Unit Plan 3 (Grade 8 Math): Approximating Irrationals & Roots
8th graders build fluency with square and cube roots by estimating irrational values, comparing magnitudes, and solving equations like x² = p and x³ = p. Students connect roots to area and volume, using bounds and precision to reason about real-world measurements.
Focus: Estimate square roots, compare magnitudes, and solve simple equations involving squares and cubes.
Grade Level: 8
Subject Area: Mathematics (The Number System • Expressions & Equations)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This week builds fluent sense-making with roots. Students will approximate irrational square roots with clear bounds, compare sizes of roots vs. familiar decimals/fractions, and solve equations of the form x^2 = p and x^3 = p. We’ll connect roots to area and volume so answers have meaning and appropriate precision.
Essential Questions
- How do I approximate a square root and explain how close my answer is?
- How can I compare √a to a decimal or fraction without a calculator?
- When do solutions to x^2 = p include ±, and when should I report only the positive value?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Approximate irrational square roots, give lower/upper bounds, and place values on a number line (8.NS.2).
- Evaluate square roots of small perfect squares and cube roots of small perfect cubes; solve x^2 = p and x^3 = p and approximate when needed (8.EE.2).
- Communicate reasoning with precision using symbols, inequality direction, units, and explicit error bounds.
Standards Alignment — CCSS Grade 8
- The Number System 8.NS.2: Use rational approximations of irrational numbers to compare size, locate values approximately on a number line, and estimate expressions (for example, by narrowing bounds for √2 to the nearest tenth, then hundredth).
- Expressions & Equations 8.EE.2: Use square root and cube root symbols to represent solutions to x^2 = p and x^3 = p (p positive rational). Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Success Criteria (student-friendly)
- I can show 2 correct bounds for a root (for example, 4.24 < √18 < 4.25) and explain how I got them.
- I can decide which is larger (for example, √50 or 7.05) and justify my choice without a calculator.
- I can solve x^2 = p as x = ±√p and x^3 = p as x = ∛p, and approximate if p is not a perfect power.
- I can state the maximum error of my approximation (for example, “within 0.01”).