math class 678 4-in-1 skill sheets
8.NS.A.2 8th Grade The Number System

Approximating Irrationals

Use rational approximations of irrational numbers to compare their sizes and to locate them approximately on a number line.

How to explain it

At this standard, students approximate irrational numbers as decimals, locate them on a number line between consecutive integers, and compare irrational values using decimal approximations — building the bridge from perfect-square root evaluation (8.EE.A.2) to number-line placement (8.NS.A.2).

The anchor students hold onto: Find the two consecutive perfect squares bracketing the radicand; their roots are the integer bounds. Use proximity to the nearer perfect square to estimate one decimal.

These approximations connect directly to 8.G.B.7 Pythagorean theorem outputs like √2 and √5, and give irrational equation solutions from 8.EE.A.2 their number-line meaning.

Worked examples

Example 1 Locate on Number Line
Locate √7 on a number line.
Step 1Find surrounding perfect squares: 4 < 7 < 9.
Step 2√4 = 2 and √9 = 3, so √7 is between 2 and 3.
Step 37 is closer to 9 than to 4, so √7 ≈ 2.6.
Answer√7 ≈ 2.6 (between 2 and 3)
Example 2 Estimate as Decimal
Estimate √20 to one decimal.
Step 1Find surrounding perfect squares: 16 < 20 < 25.
Step 2√16 = 4 and √25 = 5, so √20 is between 4 and 5.
Step 320 is closer to 16 than to 25, so √20 ≈ 4.5.
Answer√20 ≈ 4.5 (between 4 and 5)
Example 3 Compare Irrationals
Which is greater: √6 or √8?
Step 1√6: between √4=2 and √9=3 — √6 ≈ 2.4.
Step 2√8: between √4=2 and √9=3 — √8 ≈ 2.8.
Step 32.8 > 2.4, so √8 > √6.
Answer√8 > √6

Common mistakes

What students write Assuming √N ≈ N/2 — e.g., √25 ≈ 12.5 instead of 5.
The fix A square root is not half the value. Find the two perfect squares bracketing N, then take their roots.
Try this Two students estimate √17. Who is more accurate? Explain using perfect-square bounds. Ella: √17 ≈ 4.1 Max: √17 ≈ 4.2
What students write Placing √5 at exactly 2.5 because 5 is halfway between 4 and 9.
The fix 5 is closer to 4 than to 9, so √5 ≈ 2.2, not 2.5. Proximity to the nearer square determines the estimate.

Teacher tip

Head off the two predictable errors before they happen. First: A square root is not half the value. Find the two perfect squares bracketing N, then take their roots. Second: 5 is closer to 4 than to 9, so √5 ≈ 2.2, not 2.5. Proximity to the nearer square determines the estimate.