Unit Plan 2 (Grade 8 Math): Rational vs. Irrational—What’s Real?
8th graders explore rational and irrational numbers, convert repeating decimals to fractions, and approximate irrationals like √2 and π with bounds on the number line. This weeklong math unit builds precision, reasoning, and fluency within the real number system.
Focus: Differentiate rational and irrational numbers, explore decimal expansions, and place numbers on the number line.
Grade Level: 8
Subject Area: Mathematics (Number System • Real Numbers)
Total Unit Duration: 5 sessions (one week), 45–60 minutes per session
I. Introduction
This week solidifies a big idea: the real number system splits into rational and irrational numbers. Students will classify numbers, convert repeating decimals → fractions, and approximate & locate irrationals (like √2 or π) on a number line using bounds and precision.
Essential Questions
- How do I know if a number is rational or irrational?
- How can I approximate an irrational number and show how close I am?
- Why do error bounds matter when we round?
II. Objectives and Standards
Learning Objectives — Students will be able to…
- Classify numbers as rational or irrational using decimal behavior; convert repeating decimals to fractions (8.NS.1).
- Approximate irrationals with rationals, compare sizes, locate them on a number line, and estimate expressions (8.NS.2).
- Communicate reasoning with precision (clear labels, units, inequality symbols, bound statements).
Standards Alignment — CCSS Grade 8
- The Number System 8.NS.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers, show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
- The Number System 8.NS.2: Use rational approximations of irrational numbers to compare their sizes, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue to get better approximations.
Success Criteria (student-friendly)
- I can tell whether a number is rational or irrational and justify why.
- I can convert a repeating decimal like 0.(27) into a fraction and check it.
- I can give a bounded estimate (e.g., 1.41 < √2 < 1.42) and put it in the right place on a number line.
- I can explain the maximum error of my approximation (e.g., “within 0.01”).