Grade 7 Mathematics Study Notes

Place value using the abacus

An abacus is a counting frame made of a rectangular wooden or plastic border holding several vertical rods with beads that slide freely up and down. Each rod represents one position in a number — exactly like one column in the place value chart.

How the abacus shows a digit

• Each rod has up to nine beads.
• Push beads up toward the crossbar to show a digit — the number of beads pushed up equals the digit.
• Beads resting at the base are not counted.
• A rod with no beads pushed up represents the digit zero in that position.

Numbers

In this strand you will work with numbers that appear in everyday life — factory output figures, county budgets, population counts, shop prices. The strand builds skills from place value all the way up to sequences, fractions, and square roots.

What this strand covers

• Whole Numbers — place value, total value, reading, writing, rounding, operations, sequences.
• Factors — divisibility tests, prime factors, GCD, LCM.
• Fractions — comparing, ordering, and all four operations.
• Decimals — place value, total value, multiplication, division.
• Squares and Square Roots — of whole numbers, fractions, and decimals.

How to approach this strand

Work through one block at a time. Read the definition, study each worked example carefully, then try the self-test before moving on. Each block builds on the one before it.

This lesson

You will start with whole numbers — what they are and how to find the place value of any digit using a place value chart and an abacus.

Whole numbers

Whole numbers are the counting numbers starting from zero: 0, 1, 2, 3, 4, 5 … They do not include fractions or decimals.

In this sub-strand we work with whole numbers up to nine digits — from single digits all the way to hundreds of millions.

Examples of whole numbers

• Single digit: 7
• Three digits: 384
• Seven digits: 4 320 519
• Nine digits: 105 627 394

How large numbers are written

Large numbers are grouped into sets of three digits from the right, separated by spaces. This makes them easier to read.
Example: 105627394 is written as 105 627 394.

Reading an abacus

To find the place value of a digit using an abacus, count the beads pushed up on each rod, then read the label at the base of that rod.

Worked example — abacus set to 105 627 394

Read each rod from left to right:

Reading all rods together gives: 105 627 394.

Place value

The place value of a digit is the name of the position it holds in a number. The position, not the digit itself, determines its value.

The digit 5 means something different in each of these numbers:

Same digit. Three different place values — because the position changed.

The nine positions

Every whole number up to hundreds of millions has at most nine positions, listed from right to left:

1. Ones
2. Tens
3. Hundreds
4. Thousands
5. Tens of thousands
6. Hundreds of thousands
7. Millions
8. Tens of millions
9. Hundreds of millions

Key Points

• Place value is the name of the position a digit occupies in the number.
• Nine positions from ones (right) to hundreds of millions (left).
• A place value chart has nine labelled columns — one digit per column.
• An abacus has nine rods — beads pushed up equal the digit in that position.
• A zero digit holds a real position in both tools — write 0 in the chart column; leave the abacus rod empty.
• Removing or skipping a zero shifts all other digits and changes the number completely.

Place value chart vs. abacus

The place value chart

A place value chart is a table with nine labelled columns — one column for each position. You write one digit per column. The column label above any digit is its place value.

How to fill the chart

1. Write the number out in full.
2. Place the first digit of the number in the column that matches its position.
3. Continue placing each digit left to right across the chart.
4. If a position has no digit, write 0 in that column — never leave it blank.

A number with nine digits fills all nine columns. A number with fewer digits leaves the leftmost columns empty.

Reading the place value chart — worked examples

Once the chart is filled, read the column label above any digit to find its place value. That is all there is to it.

Worked example 1 — find the place value of one digit

A county government allocated sh 105 627 394 for dam construction. Find the place value of digit 1.

Fill the chart:

Column label above digit 1 → Hundreds of millions.
✔ The place value of digit 1 is hundreds of millions.

Worked example 2 — find the place value of every digit

Find the place value of each digit in 538 214 976.

• 5 → hundreds of millions
• 3 → tens of millions
• 8 → millions
• 2 → hundreds of thousands
• 1 → tens of thousands
• 4 → thousands
• 9 → hundreds
• 7 → tens
• 6 → ones

Why zeros matter

The 0 in 105 627 394 holds the tens of millions column. Without it, the number becomes 15 627 394 and digit 1 drops to tens of millions. Never remove or skip a zero.

Total value

The total value of a digit is the actual amount that digit contributes to the number. It is found by multiplying the digit by its place value.

Place value tells you the name of the position. Total value tells you the amount at that position.

The difference — place value vs. total value

Same digit 4 — but three very different total values depending on position.

Total value in real life

Total value is used whenever you need to know the exact contribution of one digit to a large number — for example, finding how many coffee trees are in one part of a farm count, or how many vaccines were used in one category of a national report.

Worked example 1 — coffee trees

A farmer had 160 279 403 coffee trees. What is the total value of digit 6?

Digit 6 → tens of millions column → 6 × 10 000 000 = 60 000 000.

There are sixty million trees accounted for by that digit.

Worked example 2 — underlined digits

Find the total value of the underlined digit in each number:

• 524 135 398 → digit 2 is in tens of millions → 2 × 10 000 000 = 20 000 000
• 346 820 172 → digit 4 is in tens of millions → 4 × 10 000 000 = 40 000 000
• 436 015 398 → digit 4 is in hundreds of millions → 4 × 100 000 000 = 400 000 000

Key Points

• Total value = digit × place value.
• Place value names the position; total value gives the amount.
• The total values of all digits in a number add up to the original number.
• The same digit has a different total value in different positions — because the place value changes.

Place value vs. total value — summary

Calculating total value

To find the total value of each digit in a number, fill the place value chart, then multiply each digit by its place value. The results are the total values.

Worked example — total value of every digit in 517 368 294

Adding all total values: 500 000 000 + 10 000 000 + 7 000 000 + 300 000 + 60 000 + 8 000 + 200 + 90 + 4 = 517 368 294. ✔ The total values sum back to the original number.

Reading and writing numbers in symbols

Numbers written using digits (0–9) are written in symbols. Reading a number in symbols means saying it aloud correctly — naming the groups from left to right.

How to read a large number in symbols

1. Identify the number of digits to know the largest group (millions, thousands, or ones).
2. Read the millions group first, then say "million".
3. Read the thousands group next, then say "thousand".
4. Read the ones group last.

Worked example

Read aloud: 124 755 212

• Millions group: 124 → "one hundred and twenty-four million"
• Thousands group: 755 → "seven hundred and fifty-five thousand"
• Ones group: 212 → "two hundred and twelve"

Full reading: "One hundred and twenty-four million, seven hundred and fifty-five thousand, two hundred and twelve."

Writing numbers in symbols from words

To convert a number from words to symbols, break the word statement into its groups (millions, thousands, ones) and write each group as digits. Use zeros to fill any empty positions within a group.

Worked example 1

Write in symbols: "One hundred and twenty-four million, seven hundred and fifty-five thousand, two hundred and twelve."

In symbols: 124 755 212 ✔

Worked example 2 — missing group

Write in symbols: "Four million, nine hundred and seventy-three thousand, one hundred and one."

• Millions: 4 (single digit → write as 4, not 004 — the millions group simply has one digit here)
• Thousands: 973
• Ones: 101

In symbols: 4 973 101 ✔

Reading and writing numbers in words

Writing a number in words means spelling out the full name of each group. The method uses the total value of each digit to build the word statement group by group.

Worked example — write 3 515 252 in words

In words: "Three million, five hundred and fifteen thousand, two hundred and fifty-two."

Practice numbers

• 268 197
• 357 128
• 961 756
• 3 184 993

Writing numbers in words on cheques

A cheque is a written order from one person to a bank to pay a stated amount to another person. By law, the amount must be written both in symbols and in words.

Writing the amount in words prevents fraud — it is much harder to alter written words than to change a digit.

How to write an amount on a cheque

1. Write the amount in symbols in the figures box.
2. Break the number into millions, thousands, and ones groups.
3. Write each group in words on the amount line.
4. Do not leave gaps — draw a line through any unused space.

Worked example

Kizito sold land worth sh 2 103 247. Write this in words for the cheque.

2 (millions) = two million
103 (thousands) = one hundred and three thousand
247 (ones) = two hundred and forty-seven

Written on cheque: "Two million, one hundred and three thousand, two hundred and forty-seven shillings."

Writing numbers in words up to millions

When writing a number in words, break it into its place value groups from left to right. Write each group in words, then combine them in order.

Worked example — 1 207 382

1. Fill place value chart: 1(mill), 2(H.thou), 0(T.thou), 7(thou), 3(hund), 8(tens), 2(ones).
2. Millions group (1): one million
3. Thousands group (207): two hundred and seven thousand
4. Ones group (382): three hundred and eighty-two

In words: "One million, two hundred and seven thousand, three hundred and eighty-two."

What to do with a zero group

If a group is 000, skip it in the words but keep it in the symbols.
Example: 5 000 008 → "Five million and eight" (the thousands group 000 is skipped).

Key Points

• Numbers written using digits are written in symbols.
• To read a large number: name the millions group first, then thousands, then ones.
• To write in words: break into groups using the place value chart, write each group in words, combine in order.
• On a cheque, always write the amount in both symbols and words to prevent fraud.
• A group of 000 is skipped in the words but kept in the symbols.

Where numbers are written in both words and symbols

• Bank cheques
• Legal documents and contracts
• Government bills and receipts
• National census reports

Rounding off numbers

Rounding off means replacing an exact number with a simpler approximate value that is close to the original. Rounded numbers are easier to work with and useful for estimating.

The two rounding rules

1. Look at the digit immediately to the right of the position you are rounding to.
2. If that digit is 0, 1, 2, 3, or 4 → keep the digit in the rounding position the same (round down). Replace all digits to its right with zeros.
3. If that digit is 5, 6, 7, 8, or 9 → increase the digit in the rounding position by 1 (round up). Replace all digits to its right with zeros.

Where rounding is used

• Estimating large quantities quickly (population, harvest figures, budgets).
• Reporting approximate results (tax totals, supermarket sales).
• Making mental calculations manageable.

Rounding off to the nearest hundreds of millions

To round to the nearest hundreds of millions, look at the digit in the tens of millions position (one step to the right of hundreds of millions).

• Tens of millions digit is 0–4 → keep the hundreds of millions digit, replace all to the right with zeros.
• Tens of millions digit is 5–9 → add 1 to the hundreds of millions digit, replace all to the right with zeros.

Worked example 1 — round 219 486 272 to nearest hundred million

Decision digit (tens of millions) = 1. Since 1 < 5 → keep 2. Result: 200 000 000 ✔

Worked example 2 — round 87 148 729 to nearest hundred million

H.millions digit = 8. Decision digit (T.millions) = 7. Since 7 ≥ 5 → add 1: 8 + 1 = 9. Result: 90 000 000 ✔

Note: 87 148 729 is an 8-digit number — its hundreds of millions digit is 0. Rounding gives 90 000 000, which rounds up to the nearest hundred million.

Key Points

• Rounding off replaces an exact number with a simpler, approximate value.
• Always look at the digit one step to the right of the rounding position.
• 0–4 → round down (keep rounding digit the same).
• 5–9 → round up (add 1 to rounding digit).
• All digits to the right of the rounding position become zero.

Rounding positions — summary

Rounding off to the nearest ten millions

To round to the nearest ten millions, look at the digit in the millions position (one step to the right of ten millions).

• Millions digit is 0–4 → keep the tens of millions digit, replace digits to the right with zeros.
• Millions digit is 5–9 → add 1 to the tens of millions digit, replace digits to the right with zeros.

Worked example 1 — round 87 148 729 to nearest ten million

Place in chart: 8(T.mill), 7(mill), 1(H.thou), 4(T.thou), 8(thou), 7(hund), 2(tens), 9(ones).

Rounding to: tens of millions (digit 8). Decision digit: millions (digit 7). Since 7 ≥ 5 → round up: 8 + 1 = 9.

Result: 90 000 000 ✔

Worked example 2 — round 14 875 629 to nearest ten million

Tens of millions digit = 1. Decision digit = 4 (millions). Since 4 < 5 → keep 1.

Result: 10 000 000 ✔