Basic 7 · Mathematics · Numeration Systems
Reading, Modelling and Comparing Numbers Beyond a Billion
In this topic you will read and write whole numbers greater than 1,000,000,000, model them using place value and expanded form, and compare and order them using the symbols >, < and =.
By the end of these notes you should be able to:
- Read and write numbers greater than one billion in words and figures
- Model large numbers using place value and expanded form
- Compare and order large numbers using >, < and =
What you already know
- Place value of numbers up to millions (Upper Primary)
Copy everything that has a blue line on its left into your notebook. The other parts are for reading.
Lesson 1
- Strand
- Number
- Sub-strand
- Numeration Systems
- Indicator
- B7.1.1.1.1 – Model number quantities more than 1,000,000,000
Introduction
Ghana's national budget, the number of mobile money transactions in a year and the population of Africa are all very large numbers. To read them, we use place value and periods.
1. Periods and Place Value
Large numbers are grouped into periods of three digits, counting from the right: ones, thousands, millions and billions. Each period has hundreds, tens and ones. Commas or spaces separate the periods so that the number is easy to read.
| Billions | Millions (H T O) | Thousands (H T O) | Ones (H T O) |
|---|---|---|---|
| 4 | 3 5 7 | 2 0 6 | 1 9 8 |
2. Reading Large Numbers
Read each period as a three-digit number and add the period name. The number 4,357,206,198 is read as "four billion, three hundred and fifty-seven million, two hundred and six thousand, one hundred and ninety-eight". Do not say the period name for the ones period.
3. Writing Numbers in Figures
Write in figures
Write "twelve billion, forty million, five hundred thousand and seven" in figures.
- Billions period: 12.
- Millions period: forty million → 040.
- Thousands period: five hundred thousand → 500.
- Ones period: seven → 007.
Answer: 12,040,500,007
4. Place Value and Value of Digits
The place value of a digit is its position; its value is the digit multiplied by its place value. In 6,284,000,315, the digit 6 is in the billions place, so its value is 6,000,000,000. The digit 2 is in the hundred-millions place, so its value is 200,000,000.
Tip: Zeros hold places. In 3,000,050,000, the zeros show that there are no millions in the hundreds, tens and ones of millions except the 50 thousand.
New words
- period
- A group of three digits in a large number.
- place value
- The position of a digit in a number.
- billion
- One thousand million: 1,000,000,000.
- value
- The digit multiplied by its place value.
Summary
- Large numbers are grouped into periods of three digits: ones, thousands, millions and billions.
- Read each period as a three-digit number followed by its name.
- The value of a digit is the digit multiplied by its place value.
Class activity: Place-Value Cards
- Use place-value cards to build three numbers greater than one billion.
- Read each number aloud and write it in words.
- Find the value of the digit in the hundred-millions place in each.
You need: Place-value cards, Ghana cedi play notes, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. How many zeros are in one billion?
- A. 9
- B. 6
- C. 12
- D. 3
Show answer
Answer: 91,000,000,000.2. The value of 7 in 7,350,000,000 is…
- A. 7,000,000,000
- B. 700,000,000
- C. 70,000,000
- D. 7,000,000
Show answer
Answer: 7,000,000,000Billions place.3. Large numbers are grouped in periods of three digits.
- A. True
- B. False
Show answer
Answer: TrueFrom the right.4. One thousand million is one ______.
Show answer
Answer: billion1,000,000,000.5. Write 2,045,300,010 in words.
Show answer
Answer: Two billion, forty-five million, three hundred thousand and ten.Read each period.6. Write in figures: "nine billion, nine thousand and nine".
Show answer
Answer: 9,000,009,009Zeros hold empty places.
Lesson 2
- Strand
- Number
- Sub-strand
- Numeration Systems
- Indicator
- B7.1.1.1.1 – Model number quantities more than 1,000,000,000
Introduction
A model helps us see the size of a number. We can model large numbers with place-value charts, expanded form and money.
1. Expanded Form
Expanded form: Writing a number as the sum of the values of its digits.
Write in expanded form
Write 3,402,150,006 in expanded form.
- 3 billions = 3,000,000,000
- 4 hundred millions = 400,000,000
- 2 millions = 2,000,000
- 1 hundred thousand = 100,000
- 5 ten thousands = 50,000
- 6 ones = 6
Answer: 3,000,000,000 + 400,000,000 + 2,000,000 + 100,000 + 50,000 + 6
2. Expanded Form with Powers of Ten
We can also write expanded form using powers of ten: 3,402,150,006 = (3 × 10⁹) + (4 × 10⁸) + (2 × 10⁶) + (1 × 10⁵) + (5 × 10⁴) + (6 × 10⁰). Here 10⁹ means 10 multiplied by itself nine times, which is one billion.
| Power | Value | Name |
|---|---|---|
| 10³ | 1,000 | thousand |
| 10⁶ | 1,000,000 | million |
| 10⁹ | 1,000,000,000 | billion |
| 10¹² | 1,000,000,000,000 | trillion |
3. Modelling with Money
Using Ghana cedi notes, 1,000,000,000 cedis could be modelled as 5,000,000 notes of 200 cedis, because 200 × 5,000,000 = 1,000,000,000. This shows how large a billion really is: if you counted one note every second, it would take you about 58 days without stopping to count five million notes.
4. Using Graph Paper
On graph paper, a 10 × 10 square shows 100 small squares. Ten of these large squares make 1,000. Imagining a thousand of these thousand-blocks gives a million. This helps us understand that each place is ten times the place to its right.
New words
- expanded form
- A number written as the sum of its digit values.
- power of ten
- A number such as 10³ or 10⁶.
- trillion
- One thousand billion.
- model
- A way of representing a number to show its size.
Summary
- Expanded form shows a number as the sum of the values of its digits.
- Powers of ten, such as 10⁹ for a billion, give a short way to write expanded form.
- Money and graph paper help us picture the size of large numbers.
Class activity: Billion Models
- Write three large numbers in expanded form, with and without powers of ten.
- Use cedi play notes to model one million cedis using different notes.
- Present your models to the class.
You need: Place-value cards, Ghana cedi play notes, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 10⁶ is equal to…
- A. 1,000,000
- B. 1,000
- C. 1,000,000,000
- D. 100
Show answer
Answer: 1,000,000One million.2. The expanded form of 5,020,000,000 is…
- A. 5,000,000,000 + 20,000,000
- B. 5,000,000 + 20,000
- C. 500,000,000 + 2,000,000
- D. 5,000,000,000 + 2,000,000
Show answer
Answer: 5,000,000,000 + 20,000,000Five billion and twenty million.3. Each place is ten times the place to its right.
- A. True
- B. False
Show answer
Answer: TrueBase ten.4. One billion written as a power of ten is 10 to the power ______.
Show answer
Answer: 910⁹.5. Write 7,006,000,450 in expanded form using powers of ten.
Show answer
Answer: (7 × 10⁹) + (6 × 10⁶) + (4 × 10²) + (5 × 10¹)Powers of ten.6. How many 100-cedi notes make one billion cedis?
Show answer
Answer: 10,000,000 notes1,000,000,000 ÷ 100.
Lesson 3
- Strand
- Number
- Sub-strand
- Numeration Systems
- Indicator
- B7.1.1.2 – Compare and order whole numbers more than 1,000,000,000 and represent the comparison using ">", "<" or "="
Introduction
Which is bigger: the population of Nigeria or the population of Ghana? To compare numbers, we look at their place values from left to right.
1. Symbols for Comparing
| Symbol | Meaning | Example |
|---|---|---|
| > | greater than | 5,000,000,001 > 4,999,999,999 |
| < | less than | 2,300,000,000 < 2,310,000,000 |
| = | equal to | 1,000,000,000 = 10⁹ |
2. Steps for Comparing
- Count the digits. The number with more digits is greater.
- If they have the same number of digits, compare digits from the left.
- The first place where the digits differ decides which is greater.
Compare two numbers
Compare 3,456,789,012 and 3,465,000,000.
- Both have 10 digits.
- Billions: 3 and 3, the same.
- Hundred millions: 4 and 4, the same.
- Ten millions: 5 and 6; 5 < 6.
Answer: 3,456,789,012 < 3,465,000,000
3. Ordering Numbers
Order from least to greatest
Arrange 2,050,000,000; 2,005,000,000; 2,500,000,000; 2,000,500,000.
- All have 10 digits and start with 2 billion.
- Hundred millions: 0, 0, 5, 0. So 2,500,000,000 is the greatest.
- Ten millions of the others: 5, 0, 0. So 2,050,000,000 is next.
- Compare 2,005,000,000 and 2,000,500,000: millions 5 and 0.
Answer: 2,000,500,000; 2,005,000,000; 2,050,000,000; 2,500,000,000
4. Real-Life Comparison
| Country | Population |
|---|---|
| Nigeria | about 220,000,000 |
| Ghana | about 33,000,000 |
| Togo | about 9,000,000 |
Comparing these, 220,000,000 > 33,000,000 > 9,000,000, so Nigeria has the largest population of the three.
5. Common Mistakes
A common mistake is to compare only the first digits without checking the number of digits. For example, 900,000,000 has a larger first digit than 1,000,000,000, but it has fewer digits, so it is smaller. Another mistake is to forget that zeros hold places. Always line up numbers by place value, period by period, before comparing.
New words
- greater than
- Larger in value (>).
- less than
- Smaller in value (<).
- ascending order
- From least to greatest.
- descending order
- From greatest to least.
Summary
- Use >, < and = to compare numbers.
- Compare by number of digits first, then digit by digit from the left.
- Ordering arranges numbers in ascending or descending order.
Class activity: Number Line-Up
- Each learner holds a number card with a large number.
- Line up in ascending order without talking, then check.
- Write comparison statements using > and < for neighbouring cards.
You need: Number cards, Greater-than, Less-than symbol cards, Bottle tops
Exercises
Answer in your exercise book first, then check your answers.
1. Which symbol means "less than"?
- A. <
- B. >
- C. =
- D. +
Show answer
Answer: <Points to the smaller number.2. Which is the greatest?
- A. 4,100,000,000
- B. 4,010,000,000
- C. 4,001,000,000
- D. 4,000,100,000
Show answer
Answer: 4,100,000,000Hundred millions digit 1.3. A 10-digit whole number is greater than a 9-digit whole number.
- A. True
- B. False
Show answer
Answer: TrueMore digits.4. Arranging from least to greatest is ______ order.
Show answer
Answer: ascendingGoing up.5. Insert > or <: 6,789,000,000 ___ 6,798,000,000.
Show answer
Answer: <Ten millions: 8 < 9.6. Arrange in descending order: 1,200,000,000; 1,020,000,000; 1,002,000,000; 1,220,000,000.
Show answer
Answer: 1,220,000,000; 1,200,000,000; 1,020,000,000; 1,002,000,000Greatest first.
Homework
- Find three large numbers in a newspaper, write them in words and order them.