Basic 8 · Mathematics · Compare and Order Whole Numbers
Comparing and Ordering Numbers and Standard Form
In this topic you will compare and order whole numbers using >, < and =, and express numbers of any size in standard form.
By the end of these notes you should be able to:
- Compare whole numbers using >, < and =
- Order whole numbers in real-life contexts
- Express large numbers in standard form
What you already know
- Place value and powers of ten
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
- Compare and Order Whole Numbers
- Indicator
- B8.1.1.1.3 – Compare and order whole numbers using “>, < and =”
Introduction
Which is more, 9,999,999,999 or 10,000,000,000? Comparing numbers correctly depends on place value, not on how the digits look.
1. Steps for Comparing
- Count the digits. The number with more digits is greater.
- If they have the same number of digits, compare from the left.
- The first place where digits differ decides which is greater.
- Use >, < or = to record your answer.
Compare
Compare 45,678,912,300 and 45,687,120,000.
- Both have 11 digits.
- 4 = 4, 5 = 5, 6 = 6.
- Next digits: 7 and 8; 7 < 8.
Answer: 45,678,912,300 < 45,687,120,000
2. Comparing Expressions
Sometimes you must calculate first. To compare 3 × 10⁹ and 28 × 10⁸, write both in full: 3,000,000,000 and 2,800,000,000. So 3 × 10⁹ > 28 × 10⁸. To compare 2.5 billion and 2,050,000,000, change 2.5 billion to 2,500,000,000.
3. Using a Number Line
On a number line, numbers increase to the right. A number to the right of another is greater. Placing large numbers between marked points, such as 2 billion and 3 billion, helps you to see their relative size.
4. Common Mistakes
Do not compare only the first digit: 8,000,000 has a larger first digit than 12,000,000, but it has fewer digits and so is smaller. Do not confuse the symbols: the wide end of > and < always faces the greater number.
New words
- greater than
- Larger in value (>).
- less than
- Smaller in value (<).
- compare
- To decide which is larger or smaller.
- expression
- A mathematical phrase such as 3 × 10⁹.
Summary
- Compare by the number of digits, then digit by digit from the left.
- Write expressions in full before comparing.
- The wide end of > or < faces the greater number.
Class activity: Compare Game
- Each partner draws two number cards to form a large number.
- Compare the numbers using >, < or =.
- The partner with the greater number scores a point.
You need: Number cards, Greater-than, Less-than symbol cards, Bottle tops
Exercises
Answer in your exercise book first, then check your answers.
1. 7,000,000 ___ 70,000,000
- A. <
- B. >
- C. =
- D. +
Show answer
Answer: <Fewer digits.2. Which is greatest?
- A. 3 × 10⁹
- B. 29 × 10⁷
- C. 2,950,000,000
- D. 2.5 billion
Show answer
Answer: 3 × 10⁹Write in full.3. A number with more digits is greater, for whole numbers.
- A. True
- B. False
Show answer
Answer: TrueDigits.4. The symbol > means greater ______.
Show answer
Answer: thanComparison.5. Compare 5.6 billion and 5,060,000,000.
Show answer
Answer: 5.6 billion = 5,600,000,000, which is greater.Convert.6. Explain why 8,000,000 is less than 12,000,000 even though 8 > 1.
Show answer
Answer: 12,000,000 has eight digits and 8,000,000 has seven, so place value makes it greater.Place value.
Lesson 2
- Strand
- Number
- Sub-strand
- Compare and Order Whole Numbers
- Indicator
- B8.1.1.1.3 – Compare and order whole numbers using “>, < and =”
Introduction
Tables of population, money and distance are usually arranged in order. Ordering numbers lets us see patterns quickly.
1. Ascending and Descending Order
Ascending order goes from least to greatest, and descending order from greatest to least. To order several numbers, line them up by place value and compare from the left.
Order the numbers
Arrange in descending order: 2,046,000,000; 2,406,000,000; 2,064,000,000; 2,460,000,000.
- All have 10 digits and begin with 2.
- Hundred-millions digits: 0, 4, 0, 4.
- Among the 4s: 2,460,000,000 > 2,406,000,000.
- Among the 0s: 2,064,000,000 > 2,046,000,000.
Answer: 2,460,000,000; 2,406,000,000; 2,064,000,000; 2,046,000,000
2. Ordering Data in Context
| City | Approximate population |
|---|---|
| Accra (metropolitan area) | over 4,000,000 |
| Kumasi (metropolitan area) | over 3,000,000 |
| Sekondi-Takoradi | about 450,000 |
| Tamale | about 370,000 |
This table is arranged in descending order, so the largest city comes first. Population figures are estimates and vary from one source to another, so always note where your data comes from.
3. Ranking
Sports tables and class results are arranged by ranking. First position has the highest score. If two scores are equal, they may share a position. Ordering numbers correctly ensures that results are fair.
4. Rounding before Ordering
Reports often round large numbers. When numbers are close, rounding can hide differences: 3,496,000 and 3,504,000 both round to 3,500,000 to the nearest hundred thousand, although one is greater. Use exact values when order matters.
New words
- ascending order
- From least to greatest.
- descending order
- From greatest to least.
- rank
- A position in an ordered list.
- estimate
- An approximate value.
Summary
- Order numbers by comparing place values from the left.
- Ordering helps to organise real-life data such as populations and scores.
- Use exact values when numbers are close.
Class activity: Data Ordering
- Collect the sizes of five districts or towns from a reference book.
- Arrange them in ascending and descending order.
- Present your table to the class.
You need: Number cards, Greater-than, Less-than symbol cards, Bottle tops
Exercises
Answer in your exercise book first, then check your answers.
1. Ordering from least to greatest is…
- A. ascending
- B. descending
- C. ranking down
- D. rounding
Show answer
Answer: ascendingGoing up.2. Which is the smallest?
- A. 1,009,000,000
- B. 1,090,000,000
- C. 1,900,000,000
- D. 1,099,000,000
Show answer
Answer: 1,009,000,000Ten millions 0.3. Rounding can hide small differences.
- A. True
- B. False
Show answer
Answer: TrueClose numbers.4. A position in an ordered list is a ______.
Show answer
Answer: rankPosition.5. Arrange in ascending order: 5,500,000; 5,050,000; 5,005,000.
Show answer
Answer: 5,005,000; 5,050,000; 5,500,000Compare.6. Explain why exact values matter when ordering close numbers.
Show answer
Answer: Rounded values may appear equal even when the true values differ.Accuracy.
Lesson 3
- Strand
- Number
- Sub-strand
- Standard Form
- Indicator
- B8.1.1.1.4 – Express integers of any size into standard form
Introduction
Scientists and engineers write very large numbers in a short way called standard form. The distance to the Sun, 150,000,000 km, becomes 1.5 × 10⁸ km.
1. What Is Standard Form?
Standard form: A way of writing a number as A × 10ⁿ, where A is at least 1 and less than 10, and n is an integer.
Standard form is also called scientific notation. It makes very large and very small numbers easier to read, compare and calculate with.
2. Writing Large Numbers in Standard Form
Write in standard form
Write 45,600,000 in standard form.
- Place the decimal point after the first non-zero digit: 4.56.
- Count how many places the digits moved: 7.
Answer: 4.56 × 10⁷
| Number | Standard form |
|---|---|
| 3,000 | 3 × 10³ |
| 72,000 | 7.2 × 10⁴ |
| 500,000 | 5 × 10⁵ |
| 86,400,000 | 8.64 × 10⁷ |
| 1,200,000,000 | 1.2 × 10⁹ |
3. Converting Back
To change standard form to an ordinary number, multiply by the power of ten. For 3.4 × 10⁶, move the digits six places to the left: 3,400,000. Fill empty places with zeros.
4. Common Errors
The first number must be between 1 and 10. So 45 × 10⁶ is not in standard form; change it to 4.5 × 10⁷. And 0.45 × 10⁸ must also become 4.5 × 10⁷. Do not confuse the index with the number of zeros in numbers such as 1.2 × 10⁹, which has eight zeros, not nine.
New words
- standard form
- A number written as A × 10ⁿ with A between 1 and 10.
- scientific notation
- Another name for standard form.
- index
- The power of ten in standard form.
- integer
- A whole number, positive, negative or zero.
Summary
- Standard form is A × 10ⁿ, where 1 ≤ A < 10.
- To write it, place the point after the first non-zero digit and count the places.
- Multiply by the power of ten to change back to an ordinary number.
Class activity: Space Numbers
- Look up distances to planets in a reference book.
- Write each in standard form.
- Order the planets from nearest to farthest.
You need: Printed power-of-ten cards, Decimal place-value chart, Graph paper
Exercises
Answer in your exercise book first, then check your answers.
1. 5,000 in standard form is…
- A. 5 × 10³
- B. 50 × 10²
- C. 0.5 × 10⁴
- D. 5 × 10⁴
Show answer
Answer: 5 × 10³A between 1 and 10.2. 2.3 × 10⁵ =
- A. 230,000
- B. 23,000
- C. 2,300,000
- D. 2,300
Show answer
Answer: 230,000Five places.3. 45 × 10⁶ is in standard form.
- A. True
- B. False
Show answer
Answer: FalseA must be below 10.4. 7,600,000 = 7.6 × 10 to the power ______.
Show answer
Answer: 6Six places.5. Write 930,000,000 in standard form.
Show answer
Answer: 9.3 × 10⁸Eight places.6. Write 0.72 × 10⁹ correctly in standard form.
Show answer
Answer: 7.2 × 10⁸Adjust A and n.
Homework
- Arrange five large numbers from the news in ascending order.
- Write ten large numbers in standard form.