Lesson 2 — Real Numbers, Roots & Radicals

Summary

Learners build the real numbers as rational and irrational together, meet roots and radicals, and simplify radicals such as √50 = 5√2 using the product property and perfect squares, justifying each step. They locate √2 and π on a number line and meet the Babylonian tablet that approximated √2 nearly four thousand years ago.

Objectives

• Work with real numbers, roots, and radicals in expressions, explaining which properties justify each step. (D05.S1.09.01)

Connection

You already know two families of numbers: rational numbers — fractions like ½ and decimals like 0.75 that end or repeat — and irrational numbers — like √2 and π, whose decimals go on forever without ever settling into a repeating pattern. Put the two families together and you have the real numbers: every number that names a place on a single number line. Today you learn to work with roots and radicals and to say why each step is legal — not by memory, but by a property you can name.

Materials

• Number-line sheet
• Worked-example card

Preparation

• Copy the number-line sheet and worked-example card.
• Retrieval: from Grade 8, recall rational vs. irrational numbers and how to approximate a square root by locating it between two integers.

Facilitator note

This lesson is written to the learner (“you”). The skill to land: a radical √a is the non-negative number whose square is a; real numbers are rational and irrational together; and radicals simplify by two properties — the product property √(ab) = √a·√b (for non-negative a, b) and the fact that √(a²) = a for non-negative a. This is a skill, so use a worked example first, then guided practice (explicit instruction carries skills for novices — S-011). Model √50 = √(25·2) = √25·√2 = 5√2 fully, naming each property, then have learners simplify their own before independent work. The critical-thinking lens: justifying each step with a named property is the seed of proof. The global lens: irrationals were discovered, not invented — Babylonian scribes approximated √2 on the clay tablet YBC 7289 (~1800 BCE) to about six decimal places, and the idea that √2 cannot be written as a ratio of whole numbers shook Greek mathematics (S-429, S-430). The ethics/egalitarian lens: the name “irrational” once meant “unsayable,” and people were uncomfortable — but an irrational number is not broken or wrong; it is a real number like any other. Mathematics is sense-making open to all; naming a property is not a gate, it is a tool everyone can pick up. The technology lens: a calculator shows √2 and π to many digits but never the whole non-repeating decimal — the machine’s display is finite while the number is not, so the human must know what the display actually means. Keep the “is it rational?” question a matter of evidence (does the decimal repeat?), not authority.

Procedure

1. Recall (5 min). From Grade 8: what is the difference between a rational and an irrational number? Name one of each.
2. Meet roots and radicals (8 min). √a is the non-negative number whose square is a. So √9 = 3 because 3² = 9. The number under the sign is the radicand; the whole sign is a radical. A perfect square (1, 4, 9, 16, 25, …) has a whole-number square root. √2 is not a perfect square, so √2 is irrational.
3. Study the worked example (12 min). Simplify √50. Step 1: write 50 as a product with the largest perfect square you can find — 50 = 25 · 2. Step 2: apply the product property — √(25·2) = √25 · √2. Step 3: replace √25 with 5 — √25 · √2 = 5√2. Check by squaring back: (5√2)² = 25 · 2 = 50. ✓
4. Guided practice (15 min). With a partner, simplify √8, √18, √32, and √75. For each, name the property you used and check by squaring back.
5. Locate on the line (10 min). On the number-line sheet, place 0, 1, 2, 3, 4, ½, and 1½. Then place √2 (about 1.41), √3 (about 1.73), and π (about 3.14). Say which are rational and which are irrational.
6. Close (5 min). One new idea to carry: an irrational number is not a mistake — it is a real place on the line that no fraction can land on exactly.

Differentiation

• Support: Give a list of perfect squares (1–144) to use while simplifying; start with √12 = √(4·3).
• Extension: Prove √2 is irrational by contradiction (assume √2 = p/q in lowest terms, square both sides, and reach a contradiction about even numbers).

Assessment

• Formative (observation): Can the learner simplify √50 to 5√2 and name the product property?
• Portfolio artifact: The completed number line with √2, √3, and π placed and labeled, kept in the portfolio.

Home connection

Find one square at home (a tile, a window, a phone screen). Measure its side; use the side length to predict its diagonal as a radical, then measure the diagonal and compare.

Resources

• On the Babylonian approximation of √2 (tablet YBC 7289): Wikipedia, “YBC 7289,” https://en.wikipedia.org/wiki/YBC_7289 (S-430).
• On the history of irrational numbers across traditions: Boyer & Merzbach, A History of Mathematics (S-429).
• On explicit instruction and worked examples: Kirschner, Sweller & Clark (2006), https://doi.org/10.1207/s15326985ep4102_1 (S-011).