Grade 8 Mathematics — Term 1 Notes

<h1 class="notes-h1">Grade 8 Mathematics — Term 1 Notes</h1>
<h2 class="notes-h2">CBC / KICD Aligned | Junior Secondary | cbcedukenya.com</h2>
<hr class="section-divider">
<p><strong>Learning Area:</strong> Mathematics</p>
<p><strong>Grade:</strong> 8</p>
<p><strong>Term:</strong> 1</p>
<p><strong>Strands Covered:</strong> Numbers, Algebra, Geometry</p>
<p><strong>Year:</strong> 2026</p>
<hr class="section-divider">
<h2 class="notes-h2">STRAND 1: NUMBERS</h2>
<h3 class="notes-h3">Sub-Strand 1.1: Integers</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Perform operations on integers (addition, subtraction, multiplication, division)</li>
<li>Apply integer operations to solve real-life problems</li>
<li>Work with order of operations (BODMAS/BEDMAS)</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.1.1 Revision of Integers</h4>
<p><strong>Integers</strong> are all whole numbers — positive, negative, and zero.</p>
<pre class="code-block"><code>
... -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, +5 ...
</code></pre>
<p><strong>Number line:</strong></p>
<pre class="code-block"><code>
←──────────────────────────────────────────→
-6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4
</code></pre>
<p><strong>Absolute value</strong> (denoted |n|) is the distance of a number from zero, always positive.</p>
<ul class="notes-list">
<li>|−7| = 7</li>
<li>|+7| = 7</li>
<li>|0| = 0</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.1.2 Operations on Integers</h4>
<p><strong>Addition and Subtraction Rules:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Operation</th><th>Rule</th><th>Example</th></tr>
</thead><tbody>
<tr><td>(+) + (+)</td><td>Add, keep positive</td><td>(+5) + (+3) = +8</td></tr>
<tr><td>(−) + (−)</td><td>Add, keep negative</td><td>(−5) + (−3) = −8</td></tr>
<tr><td>(+) + (−)</td><td>Subtract, keep sign of larger</td><td>(+8) + (−3) = +5</td></tr>
<tr><td>(−) + (+)</td><td>Subtract, keep sign of larger</td><td>(−8) + (+3) = −5</td></tr>
<tr><td>Subtracting a negative</td><td>Change to addition</td><td>7 − (−3) = 7 + 3 = 10</td></tr>
</tbody></table></div>
<p><strong>Multiplication and Division Rules:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Signs</th><th>Result</th><th>Example</th></tr>
</thead><tbody>
<tr><td>(+) × (+)</td><td>Positive</td><td>(+4) × (+3) = +12</td></tr>
<tr><td>(−) × (−)</td><td>Positive</td><td>(−4) × (−3) = +12</td></tr>
<tr><td>(+) × (−)</td><td>Negative</td><td>(+4) × (−3) = −12</td></tr>
<tr><td>(−) × (+)</td><td>Negative</td><td>(−4) × (+3) = −12</td></tr>
</tbody></table></div>
<p>The same rules apply to division:</p>
<ul class="notes-list">
<li>(−24) ÷ (−6) = +4</li>
<li>(+24) ÷ (−6) = −4</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.1.3 Order of Operations — BODMAS</h4>
<p>When an expression has multiple operations, use <strong>BODMAS</strong>:</p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Letter</th><th>Meaning</th><th>Example</th></tr>
</thead><tbody>
<tr><td>B</td><td>Brackets</td><td>( ) first</td></tr>
<tr><td>O</td><td>Orders (powers/roots)</td><td>x², √x</td></tr>
<tr><td>D</td><td>Division</td><td>÷</td></tr>
<tr><td>M</td><td>Multiplication</td><td>×</td></tr>
<tr><td>A</td><td>Addition</td><td>+</td></tr>
<tr><td>S</td><td>Subtraction</td><td>−</td></tr>
</tbody></table></div>
<p>Division and Multiplication have equal priority — work <strong>left to right</strong>.</p>
<p>Addition and Subtraction have equal priority — work <strong>left to right</strong>.</p>
<p><strong>Examples:</strong></p>
<p><em>Evaluate:</em> 3 + 4 × 2 − 1</p>
<ul class="notes-list">
<li>Multiplication first: 4 × 2 = 8</li>
<li>Then: 3 + 8 − 1 = <strong>10</strong></li>
</ul>
<p><em>Evaluate:</em> (6 + 3)² ÷ 9 − 4 × 2</p>
<ul class="notes-list">
<li>Brackets: (6 + 3) = 9</li>
<li>Orders: 9² = 81</li>
<li>Division: 81 ÷ 9 = 9</li>
<li>Multiplication: 4 × 2 = 8</li>
<li>Subtraction: 9 − 8 = <strong>1</strong></li>
</ul>
<p><em>Evaluate:</em> −3 × (−5 + 8) − (−2)²</p>
<ul class="notes-list">
<li>Brackets: (−5 + 8) = 3</li>
<li>Orders: (−2)² = 4</li>
<li>Multiplication: −3 × 3 = −9</li>
<li>Subtraction: −9 − 4 = <strong>−13</strong></li>
</ul>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 1.2: Fractions, Decimals, and Percentages</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Convert between fractions, decimals, and percentages</li>
<li>Perform operations on fractions and mixed numbers</li>
<li>Apply percentages to solve practical problems involving profit, loss, discount, and interest</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.2.1 Converting Between Forms</h4>
<p><strong>Fraction → Decimal:</strong> Divide numerator by denominator</p>
<ul class="notes-list">
<li>3/4 = 3 ÷ 4 = 0.75</li>
<li>2/3 = 0.666... = 0.6̄</li>
</ul>
<p><strong>Decimal → Fraction:</strong> Write over appropriate power of 10, simplify</p>
<ul class="notes-list">
<li>0.75 = 75/100 = <strong>3/4</strong></li>
<li>0.125 = 125/1000 = <strong>1/8</strong></li>
</ul>
<p><strong>Fraction/Decimal → Percentage:</strong> Multiply by 100</p>
<ul class="notes-list">
<li>3/4 = 0.75 × 100 = <strong>75%</strong></li>
<li>0.35 × 100 = <strong>35%</strong></li>
</ul>
<p><strong>Percentage → Decimal:</strong> Divide by 100</p>
<ul class="notes-list">
<li>45% = 45 ÷ 100 = <strong>0.45</strong></li>
</ul>
<p><strong>Percentage → Fraction:</strong> Write over 100, simplify</p>
<ul class="notes-list">
<li>60% = 60/100 = <strong>3/5</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.2.2 Operations on Fractions</h4>
<p><strong>Adding and Subtracting Fractions</strong> — find a common denominator (LCM)</p>
<p><em>Example:</em> 2/3 + 3/4</p>
<ul class="notes-list">
<li>LCM of 3 and 4 = 12</li>
<li>2/3 = 8/12</li>
<li>3/4 = 9/12</li>
<li>8/12 + 9/12 = <strong>17/12 = 1 5/12</strong></li>
</ul>
<p><strong>Multiplying Fractions</strong> — multiply numerators together, denominators together (then simplify)</p>
<p><em>Example:</em> 3/5 × 4/9 = 12/45 = <strong>4/15</strong></p>
<p>Shortcut (cross-cancellation): 3/5 × 4/9 → cancel 3 and 9 (÷3): 1/5 × 4/3 = <strong>4/15</strong></p>
<p><strong>Dividing Fractions</strong> — multiply by the reciprocal ("KCF": Keep, Change, Flip)</p>
<p><em>Example:</em> 2/3 ÷ 4/5</p>
<ul class="notes-list">
<li>Keep 2/3, Change ÷ to ×, Flip 4/5 to 5/4</li>
<li>= 2/3 × 5/4 = 10/12 = <strong>5/6</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.2.3 Percentages in Real Life</h4>
<p><strong>Profit and Loss:</strong></p>
<ul class="notes-list">
<li><strong>Profit</strong> = Selling Price − Buying Price (when SP > BP)</li>
<li><strong>Loss</strong> = Buying Price − Selling Price (when BP > SP)</li>
<li><strong>Profit %</strong> = (Profit / Buying Price) × 100</li>
<li><strong>Loss %</strong> = (Loss / Buying Price) × 100</li>
</ul>
<p><em>Example:</em> Amina buys a dress for Ksh 800 and sells it for Ksh 1,000.</p>
<ul class="notes-list">
<li>Profit = 1000 − 800 = Ksh 200</li>
<li>Profit % = (200/800) × 100 = <strong>25%</strong></li>
</ul>
<p><strong>Discount:</strong></p>
<ul class="notes-list">
<li><strong>Discount</strong> = Original Price − Sale Price</li>
<li><strong>Discount %</strong> = (Discount / Original Price) × 100</li>
</ul>
<p><em>Example:</em> A shirt costs Ksh 1,200. It is sold at 15% discount.</p>
<ul class="notes-list">
<li>Discount = 15/100 × 1,200 = Ksh 180</li>
<li>Sale price = 1,200 − 180 = <strong>Ksh 1,020</strong></li>
</ul>
<p><strong>Simple Interest:</strong></p>
<p>$$I = \frac{P \times R \times T}{100}$$</p>
<p>Where: I = interest, P = principal, R = rate (%), T = time (years)</p>
<p><em>Example:</em> Ksh 50,000 is deposited at 8% per year for 3 years.</p>
<ul class="notes-list">
<li>I = (50,000 × 8 × 3) / 100 = <strong>Ksh 12,000</strong></li>
<li>Amount = 50,000 + 12,000 = <strong>Ksh 62,000</strong></li>
</ul>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 1.3: Squares, Square Roots, Cubes, and Cube Roots</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Find squares and square roots of numbers</li>
<li>Find cubes and cube roots of numbers</li>
<li>Apply these in solving problems</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.3.1 Squares and Square Roots</h4>
<p><strong>Square</strong> of a number = number × number</p>
<ul class="notes-list">
<li>5² = 5 × 5 = 25</li>
<li>12² = 12 × 12 = 144</li>
</ul>
<p><strong>Perfect squares to memorise:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>n</th><th>n²</th><th></th><th>n</th><th>n²</th></tr>
</thead><tbody>
<tr><td>1</td><td>1</td><td></td><td>8</td><td>64</td></tr>
<tr><td>2</td><td>4</td><td></td><td>9</td><td>81</td></tr>
<tr><td>3</td><td>9</td><td></td><td>10</td><td>100</td></tr>
<tr><td>4</td><td>16</td><td></td><td>11</td><td>121</td></tr>
<tr><td>5</td><td>25</td><td></td><td>12</td><td>144</td></tr>
<tr><td>6</td><td>36</td><td></td><td>13</td><td>169</td></tr>
<tr><td>7</td><td>49</td><td></td><td>15</td><td>225</td></tr>
</tbody></table></div>
<p><strong>Square root (√)</strong> = the number that, when multiplied by itself, gives the original number</p>
<ul class="notes-list">
<li>√144 = 12 (because 12 × 12 = 144)</li>
<li>√225 = 15</li>
</ul>
<p><strong>Finding square roots using prime factorisation:</strong></p>
<p><em>Example:</em> Find √1764</p>
<ul class="notes-list">
<li>1764 = 2² × 3² × 7²</li>
<li>√1764 = 2 × 3 × 7 = <strong>42</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">1.3.2 Cubes and Cube Roots</h4>
<p><strong>Cube</strong> of a number = number × number × number = n³</p>
<ul class="notes-list">
<li>3³ = 3 × 3 × 3 = 27</li>
<li>5³ = 125</li>
</ul>
<p><strong>Perfect cubes to memorise:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>n</th><th>n³</th></tr>
</thead><tbody>
<tr><td>1</td><td>1</td></tr>
<tr><td>2</td><td>8</td></tr>
<tr><td>3</td><td>27</td></tr>
<tr><td>4</td><td>64</td></tr>
<tr><td>5</td><td>125</td></tr>
<tr><td>6</td><td>216</td></tr>
<tr><td>7</td><td>343</td></tr>
<tr><td>8</td><td>512</td></tr>
<tr><td>9</td><td>729</td></tr>
<tr><td>10</td><td>1000</td></tr>
</tbody></table></div>
<p><strong>Cube root (∛)</strong> = the number that, when cubed, gives the original number</p>
<ul class="notes-list">
<li>∛216 = 6 (because 6³ = 216)</li>
<li>∛1000 = 10</li>
</ul>
<p><strong>Using prime factorisation:</strong></p>
<p><em>Example:</em> Find ∛3375</p>
<ul class="notes-list">
<li>3375 = 3³ × 5³</li>
<li>∛3375 = 3 × 5 = <strong>15</strong></li>
</ul>
<p><strong>Review Questions 1:</strong></p>
<ol class="notes-list">
<li>Evaluate: (−4) × (−3) + 5 × (−2) − (−1)</li>
<li>Calculate: 2/5 + 3/4 − 1/2</li>
<li>Find the profit percentage when a radio bought for Ksh 3,200 is sold for Ksh 4,000.</li>
<li>Find √4096 using prime factorisation.</li>
<li>Calculate simple interest on Ksh 25,000 at 12% per annum for 2 years.</li>
</ol>
<hr class="section-divider">
<h2 class="notes-h2">STRAND 2: ALGEBRA</h2>
<h3 class="notes-h3">Sub-Strand 2.1: Algebraic Expressions</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Simplify algebraic expressions by collecting like terms</li>
<li>Expand and factorise algebraic expressions</li>
<li>Substitute values into algebraic expressions</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.1.1 Key Algebraic Terms</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Term</th><th>Definition</th><th>Example</th></tr>
</thead><tbody>
<tr><td>Variable</td><td>A letter representing an unknown number</td><td>x, y, a, b</td></tr>
<tr><td>Coefficient</td><td>The number multiplied by a variable</td><td>In 5x, the coefficient is 5</td></tr>
<tr><td>Constant</td><td>A number on its own (no variable)</td><td>In 3x + 7, the constant is 7</td></tr>
<tr><td>Term</td><td>Parts of an expression separated by + or −</td><td>3x², 5x, 7 are terms</td></tr>
<tr><td>Like terms</td><td>Terms with identical variable parts</td><td>3x and 7x; 5a² and 2a²</td></tr>
<tr><td>Expression</td><td>A combination of terms</td><td>4x² − 3x + 2</td></tr>
<tr><td>Equation</td><td>Expression set equal to something</td><td>4x − 3 = 13</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h4 class="notes-h4">2.1.2 Simplifying Expressions — Collecting Like Terms</h4>
<p>Only <strong>like terms</strong> can be added or subtracted.</p>
<p><em>Example:</em> Simplify 5a + 3b − 2a + 7b − b</p>
<ul class="notes-list">
<li>Group like terms: (5a − 2a) + (3b + 7b − b)</li>
<li>= <strong>3a + 9b</strong></li>
</ul>
<p><em>Example:</em> Simplify 3x² + 5x − 2x² − 8x + 4</p>
<ul class="notes-list">
<li>= (3x² − 2x²) + (5x − 8x) + 4</li>
<li>= <strong>x² − 3x + 4</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.1.3 Multiplying Algebraic Expressions</h4>
<p><strong>Monomial × Monomial:</strong></p>
<ul class="notes-list">
<li>3a × 4b = 12ab</li>
<li>(−5x) × 3x = −15x²</li>
<li>2x² × 3x³ = 6x⁵ (add exponents when bases are the same)</li>
</ul>
<p><strong>Expanding Brackets — Distributive Law:</strong></p>
<ul class="notes-list">
<li>a(b + c) = ab + ac</li>
</ul>
<p><em>Examples:</em></p>
<ul class="notes-list">
<li>3(2x + 5) = 6x + 15</li>
<li>−2(3a − 4b) = −6a + 8b</li>
<li>x(x² − 3x + 2) = x³ − 3x² + 2x</li>
</ul>
<p><strong>Expanding Two Brackets — FOIL:</strong></p>
<p>(a + b)(c + d) = ac + ad + bc + bd (First, Outer, Inner, Last)</p>
<p><em>Example:</em> (x + 3)(x + 5)</p>
<ul class="notes-list">
<li>First: x × x = x²</li>
<li>Outer: x × 5 = 5x</li>
<li>Inner: 3 × x = 3x</li>
<li>Last: 3 × 5 = 15</li>
<li>= <strong>x² + 8x + 15</strong></li>
</ul>
<p><em>Example:</em> (2x − 3)(x + 4)</p>
<ul class="notes-list">
<li>= 2x² + 8x − 3x − 12</li>
<li>= <strong>2x² + 5x − 12</strong></li>
</ul>
<p><strong>Special Products:</strong></p>
<ul class="notes-list">
<li><strong>(a + b)² = a² + 2ab + b²</strong> (perfect square trinomial)</li>
<li><strong>(a − b)² = a² − 2ab + b²</strong></li>
<li><strong>(a + b)(a − b) = a² − b²</strong> (difference of two squares)</li>
</ul>
<p><em>Examples:</em></p>
<ul class="notes-list">
<li>(x + 4)² = x² + 8x + 16</li>
<li>(3a − 2)² = 9a² − 12a + 4</li>
<li>(x + 5)(x − 5) = x² − 25</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.1.4 Factorisation</h4>
<p>Factorisation is the <strong>reverse of expansion</strong>.</p>
<p><strong>1. Common factor:</strong></p>
<p>Find the HCF of all terms and factor it out.</p>
<ul class="notes-list">
<li>6x² + 9x = <strong>3x(2x + 3)</strong></li>
<li>4a² − 8ab + 12a = <strong>4a(a − 2b + 3)</strong></li>
</ul>
<p><strong>2. Grouping:</strong></p>
<ul class="notes-list">
<li>2ab + 3a + 2b + 3</li>
<li>= a(2b + 3) + 1(2b + 3)</li>
<li>= <strong>(a + 1)(2b + 3)</strong></li>
</ul>
<p><strong>3. Difference of two squares:</strong></p>
<p>a² − b² = (a + b)(a − b)</p>
<ul class="notes-list">
<li>x² − 16 = <strong>(x + 4)(x − 4)</strong></li>
<li>9a² − 25 = <strong>(3a + 5)(3a − 5)</strong></li>
</ul>
<p><strong>4. Quadratic trinomial</strong> (ax² + bx + c where a = 1):</p>
<p>Find two numbers that multiply to c and add to b.</p>
<ul class="notes-list">
<li>x² + 7x + 12: numbers that multiply to 12, add to 7 → 3 and 4</li>
<li>= <strong>(x + 3)(x + 4)</strong></li>
</ul>
<ul class="notes-list">
<li>x² − 5x + 6: numbers that multiply to 6, add to −5 → −2 and −3</li>
<li>= <strong>(x − 2)(x − 3)</strong></li>
</ul>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 2.2: Linear Equations</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Solve linear equations in one variable</li>
<li>Solve simultaneous linear equations in two variables</li>
<li>Form and solve equations from word problems</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.2.1 Solving Linear Equations in One Variable</h4>
<p><strong>Goal:</strong> Isolate the variable on one side.</p>
<p><strong>Rule:</strong> Whatever you do to one side, do to the other.</p>
<p><em>Example:</em> Solve 3x − 7 = 11</p>
<ul class="notes-list">
<li>Add 7 to both sides: 3x = 18</li>
<li>Divide both sides by 3: <strong>x = 6</strong></li>
</ul>
<p><em>Example:</em> Solve 5(2x + 1) = 3x + 23</p>
<ul class="notes-list">
<li>Expand: 10x + 5 = 3x + 23</li>
<li>Subtract 3x: 7x + 5 = 23</li>
<li>Subtract 5: 7x = 18</li>
<li>Divide by 7: <strong>x = 18/7 = 2 4/7</strong></li>
</ul>
<p><em>Example (with fractions):</em> Solve x/3 + x/4 = 7</p>
<ul class="notes-list">
<li>Multiply all terms by LCM of 3 and 4 = 12</li>
<li>4x + 3x = 84</li>
<li>7x = 84</li>
<li><strong>x = 12</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.2.2 Simultaneous Equations</h4>
<p>Two equations with two unknowns, solved together.</p>
<p><strong>Method 1: Substitution</strong></p>
<p><em>Example:</em> Solve: y = 2x + 1 and 3x + y = 16</p>
<ul class="notes-list">
<li>Substitute y = 2x + 1 into the second equation:</li>
<li>3x + (2x + 1) = 16</li>
<li>5x + 1 = 16</li>
<li>5x = 15</li>
<li><strong>x = 3</strong></li>
<li>Substitute back: y = 2(3) + 1 = <strong>y = 7</strong></li>
<li>Solution: <strong>(x = 3, y = 7)</strong></li>
</ul>
<p><strong>Method 2: Elimination</strong></p>
<p><em>Example:</em> Solve: 2x + 3y = 13 and 4x − y = 5</p>
<ul class="notes-list">
<li>Multiply equation 2 by 3: 12x − 3y = 15</li>
<li>Add to equation 1: 14x = 28 → <strong>x = 2</strong></li>
<li>Substitute back: 2(2) + 3y = 13 → 4 + 3y = 13 → 3y = 9 → <strong>y = 3</strong></li>
<li>Solution: <strong>(x = 2, y = 3)</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.2.3 Word Problems Using Equations</h4>
<p><em>Example:</em> A farmer has chickens and goats. There are 30 animals in total and 86 legs. How many of each animal?</p>
<p>Let c = chickens, g = goats.</p>
<ul class="notes-list">
<li>c + g = 30 ... (i)</li>
<li>2c + 4g = 86 ... (ii) [chickens have 2 legs, goats have 4]</li>
</ul>
<p>From (i): c = 30 − g</p>
<p>Substitute into (ii): 2(30 − g) + 4g = 86</p>
<ul class="notes-list">
<li>60 − 2g + 4g = 86</li>
<li>60 + 2g = 86</li>
<li>2g = 26</li>
<li><strong>g = 13 goats</strong></li>
<li>c = 30 − 13 = <strong>17 chickens</strong></li>
</ul>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 2.3: Inequalities</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Solve linear inequalities in one variable</li>
<li>Represent solutions on a number line</li>
<li>Apply inequalities to real-life problems</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">2.3.1 Inequality Symbols</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Symbol</th><th>Meaning</th><th>Example</th></tr>
</thead><tbody>
<tr><td>></td><td>Greater than</td><td>x > 5</td></tr>
<tr><td><</td><td>Less than</td><td>x < 3</td></tr>
<tr><td>≥</td><td>Greater than or equal to</td><td>x ≥ −2</td></tr>
<tr><td>≤</td><td>Less than or equal to</td><td>x ≤ 7</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h4 class="notes-h4">2.3.2 Solving Inequalities</h4>
<p>Rules are same as equations, <strong>EXCEPT:</strong></p>
<blockquote class="notes-quote"><p>When you multiply or divide both sides by a <strong>negative number</strong>, you must <strong>reverse the inequality sign</strong>.</p></blockquote>
<p><em>Example:</em> Solve 2x + 3 > 9</p>
<ul class="notes-list">
<li>Subtract 3: 2x > 6</li>
<li>Divide by 2: <strong>x > 3</strong></li>
<li>Number line: open circle at 3, arrow pointing right →</li>
</ul>
<p><em>Example:</em> Solve −3x ≤ 12</p>
<ul class="notes-list">
<li>Divide by −3 (reverse sign!): <strong>x ≥ −4</strong></li>
<li>Number line: closed circle at −4, arrow pointing right →</li>
</ul>
<p><em>Example:</em> Solve −2 < 3x + 1 ≤ 10 (compound inequality)</p>
<ul class="notes-list">
<li>Subtract 1 from all parts: −3 < 3x ≤ 9</li>
<li>Divide all by 3: <strong>−1 < x ≤ 3</strong></li>
<li>Number line: open circle at −1, closed circle at 3, line between them</li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">STRAND 3: GEOMETRY</h2>
<h3 class="notes-h3">Sub-Strand 3.1: Angles and Lines</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Identify types of angles and lines</li>
<li>Calculate angles formed by parallel lines and transversals</li>
<li>Apply angle properties of triangles and polygons</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.1.1 Types of Angles</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Angle Type</th><th>Range</th><th>Example</th></tr>
</thead><tbody>
<tr><td>Acute</td><td>0° < x < 90°</td><td>45°</td></tr>
<tr><td>Right</td><td>Exactly 90°</td><td>Corner of a room</td></tr>
<tr><td>Obtuse</td><td>90° < x < 180°</td><td>135°</td></tr>
<tr><td>Straight</td><td>Exactly 180°</td><td>Straight line</td></tr>
<tr><td>Reflex</td><td>180° < x < 360°</td><td>270°</td></tr>
<tr><td>Full rotation</td><td>360°</td><td>Complete turn</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h4 class="notes-h4">3.1.2 Angle Relationships</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Relationship</th><th>Rule</th><th>Diagram description</th></tr>
</thead><tbody>
<tr><td>Complementary angles</td><td>Add up to 90°</td><td>Two angles in a right angle</td></tr>
<tr><td>Supplementary angles</td><td>Add up to 180°</td><td>Two angles on a straight line</td></tr>
<tr><td>Angles on a straight line</td><td>Sum = 180°</td><td>All angles on one side of a line</td></tr>
<tr><td>Angles at a point</td><td>Sum = 360°</td><td>All angles around a single point</td></tr>
<tr><td>Vertically opposite</td><td>Equal</td><td>Formed when two lines cross</td></tr>
</tbody></table></div>
<p><em>Example:</em> Find angle x if angles x, (2x + 10)° and 50° lie on a straight line.</p>
<ul class="notes-list">
<li>x + (2x + 10) + 50 = 180</li>
<li>3x + 60 = 180</li>
<li>3x = 120</li>
<li><strong>x = 40°</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.1.3 Parallel Lines and Transversals</h4>
<p>When a <strong>transversal</strong> cuts two parallel lines, special angle pairs are formed:</p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Angle Pair</th><th>Position</th><th>Relationship</th></tr>
</thead><tbody>
<tr><td>Corresponding angles</td><td>Same position at each intersection</td><td>Equal (F-shape)</td></tr>
<tr><td>Alternate angles</td><td>Opposite sides of transversal, between the lines</td><td>Equal (Z-shape)</td></tr>
<tr><td>Co-interior (same-side interior)</td><td>Same side of transversal, between the lines</td><td>Supplementary (add to 180°)</td></tr>
<tr><td>Vertically opposite</td><td>Cross-shaped</td><td>Equal</td></tr>
</tbody></table></div>
<p><em>Example:</em> Lines AB and CD are parallel. A transversal cuts them. One angle = 70°. Find all eight angles formed.</p>
<p>Using corresponding, alternate, and co-interior rules:</p>
<ul class="notes-list">
<li>Angles equal to 70°: the four corresponding/alternate pairs</li>
<li>Angles equal to 110° (= 180° − 70°): the four co-interior partners</li>
</ul>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 3.2: Triangles</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Identify types of triangles</li>
<li>Calculate missing angles and sides using triangle properties</li>
<li>Apply the Pythagorean Theorem</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.2.1 Types of Triangles</h4>
<p><strong>By sides:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Type</th><th>Property</th></tr>
</thead><tbody>
<tr><td>Equilateral</td><td>All 3 sides equal; all angles = 60°</td></tr>
<tr><td>Isosceles</td><td>Two sides equal; base angles equal</td></tr>
<tr><td>Scalene</td><td>All sides different; all angles different</td></tr>
</tbody></table></div>
<p><strong>By angles:</strong></p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Type</th><th>Property</th></tr>
</thead><tbody>
<tr><td>Acute-angled</td><td>All angles less than 90°</td></tr>
<tr><td>Right-angled</td><td>One angle = 90°</td></tr>
<tr><td>Obtuse-angled</td><td>One angle greater than 90°</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h4 class="notes-h4">3.2.2 Triangle Angle Properties</h4>
<p><strong>Property 1:</strong> The sum of interior angles of a triangle = <strong>180°</strong></p>
<p><em>Example:</em> In △ABC, angle A = 53°, angle B = 72°. Find angle C.</p>
<ul class="notes-list">
<li>A + B + C = 180°</li>
<li>53 + 72 + C = 180</li>
<li>C = 180 − 125 = <strong>55°</strong></li>
</ul>
<p><strong>Property 2:</strong> An exterior angle of a triangle = sum of the two non-adjacent interior angles</p>
<p><em>Example:</em> Exterior angle = 115°. One interior angle = 60°. Find the other interior angle.</p>
<ul class="notes-list">
<li>Other angle = 115° − 60° = <strong>55°</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.2.3 The Pythagorean Theorem</h4>
<p>In a <strong>right-angled triangle</strong>, if c is the hypotenuse (longest side, opposite the right angle):</p>
<p>$$a^2 + b^2 = c^2$$</p>
<p><strong>Finding the hypotenuse:</strong></p>
<p><em>Example:</em> a = 3 cm, b = 4 cm. Find c.</p>
<ul class="notes-list">
<li>c² = 3² + 4² = 9 + 16 = 25</li>
<li>c = √25 = <strong>5 cm</strong></li>
</ul>
<p><strong>Finding a shorter side:</strong></p>
<p><em>Example:</em> c = 13 cm, a = 5 cm. Find b.</p>
<ul class="notes-list">
<li>b² = c² − a² = 13² − 5² = 169 − 25 = 144</li>
<li>b = √144 = <strong>12 cm</strong></li>
</ul>
<p><strong>Pythagorean triples</strong> (common right-triangle sides to memorise):</p>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Triple</th></tr>
</thead><tbody>
<tr><td>3, 4, 5</td></tr>
<tr><td>5, 12, 13</td></tr>
<tr><td>8, 15, 17</td></tr>
<tr><td>7, 24, 25</td></tr>
</tbody></table></div>
<p><em>Any multiple of these is also a triple: 6, 8, 10 (×2 of 3, 4, 5)</em></p>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 3.3: Quadrilaterals and Polygons</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>State properties of quadrilaterals</li>
<li>Calculate interior and exterior angles of polygons</li>
<li>Find areas and perimeters of plane figures</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.3.1 Properties of Quadrilaterals</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Quadrilateral</th><th>Properties</th></tr>
</thead><tbody>
<tr><td>Square</td><td>4 equal sides; 4 right angles; diagonals equal, bisect at 90°</td></tr>
<tr><td>Rectangle</td><td>Opposite sides equal; 4 right angles; diagonals equal and bisect each other</td></tr>
<tr><td>Rhombus</td><td>4 equal sides; opposite angles equal; diagonals bisect at 90°</td></tr>
<tr><td>Parallelogram</td><td>Opposite sides equal and parallel; opposite angles equal; diagonals bisect each other</td></tr>
<tr><td>Trapezium</td><td>One pair of parallel sides</td></tr>
<tr><td>Kite</td><td>Two pairs of adjacent equal sides; one diagonal bisects the other at 90°</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h4 class="notes-h4">3.3.2 Angle Sum of Polygons</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Shape</th><th>Sides</th><th>Interior angle sum</th><th>Each interior angle (regular)</th></tr>
</thead><tbody>
<tr><td>Triangle</td><td>3</td><td>180°</td><td>60°</td></tr>
<tr><td>Quadrilateral</td><td>4</td><td>360°</td><td>90°</td></tr>
<tr><td>Pentagon</td><td>5</td><td>540°</td><td>108°</td></tr>
<tr><td>Hexagon</td><td>6</td><td>720°</td><td>120°</td></tr>
<tr><td>Octagon</td><td>8</td><td>1080°</td><td>135°</td></tr>
<tr><td>n-gon</td><td>n</td><td>(n−2) × 180°</td><td>(n−2) × 180° / n</td></tr>
</tbody></table></div>
<p><strong>Formula:</strong> Sum of interior angles of any polygon with n sides:</p>
<p>$$S = (n - 2) \times 180°$$</p>
<p><em>Example:</em> Find the sum of interior angles of a heptagon (7 sides).</p>
<ul class="notes-list">
<li>S = (7 − 2) × 180° = 5 × 180° = <strong>900°</strong></li>
</ul>
<p><strong>Exterior angles:</strong></p>
<ul class="notes-list">
<li>Each exterior angle of a regular polygon = 360° / n</li>
<li>Sum of exterior angles of ANY polygon = <strong>360°</strong></li>
</ul>
<p><em>Example:</em> Find the number of sides of a regular polygon with exterior angle 40°.</p>
<ul class="notes-list">
<li>n = 360° / 40° = <strong>9 sides (nonagon)</strong></li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.3.3 Perimeter and Area Formulae</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Shape</th><th>Perimeter</th><th>Area</th></tr>
</thead><tbody>
<tr><td>Square (side a)</td><td>4a</td><td>a²</td></tr>
<tr><td>Rectangle (l × w)</td><td>2(l + w)</td><td>l × w</td></tr>
<tr><td>Triangle (sides a, b, c; base b, height h)</td><td>a + b + c</td><td>½ × b × h</td></tr>
<tr><td>Parallelogram (base b, height h, sides a)</td><td>2(a + b)</td><td>b × h</td></tr>
<tr><td>Rhombus (side a, diagonals d₁, d₂)</td><td>4a</td><td>½ × d₁ × d₂</td></tr>
<tr><td>Trapezium (parallel sides a and b, height h)</td><td>Sum of all sides</td><td>½(a + b) × h</td></tr>
<tr><td>Circle (radius r)</td><td>2πr (circumference)</td><td>πr²</td></tr>
</tbody></table></div>
<p><em>Examples:</em></p>
<p><em>Area of a trapezium:</em> parallel sides 8 cm and 14 cm, height 6 cm</p>
<ul class="notes-list">
<li>A = ½ × (8 + 14) × 6 = ½ × 22 × 6 = <strong>66 cm²</strong></li>
</ul>
<p><em>Area of a rhombus:</em> diagonals 10 cm and 8 cm</p>
<ul class="notes-list">
<li>A = ½ × 10 × 8 = <strong>40 cm²</strong></li>
</ul>
<p><em>Circumference of a circle:</em> r = 7 cm, π ≈ 22/7</p>
<ul class="notes-list">
<li>C = 2 × 22/7 × 7 = <strong>44 cm</strong></li>
</ul>
<p><em>Area of a circle:</em> r = 7 cm</p>
<ul class="notes-list">
<li>A = 22/7 × 7² = 22/7 × 49 = <strong>154 cm²</strong></li>
</ul>
<hr class="section-divider">
<h3 class="notes-h3">Sub-Strand 3.4: Circles</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Identify parts of a circle</li>
<li>Apply circle theorems</li>
<li>Solve problems involving circles</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.4.1 Parts of a Circle</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Part</th><th>Definition</th></tr>
</thead><tbody>
<tr><td>Centre (O)</td><td>The middle point, equidistant from all points on the circle</td></tr>
<tr><td>Radius (r)</td><td>Distance from centre to any point on the circle</td></tr>
<tr><td>Diameter (d)</td><td>Distance across the circle through the centre; d = 2r</td></tr>
<tr><td>Circumference</td><td>The perimeter (total boundary) of the circle</td></tr>
<tr><td>Chord</td><td>A line segment with both endpoints on the circle</td></tr>
<tr><td>Arc</td><td>A part of the circumference</td></tr>
<tr><td>Sector</td><td>Region enclosed by two radii and an arc (like a pizza slice)</td></tr>
<tr><td>Segment</td><td>Region between a chord and an arc</td></tr>
<tr><td>Tangent</td><td>A line that touches the circle at exactly one point</td></tr>
<tr><td>Secant</td><td>A line that crosses the circle at two points</td></tr>
</tbody></table></div>
<hr class="section-divider">
<h4 class="notes-h4">3.4.2 Arc Length and Sector Area</h4>
<p><strong>Arc length:</strong></p>
<p>$$L = \frac{\theta}{360°} \times 2\pi r$$</p>
<p><strong>Area of sector:</strong></p>
<p>$$A = \frac{\theta}{360°} \times \pi r^2$$</p>
<p>Where θ = angle at the centre (in degrees)</p>
<p><em>Example:</em> A sector has radius 12 cm and angle 60°. Find:</p>
<p>(a) Arc length: L = (60/360) × 2 × 22/7 × 12 = (1/6) × 528/7 = <strong>12.57 cm</strong></p>
<p>(b) Area: A = (60/360) × 22/7 × 144 = (1/6) × 452.57 = <strong>75.43 cm²</strong></p>
<hr class="section-divider">
<h2 class="notes-h2">STRAND 3 — MENSURATION</h2>
<h3 class="notes-h3">Sub-Strand 3.5: Volume and Surface Area of Solids</h3>
<p><strong>Specific Learning Outcomes:</strong></p>
<p>By the end of this sub-strand, the learner should be able to:</p>
<ul class="notes-list">
<li>Calculate the volume and surface area of common solids</li>
<li>Apply these to real-life problems</li>
</ul>
<hr class="section-divider">
<h4 class="notes-h4">3.5.1 Volume and Surface Area Formulae</h4>
<div class="table-wrap"><table class="notes-table">
<thead>
<tr><th>Solid</th><th>Volume</th><th>Surface Area</th></tr>
</thead><tbody>
<tr><td>Cube (side a)</td><td>a³</td><td>6a²</td></tr>
<tr><td>Cuboid (l × w × h)</td><td>l × w × h</td><td>2(lw + lh + wh)</td></tr>
<tr><td>Cylinder (radius r, height h)</td><td>πr²h</td><td>2πr(r + h) [closed]</td></tr>
<tr><td>Cone (radius r, height h, slant l)</td><td>⅓πr²h</td><td>πr(r + l) [closed base]</td></tr>
<tr><td>Sphere (radius r)</td><td>4/3 πr³</td><td>4πr²</td></tr>
<tr><td>Pyramid (base area A, height h)</td><td>⅓ × A × h</td><td>Base + lateral faces</td></tr>
</tbody></table></div>
<p><em>Examples:</em></p>
<p><em>Volume of a cylinder:</em> r = 5 cm, h = 14 cm</p>
<ul class="notes-list">
<li>V = 22/7 × 25 × 14 = <strong>1100 cm³</strong></li>
</ul>
<p><em>Surface area of a closed cylinder:</em></p>
<ul class="notes-list">
<li>SA = 2 × 22/7 × 5 × (5 + 14) = 2 × 22/7 × 5 × 19 = <strong>596.9 cm²</strong></li>
</ul>
<p><em>Volume of a cone:</em> r = 6 cm, h = 8 cm</p>
<ul class="notes-list">
<li>V = ⅓ × 22/7 × 36 × 8 = <strong>301.7 cm³</strong></li>
</ul>
<p><em>Volume of a sphere:</em> r = 7 cm</p>
<ul class="notes-list">
<li>V = 4/3 × 22/7 × 343 = <strong>1437.3 cm³</strong></li>
</ul>
<hr class="section-divider">
<h2 class="notes-h2">TERM 1 REVIEW QUESTIONS</h2>
<h3 class="notes-h3">Section A — Fill in the Blanks</h3>
<ol class="notes-list">
<li>(−7) × (−8) = <strong>_</strong>_</li>
<li>The formula for simple interest is I = <strong>_</strong>_</li>
<li>When multiplying both sides of an inequality by a negative number, you must <strong>_</strong>_ the sign.</li>
<li>The sum of interior angles of a hexagon = <strong>_</strong>_°</li>
<li>The Pythagorean Theorem states: c² = <strong>_</strong>_</li>
</ol>
<h3 class="notes-h3">Section B — Work Out</h3>
<ol class="notes-list">
<li>Simplify: 3a − 2b + 5a − b + 3b <em>(2 marks)</em></li>
<li>Expand and simplify: (x + 4)(x − 2) + 3(x + 1) <em>(3 marks)</em></li>
<li>Solve: 4x − 3 = 2x + 9 <em>(2 marks)</em></li>
<li>Solve simultaneously: x + 2y = 8 and 2x − y = 6 <em>(4 marks)</em></li>
<li>Find the area of a trapezium with parallel sides 9 cm and 15 cm, height 7 cm. <em>(3 marks)</em></li>
</ol>
<h3 class="notes-h3">Section C — Problem Solving</h3>
<ol class="notes-list">
<li>A water tank is a cylinder with radius 3.5 m and height 2 m. Calculate:</li>
</ol>
<p>(a) The volume of water when full (in m³).</p>
<p>(b) The volume in litres (1 m³ = 1,000 litres). <em>(6 marks)</em></p>
<ol class="notes-list">
<li>Kamau buys a phone for Ksh 12,000. He sells it for Ksh 9,600.</li>
</ol>
<p>(a) Calculate his loss.</p>
<p>(b) Express the loss as a percentage. <em>(4 marks)</em></p>
<ol class="notes-list">
<li>Prove that the triangle with sides 9 cm, 40 cm, and 41 cm is right-angled. <em>(3 marks)</em></li>
</ol>
<hr class="section-divider">
<p><em>These notes cover Grade 8 Mathematics Term 1: Numbers, Algebra, and Geometry.</em></p>
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