Grade 6 Mathematics Study Notes

Welcome to Numbers

Numbers are everywhere — on price tags, population reports, cheques and budgets. This strand builds the number skills you need for everyday life.

What this strand covers

• Whole Numbers — place value, reading and writing, ordering, rounding, squares and square roots
• Multiplication — up to 4-digit × 2-digit numbers, estimating, patterns
• Division — up to 4-digit ÷ 1-, 2- and 3-digit numbers, estimating, combined operations
• Fractions — LCM, adding and subtracting fractions and mixed numbers, reciprocals, percentages
• Decimals — place value, rounding, converting, adding and subtracting
• Inequalities — inequality symbols, forming and simplifying

Lesson 1 focus

This lesson covers: place value and total value up to millions, reading numbers in symbols, reading and writing numbers in words up to 100 000, ordering, rounding to the nearest thousand, and squares and square roots.

2. Total Value up to Millions

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

Total value = digit × place value

Worked example — find the total value of each digit in 5 098 126

First put the number in a place value chart. Then multiply each digit by its place value.

Digit	Place value	Total value
5	Millions	5 × 1 000 000 = 5 000 000
0	Hundred Thousands	0 × 100 000 = 0
9	Ten Thousands	9 × 10 000 = 90 000
8	Thousands	8 × 1 000 = 8 000
1	Hundreds	1 × 100 = 100
2	Tens	2 × 10 = 20
6	Ones	6 × 1 = 6

Two things to remember

• A digit in a higher column has a larger total value, even if it is a small digit.
• Digit 0 always has a total value of zero.

1. Place Value up to Millions

The place value of a digit is its position in the number. Each position has a name.

Building up to one million

Each time you add 1 to a row of 9s, you move one place higher:

• 9 + 1 = 10 → Ten
• 99 + 1 = 100 → Hundred
• 999 + 1 = 1 000 → Thousand
• 9 999 + 1 = 10 000 → Ten Thousand
• 99 999 + 1 = 100 000 → Hundred Thousand
• 999 999 + 1 = 1 000 000 → One Million

How to find the place value of a digit

Put the number into a place value chart. The column a digit sits in is its place value.

Reading from the chart

From the chart for 2 139 408:
Digit 2 → Millions · Digit 1 → Hundred Thousands · Digit 3 → Ten Thousands · Digit 9 → Thousands · Digit 4 → Hundreds · Digit 0 → Tens · Digit 8 → Ones.

Representing a number on an abacus

An abacus is a rectangular wooden frame with vertical rods. Each rod represents one place value column, from Ones on the right to Millions on the left. Beads are slid up the rod to show the digit in that column. The number of beads slid up equals the digit.

$diagram:{description:"A front-facing rectangular wooden abacus frame, approximately 30 cm wide and 20 cm tall, with a plain light-brown wood finish. Seven thin vertical wooden rods are fixed inside the frame, evenly spaced from left to right. Each rod is labelled at the top with a place value name, from left to right: Millions, Hundred Thousands, Ten Thousands, Thousands, Hundreds, Tens, Ones. Each rod has ten round wooden beads threaded on it. Beads that represent the digit are slid upward and shown in red; beads not in use rest at the bottom and are shown in natural wood colour. The number shown is 2 139 408: on the Millions rod, 2 red beads are at the top; on the Hundred Thousands rod, 1 red bead is at the top; on the Ten Thousands rod, 3 red beads are at the top; on the Thousands rod, 9 red beads are at the top; on the Hundreds rod, 4 red beads are at the top; on the Tens rod, all beads rest at the bottom (0 beads up, digit is 0); on the Ones rod, 8 red beads are at the top. Label A — the Millions rod showing 2 red beads up. Label B — the Tens rod showing 0 beads up, with the note 'digit 0 — no beads slid up'. Label C — caption below the frame: 'Abacus showing 2 139 408'. Plain white background.", title:"Figure 1.2: Abacus showing the number 2 139 408">

4. Reading Numbers up to 9 999 999 in Symbols

To read a large number, split it into groups of three digits from the right. Read each group and say its group name.

Worked examples

• 895 401 → split: 895 | 401
  Read: Eight hundred and ninety five thousand, four hundred and one.
• 965 312 → split: 965 | 312
  Read: Nine hundred and sixty five thousand, three hundred and twelve.
• 1 045 320 → split: 1 | 045 | 320
  Read: One million, forty five thousand, three hundred and twenty.

Group names to remember

• Right group (ones group): ones, tens, hundreds.
• Middle group: thousands, ten thousands, hundred thousands.
• Left group: millions.

5. Reading Numbers up to 99 999 999 in Symbols

8-digit numbers follow the same rule: split into groups of three from the right, read each group, say the group name.

Worked examples

• 11 928 010 → split: 11

6. Reading Numbers up to 999 999 999 in Symbols

9-digit numbers reach into the hundreds of millions. Same rule: split into groups of three from the right.

Worked examples

• 325 196 000 → split: 325

7. Reading and Writing Numbers up to 50 000 in Words

Writing a number in words means expressing each part by its place value name. This is used on cheques, deposit slips and official documents.

How to write a number in words

1. Break the number into place value parts.
2. Write out the expanded form.
3. Read from left to right: thousands first, then hundreds, then tens and ones together.

Worked example 1 — Write 25 136 in words

25 136 = 2 ten thousands + 5 thousands + 1 hundred + 3 tens + 6 ones
= 20 000 + 5 000 + 100 + 30 + 6

Read: Twenty five thousand, one hundred and thirty six.

Worked example 2 — Write 40 404 in words

40 404 = 4 ten thousands + 0 thousands + 4 hundreds + 0 tens + 4 ones
= 40 000 + 0 + 400 + 0 + 4

Read: Forty thousand, four hundred and four.

Note: Thousands are read together as one group. Hundreds are read separately. Tens and ones are read together.

8. Reading and Writing Numbers from 50 001 to 100 000 in Words

The same method works for numbers from 50 001 to 100 000.

Worked examples

• 50 204 = 50 thousands + 2 hundreds + 4 ones
  Read: Fifty thousand, two hundred and four.
• 78 125 = 78 thousands + 1 hundred + 25
  Read: Seventy eight thousand, one hundred and twenty five.
• 100 000 = 1 hundred thousand
  Read: One hundred thousand.

Forming the largest number from given digits

To form the largest 5-digit number from 0, 8, 4, 6, 5: arrange digits from largest to smallest → 86 540.
Written in words: Eighty six thousand, five hundred and forty.

10. Arranging Numbers up to 100 000 from the Smallest to the Largest

Arranging numbers from smallest to largest is called ascending order.

How to arrange numbers in ascending order

1. Put each number into a place value chart.
2. Compare digits column by column, starting from the leftmost column.
3. The number with the smaller digit in the first column that differs is the smaller number.

Worked example

Arrange 20 305, 20 410, 20 154, 20 350, 20 408 from smallest to largest.

$diagram:{description:"A rectangular table with six columns and six rows on a plain white background. The column headings are: Number, Ten Thousands, Thousands, Hundreds, Tens, Ones. Row 1 data: 20 305, 2, 0, 3, 0, 5. Row 2 data: 20 410, 2, 0, 4, 1, 0. Row 3 data: 20 154, 2, 0, 1, 5, 4. Row 4 data: 20 350, 2, 0, 3, 5, 0. Row 5 data: 20 408, 2, 0, 4, 0, 8. Label A — an arrow pointing to the Ten Thousands column with the note 'all show 2 — no difference here'. Label B — an arrow pointing to the Thousands column with the note 'all show 0 — no difference here'. Label C — an arrow pointing to the Hundreds column with the note 'digits differ here: 3, 4, 1, 3, 4 — compare these'. Label D — a red circle around the digit 1 in row 3, with the note '1 is smallest → 20 154 comes first'. All borders are thin black lines, plain white background.", title:"Figure 1.3: Place value chart for arranging 5 numbers in ascending order">

From the chart: the Hundreds digits are 3, 4, 1, 3, 4. The smallest is 1 → 20 154 is first.
Hundreds digit 3 appears twice (20 305 and 20 350): compare Tens — 0 < 5, so 20 305 comes before 20 350.
Hundreds digit 4 appears twice (20 410 and 20 408): compare Tens — 0 < 1, so 20 408 comes before 20 410.

Ascending order: 20 154, 20 305, 20 350, 20 408, 20 410.

Forming and ordering numbers by rearranging digits

Rearrange the last three digits of 32 148 to form other 5-digit numbers, then arrange all six in ascending order.

Last three digits are 1, 4, 8. The six numbers are: 32 148, 32 184, 32 418, 32 481, 32 814, 32 841.

Ascending order: 32 148, 32 184, 32 418, 32 481, 32 814, 32 841.

11. Arranging Numbers up to 100 000 from the Largest to the Smallest

Arranging numbers from largest to smallest is called descending order. Use the same place value chart method, but pick the larger digit first.

Worked example

Six schools got cheques: sh 80 479, sh 84 079, sh 87 940, sh 89 470, sh 84 097, sh 89 407. Write in descending order.

All start with 8 in Ten Thousands — compare Thousands:
9 is largest → the two 89 000s come first. Among them, compare Hundreds: 4 > 0, so 89 470 > 89 407.
Next Thousands digit is 7 → 87 940.
Next is 4 → the two 84 000s. Compare Tens: 9 > 7, so 84 097 > 84 079.
Smallest Thousands digit is 0 → 80 479.

Descending order: 89 470, 89 407, 87 940, 84 097, 84 079, 80 479.

13. Rounding Off Numbers up to 10 000 to the Nearest Thousand

Rounding off means replacing a number with the nearest round number. When rounding to the nearest thousand, we find the thousand that is closest.

Using a number line

A number line is a straight line with equally spaced marks, each mark representing a number. Numbers to the left are smaller; numbers to the right are larger. To round, place the number on the line between the two thousands on either side and see which is nearer.

$diagram:{description:"A horizontal straight line drawn on plain white paper, running from left to right. The line is approximately 20 cm long. At the far left end, the number 6 000 is written and marked with a short vertical tick. At the far right end, the number 7 000 is written and marked with a short vertical tick. Along the line, ten equally spaced intermediate ticks are marked and labelled: 6 100, 6 200, 6 300, 6 400, 6 500, 6 600, 6 700, 6 800, 6 900. The midpoint tick at 6 500 is drawn slightly taller than the others and labelled '6 500 (midpoint)' with a dashed vertical line above it. A solid red dot is placed on the line at the position of 6 350 (between the 6 300 and 6 400 ticks, closer to 6 300). An arrow above this dot points left toward 6 000, labelled 'closer to 6 000'. A second solid red dot is placed at 6 850 (between 6 800 and 6 900, closer to 6 900). An arrow above this dot points right toward 7 000, labelled 'closer to 7 000'. The line, ticks and labels are black; the dots and arrows are red. Plain white background.", title:"Figure 1.4: Number line showing 6 350 is nearer 6 000 and 6 850 is nearer 7 000">

The rounding rule

Look at the digit in the Hundreds place:

• If it is less than 5 → the Thousands digit stays the same; write zeros in Hundreds, Tens and Ones. (Round down.)
• If it is 5 or more → add 1 to the Thousands digit; write zeros in Hundreds, Tens and Ones. (Round up.)

Worked examples

• 6 350: lies between 6 000 and 7 000. It is nearer to 6 000. Hundreds digit = 3 (less than 5) → rounds to 6 000.
• 6 850: lies between 6 000 and 7 000. It is nearer to 7 000. Hundreds digit = 8 (more than 5) → rounds to 7 000.
• 8 475: Hundreds digit = 4 (less than 5) → rounds to 8 000.

14. Rounding Off Numbers from 10 000 to 100 000 to the Nearest Thousand

The same rule applies to larger numbers: check the Hundreds digit, then round down or up.

Number line example

$diagram:{description:"A horizontal straight line on plain white paper, running left to right, approximately 20 cm long. At the far left end, the number 27 000 is written and marked with a short vertical tick. At the far right end, the number 28 000 is written and marked with a tick. Along the line, ten equally spaced ticks are marked and labelled: 27 100, 27 200, 27 300, 27 400, 27 500, 27 600, 27 700, 27 800, 27 900. The midpoint tick at 27 500 is drawn slightly taller and labelled '27 500 (midpoint)' with a dashed vertical line above it. A solid red dot is placed at 27 460 (between 27 400 and 27 500, clearly to the left of the midpoint). An arrow above the dot points left toward 27 000, labelled 'closer to 27 000 → rounds to 27 000'. Line and ticks are black; dot and arrow are red. Plain white background.", title:"Figure 1.5: Number line showing 27 460 rounds to 27 000">

Worked examples

• 27 460: lies between 27 000 and 28 000. Hundreds digit = 4 (less than 5) → rounds to 27 000.
• 64 715: Hundreds digit = 7 (more than 5) → rounds to 65 000.
• 78 925: Hundreds digit = 9 (more than 5) → rounds to 79 000.

Rounding using a table

Number	Hundreds digit	Rounded to nearest 1 000
18 470	4 — less than 5	18 000
42 936	9 — 5 or more	43 000
59 325	3 — less than 5	59 000
78 731	7 — 5 or more	79 000