Order Numbers

Grade 8 · mathematics · 77 practice problems · read aloud

🔊 Listen to this explanation

📊 Ordering Numbers: Mastering Numerical Relationships

Ordering numbers means arranging them from least to greatest (ascending) or greatest to least (descending). This skill is essential for comparing data, understanding number lines, and solving real-world problems like ranking scores or analyzing measurements.

🔢 Step-by-Step Guide to Ordering Numbers

  1. Identify number types: Are they integers, decimals, fractions, or mixed?
  2. Convert to same form: Change all numbers to decimals or find common denominators for fractions
  3. Compare place values: Look at digits from left to right
  4. Account for negatives: Remember negative numbers are always less than positive numbers
  5. Arrange in order: Place numbers according to the requested order

📝 Worked Examples

Example 1: Mixed Numbers & Decimals

Order: 2.75, 2⅗, 2.6, 2⅔ from least to greatest

Step 1: Convert all to decimals → 2.75, 2.6, 2.6, 2.666...

Step 2: Compare → 2.6 = 2.6 < 2.666... < 2.75

Answer: 2.6, 2⅗, 2⅔, 2.75

Example 2: Negative Numbers

Order: -3.2, 2.8, -4.1, 0, -2.9 from greatest to least

Step 1: Positives are greater than negatives → 2.8, 0, -2.9, -3.2, -4.1

Step 2: For negatives, "more negative" means smaller → -2.9 > -3.2 > -4.1

Answer: 2.8, 0, -2.9, -3.2, -4.1

🚨 Common Mistakes to Avoid

  • Negative number confusion: Thinking -5 is greater than -2
  • Decimal misreading: 0.45 is NOT greater than 0.6
  • Fraction conversion errors: Forgetting to find common denominators
  • Place value oversight: 2.35 vs. 2.4 - students often pick 2.35 as larger

💡 Tips & Tricks

  • Number line visualization: Picture where each number would fall
  • Common denominator method: Convert fractions to have same denominator
  • Place value alignment: Add zeros to compare decimals: 0.6 = 0.60
  • Negative number trick: The "more negative" a number is, the smaller its value

🎯 Practice Suggestions

  • Create your own number sets with mixed types (fractions, decimals, negatives)
  • Use online ordering games and interactive number lines
  • Practice with real-world data: sports statistics, weather temperatures, or prices
  • Work with a partner and check each other's ordering
  • Time yourself to build speed and accuracy

Practice problems

6 of the 77, worked through step by step — try them before opening the answer.

1 √(2² + 6²) = ?

Hint: This involves finding the square root of a sum of squares. First calculate the squares of the numbers, then add them, and finally find the square root of the result.

Show the answer

Answer: √40

  1. Identify the operations inside the square root. We have 2 squared plus 6 squared: 2² + 6².
  2. Calculate each square. 2² = 4 6² = 36
  3. Add the results. 4 + 36 = 40
  4. Apply the square root. √(2² + 6²) = √(40)
  5. Final answer. The expression simplifies to √40.

2 √(2.25 × 10²) = ?

Hint: Consider simplifying the expression inside the square root first, then take the square root of the result.

Show the answer

Answer: 15

  1. Understand the expression We have √(2.25 × 10²). This means: square root of (2.25 multiplied by 10 squared).
  2. Compute 10² 10² = 10 × 10 = 100.
  3. Multiply 2.25 by 100 2.25 × 100 = 225. So the expression becomes √225.
  4. Find the square root of 225 We ask: which number multiplied by itself gives 225? 15 × 15 = 225. So √225 = 15.
  5. Final answer Thus, √(2.25 × 10²) = 15.

Let's solve step by step.

3 √(49) + ∛(64) = ?

Hint: Remember that square root and cube root are inverse operations of squaring and cubing numbers. For example, √(36) = 6 because 6² = 36.

Show the answer

Answer: 11

  1. Understand the problem We need to compute √(49) + ∛(64).
  2. Evaluate √(49) The square root of 49 is the number that, when multiplied by itself, gives 49. Since 7 × 7 = 49, we have √(49) = 7.
  3. Evaluate ∛(64) The cube root of 64 is the number that, when multiplied by itself three times, gives 64. Since 4 × 4 × 4 = 64, we have ∛(64) = 4.
  4. Add the results √(49) + ∛(64) = 7 + 4 = 11.
  5. Final answer

Let's solve step by step. The correct answer is 11.

4 √(72) - 5√(2) = ?

Hint: Try simplifying the square root by factoring out perfect squares before combining like terms.

Show the answer

Answer: √2

  1. Simplify √(72) by factoring: √(72) = √(36 × 2) = √(36) × √(2) = 6√(2)
  2. Substitute back into the expression: 6√(2) - 5√(2)
  3. Combine like terms: (6 - 5)√(2) = 1√(2)
  4. Simplify: 1√(2) = √(2)

The answer is √2.

5 √(2.25 × 10⁻²) = ?

Hint: First simplify the multiplication inside the square root, then find the square root of the result.

Show the answer

Answer: 0.15

  1. Understand the expression** The square root applies to the entire product: √(2.25 × 10⁻²) = √2.25 × √(10⁻²) --- **
  2. Simplify √2.25** 2.25 = 225/100 √(225/100) = √225 / √100 = 15/10 = 1.5 So √2.25 = 1.5 --- **
  3. Simplify √(10⁻²)** 10⁻² = 1/100 √(1/100) = 1/√100 = 1/10 = 0.1 Alternatively: √(10⁻²) = (10⁻²)^(1/2) = 10⁻¹ = 0.1 --- **
  4. Multiply the results** 1.5 × 0.1 = 0.15 --- **
  5. Final answer** √(2.25 × 10⁻²) = 0.15

Let's solve step by step. We are given: √(2.25 × 10⁻²) --- **

6 √(2.25 × 10⁻⁴) = ?

Hint: Consider simplifying the expression inside the square root first by handling the scientific notation and decimal separately.

Show the answer

Answer: 0.015

  1. Write the expression: √(2.25 × 10⁻⁴)
  2. Separate the square root: √(2.25) × √(10⁻⁴)
  3. Calculate √(2.25): Since 1.5 × 1.5 = 2.25, √(2.25) = 1.5
  4. Calculate √(10⁻⁴): √(10⁻⁴) = 10⁻² = 0.01
  5. Multiply the results: 1.5 × 0.01 = 0.015

The answer is 0.015.

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