Square Roots Guided Notes
Pre-Algebra • Real Number System
Section 2.3: Estimating Square Roots
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Date:
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THE FIRST 15 PERFECT SQUARES REFERENCE BANK
Complete each square value:
\(1^2\)
\(2^2\)
\(3^2\)
\(4^2\)
\(5^2\)
\(6^2\)
\(7^2\)
\(8^2\)
\(9^2\)
\(10^2\)
\(11^2\)
\(12^2\)
\(13^2\)
\(14^2\)
\(15^2\)
THE 4-STEP HALFWAY STRATEGY
Rule: Integers go on TOP • Square Roots go on BOTTOM
1. Bounding Squares \(\sqrt{a} < \sqrt{x} < \sqrt{b}\) (roots below, integers above)
2. Find Halfway Spot Radicand: \(\frac{a+b}{2}\) • Midpoint: \(\_\_.5\)
3. Compare to Halfway Is \(x\) greater or less than halfway radicand?
4. Decimal Estimate Decide if root is \(> \_\_.5\) or \(< \_\_.5\) to tenth
1 Guided Model: Estimate the value of \(\sqrt{50}\)
Worked Example
7 \(\sqrt{49}\) 7.5 \(\approx \sqrt{56}\) 8 \(\sqrt{64}\) \(\sqrt{50}\)
Step 1: \(\sqrt{49} < \sqrt{50} < \sqrt{64}\) • Integers: 7 and 8
Step 2 (Halfway Spot): \(\frac{49 + 64}{2} \approx 56\) • Halfway integer: 7.5
Step 3 (Compare): Since \(50 < 56\), \(\sqrt{50}\) is less than 7.5 (closer to 7)
Step 4: Estimate: \(\sqrt{50} \approx 7.1\)
2 Estimate the value of \(\sqrt{22}\)
4 \(\sqrt{16}\) 4.5 \(\approx \sqrt{20.5}\) 5 \(\sqrt{25}\)
• Bounding integers: between and
• Halfway spot: \(\frac{16 + 25}{2} =\) (benchmark 4.5)
• Compare: \(22\) is ( greater / less ) than halfway.
Decimal estimate: \(\sqrt{22} \approx\)
3 Estimate the value of \(\sqrt{103}\)
Fill in line
• Bounding integers: between and
• Halfway spot: \(\frac{\text{\_\_\_} + \text{\_\_\_}}{2} =\) (benchmark \(\_\_.5\))
• Compare: \(103\) is ( greater / less ) than halfway.
Decimal estimate: \(\sqrt{103} \approx\)
4 Estimate the value of \(\sqrt{39}\)
Fill in line
• Bounding integers: between and
• Halfway spot: \(\frac{\text{\_\_\_} + \text{\_\_\_}}{2} =\) (benchmark \(\_\_.5\))
• Compare: \(39\) is ( greater / less ) than halfway.
Decimal estimate: \(\sqrt{39} \approx\)
5 Estimate the value of \(\sqrt{78}\)
Fill in line
• Bounding integers: between and
• Halfway spot: \(\frac{\text{\_\_\_} + \text{\_\_\_}}{2} =\) (benchmark \(\_\_.5\))
• Compare: \(78\) is ( greater / less ) than halfway.
Decimal estimate: \(\sqrt{78} \approx\)
Unit 2: Real Number System • Section 2.3 Notes Page 1 of 2
Practice
Section 2.3 Independent & Application Practice
Name:
6 Rapid Estimates: Sketch number line and estimate to nearest tenth:
No calculator
a. Estimate \(\sqrt{14}\) Workspace
Integers: & \(\sqrt{14} \approx\)
b. Estimate \(\sqrt{60}\) Workspace
Integers: & \(\sqrt{60} \approx\)
c. Estimate \(\sqrt{115}\) Workspace
Integers: & \(\sqrt{115} \approx\)
d. Estimate \(\sqrt{170}\) Workspace
Integers: & \(\sqrt{170} \approx\)
7 Between which two consecutive integers does each lie?
a. \(\sqrt{90}\) between and
b. \(\sqrt{175}\) between and
c. \(-\sqrt{12}\) between and
d. \(\sqrt{250}\) between and
8 Plot and label each point on the number line below:
Integers Top • Roots Bottom
\(A: \sqrt{5}\) \(B: \sqrt{88}\) \(C: \sqrt{17}\) \(D: \sqrt{118}\) \(E: \sqrt{56}\)
1 \(\sqrt{1}\) 2 \(\sqrt{4}\) 3 \(\sqrt{9}\) 4 \(\sqrt{16}\) 5 \(\sqrt{25}\) 6 \(\sqrt{36}\) 7 \(\sqrt{49}\) 8 \(\sqrt{64}\) 9 \(\sqrt{81}\) 10 \(\sqrt{100}\) 11 \(\sqrt{121}\)
Decimal Estimates: \(A \approx 2.\) \(B \approx 9.\) \(C \approx 4.\) \(D \approx 10.\) \(E \approx 7.\)
9 Plot and label each point on the number line below:
Integers Top • Roots Bottom
\(F: \sqrt{10}\) \(G: \sqrt{32}\) \(H: \sqrt{70}\) \(I: \sqrt{95}\) \(J: \sqrt{45}\)
1 \(\sqrt{1}\) 2 \(\sqrt{4}\) 3 \(\sqrt{9}\) 4 \(\sqrt{16}\) 5 \(\sqrt{25}\) 6 \(\sqrt{36}\) 7 \(\sqrt{49}\) 8 \(\sqrt{64}\) 9 \(\sqrt{81}\) 10 \(\sqrt{100}\) 11 \(\sqrt{121}\)
Decimal Estimates: \(F \approx 3.\) \(G \approx 5.\) \(H \approx 8.\) \(I \approx 9.\) \(J \approx 6.\)
10 Mark True or False. If false, rewrite or explain:
a. \(\sqrt{202}\) is between 14 and 15, but closer to 15.
TRUE FALSE
Correction:
b. \(\sqrt{7}\) is between 3 and 4, but closer to 3.
TRUE FALSE
Correction:
c. \(\sqrt{99}\) is greater than 9.5.
TRUE FALSE
Correction:
Lesson Synthesis
Summarize today's lesson: How does using the halfway benchmark make estimating square roots faster and more accurate?
Unit 2: Real Number System • Section 2.3 Notes Page 2 of 2
