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Year 5 Maths Whole Numbers & Place Value

Partition & rename numbers (expanded notation)

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Theory

Partitioning and renaming numbers is a Year 5 place value skill: writing a number in expanded notation as the sum of the value of each digit, splitting it into other parts that add to the same total, and renaming it as so many thousands, hundreds or tens. This Year 5 maths page explains each step with six- and seven-digit numbers, the role of zero, and how to find missing parts.

Expanded notation writes a number as the sum of the value of each digit: \(52\,407 = 50\,000 + 2000 + 400 + 7\).

In charts and working, HTh means hundred thousands, TTh ten thousands, Th thousands, and H, T, O hundreds, tens and ones.

To partition a number is to split it into parts that add back to the number. Standard partitioning uses place value parts, but you can also use other parts: \(437\,215 = 400\,000 + 37\,215\) or \(437\,215 = 430\,000 + 7215\).

To rename a number is to say it using a different unit. \(437\,215\) is \(437\) thousands, \(2\) hundreds, \(1\) ten and \(5\) ones. It is also \(4372\) hundreds and \(15\) ones.

A zero digit is worth nothing, so it gets no term in expanded notation: \(708\,040 = 700\,000 + 8000 + 40\).

Place value chart for 437 215 with the value of each digit A place value chart with columns HTh, TTh, Th, H, T and O holding 4, 3, 7, 2, 1 and 5. Under each digit is its value: 400 000, 30 000, 7000, 200, 10 and 5. HThTThThHTO 437215 400 00030 0007000200105
Each digit of \(437\,215\) has a value. Adding the values gives the expanded notation.
Place value chart for 708 040 with zero digits in red A place value chart for 708 040. The zeros in the ten thousands, hundreds and ones places are red; the values are 700 000, 8000 and 40. HThTThThHTO 708040 700 000080000400
The red zeros in \(708\,040\) are worth nothing, so \(708\,040 = 700\,000 + 8000 + 40\).

Expanded notation and renaming both use place value:

\[ 437\,000 = 400\,000 + 30\,000 + 7000 \]
437000=400000+30000+7000
\[ 215 = 200 + 10 + 5 \]
215=200+10+5
\[ 437\,215 = 437 \text{ thousands} + 215 \]
437215=437 thousands +215
\[ 1 \text{ million} = 1000 \text{ thousands} \]
1 million =1000 thousands
\[ 1 \text{ thousand} = 10 \text{ hundreds} = 100 \text{ tens} \]
1 thousand =10 hundreds =100 tens
Counting thousands or hundreds. To find how many thousands are in a number, cover the last three digits: \(8\,350\,000\) has \(8350\) thousands. To count hundreds, cover the last two digits: \(61\,900\) has \(619\) hundreds.
Missing parts. If \(437\,215 = 400\,000 + \underline{\quad}\), the missing part is \(437\,215 - 400\,000 = 37\,215\).

How to write expanded notation

  1. Name the place of each digit, starting from the left.
  2. Write its value: the digit followed by the right number of zeros.
  3. Add the values, leaving out any zero digits.

How to write a numeral from expanded notation

  1. Find the largest place used, so you know how many digits the number has.
  2. Place each value in its column.
  3. Fill empty places with zeros.

How to complete a partition

  1. Subtract the part you know from the whole number.
  2. Check by adding the two parts back together.
Example 1 — Expanded notation
Write \(651\,384\) in expanded notation.
Solution
\(6 \text{ HTh}\)\(=\)\(600\,000\)
\(5 \text{ TTh}\)\(=\)\(50\,000\)
\(1 \text{ Th}\)\(=\)\(1000\)
\(384\)\(=\)\(300 + 80 + 4\)
651384=600000+50000+1000+300+80+4

Add the four lines: \(651\,384\) is the sum of the value of each digit.

Example 2 — Zeros in the number
Write \(308\,090\) in expanded notation.
Solution
\(308\,090\)\(=\)\(300\,000 + 8000 + 90\)
308090=300000+8000+90

The ten thousands, hundreds and ones digits are zero, so they have no term.

Example 3 — Expanded notation to a numeral
Write the numeral for \(200\,000 + 9000 + 70 + 1\).
Solution
\(\text{HTh, TTh, Th}\)\(:\)\(2,\ 0,\ 9\)
\(\text{H, T, O}\)\(:\)\(0,\ 7,\ 1\)
200000+9000+70+1=209071

The number is \(209\,071\). Zeros hold the empty ten thousands and hundreds places.

Example 4 — Missing part
Complete: \(519\,600 = 400\,000 + \underline{\qquad}\)
Solution
\(519\,600 - 400\,000\)\(=\)\(119\,600\)
519600−400000=119600

The missing part is \(119\,600\). Check: \(400\,000 + 119\,600 = 519\,600\).

Example 5 — Different partitions
Complete each partition of \(357\,400\).
(i) \(357\,400 = 300\,000 + \underline{\qquad}\)
(ii) \(357\,400 = 350\,000 + \underline{\qquad}\)
(iii) \(357\,400 = 357\,000 + \underline{\qquad}\)
Solution
\(357\,400 - 300\,000\)\(=\)\(57\,400\)
\(357\,400 - 350\,000\)\(=\)\(7400\)
\(357\,400 - 357\,000\)\(=\)\(400\)
357400−350000=7400

(i) \(57\,400\).
(ii) \(7400\).
(iii) \(400\).

Example 6 — Rename
(i) How many thousands are in \(8\,350\,000\)?
(ii) How many hundreds are in \(61\,900\)?
(iii) How many thousands make \(9\) million?
Solution
\(8\,350\,000\)\(=\)\(8350 \text{ thousands}\)
\(61\,900\)\(=\)\(619 \text{ hundreds}\)
\(9 \text{ million}\)\(=\)\(9000 \text{ thousands}\)
8350000=8350 thousands

(i) \(8350\).
(ii) \(619\).
(iii) \(9000\).

Example 7 — Thousands, hundreds, tens and ones
Write \(283\,560\) as thousands, hundreds and tens.
Solution
\(283\,560\)\(=\)\(283\,000 + 500 + 60\)
283560=283000+500+60

\(283\) thousands, \(5\) hundreds and \(6\) tens.

Example 8 — Find the error
Noah wrote the expanded notation of \(603\,200\) as \(600\,000 + 30\,000 + 200\). Explain the mistake.
Solution
\(\text{the } 3\)\(=\)\(3 \text{ thousands}\)
\(\text{Noah's sum}\)\(=\)\(630\,200\)
603200=600000+3000+200

The \(3\) is worth \(3000\), not \(30\,000\), because a zero holds the ten thousands place. The correct expanded notation is \(600\,000 + 3000 + 200\).

Common pitfalls

Wrong number of zeros. The value of a digit has one zero for each place to its right. The \(7\) in \(437\,215\) is \(7000\), not \(700\).
Writing terms for zeros. A zero digit is worth nothing, so \(708\,040\) has no ten thousands, hundreds or ones term.
Losing a zero when going back. \(200\,000 + 9000 + 70 + 1\) is \(209\,071\), not \(29\,071\). Fill every empty place with a zero.
Hundreds is not the hundreds digit. The hundreds digit of \(61\,900\) is \(9\), but the number of hundreds altogether is \(619\).

Frequently asked questions

What is expanded notation?

Expanded notation writes a number as the sum of the values of its digits. For example, \(84\,036 = 80\,000 + 4000 + 30 + 6\).

What happens to zeros in expanded notation?

A zero digit is worth nothing, so it has no term. \(308\,090\) is \(300\,000 + 8000 + 90\).

What does partition mean?

To partition a number is to split it into parts that add back to the whole. \(357\,400\) can be split as \(300\,000 + 57\,400\) or \(350\,000 + 7400\).

How do you find a missing part of a partition?

Subtract the part you know from the whole number. For \(519\,600 = 400\,000 + \text{?}\), the missing part is \(119\,600\).

How many thousands are in a number?

Cover the last three digits and read what is left. \(8\,350\,000\) has \(8350\) thousands.

How many thousands make a million?

One million is \(1000\) thousands, so \(9\) million is \(9000\) thousands.

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