Grade 5 Mathematics Study Notes

Welcome to Numbers

This strand is about reading, writing and working with whole numbers. By the end, you will handle numbers up to hundreds of thousands — useful for counting people, harvests, money and distances.

What this strand covers

• Whole Numbers: place value, reading and writing, ordering, rounding off, divisibility, HCF/GCD, LCM.
• Addition and Subtraction: up to 6-digit numbers.
• Multiplication and Division: up to 3-digit by 2-digit numbers.
• Fractions and Decimals: simplifying, comparing, adding, subtracting.
• Simple Equations: one unknown.

Lesson 1 focus

This lesson covers Place value of digits up to hundreds of thousands using a place value chart.

Place value of digits in up to 6-digit numbers

In this lesson you will write any 6-digit number, name the place value of each digit, and then show the number on an abacus.

Total value of digits up to hundreds of thousands

You already know that place value names the position of a digit. The total value tells you the amount the digit stands for in the whole number.

Place value of digits up to hundreds of thousands

Each time you add 1 to a row of 9s, a new place value opens up:

• 9 + 1 = 10
• 99 + 1 = 100
• 999 + 1 = 1 000
• 9 999 + 1 = 10 000
• 99 999 + 1 = 100 000

100 000 is read as one hundred thousand. It has six digits, and opens a new place value — hundreds of thousands. Every 6-digit number sits between 100 000 and 999 999.

You meet such numbers in real life — county populations, harvests in kilograms, fuel in litres, money in a bank account.

What is an abacus

An abacus is a counting tool with rods standing side by side, each rod holding small rings or beads. Every rod stands for one place value.

For 6-digit numbers, the abacus has six rods. Reading from right to left, the rods are labelled: O, T, H, Th, TTh, HTh — the same six place values as on a place value chart.

The number of rings on a rod tells you the digit in that place.

What is total value

The total value of a digit is the digit multiplied by its place value.

For example, in 495 128:

• Digit 4 sits in HTh. Total value = 4 × 100 000 = 400 000.
• Digit 2 sits in T. Total value = 2 × 10 = 20.

Total value answers the question: how much does this digit stand for in the number?

The six place values and their short forms

Every digit has a place value — the name of its position in the number. A 6-digit number has six place values. Each has a short form used as the column heading on a chart.

The number 100 000 placed on a chart looks like this:

Finding total value step by step

The book uses this method:

Worked example

The population of a county was 495 128. How many people are represented by the digit 4?

Place the digits on a chart:

• Place value of 4 is hundred thousands.
• Total value of 4 = 4 × 100 000 = 400 000.

So the digit 4 represents 400 000 people.

For the digit 2:

• Place value of 2 is tens.
• Total value of 2 = 2 × 10 = 20.

How to put a number on an abacus

To show a number on an abacus, put the same number of rings on each rod as the digit for that place value.

Worked example

Represent 253 918 on an abacus.

First, write the short forms under each digit:

Then put rings on each rod:

• HTh rod: 2 rings
• TTh rod: 5 rings
• Th rod: 3 rings
• H rod: 9 rings
• T rod: 1 ring
• O rod: 8 rings

Reading a number from an abacus

To read a number from an abacus, count the rings on each rod and write the count under the rod's short form. Then read the digits from HTh to O.

Worked example

An abacus shows: 6 rings on HTh, 4 rings on TTh, 0 rings on Th, 2 rings on H, 7 rings on T, 3 rings on O.

Reading left to right: the number is 640 273.

Total value of zero

A digit of 0 always has a total value of 0, no matter what place value it sits in.

For example, in 500 000:

• The 5 is in HTh. Total value = 5 × 100 000 = 500 000.
• Every 0 has total value = 0.

This is because 0 × any number = 0.

The same digit, different place values

The position of a digit decides its place value, not the digit itself. The same digit can have a different place value in different positions.

Look at the digit 3 in each of these numbers:

The digit is always 3, but the place value changes because the position changes.

Key Points

• An abacus is a counting tool with rods, each rod standing for one place value.
• For a 6-digit number, the abacus has six rods labelled HTh, TTh, Th, H, T, O from left to right.
• The number of rings on a rod = the digit in that place value.
• A rod with zero rings means the digit is 0.
• To read an abacus, count the rings on each rod from HTh to O.

Finding the place value of a digit

To find the place value of a digit, write the short forms directly under each digit, then read off the short form under the digit you want.

Worked example

Find the place value of the digits 5, 0 and 9 in 940 758.

Write the digits in a row. Underneath each digit, write its place value short form, starting from O on the right:

Now read off:

• The digit 5 sits above T → place value is tens.
• The digit 0 sits above Th → place value is thousands.
• The digit 9 sits above HTh → place value is hundred thousands.

Place value vs total value

Same digits, but different ideas. Place value names the column. Total value tells how much.

Key Points

• Adding 1 to 99 999 gives 100 000, opening the hundreds of thousands place.
• A 6-digit number has six place values: HTh, TTh, Th, H, T, O.
• Place value is the name of the position of a digit in a number.
• The same digit can have different place values in different positions.
• To find a digit's place value, write the short forms under each digit from the right, then read off.

Key Points

• Total value = digit × place value.
• Total value tells how much a digit stands for in the whole number.
• A digit of 0 always has total value 0.
• Place value names the position; total value gives the amount.

Reading and writing numbers up to 500 000 in symbols

You will learn to read 6-digit numbers up to 500 000 and to write them in symbols (using digits) when the parts are given to you.

Reading and writing numbers from 500 001 to 999 999 in symbols

This lesson covers larger 6-digit numbers — those greater than 500 000 but less than one million.

Reading a 6-digit number

To read a 6-digit number, relate each digit to its place value.

The book reads the number aloud as: "___ hundred and ___ thousand, ___ hundred ___."

Worked example

Read 196 125.

Place the digits on a chart:

Read as: One hundred and ninety-six thousand, one hundred and twenty-five.

Forming numbers greater than 500 000

To form a 6-digit number greater than 500 000, the digit in the HTh place must be 5, 6, 7, 8 or 9.

If the HTh digit is 5, the number must be greater than 500 000 — so it cannot be 500 000 itself.

Example

Using the digits 9, 5, 8, 6, 3 and 7, here are some 6-digit numbers greater than 500 000:

• 985 637
• 678 359
• 596 873

Writing a number in symbols from its parts

When the parts of a number are given, place each part in its column on a chart, then read the digits left to right.

Worked example

Write in symbols: 2 hundred thousands, 6 ten thousands, 4 thousands, 7 hundreds, 3 tens, 5 ones.

Place each digit in its column:

The number in symbols is 264 735.

Reading from a place value chart

When a number is given on a place value chart, read it by relating each digit to its place value, the same way as before.

Worked example

The number of camels exported in a year is shown:

The number in symbols is 645 729.

Read as: Six hundred and forty-five thousand, seven hundred and twenty-nine.

Reading numbers from charts and cards

Numbers are often given on a place value chart or on number cards. Read them by relating each digit to its place value.

Worked example

A tea factory produced this many kg of tea: 478 159. Read the number.

Place the digits on a chart:

Read as: Four hundred and seventy-eight thousand, one hundred and fifty-nine.

Writing numbers in symbols from parts

Worked examples

1. 5 hundred thousands, 9 ten thousands, 4 thousands, 3 hundreds, 2 tens, 6 ones → 5-HTh, 9-TTh, 4-Th, 3-H, 2-T, 6-O → 594 326
2. 6 hundred thousands, 1 ten thousand, 8 thousands, 5 hundreds, 3 tens, 7 ones → 6-HTh, 1-TTh, 8-Th, 5-H, 3-T, 7-O → 618 537
3. 7 hundred thousands, 0 ten thousands, 5 thousands, 3 hundreds, 0 tens, 8 ones → 7-HTh, 0-TTh, 5-Th, 3-H, 0-T, 8-O → 705 308

Key Points

• To read a 6-digit number, group the first three digits as "___ thousand", then read the last three digits.
• To write a number in symbols from its parts, place each part in its column on a place value chart.
• The chart turns the parts into the correct number from HTh to O.

Key Points

• Numbers greater than 500 000 have HTh between 5 and 9.
• The largest 6-digit number is 999 999.
• The reading rule is the same: group HTh-TTh-Th as "___ thousand", then read H-T-O.
• To write from parts, place each part in its column.

Reading and writing numbers up to 50 000 in words

You will learn to write 5-digit numbers up to 50 000 in words — useful for cheques, signs and formal writing.

Reading and writing numbers from 50 001 to 99 999 in words

This lesson covers 5-digit numbers larger than 50 000 — going up to 99 999.

Relating numbers in symbols and words

You can move both ways — from symbols to words, and from words to symbols. Both are the same number, just written differently.

Breaking a number into expanded form

Before writing in words, break the number into its expanded form — show the total value of each digit added together.

Worked example

Take 36 425.

Place the digits on a chart:

Expanded form: 30 000 + 6 000 + 400 + 20 + 5.

Forming numbers greater than 50 000

For a 5-digit number to be greater than 50 000, the digit in the TTh place must be 5, 6, 7, 8 or 9.

Example

Using the digits 4, 3, 6, 0 and 5, here are 5-digit numbers greater than 50 000:

• 64 305
• 53 460
• 65 430

From symbols to words

Use the same method as before — break into expanded form, then write in words.

Worked example

Write 27 408 in words.

Expanded form: 20 000 + 7 000 + 400 + 8

In words: Twenty-seven thousand, four hundred and eight.

Writing in words

After expanding, read each part in words and join with "thousand" between the thousands part and the hundreds part.

Worked example

36 425 = 30 000 + 6 000 + 400 + 20 + 5

In words: Thirty-six thousand, four hundred and twenty-five.

More examples

• 5 720 → Five thousand, seven hundred and twenty.
• 17 492 → Seventeen thousand, four hundred and ninety-two.
• 20 002 → Twenty thousand and two.
• 40 440 → Forty thousand, four hundred and forty.
• 45 386 → Forty-five thousand, three hundred and eighty-six.

From words to symbols

Listen to the parts in the words, write each part in its column on a place value chart, then read the digits.

Worked example

Convert "Forty-eight thousand, two hundred and fifty-one" to symbols.

Break the words into parts:

• Forty-eight thousand = 48 000
• Two hundred = 200
• Fifty-one = 51

Add: 48 000 + 200 + 51 = 48 251.

Writing larger 5-digit numbers in words

The method is the same: expanded form first, then write in words.

Worked examples

• 61 052 → 60 000 + 1 000 + 50 + 2 → Sixty-one thousand and fifty-two
• 81 734 → 80 000 + 1 000 + 700 + 30 + 4 → Eighty-one thousand, seven hundred and thirty-four
• 75 756 → 70 000 + 5 000 + 700 + 50 + 6 → Seventy-five thousand, seven hundred and fifty-six
• 90 105 → 90 000 + 100 + 5 → Ninety thousand, one hundred and five
• 99 999 → 90 000 + 9 000 + 900 + 90 + 9 → Ninety-nine thousand, nine hundred and ninety-nine

Largest and smallest 5-digit numbers

To make the largest 5-digit number from given digits, place the largest digit on the left (highest place value), then continue in decreasing order.

To make the smallest, place the smallest digit (that is not 0) on the left, then continue in increasing order.

Worked example

Using the digits 0, 1, 3, 2 and 4:

• Largest = 43 210
• Smallest = 10 234 (you cannot start with 0, so put 1 first)

Key Points

• To write a number in words, first break it into expanded form.
• Read the thousands part first, then the word "thousand", then the rest.
• Use a comma after the thousands part when writing.
• Use the word "and" before the tens or ones (e.g. "three hundred and five").

Why we need both forms

In real life:

• Symbols are used in calculations, signs, prices on tags and on number plates.
• Words are used on cheques, in formal letters and to avoid mistakes (e.g. a cheque written in words is harder to change).

Key Points

• A number can be written in symbols (digits) or in words — both mean the same thing.
• From symbols to words: break into expanded form, then write in words.
• From words to symbols: break the phrase into parts and add them.

Key Points

• For numbers greater than 50 000, the TTh digit is 5, 6, 7, 8 or 9.
• The largest 5-digit number is 99 999.
• To find the largest number from given digits, arrange them in decreasing order.
• To find the smallest, arrange in increasing order (and don't start with 0).

Arranging numbers from smallest to largest

When you arrange numbers in order, you compare them using their place values.

Arranging numbers from largest to smallest

This is the reverse of the last lesson — same method, opposite direction.

How to arrange from largest to smallest

Use the same comparison method:

1. Compare digits in the highest place value first.
2. If equal, move right one column and compare again.
3. Put the number with the larger digit first (largest on top).

Worked example

Arrange from largest to smallest: 7 126, 7 612, 7 621, 7 162, 7 261, 7 651.

All have 7 in Th. Compare H:

• H = 6: 7 612, 7 621, 7 651 (top group — largest)
• H = 2: 7 261
• H = 1: 7 126, 7 162 (bottom group — smallest)

Within the top group, compare T and O. Within the bottom group, do the same.

Final order: 7 651, 7 621, 7 612, 7 261, 7 162, 7 126.

How to compare two numbers

Compare digits in the highest place value first. If they are equal, move one place to the right and compare again.

The number with the larger highest-place digit is larger.

Example

Compare 25 050 and 20 550.

TTh column: both have 2 (equal). Th column: 5 vs 0 — so 25 050 is larger.

Arranging from smallest to largest

To arrange a list, compare pairs by place value and put them in increasing order.

Worked example

Arrange from smallest to largest: 20 505, 25 050, 25 005, 20 055, 20 550, 25 500.

All have 2 in TTh — equal. So compare the Th column:

• Numbers with 0 in Th: 20 505, 20 055, 20 550
• Numbers with 5 in Th: 25 050, 25 005, 25 500

Within each group, compare H, then T, then O.

Final order: 20 055, 20 505, 20 550, 25 005, 25 050, 25 500.

Why ordering matters in real life

Ordering helps us answer questions like:

• Which candidate got the most votes?
• Which farmer delivered the most bags?
• Which car finished the race fastest?

It also helps us find the highest or lowest in any set of numbers.

Key Points

• Compare digits in the highest place value first.
• If the highest digits are equal, move one place to the right and compare.
• The number with the larger digit in the first different column is larger.

Key Points

• Arranging from largest to smallest uses the same comparison rule as smallest to largest.
• Compare from the highest place value first; move right only when digits are equal.
• Real-life uses: ranking votes, sales, race times, populations.

Rounding off numbers to the nearest hundred

Rounding off means replacing a number with another one nearby that ends in 0s. It makes numbers easier to talk about.

Rounding off to the nearest thousand

The idea is the same as rounding to the nearest hundred — just one place value higher.

The number line picture

Between any two hundreds — say 70 400 and 70 500 — the middle point is 70 450. Any number smaller than 70 450 is closer to 70 400. Any number larger is closer to 70 500.

Numbers equal to 70 450 are taken to be closer to 70 500 (we round up).

The rounding rule for the nearest thousand

To round off to the nearest thousand, look at the digit in the hundreds place value:

• If the hundreds digit is 5 or more: the thousands digit goes up by 1. Put 0 in hundreds, tens and ones.
• If the hundreds digit is less than 5: the thousands digit stays the same. Put 0 in hundreds, tens and ones.

The rounding rule for the nearest hundred

To round off to the nearest hundred, look at the digit in the tens place value:

• If the tens digit is 5 or more: the hundreds digit goes up by 1. Put 0 in tens and ones.
• If the tens digit is less than 5: the hundreds digit stays the same. Put 0 in tens and ones.

Worked example

Round off 84 869 to the nearest thousand.

The hundreds digit is 8 (which is ≥5). So the thousands digit (4) goes up by 1 to become 5. Put 0 in hundreds, tens and ones.

Answer: 85 000

Worked examples — rounding to the nearest hundred

(a) Round 8 672

Tens digit is 7 (which is ≥5). So hundreds digit (6) goes up to 7. Put 0 in tens and ones.

Answer: 8 700

(b) Round 38 426

Tens digit is 2 (which is <5). So hundreds digit (4) stays. Put 0 in tens and ones.

Answer: 38 400

(c) Round 72 193

Tens digit is 9 (which is ≥5). So hundreds digit (1) goes up to 2. Put 0 in tens and ones.

Answer: 72 200

Key Points

• To round to the nearest thousand, check the hundreds digit.
• Hundreds digit 5 or more → round up.
• Hundreds digit less than 5 → round down.
• Put 0 in hundreds, tens and ones.

Key Points

• Rounding off makes a number simpler by replacing it with a nearby number ending in 0s.
• To round to the nearest hundred, check the tens digit.
• Tens digit 5 or more → round up. Tens digit less than 5 → round down.
• Put 0 in the tens and ones after rounding.