C) can be used as subtractive prefixes, and each can only be paired with the next two higher symbols:

Exam Trap: You cannot write $\text{IL}$ for $49$ or $\text{IC}$ for $99$. $\text{I}$ can only be subtracted from $\text{V}$ and $\text{X}$. To write $99$, decompose into tens and ones: $90 + 9 = \text{XC} + \text{IX} = \text{XCIX}$.

Systematic Decomposition for Conversions

To convert any Hindu-Arabic number into Roman numerals, partition it into thousands, hundreds, tens, and units:

$$\begin{aligned} 1,984 &= 1,000 + 900 + 80 + 4 \\ &= \text{M} + \text{CM} + \text{LXXX} + \text{IV} \\ &= \mathbf{\text{MCMLXXXIV}} \end{aligned}$$ $$\begin{aligned} 847 &= 800 + 40 + 7 \\ &= \text{DCCC} + \text{XL} + \text{VII} \\ &= \mathbf{\text{DCCCXLVII}} \end{aligned}$$

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Core rounding and counting checks

To round 6.37 to the nearest whole number, inspect the tenths digit 3: the result is 6. To round the same number to one decimal place, inspect the hundredths digit 7: the result is 6.4. The requested precision changes which digit decides. For a whole-number example, 346,728 rounds to 350,000 to the nearest ten thousand because the thousands digit is 6.

Counting in steps is different from multiplying by the step size. Starting at 45,630 and adding 1,000 each time gives 46,630, 47,630, 48,630. Counting backwards by 100 gives 45,530, 45,430, 45,330. A decimal sequence 1.25, 1.50, 1.75, 2.00 adds 0.25 each time; its fraction version adds one quarter. Describe the change in words and verify every supplied jump before extending the sequence.