Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Surds
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Surds
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Subject: Further Mathematics
Class: Senior Secondary 1
Term: 3rd Term
Week: 2
Theme: Pure Mathematics
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Define surds Use the rules of surds in manipulating surds Rationalize the denominators of surds
Rule 1: Simplification of Surds A surd can often be simplified by extracting perfect square factors from the number under the radical.
Principle: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ (where $a$ and $b$ are non-negative) and $\sqrt{a^2} = a$.
Method: Find the largest perfect square factor of the number under the radical.
Worked Example 1: Simplify $\sqrt{72}$.
Step 1: Find perfect square factors of
7
2. Factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36,
7
2. The largest perfect square factor is
3
6. Step 2: Rewrite 72 as a product of 36 and another number: $72 = 36 \times 2$.
Step 3: Apply the rule $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$: $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2}$.
Step 4: Simplify the perfect square: $\sqrt{36} = 6$.
Step 5: Combine: $\sqrt{72} = 6\sqrt{2}$.
Rule 2: Multiplication of Surds Principle: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ (where $a$ and $b$ are non-negative).
Extension: $x\sqrt{a} \times y\sqrt{b} = xy\sqrt{ab}$.
Worked Example 2: Multiply $3\sqrt{5}$ by $2\sqrt{10}$.
Step 1: Multiply the coefficients outside the surds: $3 \times 2 = 6$.
Step 2: Multiply the numbers inside the surds: $\sqrt{5} \times \sqrt{10} = \sqrt{5 \times 10} = \sqrt{50}$.
Step 3: Combine and simplify the resulting surd if possible: $6\sqrt{50}$.
Step 4: Simplify $\sqrt{50}$: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$.
Step 5: Substitute back: $6 \times 5\sqrt{2} = 30\sqrt{2}$.
Rule 3: Division of Surds Principle: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ (where $a \ge 0$ and $b > 0$).
Extension: $\frac{x\sqrt{a}}{y\sqrt{b}} = \frac{x}{y}\sqrt{\frac{a}{b}}$.
Worked Example 3: Divide $\sqrt{48}$ by $\sqrt{3}$.
Method 1 (Using division rule directly): Step 1: Apply the division rule: $\frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}}$.
Step 2: Perform the division inside the radical: $\sqrt{16}$.
Step 3: Simplify the perfect root: $\sqrt{16} = 4$.
Method 2 (Simplifying first): Step 1: Simplify $\sqrt{48}$: $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$.
Step 2: Divide: $\frac{4\sqrt{3}}{\sqrt{3}}$.
Step 3: Cancel out the common surd $\sqrt{3}$: $4$.
Rule 4: Addition and Subtraction of Surds Principle: Surds can only be added or subtracted if they are like surds (i.e., they have the same number under the radical sign after simplification). This is analogous to adding or subtracting like terms in algebra ($x$ and $x$).
Method: Simplify all surds first, then combine like surds by adding or subtracting their rational coefficients.
Worked Example 4: Simplify $2\sqrt{3} + 5\sqrt{3} - \sqrt{3}$.
Step 1: All surds are like surds ($\sqrt{3}$).
Step 2: Combine the coefficients: $(2 + 5 - 1)\sqrt{3}$.
Step 3: Calculate the result: $6\sqrt{3}$.
Worked Example 5: Simplify $3\sqrt{12} + \sqrt{75} - 2\sqrt{3}$.
Step 1: Simplify each surd: $3\sqrt{12} = 3\sqrt{4 \times 3} = 3 \times 2\sqrt{3} = 6\sqrt{3}$. $\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$. $2\sqrt{3}$ is already in its simplest form.
Step 2: Substitute the simplified forms back into the expression: $6\sqrt{3} + 5\sqrt{3} - 2\sqrt{3}$.
Step 3: Combine the coefficients of the like surds: $(6 + 5 - 2)\sqrt{3}$.
Step 4: Calculate the result: $9\sqrt{3}$. Rationalization is the process of eliminating surds from the denominator of a fraction. This is done because it is generally considered good mathematical practice to express fractions with rational denominators.
Case 1: Monomial Denominator (e.g., $\frac{a}{\sqrt{b}}$)
Method: Multiply both the numerator and the denominator by the surd in the denominator.
Principle: $\sqrt{b} \times \sqrt{b} = b$. This makes the denominator rational.
Worked Example 6: Rationalize $\frac{6}{\sqrt{3}}$.
Step 1: Identify the surd in the denominator: $\sqrt{3}$.
Step 2: Multiply numerator and denominator by $\sqrt{3}$: $\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$.
Step 3: Perform the multiplication: $\frac{6\sqrt{3}}{3}$.
Step 4: Simplify the fraction: $2\sqrt{3}$.
Case 2: Binomial Denominator (e.g., $\frac{a}{b+\sqrt{c}}$ or $\frac{a}{b-\sqrt{c}}$)
Method: Multiply both the numerator and the denominator by the conjugate of the denominator.
Conjugate: The conjugate of $(b+\sqrt{c})$ is $(b-\sqrt{c})$, and the conjugate of $(b-\sqrt{c})$ is $(b+\sqrt{c})$. The sign between the terms is changed.
Principle: This uses the "difference of squares" formula: $(x+y)(x-y) = x^2 - y^2$. When applied to surds, $(b+\sqrt{c})(b-\sqrt{c}) = b^2 - (\sqrt{c})^2 = b^2 - c$, which results in a rational number.
Worked Example 7: Rationalize $\frac{4}{3+\sqrt{2}}$.
Step 1: Identify the denominator: $3+\sqrt{2}$.
Step 2: Determine its conjugate: $3-\sqrt{2}$.
Step 3: Multiply numerator and denominator by the conjugate: $\frac{4}{3+\sqrt{2}} \times \frac{3-\sqrt{2}}{3-\sqrt{2}}$.
Step 4: Expand the numerator: $4(3-\sqrt{2}) = 12 - 4\sqrt{2}$.
Step 5: Expand the denominator using the difference of squares: $(3+\sqrt{2})(3-\sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7$.
Step 6: Combine the simplified numerator and denominator: $\frac{12 - 4\sqrt{2}}{7}$.
Worked Example 8: Rationalize $\frac{\sqrt{5}}{\sqrt{7}-\sqrt{3}}$.
Step 1: Identify the denominator: $\sqrt{7}-\sqrt{3}$.
Step 2: Determine its conjugate: $\sqrt{7}+\sqrt{3}$.
Step 3: Multiply numerator and denominator by the conjugate: $\frac{\sqrt{5}}{\sqrt{7}-\sqrt{3}} \times \frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}+\sqrt{3}}$.
Step 4: Expand the numerator: $\sqrt{5}(\sqrt{7}+\sqrt{3}) = \sqrt{35} + \sqrt{15}$.
Step 5: Expand the denominator: $(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3}) = (\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$.
Step 6: Combine: $\frac{\sqrt{35} + \sqrt{15}}{4}$. This section provides a detailed explanation of surds, their properties, and methods for their manipulation, including rationalization. This section outlines the pedagogical steps for delivering the lesson, focusing on teacher actions and corresponding student engagement.
Phase 1: Introduction and Definition (15 minutes)
Teacher Activity: Begin by reviewing rational and irrational numbers. Provide examples like $\frac{1}{2}$, $0.75$, $5$ (rational) and $\pi$, $e$ (irrational). Introduce the concept of roots. Ask students to calculate $\sqrt{4}$, $\sqrt{9}$, $\sqrt{25}$.
Pose the question: "What about $\sqrt{2}$ or $\sqrt{3}$? Can they be simplified to whole numbers or simple fractions?" Formally define a surd, emphasizing that it's an irrational number expressed as a root of a non-perfect square. Provide a list of numbers and ask students to identify which are surds and which are not.
Student Activity: Participate in the review of rational and irrational numbers. Calculate simple perfect square roots. Attempt to simplify $\sqrt{2}$ and $\sqrt{3}$, observing that they don't yield exact rational numbers. Actively listen to the definition of surds. Engage in identifying surds from a provided list (e.g., $\sqrt{16}, \sqrt{18}, \sqrt{49}, \sqrt{50}, \sqrt[3]{8}, \sqrt[3]{10}$).
Phase 2: Rules of Surds (30 minutes)
Teacher Activity: Introduce the rule for simplifying surds ($\sqrt{ab} = \sqrt{a}\sqrt{b}$). Demonstrate with $\sqrt{72}$ and $\sqrt{50}$, emphasizing finding the largest perfect square factor. Introduce multiplication rule ($x\sqrt{a} \times y\sqrt{b} = xy\sqrt{ab}$). Provide examples like $3\sqrt{5} \times 2\sqrt{10}$. Introduce division rule ($\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$). Demonstrate with $\frac{\sqrt{48}}{\sqrt{3}}$. Discuss simplifying before or after division. Introduce addition and subtraction of surds, highlighting the concept of "like surds" (similar to like terms in algebra). Demonstrate with $2\sqrt{3} + 5\sqrt{3} - \sqrt{3}$ and $3\sqrt{12} + \sqrt{75} - 2\sqrt{3}$. Provide scaffolded practice problems for each rule, guiding students through step-by-step solutions on the board.
Student Activity: Work along with the teacher, taking notes on the rules and examples. Practice simplifying surds, multiplying, and dividing surds individually or in pairs. Identify like surds and practice adding/subtracting them. Ask clarifying questions regarding the application of the rules. Collaborate on solving practice problems, sharing their methods and solutions.
Phase 3: Rationalization of Denominators (25 minutes)
Teacher Activity: Explain the purpose of rationalization: to remove surds from the denominator for mathematical convention and ease of calculation. Demonstrate rationalization of monomial denominators (e.g., $\frac{6}{\sqrt{3}}$), explaining why multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$ works. Introduce the concept of a conjugate for binomial denominators. Explain how $(a+b)(a-b) = a^2-b^2$ is applied to remove the surd. Demonstrate rationalization of binomial denominators (e.g., $\frac{4}{3+\sqrt{2}}$ and $\frac{\sqrt{5}}{\sqrt{7}-\sqrt{3}}$). Provide further examples and encourage students to attempt them.
Student Activity: Understand the rationale behind rationalization. Practice rationalizing denominators with monomial surds. Learn how to find the conjugate of a binomial surd expression. Apply the conjugate method to rationalize denominators with binomial surds. Work through examples, presenting their solutions and explaining their steps.
Phase 4: Consolidation and Practice (10 minutes)
Teacher Activity: Summarize the key concepts covered: definition of surds, rules for simplification, multiplication, division, addition/subtraction, and rationalization. Assign a few mixed practice problems covering all objectives for immediate in-class work. Circulate among students, providing individual support and feedback.
Student Activity: Attempt the mixed practice problems. Ask for help when encountering difficulties. Review their understanding of the different operations with surds.
Construction and Architecture (Nigerian Context): When designing houses or structures in Nigerian cities like Lagos or Port Harcourt, architects and structural engineers often need precise measurements. For instance, if a square room has a diagonal length that is exactly $5\sqrt{2}$ meters, understanding surds allows them to work with this exact value for calculating material requirements (e.g., length of electrical conduits, roofing trusses, or decorative cornices) rather than rounding off, which could lead to errors in fitting or material wastage. Similarly, calculating the exact length of a ramp or the hypotenuse of a right-angled structure using the Pythagorean theorem often results in surds.
Physics and Engineering Calculations: In various fields of engineering relevant to Nigeria's development (e.g., civil, electrical, mechanical), exact values are critical. For example, in a physics experiment carried out in a Nigerian university, a student might be calculating the period of a simple pendulum with a length $L = \sqrt{2}$ meters using the formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $g$ is acceleration due to gravity. The exact value of $T$ will involve surds, and manipulating these surds ensures that calculations maintain accuracy crucial for research or industrial applications. Land Surveying and Geographical Information Systems (GIS): In land allocation and planning across Nigeria, especially in rural-urban fringe areas, surveyors determine exact distances and areas. When dealing with irregular plots or measurements derived from satellite imagery, coordinates can lead to distances expressed as surds. Being able to manipulate and rationalize these surds ensures that land records are accurate and legally sound, preventing disputes over boundary lines or land size.