mirror of
https://github.com/HananoshikaYomaru/openfreebooks.git
synced 2026-11-12 20:45:52 +00:00
Update curriculum content and improve site metadata: Change the status of multiple math chapters from 'planned' to 'live' in the JSON curriculum file. Enhance the chapter content in spec.md with updated best practices for chapter creation. Revise the feed and sitemap XML files to include new chapters and their last modified timestamps for better SEO and navigation. Update the section title in the why-sections template for clarity.
This commit is contained in:
Hananoshika Yomaru
parent
eee5d109d5
commit
81c6614722
53 changed files with 3903 additions and 53 deletions
134
content/math/advanced-trigonometry/_index.md
Normal file
134
content/math/advanced-trigonometry/_index.md
Normal file
|
|
@ -0,0 +1,134 @@
|
|||
+++
|
||||
title = "Advanced trigonometry"
|
||||
description = "Use sine and cosine rules, radians, and trigonometric graphs in 2D and 3D applications."
|
||||
template = "chapter.html"
|
||||
weight = 27
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "advanced-trigonometry"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Advanced trigonometry extends right-triangle work to non-right triangles, radians, and trig
|
||||
graphs used in periodic modeling.
|
||||
</p>
|
||||
|
||||
<h2 id="radian-measure" class="book-prose__heading">Radians and arc length</h2>
|
||||
<p>
|
||||
Convert degrees/radians:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
180^\circ=\pi\ \text{rad}.
|
||||
\]</p>
|
||||
<p>
|
||||
Arc length and sector area:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
s=r\theta,\qquad A=\frac12r^2\theta
|
||||
\]</p>
|
||||
<p>with \(\theta\) in radians.</p>
|
||||
|
||||
<h2 id="sine-cosine-rules" class="book-prose__heading">Sine and cosine rules</h2>
|
||||
<p class="book-formula">\[
|
||||
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
|
||||
\]</p>
|
||||
<p class="book-formula">\[
|
||||
c^2=a^2+b^2-2ab\cos C.
|
||||
\]</p>
|
||||
<p>
|
||||
Use sine rule for side-angle opposite pairs; cosine rule for SAS or SSS cases.
|
||||
</p>
|
||||
|
||||
<h2 id="area-triangle" class="book-prose__heading">Triangle area formula</h2>
|
||||
<p class="book-formula">\[
|
||||
\text{Area}=\frac12ab\sin C.
|
||||
\]</p>
|
||||
<p>
|
||||
Useful when two sides and included angle are known.
|
||||
</p>
|
||||
|
||||
<h2 id="identities" class="book-prose__heading">Core identities</h2>
|
||||
<p class="book-formula">\[
|
||||
\sin^2x+\cos^2x=1,\qquad
|
||||
\tan x=\frac{\sin x}{\cos x}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="trig-equations" class="book-prose__heading">Solving basic trig equations</h2>
|
||||
<p>
|
||||
Solve in interval (e.g., \(0^\circ\le x<360^\circ\)) using unit-circle symmetry.
|
||||
</p>
|
||||
<p><strong>Example:</strong> \(\sin x=\frac12\Rightarrow x=30^\circ,150^\circ\).</p>
|
||||
|
||||
<h2 id="graphs" class="book-prose__heading">Trigonometric graphs</h2>
|
||||
<p>
|
||||
For \(y=a\sin(bx)+d\) (similarly cosine):
|
||||
</p>
|
||||
<ul>
|
||||
<li>Amplitude \(=|a|\)</li>
|
||||
<li>Period \(=\frac{2\pi}{|b|}\) (or \(\frac{360^\circ}{|b|}\) in degrees mode)</li>
|
||||
<li>Vertical shift \(=d\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Check calculator mode (degree/radian) before solving.</li>
|
||||
<li>State interval when listing trig equation solutions.</li>
|
||||
<li>Draw quick triangle sketch for sine/cosine rule setup.</li>
|
||||
<li>Keep exact radian forms when possible.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Advanced trigonometric methods were developed for astronomy, navigation, and surveying. Sine and
|
||||
cosine rule tables were essential long before modern calculators.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Convert \(150^\circ\) to radians.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{5\pi}{6}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find arc length for \(r=8\), \(\theta=\frac{\pi}{3}\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(s=r\theta=\frac{8\pi}{3}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In triangle, \(a=10,b=7,C=60^\circ\). Find \(c\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(c^2=10^2+7^2-2(10)(7)\cos60^\circ=79\), so \(c=\sqrt{79}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Solve \(2\sin x=1\) for \(0^\circ\le x<360^\circ\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(x=30^\circ,150^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">For \(y=3\cos(2x)-1\), state amplitude and period.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Amplitude \(3\), period \(\pi\) radians (or \(180^\circ\)).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Wave modeling:</strong> sound, light, and AC signals.</li>
|
||||
<li><strong>Navigation:</strong> triangulation in maps and GPS support calculations.</li>
|
||||
<li><strong>Engineering:</strong> periodic motion and oscillation analysis.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school advanced trigonometry syllabi (IB, A-Level, DSE extension).</li>
|
||||
<li>Precalculus texts on trig identities, equations, and graphs.</li>
|
||||
</ul>
|
||||
</div>
|
||||
70
content/math/advanced-trigonometry/core.html
Normal file
70
content/math/advanced-trigonometry/core.html
Normal file
|
|
@ -0,0 +1,70 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Advanced trigonometry extends right-triangle work to non-right triangles, radians, and trig
|
||||
graphs used in periodic modeling.
|
||||
</p>
|
||||
|
||||
<h2 id="radian-measure" class="book-prose__heading">Radians and arc length</h2>
|
||||
<p>
|
||||
Convert degrees/radians:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
180^\circ=\pi\ \text{rad}.
|
||||
\]</p>
|
||||
<p>
|
||||
Arc length and sector area:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
s=r\theta,\qquad A=\frac12r^2\theta
|
||||
\]</p>
|
||||
<p>with \(\theta\) in radians.</p>
|
||||
|
||||
<h2 id="sine-cosine-rules" class="book-prose__heading">Sine and cosine rules</h2>
|
||||
<p class="book-formula">\[
|
||||
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
|
||||
\]</p>
|
||||
<p class="book-formula">\[
|
||||
c^2=a^2+b^2-2ab\cos C.
|
||||
\]</p>
|
||||
<p>
|
||||
Use sine rule for side-angle opposite pairs; cosine rule for SAS or SSS cases.
|
||||
</p>
|
||||
|
||||
<h2 id="area-triangle" class="book-prose__heading">Triangle area formula</h2>
|
||||
<p class="book-formula">\[
|
||||
\text{Area}=\frac12ab\sin C.
|
||||
\]</p>
|
||||
<p>
|
||||
Useful when two sides and included angle are known.
|
||||
</p>
|
||||
|
||||
<h2 id="identities" class="book-prose__heading">Core identities</h2>
|
||||
<p class="book-formula">\[
|
||||
\sin^2x+\cos^2x=1,\qquad
|
||||
\tan x=\frac{\sin x}{\cos x}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="trig-equations" class="book-prose__heading">Solving basic trig equations</h2>
|
||||
<p>
|
||||
Solve in interval (e.g., \(0^\circ\le x<360^\circ\)) using unit-circle symmetry.
|
||||
</p>
|
||||
<p><strong>Example:</strong> \(\sin x=\frac12\Rightarrow x=30^\circ,150^\circ\).</p>
|
||||
|
||||
<h2 id="graphs" class="book-prose__heading">Trigonometric graphs</h2>
|
||||
<p>
|
||||
For \(y=a\sin(bx)+d\) (similarly cosine):
|
||||
</p>
|
||||
<ul>
|
||||
<li>Amplitude \(=|a|\)</li>
|
||||
<li>Period \(=\frac{2\pi}{|b|}\) (or \(\frac{360^\circ}{|b|}\) in degrees mode)</li>
|
||||
<li>Vertical shift \(=d\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Check calculator mode (degree/radian) before solving.</li>
|
||||
<li>State interval when listing trig equation solutions.</li>
|
||||
<li>Draw quick triangle sketch for sine/cosine rule setup.</li>
|
||||
<li>Keep exact radian forms when possible.</li>
|
||||
</ul>
|
||||
</div>
|
||||
52
content/math/advanced-trigonometry/supplement.html
Normal file
52
content/math/advanced-trigonometry/supplement.html
Normal file
|
|
@ -0,0 +1,52 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Advanced trigonometric methods were developed for astronomy, navigation, and surveying. Sine and
|
||||
cosine rule tables were essential long before modern calculators.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Convert \(150^\circ\) to radians.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{5\pi}{6}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find arc length for \(r=8\), \(\theta=\frac{\pi}{3}\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(s=r\theta=\frac{8\pi}{3}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In triangle, \(a=10,b=7,C=60^\circ\). Find \(c\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(c^2=10^2+7^2-2(10)(7)\cos60^\circ=79\), so \(c=\sqrt{79}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Solve \(2\sin x=1\) for \(0^\circ\le x<360^\circ\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(x=30^\circ,150^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">For \(y=3\cos(2x)-1\), state amplitude and period.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Amplitude \(3\), period \(\pi\) radians (or \(180^\circ\)).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Wave modeling:</strong> sound, light, and AC signals.</li>
|
||||
<li><strong>Navigation:</strong> triangulation in maps and GPS support calculations.</li>
|
||||
<li><strong>Engineering:</strong> periodic motion and oscillation analysis.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school advanced trigonometry syllabi (IB, A-Level, DSE extension).</li>
|
||||
<li>Precalculus texts on trig identities, equations, and graphs.</li>
|
||||
</ul>
|
||||
</div>
|
||||
82
content/math/basic-probability/_index.md
Normal file
82
content/math/basic-probability/_index.md
Normal file
|
|
@ -0,0 +1,82 @@
|
|||
+++
|
||||
title = "Basic probability"
|
||||
description = "Calculate theoretical and experimental probabilities for simple events and complements."
|
||||
template = "chapter.html"
|
||||
weight = 32
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "basic-probability"
|
||||
strand = "Data Handling"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Probability measures how likely an event is, from 0 (impossible) to 1 (certain).
|
||||
</p>
|
||||
|
||||
<h2 id="fundamentals" class="book-prose__heading">Fundamentals</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}.
|
||||
\]</p>
|
||||
<p>
|
||||
Also: \(0\le P(E)\le1\).
|
||||
</p>
|
||||
|
||||
<h2 id="complement" class="book-prose__heading">Complement rule</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E')=1-P(E).
|
||||
\]</p>
|
||||
|
||||
<h2 id="multiple-events" class="book-prose__heading">Combined events</h2>
|
||||
<ul>
|
||||
<li>\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)</li>
|
||||
<li>If independent: \(P(A\cap B)=P(A)P(B)\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="experimental" class="book-prose__heading">Experimental probability</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="sample-space" class="book-prose__heading">Sample spaces and trees</h2>
|
||||
<p>
|
||||
Use sample-space tables or tree diagrams for multi-step outcomes.
|
||||
Multiply along branches, add across valid branches.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Define event clearly before calculating.</li>
|
||||
<li>Use fractions first; round at end if needed.</li>
|
||||
<li>Check complement and total probability sanity.</li>
|
||||
<li>Use diagrams to avoid missing outcomes.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A fair die is rolled. Find \(P(\text{prime})\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{3}{6}=\frac12\) (2,3,5).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A bag has 4 red and 6 blue balls. Find \(P(\text{not red})\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{6}{10}=\frac35\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">For independent events, \(P(A)=0.3,P(B)=0.4\). Find \(P(A\cap B)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(0.12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">An event occurs 18 times in 50 trials. Estimate probability.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{18}{50}=0.36\).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
43
content/math/basic-probability/core.html
Normal file
43
content/math/basic-probability/core.html
Normal file
|
|
@ -0,0 +1,43 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Probability measures how likely an event is, from 0 (impossible) to 1 (certain).
|
||||
</p>
|
||||
|
||||
<h2 id="fundamentals" class="book-prose__heading">Fundamentals</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}.
|
||||
\]</p>
|
||||
<p>
|
||||
Also: \(0\le P(E)\le1\).
|
||||
</p>
|
||||
|
||||
<h2 id="complement" class="book-prose__heading">Complement rule</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E')=1-P(E).
|
||||
\]</p>
|
||||
|
||||
<h2 id="multiple-events" class="book-prose__heading">Combined events</h2>
|
||||
<ul>
|
||||
<li>\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)</li>
|
||||
<li>If independent: \(P(A\cap B)=P(A)P(B)\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="experimental" class="book-prose__heading">Experimental probability</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="sample-space" class="book-prose__heading">Sample spaces and trees</h2>
|
||||
<p>
|
||||
Use sample-space tables or tree diagrams for multi-step outcomes.
|
||||
Multiply along branches, add across valid branches.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Define event clearly before calculating.</li>
|
||||
<li>Use fractions first; round at end if needed.</li>
|
||||
<li>Check complement and total probability sanity.</li>
|
||||
<li>Use diagrams to avoid missing outcomes.</li>
|
||||
</ul>
|
||||
</div>
|
||||
27
content/math/basic-probability/supplement.html
Normal file
27
content/math/basic-probability/supplement.html
Normal file
|
|
@ -0,0 +1,27 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A fair die is rolled. Find \(P(\text{prime})\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{3}{6}=\frac12\) (2,3,5).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A bag has 4 red and 6 blue balls. Find \(P(\text{not red})\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{6}{10}=\frac35\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">For independent events, \(P(A)=0.3,P(B)=0.4\). Find \(P(A\cap B)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(0.12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">An event occurs 18 times in 50 trials. Estimate probability.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{18}{50}=0.36\).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
114
content/math/circle-properties/_index.md
Normal file
114
content/math/circle-properties/_index.md
Normal file
|
|
@ -0,0 +1,114 @@
|
|||
+++
|
||||
title = "Properties of circles"
|
||||
description = "Apply chord, tangent, and angle theorems in circles to prove results and solve geometry problems."
|
||||
template = "chapter.html"
|
||||
weight = 28
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "circle-properties"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs
|
||||
and exam reasoning.
|
||||
</p>
|
||||
|
||||
<h2 id="key-theorems" class="book-prose__heading">Key circle theorems</h2>
|
||||
<ul>
|
||||
<li>Angle at center is twice angle at circumference on same arc.</li>
|
||||
<li>Angles in the same segment are equal.</li>
|
||||
<li>Angle in a semicircle is \(90^\circ\).</li>
|
||||
<li>Opposite angles of cyclic quadrilateral sum to \(180^\circ\).</li>
|
||||
<li>Tangent is perpendicular to radius at point of contact.</li>
|
||||
<li>Two tangents from same external point are equal in length.</li>
|
||||
<li>Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="proof-usage" class="book-prose__heading">Using theorems in proofs</h2>
|
||||
<ol>
|
||||
<li>Mark known equal angles/lengths clearly.</li>
|
||||
<li>Name theorem used at each step.</li>
|
||||
<li>Chain results to target statement.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="cyclic-quadrilateral" class="book-prose__heading">Cyclic quadrilaterals</h2>
|
||||
<p>
|
||||
A quadrilateral is cyclic if all vertices lie on one circle.
|
||||
</p>
|
||||
<p>
|
||||
Useful tests:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Opposite angles supplementary.</li>
|
||||
<li>Exterior angle equals opposite interior angle.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="tangent-chord" class="book-prose__heading">Tangent-chord applications</h2>
|
||||
<p>
|
||||
When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle
|
||||
with an inscribed angle subtending that chord.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Label all equal angles immediately.</li>
|
||||
<li>Write theorem names, not just numbers.</li>
|
||||
<li>Avoid assuming symmetry unless given or proved.</li>
|
||||
<li>Finish with a clear conclusion sentence.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Circle theorems were developed in classical geometry and refined through Euclidean proofs. They
|
||||
remain central in modern school geometry due to their structured reasoning style.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If central angle is \(100^\circ\), what is inscribed angle on same arc?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(50^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In a cyclic quadrilateral, one angle is \(72^\circ\). Find opposite angle.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(108^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">What angle is formed by radius and tangent at point of contact?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(90^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Two tangents from external point \(P\) touch at \(A,B\). Compare \(PA\) and \(PB\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(PA=PB\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why does one theorem name per step improve proof quality?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> It makes logical justification explicit and verifiable.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Design:</strong> arcs and tangency in mechanical/architectural drawings.</li>
|
||||
<li><strong>Optics:</strong> circular geometry in lenses and reflective paths.</li>
|
||||
<li><strong>CAD tools:</strong> tangent/chord constraints in modeling software.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry materials on circle theorems.</li>
|
||||
<li>Classical Euclidean geometry references.</li>
|
||||
</ul>
|
||||
</div>
|
||||
50
content/math/circle-properties/core.html
Normal file
50
content/math/circle-properties/core.html
Normal file
|
|
@ -0,0 +1,50 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs
|
||||
and exam reasoning.
|
||||
</p>
|
||||
|
||||
<h2 id="key-theorems" class="book-prose__heading">Key circle theorems</h2>
|
||||
<ul>
|
||||
<li>Angle at center is twice angle at circumference on same arc.</li>
|
||||
<li>Angles in the same segment are equal.</li>
|
||||
<li>Angle in a semicircle is \(90^\circ\).</li>
|
||||
<li>Opposite angles of cyclic quadrilateral sum to \(180^\circ\).</li>
|
||||
<li>Tangent is perpendicular to radius at point of contact.</li>
|
||||
<li>Two tangents from same external point are equal in length.</li>
|
||||
<li>Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="proof-usage" class="book-prose__heading">Using theorems in proofs</h2>
|
||||
<ol>
|
||||
<li>Mark known equal angles/lengths clearly.</li>
|
||||
<li>Name theorem used at each step.</li>
|
||||
<li>Chain results to target statement.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="cyclic-quadrilateral" class="book-prose__heading">Cyclic quadrilaterals</h2>
|
||||
<p>
|
||||
A quadrilateral is cyclic if all vertices lie on one circle.
|
||||
</p>
|
||||
<p>
|
||||
Useful tests:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Opposite angles supplementary.</li>
|
||||
<li>Exterior angle equals opposite interior angle.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="tangent-chord" class="book-prose__heading">Tangent-chord applications</h2>
|
||||
<p>
|
||||
When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle
|
||||
with an inscribed angle subtending that chord.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Label all equal angles immediately.</li>
|
||||
<li>Write theorem names, not just numbers.</li>
|
||||
<li>Avoid assuming symmetry unless given or proved.</li>
|
||||
<li>Finish with a clear conclusion sentence.</li>
|
||||
</ul>
|
||||
</div>
|
||||
52
content/math/circle-properties/supplement.html
Normal file
52
content/math/circle-properties/supplement.html
Normal file
|
|
@ -0,0 +1,52 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Circle theorems were developed in classical geometry and refined through Euclidean proofs. They
|
||||
remain central in modern school geometry due to their structured reasoning style.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If central angle is \(100^\circ\), what is inscribed angle on same arc?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(50^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In a cyclic quadrilateral, one angle is \(72^\circ\). Find opposite angle.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(108^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">What angle is formed by radius and tangent at point of contact?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(90^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Two tangents from external point \(P\) touch at \(A,B\). Compare \(PA\) and \(PB\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(PA=PB\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why does one theorem name per step improve proof quality?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> It makes logical justification explicit and verifiable.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Design:</strong> arcs and tangency in mechanical/architectural drawings.</li>
|
||||
<li><strong>Optics:</strong> circular geometry in lenses and reflective paths.</li>
|
||||
<li><strong>CAD tools:</strong> tangent/chord constraints in modeling software.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry materials on circle theorems.</li>
|
||||
<li>Classical Euclidean geometry references.</li>
|
||||
</ul>
|
||||
</div>
|
||||
154
content/math/congruence-similarity/_index.md
Normal file
154
content/math/congruence-similarity/_index.md
Normal file
|
|
@ -0,0 +1,154 @@
|
|||
+++
|
||||
title = "Congruence & similarity"
|
||||
description = "Apply congruence and similarity criteria to justify properties of triangles and scale figures."
|
||||
template = "chapter.html"
|
||||
weight = 22
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "congruence-similarity"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Congruence and similarity compare shapes by size and shape. They are central for triangle proofs,
|
||||
scale drawings, and indirect measurement.
|
||||
</p>
|
||||
|
||||
<h2 id="congruence" class="book-prose__heading">Congruence</h2>
|
||||
<p>
|
||||
Congruent figures have exactly the same shape and size. Corresponding sides and angles are equal.
|
||||
</p>
|
||||
<p>
|
||||
For triangles, common criteria:
|
||||
</p>
|
||||
<ul>
|
||||
<li>SSS (three sides equal)</li>
|
||||
<li>SAS (two sides and included angle equal)</li>
|
||||
<li>ASA / AAS (two angles and one corresponding side equal)</li>
|
||||
<li>RHS (right triangle: hypotenuse and one side equal)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="similarity" class="book-prose__heading">Similarity</h2>
|
||||
<p>
|
||||
Similar figures have the same shape but possibly different sizes.
|
||||
</p>
|
||||
<ul>
|
||||
<li>Corresponding angles are equal.</li>
|
||||
<li>Corresponding sides are in one constant ratio (scale factor).</li>
|
||||
</ul>
|
||||
<p>
|
||||
Triangle similarity criteria:
|
||||
</p>
|
||||
<ul>
|
||||
<li>AA (two angles equal)</li>
|
||||
<li>SAS in proportion</li>
|
||||
<li>SSS in proportion</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="scale-factor" class="book-prose__heading">Scale factor relationships</h2>
|
||||
<p>
|
||||
If linear scale factor from shape A to B is \(k\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>Lengths multiply by \(k\).</li>
|
||||
<li>Areas multiply by \(k^2\).</li>
|
||||
<li>Volumes multiply by \(k^3\).</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="using-similarity" class="book-prose__heading">Using similarity to solve problems</h2>
|
||||
<ol>
|
||||
<li>Identify corresponding vertices in correct order.</li>
|
||||
<li>Write ratio equation using matching sides.</li>
|
||||
<li>Solve unknown lengths/areas.</li>
|
||||
<li>Check ratio consistency across all pairs.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="proofs" class="book-prose__heading">Congruence and similarity in proofs</h2>
|
||||
<p>
|
||||
Common proof flow:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Show two triangles are congruent/similar by criteria.</li>
|
||||
<li>Conclude equal angles/sides or proportional segments.</li>
|
||||
<li>Use result to prove target statement.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>State criterion explicitly (e.g., “\(\triangle ABC\cong\triangle DEF\) by SAS”).</li>
|
||||
<li>Match vertices in correct correspondence order.</li>
|
||||
<li>Do not mix congruence and similarity conditions.</li>
|
||||
<li>For ratio problems, keep fractions in consistent side order.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Similarity ideas appear in ancient surveying and architecture. Greek geometers formalized triangle
|
||||
relationships, enabling rigorous methods for unknown lengths and indirect measurements.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="aa-why" class="book-prose__heading">Why AA implies triangle similarity</h3>
|
||||
<p>
|
||||
In triangles, angle sum is \(180^\circ\). If two angles match, the third angle must also match.
|
||||
Matching angles force proportional corresponding sides, so triangles are similar.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Triangles have side sets \((5,7,9)\) and \((5,7,9)\). Congruent or similar?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Congruent (SSS, identical side lengths).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Two triangles have angles \(40^\circ,60^\circ,80^\circ\) and \(40^\circ,60^\circ,80^\circ\), but one is larger. What relation?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Similar (AA), not necessarily congruent.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Similar triangles have scale factor \(k=3\). If a side is \(4\) in the small triangle, what is matching side in large triangle?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(4\times3=12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
If two similar figures have area ratio \(25:9\), what is linear scale factor ratio?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(5:3\) (square root of area ratio).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Right triangles share one acute angle and each has a right angle. What criterion gives similarity?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> AA similarity.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Scale maps/drawings:</strong> convert plan dimensions to real distances.</li>
|
||||
<li><strong>Shadow methods:</strong> estimate object heights using similar triangles.</li>
|
||||
<li><strong>Engineering design:</strong> preserve shape while resizing components.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry syllabi on triangle congruence and similarity.</li>
|
||||
<li>Geometry textbooks on ratio, scale factor, and proof applications.</li>
|
||||
</ul>
|
||||
</div>
|
||||
73
content/math/congruence-similarity/core.html
Normal file
73
content/math/congruence-similarity/core.html
Normal file
|
|
@ -0,0 +1,73 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Congruence and similarity compare shapes by size and shape. They are central for triangle proofs,
|
||||
scale drawings, and indirect measurement.
|
||||
</p>
|
||||
|
||||
<h2 id="congruence" class="book-prose__heading">Congruence</h2>
|
||||
<p>
|
||||
Congruent figures have exactly the same shape and size. Corresponding sides and angles are equal.
|
||||
</p>
|
||||
<p>
|
||||
For triangles, common criteria:
|
||||
</p>
|
||||
<ul>
|
||||
<li>SSS (three sides equal)</li>
|
||||
<li>SAS (two sides and included angle equal)</li>
|
||||
<li>ASA / AAS (two angles and one corresponding side equal)</li>
|
||||
<li>RHS (right triangle: hypotenuse and one side equal)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="similarity" class="book-prose__heading">Similarity</h2>
|
||||
<p>
|
||||
Similar figures have the same shape but possibly different sizes.
|
||||
</p>
|
||||
<ul>
|
||||
<li>Corresponding angles are equal.</li>
|
||||
<li>Corresponding sides are in one constant ratio (scale factor).</li>
|
||||
</ul>
|
||||
<p>
|
||||
Triangle similarity criteria:
|
||||
</p>
|
||||
<ul>
|
||||
<li>AA (two angles equal)</li>
|
||||
<li>SAS in proportion</li>
|
||||
<li>SSS in proportion</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="scale-factor" class="book-prose__heading">Scale factor relationships</h2>
|
||||
<p>
|
||||
If linear scale factor from shape A to B is \(k\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>Lengths multiply by \(k\).</li>
|
||||
<li>Areas multiply by \(k^2\).</li>
|
||||
<li>Volumes multiply by \(k^3\).</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="using-similarity" class="book-prose__heading">Using similarity to solve problems</h2>
|
||||
<ol>
|
||||
<li>Identify corresponding vertices in correct order.</li>
|
||||
<li>Write ratio equation using matching sides.</li>
|
||||
<li>Solve unknown lengths/areas.</li>
|
||||
<li>Check ratio consistency across all pairs.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="proofs" class="book-prose__heading">Congruence and similarity in proofs</h2>
|
||||
<p>
|
||||
Common proof flow:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Show two triangles are congruent/similar by criteria.</li>
|
||||
<li>Conclude equal angles/sides or proportional segments.</li>
|
||||
<li>Use result to prove target statement.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>State criterion explicitly (e.g., “\(\triangle ABC\cong\triangle DEF\) by SAS”).</li>
|
||||
<li>Match vertices in correct correspondence order.</li>
|
||||
<li>Do not mix congruence and similarity conditions.</li>
|
||||
<li>For ratio problems, keep fractions in consistent side order.</li>
|
||||
</ul>
|
||||
</div>
|
||||
69
content/math/congruence-similarity/supplement.html
Normal file
69
content/math/congruence-similarity/supplement.html
Normal file
|
|
@ -0,0 +1,69 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Similarity ideas appear in ancient surveying and architecture. Greek geometers formalized triangle
|
||||
relationships, enabling rigorous methods for unknown lengths and indirect measurements.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="aa-why" class="book-prose__heading">Why AA implies triangle similarity</h3>
|
||||
<p>
|
||||
In triangles, angle sum is \(180^\circ\). If two angles match, the third angle must also match.
|
||||
Matching angles force proportional corresponding sides, so triangles are similar.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Triangles have side sets \((5,7,9)\) and \((5,7,9)\). Congruent or similar?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Congruent (SSS, identical side lengths).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Two triangles have angles \(40^\circ,60^\circ,80^\circ\) and \(40^\circ,60^\circ,80^\circ\), but one is larger. What relation?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Similar (AA), not necessarily congruent.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Similar triangles have scale factor \(k=3\). If a side is \(4\) in the small triangle, what is matching side in large triangle?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(4\times3=12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
If two similar figures have area ratio \(25:9\), what is linear scale factor ratio?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(5:3\) (square root of area ratio).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Right triangles share one acute angle and each has a right angle. What criterion gives similarity?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> AA similarity.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Scale maps/drawings:</strong> convert plan dimensions to real distances.</li>
|
||||
<li><strong>Shadow methods:</strong> estimate object heights using similar triangles.</li>
|
||||
<li><strong>Engineering design:</strong> preserve shape while resizing components.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry syllabi on triangle congruence and similarity.</li>
|
||||
<li>Geometry textbooks on ratio, scale factor, and proof applications.</li>
|
||||
</ul>
|
||||
</div>
|
||||
136
content/math/coordinate-circles/_index.md
Normal file
136
content/math/coordinate-circles/_index.md
Normal file
|
|
@ -0,0 +1,136 @@
|
|||
+++
|
||||
title = "Coordinate geometry of circles"
|
||||
description = "Find equations of circles, tangents, and intersections using coordinate methods and distance."
|
||||
template = "chapter.html"
|
||||
weight = 26
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "coordinate-circles"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Coordinate geometry of circles combines algebra with geometric properties such as radius, center,
|
||||
tangents, and intersections.
|
||||
</p>
|
||||
|
||||
<h2 id="standard-equation" class="book-prose__heading">Standard equation of a circle</h2>
|
||||
<p>
|
||||
A circle with center \((h,k)\) and radius \(r\) has equation:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
(x-h)^2+(y-k)^2=r^2.
|
||||
\]</p>
|
||||
<p>
|
||||
If center is origin, this simplifies to \(x^2+y^2=r^2\).
|
||||
</p>
|
||||
|
||||
<h2 id="from-equation-to-geometry" class="book-prose__heading">Reading center and radius</h2>
|
||||
<p><strong>Example:</strong> \((x-3)^2+(y+2)^2=25\)</p>
|
||||
<ul>
|
||||
<li>Center: \((3,-2)\)</li>
|
||||
<li>Radius: \(5\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="general-form" class="book-prose__heading">General form</h2>
|
||||
<p>
|
||||
A circle can also appear as:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
x^2+y^2+Dx+Ey+F=0.
|
||||
\]</p>
|
||||
<p>
|
||||
Complete the square in \(x\) and \(y\) to convert to standard form.
|
||||
</p>
|
||||
|
||||
<h2 id="tangent-condition" class="book-prose__heading">Tangents and radii</h2>
|
||||
<ul>
|
||||
<li>A tangent touches a circle at exactly one point.</li>
|
||||
<li>Radius to tangent point is perpendicular to the tangent.</li>
|
||||
</ul>
|
||||
<p>
|
||||
If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined).
|
||||
</p>
|
||||
|
||||
<h2 id="point-position" class="book-prose__heading">Point position relative to circle</h2>
|
||||
<p>
|
||||
For circle center \(C\), radius \(r\), and point \(P\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(CP<r\): inside</li>
|
||||
<li>\(CP=r\): on circle</li>
|
||||
<li>\(CP>r\): outside</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="line-circle-intersection" class="book-prose__heading">Line-circle intersections</h2>
|
||||
<p>
|
||||
Substitute line equation into circle equation.
|
||||
</p>
|
||||
<ul>
|
||||
<li>Two solutions: secant intersects twice.</li>
|
||||
<li>One solution: tangent.</li>
|
||||
<li>No real solution: no intersection.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Write center and radius before computation.</li>
|
||||
<li>Use exact values first; round at final step.</li>
|
||||
<li>For tangents, use radius-perpendicular fact explicitly.</li>
|
||||
<li>Check whether solutions satisfy geometric context.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Coordinate methods made circle geometry algebraic: geometric questions about tangency and
|
||||
intersection can be solved by equations. This approach grew from analytic geometry.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find center and radius of \((x+4)^2+(y-1)^2=36\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Center \((-4,1)\), radius \(6\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Write equation of circle with center \((2,-3)\), radius \(7\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \((x-2)^2+(y+3)^2=49\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Is point \((5,1)\) on circle \((x-1)^2+(y-1)^2=16\)?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Yes, \((5-1)^2+(1-1)^2=16\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Convert \(x^2+y^2-6x+4y-12=0\) to standard form.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \((x-3)^2+(y+2)^2=25\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A line through tangent point has slope 2. What is slope of radius to that point?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(-\frac12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Engineering:</strong> circular component alignment and tolerance checks.</li>
|
||||
<li><strong>Computer graphics:</strong> collision and boundary detection.</li>
|
||||
<li><strong>Robotics:</strong> turning path and sensor range modeling.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school coordinate geometry materials on circles and tangents.</li>
|
||||
<li>Analytic geometry references on conic sections (intro level).</li>
|
||||
</ul>
|
||||
</div>
|
||||
72
content/math/coordinate-circles/core.html
Normal file
72
content/math/coordinate-circles/core.html
Normal file
|
|
@ -0,0 +1,72 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Coordinate geometry of circles combines algebra with geometric properties such as radius, center,
|
||||
tangents, and intersections.
|
||||
</p>
|
||||
|
||||
<h2 id="standard-equation" class="book-prose__heading">Standard equation of a circle</h2>
|
||||
<p>
|
||||
A circle with center \((h,k)\) and radius \(r\) has equation:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
(x-h)^2+(y-k)^2=r^2.
|
||||
\]</p>
|
||||
<p>
|
||||
If center is origin, this simplifies to \(x^2+y^2=r^2\).
|
||||
</p>
|
||||
|
||||
<h2 id="from-equation-to-geometry" class="book-prose__heading">Reading center and radius</h2>
|
||||
<p><strong>Example:</strong> \((x-3)^2+(y+2)^2=25\)</p>
|
||||
<ul>
|
||||
<li>Center: \((3,-2)\)</li>
|
||||
<li>Radius: \(5\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="general-form" class="book-prose__heading">General form</h2>
|
||||
<p>
|
||||
A circle can also appear as:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
x^2+y^2+Dx+Ey+F=0.
|
||||
\]</p>
|
||||
<p>
|
||||
Complete the square in \(x\) and \(y\) to convert to standard form.
|
||||
</p>
|
||||
|
||||
<h2 id="tangent-condition" class="book-prose__heading">Tangents and radii</h2>
|
||||
<ul>
|
||||
<li>A tangent touches a circle at exactly one point.</li>
|
||||
<li>Radius to tangent point is perpendicular to the tangent.</li>
|
||||
</ul>
|
||||
<p>
|
||||
If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined).
|
||||
</p>
|
||||
|
||||
<h2 id="point-position" class="book-prose__heading">Point position relative to circle</h2>
|
||||
<p>
|
||||
For circle center \(C\), radius \(r\), and point \(P\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(CP<r\): inside</li>
|
||||
<li>\(CP=r\): on circle</li>
|
||||
<li>\(CP>r\): outside</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="line-circle-intersection" class="book-prose__heading">Line-circle intersections</h2>
|
||||
<p>
|
||||
Substitute line equation into circle equation.
|
||||
</p>
|
||||
<ul>
|
||||
<li>Two solutions: secant intersects twice.</li>
|
||||
<li>One solution: tangent.</li>
|
||||
<li>No real solution: no intersection.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Write center and radius before computation.</li>
|
||||
<li>Use exact values first; round at final step.</li>
|
||||
<li>For tangents, use radius-perpendicular fact explicitly.</li>
|
||||
<li>Check whether solutions satisfy geometric context.</li>
|
||||
</ul>
|
||||
</div>
|
||||
52
content/math/coordinate-circles/supplement.html
Normal file
52
content/math/coordinate-circles/supplement.html
Normal file
|
|
@ -0,0 +1,52 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Coordinate methods made circle geometry algebraic: geometric questions about tangency and
|
||||
intersection can be solved by equations. This approach grew from analytic geometry.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find center and radius of \((x+4)^2+(y-1)^2=36\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Center \((-4,1)\), radius \(6\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Write equation of circle with center \((2,-3)\), radius \(7\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \((x-2)^2+(y+3)^2=49\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Is point \((5,1)\) on circle \((x-1)^2+(y-1)^2=16\)?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Yes, \((5-1)^2+(1-1)^2=16\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Convert \(x^2+y^2-6x+4y-12=0\) to standard form.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \((x-3)^2+(y+2)^2=25\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A line through tangent point has slope 2. What is slope of radius to that point?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(-\frac12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Engineering:</strong> circular component alignment and tolerance checks.</li>
|
||||
<li><strong>Computer graphics:</strong> collision and boundary detection.</li>
|
||||
<li><strong>Robotics:</strong> turning path and sensor range modeling.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school coordinate geometry materials on circles and tangents.</li>
|
||||
<li>Analytic geometry references on conic sections (intro level).</li>
|
||||
</ul>
|
||||
</div>
|
||||
159
content/math/coordinate-geometry-foundations/_index.md
Normal file
159
content/math/coordinate-geometry-foundations/_index.md
Normal file
|
|
@ -0,0 +1,159 @@
|
|||
+++
|
||||
title = "Coordinate geometry foundations"
|
||||
description = "Plot points, find distances and midpoints, and work with slopes of lines in the Cartesian plane."
|
||||
template = "chapter.html"
|
||||
weight = 20
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "coordinate-geometry-foundations"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Coordinate geometry connects algebra and geometry by representing points with ordered pairs
|
||||
\((x,y)\) on the Cartesian plane.
|
||||
</p>
|
||||
|
||||
<h2 id="cartesian-plane" class="book-prose__heading">The Cartesian plane</h2>
|
||||
<p>
|
||||
Horizontal axis is \(x\)-axis, vertical axis is \(y\)-axis, and intersection is origin \((0,0)\).
|
||||
</p>
|
||||
<p>
|
||||
Quadrants:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Q1: \((+,+)\)</li>
|
||||
<li>Q2: \((-,+)\)</li>
|
||||
<li>Q3: \((-,-)\)</li>
|
||||
<li>Q4: \((+,-)\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="plot-points" class="book-prose__heading">Plotting and reading points</h2>
|
||||
<p>
|
||||
Ordered pair \((a,b)\) means move \(a\) units along \(x\), then \(b\) units along \(y\).
|
||||
</p>
|
||||
<p><strong>Example:</strong> \((-3,2)\) lies in Quadrant II.</p>
|
||||
|
||||
<h2 id="distance-formula" class="book-prose__heading">Distance between two points</h2>
|
||||
<p>
|
||||
For \(A(x_1,y_1)\), \(B(x_2,y_2)\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.
|
||||
\]</p>
|
||||
<p>
|
||||
This comes from Pythagoras' theorem on horizontal and vertical differences.
|
||||
</p>
|
||||
|
||||
<h2 id="midpoint-formula" class="book-prose__heading">Midpoint of a segment</h2>
|
||||
<p class="book-formula">\[
|
||||
M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).
|
||||
\]</p>
|
||||
<p><strong>Example:</strong> Midpoint of \((2,5)\) and \((8,-1)\) is \((5,2)\).</p>
|
||||
|
||||
<h2 id="gradient-slope" class="book-prose__heading">Gradient (slope) of a line</h2>
|
||||
<p>
|
||||
For points \(A(x_1,y_1)\), \(B(x_2,y_2)\) with \(x_2\ne x_1\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
m=\frac{y_2-y_1}{x_2-x_1}.
|
||||
\]</p>
|
||||
<ul>
|
||||
<li>\(m>0\): line rises left to right.</li>
|
||||
<li>\(m<0\): line falls left to right.</li>
|
||||
<li>\(m=0\): horizontal line.</li>
|
||||
<li>Vertical line has undefined slope.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="line-equations" class="book-prose__heading">Equations of lines</h2>
|
||||
<p>
|
||||
Slope-intercept form:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
y=mx+c
|
||||
\]</p>
|
||||
<p>
|
||||
where \(m\) is slope and \(c\) is \(y\)-intercept.
|
||||
</p>
|
||||
<p>
|
||||
Point-slope form:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
y-y_1=m(x-x_1).
|
||||
\]</p>
|
||||
|
||||
<h2 id="parallel-perpendicular" class="book-prose__heading">Parallel and perpendicular lines</h2>
|
||||
<ul>
|
||||
<li>Parallel lines have equal slopes: \(m_1=m_2\).</li>
|
||||
<li>Perpendicular lines satisfy \(m_1m_2=-1\) (when both slopes are defined).</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Write coordinates clearly before substituting formulas.</li>
|
||||
<li>Keep exact form (roots/fractions) unless decimal is requested.</li>
|
||||
<li>Check whether slope is undefined for vertical lines.</li>
|
||||
<li>Use diagrams to avoid sign mistakes in differences.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Coordinate geometry was formalized by Rene Descartes and Pierre de Fermat. Their work connected
|
||||
algebraic equations to geometric curves, creating a major foundation for modern mathematics.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="distance-derivation" class="book-prose__heading">Why the distance formula works</h3>
|
||||
<p>
|
||||
Horizontal change is \(\Delta x=x_2-x_1\), vertical change is \(\Delta y=y_2-y_1\). These form
|
||||
a right triangle, so distance is \(\sqrt{(\Delta x)^2+(\Delta y)^2}\).
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">State the quadrant of \((-4,-7)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Quadrant III.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find distance between \((1,2)\) and \((7,10)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{6^2+8^2}=\sqrt{100}=10\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find midpoint of \((-3,5)\) and \((9,-1)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\left(3,2\right)\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find slope through \((2,-1)\) and \((6,7)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(m=\frac{7-(-1)}{6-2}=\frac84=2\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Line with slope \(3\) through \((1,4)\): find equation.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(y-4=3(x-1)\Rightarrow y=3x+1\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Maps/GPS:</strong> locations represented using coordinate systems.</li>
|
||||
<li><strong>Computer graphics:</strong> points and lines placed on pixel coordinate planes.</li>
|
||||
<li><strong>Physics:</strong> trajectories and motion graphs depend on coordinates and slope.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school coordinate geometry units.</li>
|
||||
<li>Analytic geometry references on points, lines, distance, and midpoint.</li>
|
||||
</ul>
|
||||
</div>
|
||||
88
content/math/coordinate-geometry-foundations/core.html
Normal file
88
content/math/coordinate-geometry-foundations/core.html
Normal file
|
|
@ -0,0 +1,88 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Coordinate geometry connects algebra and geometry by representing points with ordered pairs
|
||||
\((x,y)\) on the Cartesian plane.
|
||||
</p>
|
||||
|
||||
<h2 id="cartesian-plane" class="book-prose__heading">The Cartesian plane</h2>
|
||||
<p>
|
||||
Horizontal axis is \(x\)-axis, vertical axis is \(y\)-axis, and intersection is origin \((0,0)\).
|
||||
</p>
|
||||
<p>
|
||||
Quadrants:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Q1: \((+,+)\)</li>
|
||||
<li>Q2: \((-,+)\)</li>
|
||||
<li>Q3: \((-,-)\)</li>
|
||||
<li>Q4: \((+,-)\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="plot-points" class="book-prose__heading">Plotting and reading points</h2>
|
||||
<p>
|
||||
Ordered pair \((a,b)\) means move \(a\) units along \(x\), then \(b\) units along \(y\).
|
||||
</p>
|
||||
<p><strong>Example:</strong> \((-3,2)\) lies in Quadrant II.</p>
|
||||
|
||||
<h2 id="distance-formula" class="book-prose__heading">Distance between two points</h2>
|
||||
<p>
|
||||
For \(A(x_1,y_1)\), \(B(x_2,y_2)\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.
|
||||
\]</p>
|
||||
<p>
|
||||
This comes from Pythagoras' theorem on horizontal and vertical differences.
|
||||
</p>
|
||||
|
||||
<h2 id="midpoint-formula" class="book-prose__heading">Midpoint of a segment</h2>
|
||||
<p class="book-formula">\[
|
||||
M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).
|
||||
\]</p>
|
||||
<p><strong>Example:</strong> Midpoint of \((2,5)\) and \((8,-1)\) is \((5,2)\).</p>
|
||||
|
||||
<h2 id="gradient-slope" class="book-prose__heading">Gradient (slope) of a line</h2>
|
||||
<p>
|
||||
For points \(A(x_1,y_1)\), \(B(x_2,y_2)\) with \(x_2\ne x_1\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
m=\frac{y_2-y_1}{x_2-x_1}.
|
||||
\]</p>
|
||||
<ul>
|
||||
<li>\(m>0\): line rises left to right.</li>
|
||||
<li>\(m<0\): line falls left to right.</li>
|
||||
<li>\(m=0\): horizontal line.</li>
|
||||
<li>Vertical line has undefined slope.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="line-equations" class="book-prose__heading">Equations of lines</h2>
|
||||
<p>
|
||||
Slope-intercept form:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
y=mx+c
|
||||
\]</p>
|
||||
<p>
|
||||
where \(m\) is slope and \(c\) is \(y\)-intercept.
|
||||
</p>
|
||||
<p>
|
||||
Point-slope form:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
y-y_1=m(x-x_1).
|
||||
\]</p>
|
||||
|
||||
<h2 id="parallel-perpendicular" class="book-prose__heading">Parallel and perpendicular lines</h2>
|
||||
<ul>
|
||||
<li>Parallel lines have equal slopes: \(m_1=m_2\).</li>
|
||||
<li>Perpendicular lines satisfy \(m_1m_2=-1\) (when both slopes are defined).</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Write coordinates clearly before substituting formulas.</li>
|
||||
<li>Keep exact form (roots/fractions) unless decimal is requested.</li>
|
||||
<li>Check whether slope is undefined for vertical lines.</li>
|
||||
<li>Use diagrams to avoid sign mistakes in differences.</li>
|
||||
</ul>
|
||||
</div>
|
||||
59
content/math/coordinate-geometry-foundations/supplement.html
Normal file
59
content/math/coordinate-geometry-foundations/supplement.html
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Coordinate geometry was formalized by Rene Descartes and Pierre de Fermat. Their work connected
|
||||
algebraic equations to geometric curves, creating a major foundation for modern mathematics.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="distance-derivation" class="book-prose__heading">Why the distance formula works</h3>
|
||||
<p>
|
||||
Horizontal change is \(\Delta x=x_2-x_1\), vertical change is \(\Delta y=y_2-y_1\). These form
|
||||
a right triangle, so distance is \(\sqrt{(\Delta x)^2+(\Delta y)^2}\).
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">State the quadrant of \((-4,-7)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Quadrant III.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find distance between \((1,2)\) and \((7,10)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{6^2+8^2}=\sqrt{100}=10\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find midpoint of \((-3,5)\) and \((9,-1)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\left(3,2\right)\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find slope through \((2,-1)\) and \((6,7)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(m=\frac{7-(-1)}{6-2}=\frac84=2\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Line with slope \(3\) through \((1,4)\): find equation.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(y-4=3(x-1)\Rightarrow y=3x+1\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Maps/GPS:</strong> locations represented using coordinate systems.</li>
|
||||
<li><strong>Computer graphics:</strong> points and lines placed on pixel coordinate planes.</li>
|
||||
<li><strong>Physics:</strong> trajectories and motion graphs depend on coordinates and slope.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school coordinate geometry units.</li>
|
||||
<li>Analytic geometry references on points, lines, distance, and midpoint.</li>
|
||||
</ul>
|
||||
</div>
|
||||
161
content/math/deductive-geometry/_index.md
Normal file
161
content/math/deductive-geometry/_index.md
Normal file
|
|
@ -0,0 +1,161 @@
|
|||
+++
|
||||
title = "Deductive geometry"
|
||||
description = "Prove angle and side relationships for intersecting lines, triangles, and polygons."
|
||||
template = "chapter.html"
|
||||
weight = 21
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "deductive-geometry"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Deductive geometry uses known facts and logical steps to prove new geometric results.
|
||||
A valid proof must justify each statement with a theorem, definition, or given condition.
|
||||
</p>
|
||||
|
||||
<h2 id="reasoning-language" class="book-prose__heading">Reasoning language</h2>
|
||||
<ul>
|
||||
<li><strong>Given</strong>: information provided in the question.</li>
|
||||
<li><strong>To prove</strong>: target statement.</li>
|
||||
<li><strong>Because / therefore</strong>: links each conclusion to a reason.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="line-angle-facts" class="book-prose__heading">Core angle facts</h2>
|
||||
<ul>
|
||||
<li>Angles on a straight line sum to \(180^\circ\).</li>
|
||||
<li>Angles around a point sum to \(360^\circ\).</li>
|
||||
<li>Vertically opposite angles are equal.</li>
|
||||
<li>In parallel lines:
|
||||
<ul>
|
||||
<li>alternate interior angles are equal,</li>
|
||||
<li>corresponding angles are equal,</li>
|
||||
<li>co-interior angles sum to \(180^\circ\).</li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="triangle-facts" class="book-prose__heading">Triangle facts</h2>
|
||||
<ul>
|
||||
<li>Interior angles sum to \(180^\circ\).</li>
|
||||
<li>Exterior angle equals sum of two opposite interior angles.</li>
|
||||
<li>Isosceles triangle: equal sides oppose equal angles.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="polygon-facts" class="book-prose__heading">Polygon facts</h2>
|
||||
<p>
|
||||
Sum of interior angles of \(n\)-gon:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
(n-2)\times180^\circ.
|
||||
\]</p>
|
||||
<p>
|
||||
Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\).
|
||||
</p>
|
||||
|
||||
<h2 id="proof-structure" class="book-prose__heading">Proof structure</h2>
|
||||
<ol>
|
||||
<li>Write what is given and what must be shown.</li>
|
||||
<li>Mark known equal angles/lengths on diagram.</li>
|
||||
<li>Use relevant theorems in short logical chain.</li>
|
||||
<li>End with clear conclusion sentence.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="example-proof" class="book-prose__heading">Example proof pattern</h2>
|
||||
<p>
|
||||
If \(AB\parallel CD\) and line \(AD\) intersects them, prove one pair of alternate interior angles
|
||||
is equal.
|
||||
</p>
|
||||
<p>
|
||||
Since \(AB\parallel CD\), alternate interior angles are equal (parallel-line theorem). Therefore
|
||||
required pair is equal.
|
||||
</p>
|
||||
|
||||
<h2 id="common-errors" class="book-prose__heading">Common proof errors</h2>
|
||||
<ul>
|
||||
<li>Stating result without reason.</li>
|
||||
<li>Using theorem conditions not satisfied by diagram.</li>
|
||||
<li>Assuming diagram scale implies equality.</li>
|
||||
<li>Skipping steps between statements.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Use short statements with explicit reasons.</li>
|
||||
<li>Reference standard theorem names precisely.</li>
|
||||
<li>Keep symbols consistent with question labels.</li>
|
||||
<li>Do not rely on appearance; rely on given facts and proven steps.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Deductive geometry traces to Euclid's <em>Elements</em>, where geometric knowledge was organized
|
||||
into axioms, definitions, and logically derived propositions. This method shaped modern proof.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="why-proof" class="book-prose__heading">Why proof matters</h3>
|
||||
<p>
|
||||
Measurement drawings can suggest answers, but proofs guarantee truth for all valid cases. Deductive
|
||||
reasoning removes uncertainty caused by drawing scale or rounding.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Two angles on a straight line are \(3x\) and \(x+40\). Find \(x\).
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(3x+(x+40)=180\Rightarrow 4x=140\Rightarrow x=35\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
In a triangle, angles are \(x,2x,3x\). Find all three angles.
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(x+2x+3x=180\Rightarrow x=30\).</p>
|
||||
<p>Angles: \(30^\circ,60^\circ,90^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
If corresponding angles are equal for a transversal, what can you conclude?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> The two lines are parallel (converse theorem).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Find interior-angle sum of a decagon.
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \((10-2)\times180=1440^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Explain one reason why “it looks equal” is not valid proof.
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Diagram scale can be inaccurate; only theorem-based logic is reliable.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Engineering drawings:</strong> design correctness depends on provable geometric constraints.</li>
|
||||
<li><strong>Architecture:</strong> angle and parallel-line reasoning ensures structural fit and symmetry.</li>
|
||||
<li><strong>Computer-aided design:</strong> geometric rule engines rely on deductive relationships.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Euclid, <em>Elements</em> (classical deductive geometry foundation).</li>
|
||||
<li>Standard high-school geometry syllabi on angle and polygon proofs.</li>
|
||||
</ul>
|
||||
</div>
|
||||
79
content/math/deductive-geometry/core.html
Normal file
79
content/math/deductive-geometry/core.html
Normal file
|
|
@ -0,0 +1,79 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Deductive geometry uses known facts and logical steps to prove new geometric results.
|
||||
A valid proof must justify each statement with a theorem, definition, or given condition.
|
||||
</p>
|
||||
|
||||
<h2 id="reasoning-language" class="book-prose__heading">Reasoning language</h2>
|
||||
<ul>
|
||||
<li><strong>Given</strong>: information provided in the question.</li>
|
||||
<li><strong>To prove</strong>: target statement.</li>
|
||||
<li><strong>Because / therefore</strong>: links each conclusion to a reason.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="line-angle-facts" class="book-prose__heading">Core angle facts</h2>
|
||||
<ul>
|
||||
<li>Angles on a straight line sum to \(180^\circ\).</li>
|
||||
<li>Angles around a point sum to \(360^\circ\).</li>
|
||||
<li>Vertically opposite angles are equal.</li>
|
||||
<li>In parallel lines:
|
||||
<ul>
|
||||
<li>alternate interior angles are equal,</li>
|
||||
<li>corresponding angles are equal,</li>
|
||||
<li>co-interior angles sum to \(180^\circ\).</li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="triangle-facts" class="book-prose__heading">Triangle facts</h2>
|
||||
<ul>
|
||||
<li>Interior angles sum to \(180^\circ\).</li>
|
||||
<li>Exterior angle equals sum of two opposite interior angles.</li>
|
||||
<li>Isosceles triangle: equal sides oppose equal angles.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="polygon-facts" class="book-prose__heading">Polygon facts</h2>
|
||||
<p>
|
||||
Sum of interior angles of \(n\)-gon:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
(n-2)\times180^\circ.
|
||||
\]</p>
|
||||
<p>
|
||||
Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\).
|
||||
</p>
|
||||
|
||||
<h2 id="proof-structure" class="book-prose__heading">Proof structure</h2>
|
||||
<ol>
|
||||
<li>Write what is given and what must be shown.</li>
|
||||
<li>Mark known equal angles/lengths on diagram.</li>
|
||||
<li>Use relevant theorems in short logical chain.</li>
|
||||
<li>End with clear conclusion sentence.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="example-proof" class="book-prose__heading">Example proof pattern</h2>
|
||||
<p>
|
||||
If \(AB\parallel CD\) and line \(AD\) intersects them, prove one pair of alternate interior angles
|
||||
is equal.
|
||||
</p>
|
||||
<p>
|
||||
Since \(AB\parallel CD\), alternate interior angles are equal (parallel-line theorem). Therefore
|
||||
required pair is equal.
|
||||
</p>
|
||||
|
||||
<h2 id="common-errors" class="book-prose__heading">Common proof errors</h2>
|
||||
<ul>
|
||||
<li>Stating result without reason.</li>
|
||||
<li>Using theorem conditions not satisfied by diagram.</li>
|
||||
<li>Assuming diagram scale implies equality.</li>
|
||||
<li>Skipping steps between statements.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Use short statements with explicit reasons.</li>
|
||||
<li>Reference standard theorem names precisely.</li>
|
||||
<li>Keep symbols consistent with question labels.</li>
|
||||
<li>Do not rely on appearance; rely on given facts and proven steps.</li>
|
||||
</ul>
|
||||
</div>
|
||||
70
content/math/deductive-geometry/supplement.html
Normal file
70
content/math/deductive-geometry/supplement.html
Normal file
|
|
@ -0,0 +1,70 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Deductive geometry traces to Euclid's <em>Elements</em>, where geometric knowledge was organized
|
||||
into axioms, definitions, and logically derived propositions. This method shaped modern proof.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="why-proof" class="book-prose__heading">Why proof matters</h3>
|
||||
<p>
|
||||
Measurement drawings can suggest answers, but proofs guarantee truth for all valid cases. Deductive
|
||||
reasoning removes uncertainty caused by drawing scale or rounding.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Two angles on a straight line are \(3x\) and \(x+40\). Find \(x\).
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(3x+(x+40)=180\Rightarrow 4x=140\Rightarrow x=35\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
In a triangle, angles are \(x,2x,3x\). Find all three angles.
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(x+2x+3x=180\Rightarrow x=30\).</p>
|
||||
<p>Angles: \(30^\circ,60^\circ,90^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
If corresponding angles are equal for a transversal, what can you conclude?
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> The two lines are parallel (converse theorem).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Find interior-angle sum of a decagon.
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \((10-2)\times180=1440^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">
|
||||
Explain one reason why “it looks equal” is not valid proof.
|
||||
</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Diagram scale can be inaccurate; only theorem-based logic is reliable.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Engineering drawings:</strong> design correctness depends on provable geometric constraints.</li>
|
||||
<li><strong>Architecture:</strong> angle and parallel-line reasoning ensures structural fit and symmetry.</li>
|
||||
<li><strong>Computer-aided design:</strong> geometric rule engines rely on deductive relationships.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Euclid, <em>Elements</em> (classical deductive geometry foundation).</li>
|
||||
<li>Standard high-school geometry syllabi on angle and polygon proofs.</li>
|
||||
</ul>
|
||||
</div>
|
||||
132
content/math/introduction-trigonometry/_index.md
Normal file
132
content/math/introduction-trigonometry/_index.md
Normal file
|
|
@ -0,0 +1,132 @@
|
|||
+++
|
||||
title = "Introduction to trigonometry"
|
||||
description = "Define sine, cosine, and tangent ratios and solve right-angled triangle problems."
|
||||
template = "chapter.html"
|
||||
weight = 24
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "introduction-trigonometry"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Trigonometry relates angles and side lengths in right triangles. The three core ratios are sine,
|
||||
cosine, and tangent.
|
||||
</p>
|
||||
|
||||
<h2 id="right-triangle-ratios" class="book-prose__heading">Right-triangle ratios</h2>
|
||||
<p>
|
||||
For angle \(\theta\) in a right triangle:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad
|
||||
\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad
|
||||
\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.
|
||||
\]</p>
|
||||
<p>
|
||||
Remember: opposite/adjacent depend on the chosen angle.
|
||||
</p>
|
||||
|
||||
<h2 id="basic-values" class="book-prose__heading">Common exact values</h2>
|
||||
<p>
|
||||
For \(30^\circ,45^\circ,60^\circ\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(\sin30^\circ=\frac12,\ \cos30^\circ=\frac{\sqrt3}{2},\ \tan30^\circ=\frac1{\sqrt3}\)</li>
|
||||
<li>\(\sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2},\ \tan45^\circ=1\)</li>
|
||||
<li>\(\sin60^\circ=\frac{\sqrt3}{2},\ \cos60^\circ=\frac12,\ \tan60^\circ=\sqrt3\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="solve-sides" class="book-prose__heading">Finding missing sides</h2>
|
||||
<p><strong>Example.</strong> Hypotenuse \(=10\), angle \(=37^\circ\). Find opposite side \(x\):</p>
|
||||
<p class="book-formula">\[
|
||||
\sin37^\circ=\frac{x}{10}\Rightarrow x=10\sin37^\circ\approx 6.02.
|
||||
\]</p>
|
||||
|
||||
<h2 id="solve-angles" class="book-prose__heading">Finding missing angles</h2>
|
||||
<p><strong>Example.</strong> Opposite \(=5\), adjacent \(=12\). Find \(\theta\):</p>
|
||||
<p class="book-formula">\[
|
||||
\tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 22.6^\circ.
|
||||
\]</p>
|
||||
|
||||
<h2 id="elevation-depression" class="book-prose__heading">Angles of elevation and depression</h2>
|
||||
<p>
|
||||
Angle of elevation looks upward from horizontal; angle of depression looks downward. Draw a clear
|
||||
diagram and label horizontal lines before forming trig equations.
|
||||
</p>
|
||||
|
||||
<h2 id="pythagoras-link" class="book-prose__heading">Link with Pythagoras</h2>
|
||||
<p>
|
||||
In right triangles, trigonometry and Pythagoras work together: first find one side/angle, then use
|
||||
the other method to complete the triangle.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Choose ratio from known and required sides (SOH-CAH-TOA).</li>
|
||||
<li>Set calculator mode to degrees unless stated otherwise.</li>
|
||||
<li>Round only at final step unless instructed.</li>
|
||||
<li>Include units and context in word-problem answers.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Trigonometry grew from astronomy, navigation, and surveying. Mathematicians in ancient Greece,
|
||||
India, and the Islamic Golden Age developed angle and chord/sine tables for practical measurement.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="similarity-idea" class="book-prose__heading">Why trig ratios depend only on angle</h3>
|
||||
<p>
|
||||
Any two right triangles with the same acute angle are similar, so corresponding side ratios are
|
||||
equal. That is why \(\sin\theta,\cos\theta,\tan\theta\) are functions of \(\theta\) only.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find \(\sin30^\circ\) and \(\cos60^\circ\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Both are \(\frac12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In a right triangle, opposite to \(\theta\) is 9 and hypotenuse is 15. Find \(\sin\theta\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sin\theta=\frac{9}{15}=\frac35\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Adjacent is 7, hypotenuse is 25. Find \(\cos\theta\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\cos\theta=\frac{7}{25}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(\tan\theta=0.75\), find \(\theta\) (to 1 d.p.).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A ladder leans against a wall. Foot is 4 m from wall and ladder length is 6 m. Find angle with ground.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\cos\theta=\frac{4}{6}=\frac23\Rightarrow \theta\approx 48.2^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Surveying:</strong> height and distance estimation without direct measurement.</li>
|
||||
<li><strong>Navigation:</strong> direction and angle-based positioning.</li>
|
||||
<li><strong>Engineering:</strong> force components and incline analysis.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school trigonometry curricula (right-triangle ratios).</li>
|
||||
<li>Geometry and trigonometry texts on SOH-CAH-TOA applications.</li>
|
||||
</ul>
|
||||
</div>
|
||||
61
content/math/introduction-trigonometry/core.html
Normal file
61
content/math/introduction-trigonometry/core.html
Normal file
|
|
@ -0,0 +1,61 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Trigonometry relates angles and side lengths in right triangles. The three core ratios are sine,
|
||||
cosine, and tangent.
|
||||
</p>
|
||||
|
||||
<h2 id="right-triangle-ratios" class="book-prose__heading">Right-triangle ratios</h2>
|
||||
<p>
|
||||
For angle \(\theta\) in a right triangle:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad
|
||||
\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad
|
||||
\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.
|
||||
\]</p>
|
||||
<p>
|
||||
Remember: opposite/adjacent depend on the chosen angle.
|
||||
</p>
|
||||
|
||||
<h2 id="basic-values" class="book-prose__heading">Common exact values</h2>
|
||||
<p>
|
||||
For \(30^\circ,45^\circ,60^\circ\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(\sin30^\circ=\frac12,\ \cos30^\circ=\frac{\sqrt3}{2},\ \tan30^\circ=\frac1{\sqrt3}\)</li>
|
||||
<li>\(\sin45^\circ=\cos45^\circ=\frac{\sqrt2}{2},\ \tan45^\circ=1\)</li>
|
||||
<li>\(\sin60^\circ=\frac{\sqrt3}{2},\ \cos60^\circ=\frac12,\ \tan60^\circ=\sqrt3\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="solve-sides" class="book-prose__heading">Finding missing sides</h2>
|
||||
<p><strong>Example.</strong> Hypotenuse \(=10\), angle \(=37^\circ\). Find opposite side \(x\):</p>
|
||||
<p class="book-formula">\[
|
||||
\sin37^\circ=\frac{x}{10}\Rightarrow x=10\sin37^\circ\approx 6.02.
|
||||
\]</p>
|
||||
|
||||
<h2 id="solve-angles" class="book-prose__heading">Finding missing angles</h2>
|
||||
<p><strong>Example.</strong> Opposite \(=5\), adjacent \(=12\). Find \(\theta\):</p>
|
||||
<p class="book-formula">\[
|
||||
\tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 22.6^\circ.
|
||||
\]</p>
|
||||
|
||||
<h2 id="elevation-depression" class="book-prose__heading">Angles of elevation and depression</h2>
|
||||
<p>
|
||||
Angle of elevation looks upward from horizontal; angle of depression looks downward. Draw a clear
|
||||
diagram and label horizontal lines before forming trig equations.
|
||||
</p>
|
||||
|
||||
<h2 id="pythagoras-link" class="book-prose__heading">Link with Pythagoras</h2>
|
||||
<p>
|
||||
In right triangles, trigonometry and Pythagoras work together: first find one side/angle, then use
|
||||
the other method to complete the triangle.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Choose ratio from known and required sides (SOH-CAH-TOA).</li>
|
||||
<li>Set calculator mode to degrees unless stated otherwise.</li>
|
||||
<li>Round only at final step unless instructed.</li>
|
||||
<li>Include units and context in word-problem answers.</li>
|
||||
</ul>
|
||||
</div>
|
||||
59
content/math/introduction-trigonometry/supplement.html
Normal file
59
content/math/introduction-trigonometry/supplement.html
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Trigonometry grew from astronomy, navigation, and surveying. Mathematicians in ancient Greece,
|
||||
India, and the Islamic Golden Age developed angle and chord/sine tables for practical measurement.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="similarity-idea" class="book-prose__heading">Why trig ratios depend only on angle</h3>
|
||||
<p>
|
||||
Any two right triangles with the same acute angle are similar, so corresponding side ratios are
|
||||
equal. That is why \(\sin\theta,\cos\theta,\tan\theta\) are functions of \(\theta\) only.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find \(\sin30^\circ\) and \(\cos60^\circ\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Both are \(\frac12\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In a right triangle, opposite to \(\theta\) is 9 and hypotenuse is 15. Find \(\sin\theta\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sin\theta=\frac{9}{15}=\frac35\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Adjacent is 7, hypotenuse is 25. Find \(\cos\theta\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\cos\theta=\frac{7}{25}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(\tan\theta=0.75\), find \(\theta\) (to 1 d.p.).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A ladder leans against a wall. Foot is 4 m from wall and ladder length is 6 m. Find angle with ground.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\cos\theta=\frac{4}{6}=\frac23\Rightarrow \theta\approx 48.2^\circ\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Surveying:</strong> height and distance estimation without direct measurement.</li>
|
||||
<li><strong>Navigation:</strong> direction and angle-based positioning.</li>
|
||||
<li><strong>Engineering:</strong> force components and incline analysis.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school trigonometry curricula (right-triangle ratios).</li>
|
||||
<li>Geometry and trigonometry texts on SOH-CAH-TOA applications.</li>
|
||||
</ul>
|
||||
</div>
|
||||
119
content/math/loci/_index.md
Normal file
119
content/math/loci/_index.md
Normal file
|
|
@ -0,0 +1,119 @@
|
|||
+++
|
||||
title = "Loci"
|
||||
description = "Describe and construct loci in the plane, linking geometric constraints to algebraic equations."
|
||||
template = "chapter.html"
|
||||
weight = 29
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "loci"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
A <strong>locus</strong> is the set of all points that satisfy a given condition. Loci connect
|
||||
geometric constructions with equations and constraints.
|
||||
</p>
|
||||
|
||||
<h2 id="basic-loci" class="book-prose__heading">Basic loci patterns</h2>
|
||||
<ul>
|
||||
<li>Fixed distance \(r\) from point \(A\): a circle center \(A\), radius \(r\).</li>
|
||||
<li>Fixed distance from line \(l\): two parallel lines on each side of \(l\).</li>
|
||||
<li>Equidistant from points \(A,B\): perpendicular bisector of \(AB\).</li>
|
||||
<li>Equidistant from two intersecting lines: angle bisectors.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="constructing-loci" class="book-prose__heading">Constructing loci</h2>
|
||||
<ol>
|
||||
<li>Translate condition into known locus type.</li>
|
||||
<li>Construct each required locus accurately.</li>
|
||||
<li>Find intersection of loci for final solution points/region.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="inequality-regions" class="book-prose__heading">Loci and inequalities</h2>
|
||||
<p>
|
||||
Conditions like “distance at most 3 cm from line” describe a region, not a single line.
|
||||
</p>
|
||||
<p>
|
||||
Boundary inclusion matters:
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(\le\): boundary included (solid)</li>
|
||||
<li>\(<\): boundary excluded</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="equation-links" class="book-prose__heading">Algebra links</h2>
|
||||
<p>
|
||||
Coordinate forms:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Circle: \((x-h)^2+(y-k)^2=r^2\)</li>
|
||||
<li>Perpendicular bisector from equal-distance condition \(PA=PB\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="real-world-models" class="book-prose__heading">Modeling with loci</h2>
|
||||
<p>
|
||||
Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Underline keywords: equidistant, fixed distance, within, at least.</li>
|
||||
<li>Draw boundaries first, then shade valid region.</li>
|
||||
<li>State whether endpoints/boundaries are included.</li>
|
||||
<li>Use intersections of loci to justify final points.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Locus ideas emerged from classical compass-and-straightedge geometry and later became a core part
|
||||
of analytic geometry and optimization problems.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Describe the locus of points 5 cm from point \(A\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> A circle center \(A\), radius 5 cm.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">What is locus of points equidistant from \(A\) and \(B\)?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Perpendicular bisector of segment \(AB\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Points within 2 m of line \(l\): describe region.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Strip between two parallels 2 m from \(l\), including boundaries if “within or equal to”.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Points equidistant from intersecting lines \(p,q\): what locus?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> The two angle bisectors of lines \(p,q\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why can a locus answer be a region instead of a curve?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Inequality conditions describe many points satisfying a distance range.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Cell/Wi-Fi planning:</strong> coverage zones around transmitters.</li>
|
||||
<li><strong>Urban design:</strong> buffer distances from roads or facilities.</li>
|
||||
<li><strong>Robotics:</strong> safe navigation boundaries and target zones.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry content on loci and construction.</li>
|
||||
<li>Analytic geometry materials linking loci to equations.</li>
|
||||
</ul>
|
||||
</div>
|
||||
55
content/math/loci/core.html
Normal file
55
content/math/loci/core.html
Normal file
|
|
@ -0,0 +1,55 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
A <strong>locus</strong> is the set of all points that satisfy a given condition. Loci connect
|
||||
geometric constructions with equations and constraints.
|
||||
</p>
|
||||
|
||||
<h2 id="basic-loci" class="book-prose__heading">Basic loci patterns</h2>
|
||||
<ul>
|
||||
<li>Fixed distance \(r\) from point \(A\): a circle center \(A\), radius \(r\).</li>
|
||||
<li>Fixed distance from line \(l\): two parallel lines on each side of \(l\).</li>
|
||||
<li>Equidistant from points \(A,B\): perpendicular bisector of \(AB\).</li>
|
||||
<li>Equidistant from two intersecting lines: angle bisectors.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="constructing-loci" class="book-prose__heading">Constructing loci</h2>
|
||||
<ol>
|
||||
<li>Translate condition into known locus type.</li>
|
||||
<li>Construct each required locus accurately.</li>
|
||||
<li>Find intersection of loci for final solution points/region.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="inequality-regions" class="book-prose__heading">Loci and inequalities</h2>
|
||||
<p>
|
||||
Conditions like “distance at most 3 cm from line” describe a region, not a single line.
|
||||
</p>
|
||||
<p>
|
||||
Boundary inclusion matters:
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(\le\): boundary included (solid)</li>
|
||||
<li>\(<\): boundary excluded</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="equation-links" class="book-prose__heading">Algebra links</h2>
|
||||
<p>
|
||||
Coordinate forms:
|
||||
</p>
|
||||
<ul>
|
||||
<li>Circle: \((x-h)^2+(y-k)^2=r^2\)</li>
|
||||
<li>Perpendicular bisector from equal-distance condition \(PA=PB\)</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="real-world-models" class="book-prose__heading">Modeling with loci</h2>
|
||||
<p>
|
||||
Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Underline keywords: equidistant, fixed distance, within, at least.</li>
|
||||
<li>Draw boundaries first, then shade valid region.</li>
|
||||
<li>State whether endpoints/boundaries are included.</li>
|
||||
<li>Use intersections of loci to justify final points.</li>
|
||||
</ul>
|
||||
</div>
|
||||
52
content/math/loci/supplement.html
Normal file
52
content/math/loci/supplement.html
Normal file
|
|
@ -0,0 +1,52 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Locus ideas emerged from classical compass-and-straightedge geometry and later became a core part
|
||||
of analytic geometry and optimization problems.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Describe the locus of points 5 cm from point \(A\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> A circle center \(A\), radius 5 cm.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">What is locus of points equidistant from \(A\) and \(B\)?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Perpendicular bisector of segment \(AB\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Points within 2 m of line \(l\): describe region.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Strip between two parallels 2 m from \(l\), including boundaries if “within or equal to”.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Points equidistant from intersecting lines \(p,q\): what locus?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> The two angle bisectors of lines \(p,q\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why can a locus answer be a region instead of a curve?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Inequality conditions describe many points satisfying a distance range.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Cell/Wi-Fi planning:</strong> coverage zones around transmitters.</li>
|
||||
<li><strong>Urban design:</strong> buffer distances from roads or facilities.</li>
|
||||
<li><strong>Robotics:</strong> safe navigation boundaries and target zones.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry content on loci and construction.</li>
|
||||
<li>Analytic geometry materials linking loci to equations.</li>
|
||||
</ul>
|
||||
</div>
|
||||
84
content/math/measures-dispersion/_index.md
Normal file
84
content/math/measures-dispersion/_index.md
Normal file
|
|
@ -0,0 +1,84 @@
|
|||
+++
|
||||
title = "Measures of dispersion"
|
||||
description = "Interpret range, interquartile range, variance, and standard deviation for comparing data sets."
|
||||
template = "chapter.html"
|
||||
weight = 31
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "measures-dispersion"
|
||||
strand = "Data Handling"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Dispersion describes how spread out data is. Two datasets can have the same average but very
|
||||
different variability.
|
||||
</p>
|
||||
|
||||
<h2 id="range-iqr" class="book-prose__heading">Range and interquartile range</h2>
|
||||
<p class="book-formula">\[
|
||||
\text{Range}=\text{max}-\text{min},\qquad
|
||||
\text{IQR}=Q_3-Q_1.
|
||||
\]</p>
|
||||
<p>
|
||||
IQR is less affected by outliers than range.
|
||||
</p>
|
||||
|
||||
<h2 id="variance-sd" class="book-prose__heading">Variance and standard deviation</h2>
|
||||
<p>
|
||||
Variance measures average squared distance from mean.
|
||||
Standard deviation is square root of variance and has same unit as data.
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}}
|
||||
\]</p>
|
||||
|
||||
<h2 id="comparing-data" class="book-prose__heading">Comparing datasets</h2>
|
||||
<ul>
|
||||
<li>Smaller spread statistics \(\Rightarrow\) more consistent data.</li>
|
||||
<li>Use median + IQR for skewed data.</li>
|
||||
<li>Use mean + standard deviation for roughly symmetric data.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="boxplot-reading" class="book-prose__heading">Box plot interpretation</h2>
|
||||
<p>
|
||||
Box shows middle 50% (\(Q_1\) to \(Q_3\)); median line marks center.
|
||||
Longer whisker/box indicates larger spread.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>State both center and spread when comparing sets.</li>
|
||||
<li>Do not confuse variance with standard deviation.</li>
|
||||
<li>Check if grouped-data values are estimates.</li>
|
||||
<li>Interpret spread in context (consistency/risk).</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find range of 12, 9, 15, 20, 13.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(20-9=11\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(Q_1=18\), \(Q_3=31\), find IQR.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> 13.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Dataset A and B have same mean. A has SD 2, B has SD 8. Which is more consistent?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> A (smaller SD).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why is IQR preferred over range when outliers exist?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> IQR focuses on middle 50% and is less distorted by extreme values.</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
45
content/math/measures-dispersion/core.html
Normal file
45
content/math/measures-dispersion/core.html
Normal file
|
|
@ -0,0 +1,45 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Dispersion describes how spread out data is. Two datasets can have the same average but very
|
||||
different variability.
|
||||
</p>
|
||||
|
||||
<h2 id="range-iqr" class="book-prose__heading">Range and interquartile range</h2>
|
||||
<p class="book-formula">\[
|
||||
\text{Range}=\text{max}-\text{min},\qquad
|
||||
\text{IQR}=Q_3-Q_1.
|
||||
\]</p>
|
||||
<p>
|
||||
IQR is less affected by outliers than range.
|
||||
</p>
|
||||
|
||||
<h2 id="variance-sd" class="book-prose__heading">Variance and standard deviation</h2>
|
||||
<p>
|
||||
Variance measures average squared distance from mean.
|
||||
Standard deviation is square root of variance and has same unit as data.
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}}
|
||||
\]</p>
|
||||
|
||||
<h2 id="comparing-data" class="book-prose__heading">Comparing datasets</h2>
|
||||
<ul>
|
||||
<li>Smaller spread statistics \(\Rightarrow\) more consistent data.</li>
|
||||
<li>Use median + IQR for skewed data.</li>
|
||||
<li>Use mean + standard deviation for roughly symmetric data.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="boxplot-reading" class="book-prose__heading">Box plot interpretation</h2>
|
||||
<p>
|
||||
Box shows middle 50% (\(Q_1\) to \(Q_3\)); median line marks center.
|
||||
Longer whisker/box indicates larger spread.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>State both center and spread when comparing sets.</li>
|
||||
<li>Do not confuse variance with standard deviation.</li>
|
||||
<li>Check if grouped-data values are estimates.</li>
|
||||
<li>Interpret spread in context (consistency/risk).</li>
|
||||
</ul>
|
||||
</div>
|
||||
27
content/math/measures-dispersion/supplement.html
Normal file
27
content/math/measures-dispersion/supplement.html
Normal file
|
|
@ -0,0 +1,27 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find range of 12, 9, 15, 20, 13.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(20-9=11\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(Q_1=18\), \(Q_3=31\), find IQR.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> 13.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Dataset A and B have same mean. A has SD 2, B has SD 8. Which is more consistent?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> A (smaller SD).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why is IQR preferred over range when outliers exist?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> IQR focuses on middle 50% and is less distorted by extreme values.</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
148
content/math/mensuration-3d/_index.md
Normal file
148
content/math/mensuration-3d/_index.md
Normal file
|
|
@ -0,0 +1,148 @@
|
|||
+++
|
||||
title = "Mensuration in 3D"
|
||||
description = "Find volumes and surface areas of prisms, cylinders, pyramids, cones, and spheres."
|
||||
template = "chapter.html"
|
||||
weight = 25
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "mensuration-3d"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Mensuration in 3D focuses on volume (space inside solids) and surface area (total outer area).
|
||||
</p>
|
||||
|
||||
<h2 id="units" class="book-prose__heading">Units and interpretation</h2>
|
||||
<ul>
|
||||
<li>Length: cm, m</li>
|
||||
<li>Area: cm\(^2\), m\(^2\)</li>
|
||||
<li>Volume: cm\(^3\), m\(^3\)</li>
|
||||
</ul>
|
||||
<p>
|
||||
Always keep units consistent before substituting values.
|
||||
</p>
|
||||
|
||||
<h2 id="prisms-cylinders" class="book-prose__heading">Prisms and cylinders</h2>
|
||||
<p>
|
||||
For any prism/cylinder:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\text{Volume}=\text{base area}\times\text{height}.
|
||||
\]</p>
|
||||
<p>
|
||||
Cylinder formulas:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
V=\pi r^2h,\qquad
|
||||
\text{TSA}=2\pi r^2+2\pi rh.
|
||||
\]</p>
|
||||
|
||||
<h2 id="pyramids-cones" class="book-prose__heading">Pyramids and cones</h2>
|
||||
<p class="book-formula">\[
|
||||
V_{\text{pyramid}}=\frac13(\text{base area})h,\qquad
|
||||
V_{\text{cone}}=\frac13\pi r^2h.
|
||||
\]</p>
|
||||
<p>
|
||||
Cone curved surface area:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\text{CSA}_{\text{cone}}=\pi rl
|
||||
\]</p>
|
||||
<p>
|
||||
where \(l\) is slant height. Total surface area adds base area \(\pi r^2\).
|
||||
</p>
|
||||
|
||||
<h2 id="spheres" class="book-prose__heading">Spheres</h2>
|
||||
<p class="book-formula">\[
|
||||
V=\frac43\pi r^3,\qquad
|
||||
\text{Surface area}=4\pi r^2.
|
||||
\]</p>
|
||||
|
||||
<h2 id="composite-solids" class="book-prose__heading">Composite solids</h2>
|
||||
<p>
|
||||
Split complex shapes into known solids, then add/subtract volumes or areas appropriately.
|
||||
</p>
|
||||
<ul>
|
||||
<li>Add volumes when parts are joined.</li>
|
||||
<li>Subtract volumes for holes/cut-outs.</li>
|
||||
<li>For surface area, remove hidden contact faces.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="density-context" class="book-prose__heading">Volume in context</h2>
|
||||
<p>
|
||||
Common contexts include capacity, material usage, and mass:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\text{mass}=\text{density}\times\text{volume}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Sketch and label all dimensions before formula use.</li>
|
||||
<li>Choose exact \(\pi\) form first; round at final step if required.</li>
|
||||
<li>Convert units early (e.g., cm to m).</li>
|
||||
<li>For composite shapes, explain add/subtract logic briefly.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Mensuration developed from practical needs in architecture, agriculture, and trade. Ancient
|
||||
mathematicians studied area and volume formulas, with later refinement by Archimedes and others.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="why-one-third" class="book-prose__heading">Why pyramid and cone have one-third factor</h3>
|
||||
<p>
|
||||
A pyramid (or cone) with same base area and height as a prism (or cylinder) has one-third the
|
||||
volume. This can be shown by dissection arguments or calculus in advanced courses.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find volume of a cylinder with \(r=3\) cm, \(h=10\) cm.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(V=\pi r^2h=90\pi\) cm\(^3\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find total surface area of a closed cylinder with \(r=2\) cm, \(h=7\) cm.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(2\pi r^2+2\pi rh=8\pi+28\pi=36\pi\) cm\(^2\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find volume of a cone with \(r=6\) cm, \(h=9\) cm.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac13\pi r^2h=\frac13\pi(36)(9)=108\pi\) cm\(^3\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Sphere has radius 5 cm. Find its volume.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(V=\frac43\pi(5^3)=\frac{500}{3}\pi\) cm\(^3\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A block is \(4\times3\times2\) m. Density is \(2500\) kg/m\(^3\). Find mass.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Volume \(=24\) m\(^3\). Mass \(=2500\times24=60000\) kg.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Construction:</strong> concrete volume and paint area estimation.</li>
|
||||
<li><strong>Packaging:</strong> minimizing material for fixed volume containers.</li>
|
||||
<li><strong>Engineering:</strong> tank capacity and material mass calculations.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school mensuration units on 3D solids.</li>
|
||||
<li>Geometry texts on prism, cylinder, cone, pyramid, and sphere formulas.</li>
|
||||
</ul>
|
||||
</div>
|
||||
77
content/math/mensuration-3d/core.html
Normal file
77
content/math/mensuration-3d/core.html
Normal file
|
|
@ -0,0 +1,77 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Mensuration in 3D focuses on volume (space inside solids) and surface area (total outer area).
|
||||
</p>
|
||||
|
||||
<h2 id="units" class="book-prose__heading">Units and interpretation</h2>
|
||||
<ul>
|
||||
<li>Length: cm, m</li>
|
||||
<li>Area: cm\(^2\), m\(^2\)</li>
|
||||
<li>Volume: cm\(^3\), m\(^3\)</li>
|
||||
</ul>
|
||||
<p>
|
||||
Always keep units consistent before substituting values.
|
||||
</p>
|
||||
|
||||
<h2 id="prisms-cylinders" class="book-prose__heading">Prisms and cylinders</h2>
|
||||
<p>
|
||||
For any prism/cylinder:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\text{Volume}=\text{base area}\times\text{height}.
|
||||
\]</p>
|
||||
<p>
|
||||
Cylinder formulas:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
V=\pi r^2h,\qquad
|
||||
\text{TSA}=2\pi r^2+2\pi rh.
|
||||
\]</p>
|
||||
|
||||
<h2 id="pyramids-cones" class="book-prose__heading">Pyramids and cones</h2>
|
||||
<p class="book-formula">\[
|
||||
V_{\text{pyramid}}=\frac13(\text{base area})h,\qquad
|
||||
V_{\text{cone}}=\frac13\pi r^2h.
|
||||
\]</p>
|
||||
<p>
|
||||
Cone curved surface area:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\text{CSA}_{\text{cone}}=\pi rl
|
||||
\]</p>
|
||||
<p>
|
||||
where \(l\) is slant height. Total surface area adds base area \(\pi r^2\).
|
||||
</p>
|
||||
|
||||
<h2 id="spheres" class="book-prose__heading">Spheres</h2>
|
||||
<p class="book-formula">\[
|
||||
V=\frac43\pi r^3,\qquad
|
||||
\text{Surface area}=4\pi r^2.
|
||||
\]</p>
|
||||
|
||||
<h2 id="composite-solids" class="book-prose__heading">Composite solids</h2>
|
||||
<p>
|
||||
Split complex shapes into known solids, then add/subtract volumes or areas appropriately.
|
||||
</p>
|
||||
<ul>
|
||||
<li>Add volumes when parts are joined.</li>
|
||||
<li>Subtract volumes for holes/cut-outs.</li>
|
||||
<li>For surface area, remove hidden contact faces.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="density-context" class="book-prose__heading">Volume in context</h2>
|
||||
<p>
|
||||
Common contexts include capacity, material usage, and mass:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
\text{mass}=\text{density}\times\text{volume}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Sketch and label all dimensions before formula use.</li>
|
||||
<li>Choose exact \(\pi\) form first; round at final step if required.</li>
|
||||
<li>Convert units early (e.g., cm to m).</li>
|
||||
<li>For composite shapes, explain add/subtract logic briefly.</li>
|
||||
</ul>
|
||||
</div>
|
||||
59
content/math/mensuration-3d/supplement.html
Normal file
59
content/math/mensuration-3d/supplement.html
Normal file
|
|
@ -0,0 +1,59 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
Mensuration developed from practical needs in architecture, agriculture, and trade. Ancient
|
||||
mathematicians studied area and volume formulas, with later refinement by Archimedes and others.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="why-one-third" class="book-prose__heading">Why pyramid and cone have one-third factor</h3>
|
||||
<p>
|
||||
A pyramid (or cone) with same base area and height as a prism (or cylinder) has one-third the
|
||||
volume. This can be shown by dissection arguments or calculus in advanced courses.
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find volume of a cylinder with \(r=3\) cm, \(h=10\) cm.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(V=\pi r^2h=90\pi\) cm\(^3\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find total surface area of a closed cylinder with \(r=2\) cm, \(h=7\) cm.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(2\pi r^2+2\pi rh=8\pi+28\pi=36\pi\) cm\(^2\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find volume of a cone with \(r=6\) cm, \(h=9\) cm.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac13\pi r^2h=\frac13\pi(36)(9)=108\pi\) cm\(^3\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Sphere has radius 5 cm. Find its volume.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(V=\frac43\pi(5^3)=\frac{500}{3}\pi\) cm\(^3\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A block is \(4\times3\times2\) m. Density is \(2500\) kg/m\(^3\). Find mass.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Volume \(=24\) m\(^3\). Mass \(=2500\times24=60000\) kg.</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Construction:</strong> concrete volume and paint area estimation.</li>
|
||||
<li><strong>Packaging:</strong> minimizing material for fixed volume containers.</li>
|
||||
<li><strong>Engineering:</strong> tank capacity and material mass calculations.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school mensuration units on 3D solids.</li>
|
||||
<li>Geometry texts on prism, cylinder, cone, pyramid, and sphere formulas.</li>
|
||||
</ul>
|
||||
</div>
|
||||
85
content/math/normal-distribution-z-scores/_index.md
Normal file
85
content/math/normal-distribution-z-scores/_index.md
Normal file
|
|
@ -0,0 +1,85 @@
|
|||
+++
|
||||
title = "Normal distribution & z-scores"
|
||||
description = "Model continuous data with the normal curve, standardise with z-scores, and read probabilities from tables."
|
||||
template = "chapter.html"
|
||||
weight = 34
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "normal-distribution-z-scores"
|
||||
strand = "Data Handling"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
The normal distribution is a symmetric bell-shaped model for many real measurements.
|
||||
</p>
|
||||
|
||||
<h2 id="normal-model" class="book-prose__heading">Normal model basics</h2>
|
||||
<p>
|
||||
A normal variable \(X\sim N(\mu,\sigma^2)\) has mean \(\mu\) and standard deviation \(\sigma\).
|
||||
</p>
|
||||
<ul>
|
||||
<li>About 68% of values lie within \(\mu\pm\sigma\).</li>
|
||||
<li>About 95% lie within \(\mu\pm2\sigma\).</li>
|
||||
<li>About 99.7% lie within \(\mu\pm3\sigma\).</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="z-score" class="book-prose__heading">Standardizing with z-score</h2>
|
||||
<p class="book-formula">\[
|
||||
z=\frac{x-\mu}{\sigma}.
|
||||
\]</p>
|
||||
<p>
|
||||
This converts values to the standard normal \(Z\sim N(0,1)\).
|
||||
</p>
|
||||
|
||||
<h2 id="probabilities" class="book-prose__heading">Finding probabilities</h2>
|
||||
<ol>
|
||||
<li>Convert boundary value(s) to z-score(s).</li>
|
||||
<li>Use normal table/calculator for area.</li>
|
||||
<li>Apply complement or subtraction if needed.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="reverse-normal" class="book-prose__heading">Finding values from probabilities</h2>
|
||||
<p>
|
||||
If \(P(X\le x)=p\), find corresponding \(z_p\), then:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
x=\mu+z_p\sigma.
|
||||
\]</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Draw a quick bell curve and shade required region.</li>
|
||||
<li>Label mean, SD marks, and inequality direction.</li>
|
||||
<li>Use continuity correction when question requires discrete-to-normal approximation.</li>
|
||||
<li>Round only at final step.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(\mu=70,\sigma=10\), find z-score of \(x=85\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(z=\frac{85-70}{10}=1.5\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(z=-0.8\), \(\mu=50,\sigma=5\), find \(x\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(x=50+(-0.8)(5)=46\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In a normal model, what does \(P(Z>2)\) represent?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Area to the right of z=2 (upper-tail probability).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why do we standardize before using z-tables?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Tables are for \(N(0,1)\), so we convert \(X\) to \(Z\).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
46
content/math/normal-distribution-z-scores/core.html
Normal file
46
content/math/normal-distribution-z-scores/core.html
Normal file
|
|
@ -0,0 +1,46 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
The normal distribution is a symmetric bell-shaped model for many real measurements.
|
||||
</p>
|
||||
|
||||
<h2 id="normal-model" class="book-prose__heading">Normal model basics</h2>
|
||||
<p>
|
||||
A normal variable \(X\sim N(\mu,\sigma^2)\) has mean \(\mu\) and standard deviation \(\sigma\).
|
||||
</p>
|
||||
<ul>
|
||||
<li>About 68% of values lie within \(\mu\pm\sigma\).</li>
|
||||
<li>About 95% lie within \(\mu\pm2\sigma\).</li>
|
||||
<li>About 99.7% lie within \(\mu\pm3\sigma\).</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="z-score" class="book-prose__heading">Standardizing with z-score</h2>
|
||||
<p class="book-formula">\[
|
||||
z=\frac{x-\mu}{\sigma}.
|
||||
\]</p>
|
||||
<p>
|
||||
This converts values to the standard normal \(Z\sim N(0,1)\).
|
||||
</p>
|
||||
|
||||
<h2 id="probabilities" class="book-prose__heading">Finding probabilities</h2>
|
||||
<ol>
|
||||
<li>Convert boundary value(s) to z-score(s).</li>
|
||||
<li>Use normal table/calculator for area.</li>
|
||||
<li>Apply complement or subtraction if needed.</li>
|
||||
</ol>
|
||||
|
||||
<h2 id="reverse-normal" class="book-prose__heading">Finding values from probabilities</h2>
|
||||
<p>
|
||||
If \(P(X\le x)=p\), find corresponding \(z_p\), then:
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
x=\mu+z_p\sigma.
|
||||
\]</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Draw a quick bell curve and shade required region.</li>
|
||||
<li>Label mean, SD marks, and inequality direction.</li>
|
||||
<li>Use continuity correction when question requires discrete-to-normal approximation.</li>
|
||||
<li>Round only at final step.</li>
|
||||
</ul>
|
||||
</div>
|
||||
27
content/math/normal-distribution-z-scores/supplement.html
Normal file
27
content/math/normal-distribution-z-scores/supplement.html
Normal file
|
|
@ -0,0 +1,27 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(\mu=70,\sigma=10\), find z-score of \(x=85\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(z=\frac{85-70}{10}=1.5\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">If \(z=-0.8\), \(\mu=50,\sigma=5\), find \(x\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(x=50+(-0.8)(5)=46\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">In a normal model, what does \(P(Z>2)\) represent?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Area to the right of z=2 (upper-tail probability).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Why do we standardize before using z-tables?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Tables are for \(N(0,1)\), so we convert \(X\) to \(Z\).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
90
content/math/permutations-combinations-probability/_index.md
Normal file
90
content/math/permutations-combinations-probability/_index.md
Normal file
|
|
@ -0,0 +1,90 @@
|
|||
+++
|
||||
title = "Permutations, combinations & probability"
|
||||
description = "Count outcomes with permutations and combinations, then compute probabilities in finite sample spaces."
|
||||
template = "chapter.html"
|
||||
weight = 33
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "permutations-combinations-probability"
|
||||
strand = "Data Handling"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Counting methods make probability efficient when listing all outcomes is impractical.
|
||||
</p>
|
||||
|
||||
<h2 id="factorial" class="book-prose__heading">Factorial</h2>
|
||||
<p class="book-formula">\[
|
||||
n!=n(n-1)(n-2)\cdots 1,\quad 0!=1.
|
||||
\]</p>
|
||||
|
||||
<h2 id="permutations" class="book-prose__heading">Permutations (order matters)</h2>
|
||||
<p class="book-formula">\[
|
||||
{}_nP_r=\frac{n!}{(n-r)!}.
|
||||
\]</p>
|
||||
<p>
|
||||
Use for arrangements where position/order is important.
|
||||
</p>
|
||||
|
||||
<h2 id="combinations" class="book-prose__heading">Combinations (order not important)</h2>
|
||||
<p class="book-formula">\[
|
||||
{}_nC_r=\frac{n!}{r!(n-r)!}.
|
||||
\]</p>
|
||||
<p>
|
||||
Use for selections/groups.
|
||||
</p>
|
||||
|
||||
<h2 id="probability-counting" class="book-prose__heading">Probability with counting</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E)=\frac{\text{count of favorable outcomes}}{\text{count of all outcomes}}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="with-without-replacement" class="book-prose__heading">With/without replacement</h2>
|
||||
<ul>
|
||||
<li><strong>With replacement:</strong> probabilities stay same each draw.</li>
|
||||
<li><strong>Without replacement:</strong> probabilities change each draw.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>First decide: arrangement or selection?</li>
|
||||
<li>Check if repetitions are allowed.</li>
|
||||
<li>Use structured counting (tree/cases) for restrictions.</li>
|
||||
<li>Simplify probability fractions at end.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Compute \({}_7P_3\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(7\times6\times5=210\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Compute \({}_8C_2\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{8\cdot7}{2}=28\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">How many 3-letter arrangements from A,B,C,D (no repeats)?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \({}_4P_3=24\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">How many ways to choose 3 students from 10?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \({}_{10}C_3=120\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A 5-card hand from 52 cards: probability all hearts.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{{}_{13}C_5}{{}_{52}C_5}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
45
content/math/permutations-combinations-probability/core.html
Normal file
45
content/math/permutations-combinations-probability/core.html
Normal file
|
|
@ -0,0 +1,45 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Counting methods make probability efficient when listing all outcomes is impractical.
|
||||
</p>
|
||||
|
||||
<h2 id="factorial" class="book-prose__heading">Factorial</h2>
|
||||
<p class="book-formula">\[
|
||||
n!=n(n-1)(n-2)\cdots 1,\quad 0!=1.
|
||||
\]</p>
|
||||
|
||||
<h2 id="permutations" class="book-prose__heading">Permutations (order matters)</h2>
|
||||
<p class="book-formula">\[
|
||||
{}_nP_r=\frac{n!}{(n-r)!}.
|
||||
\]</p>
|
||||
<p>
|
||||
Use for arrangements where position/order is important.
|
||||
</p>
|
||||
|
||||
<h2 id="combinations" class="book-prose__heading">Combinations (order not important)</h2>
|
||||
<p class="book-formula">\[
|
||||
{}_nC_r=\frac{n!}{r!(n-r)!}.
|
||||
\]</p>
|
||||
<p>
|
||||
Use for selections/groups.
|
||||
</p>
|
||||
|
||||
<h2 id="probability-counting" class="book-prose__heading">Probability with counting</h2>
|
||||
<p class="book-formula">\[
|
||||
P(E)=\frac{\text{count of favorable outcomes}}{\text{count of all outcomes}}.
|
||||
\]</p>
|
||||
|
||||
<h2 id="with-without-replacement" class="book-prose__heading">With/without replacement</h2>
|
||||
<ul>
|
||||
<li><strong>With replacement:</strong> probabilities stay same each draw.</li>
|
||||
<li><strong>Without replacement:</strong> probabilities change each draw.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>First decide: arrangement or selection?</li>
|
||||
<li>Check if repetitions are allowed.</li>
|
||||
<li>Use structured counting (tree/cases) for restrictions.</li>
|
||||
<li>Simplify probability fractions at end.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
|
@ -0,0 +1,33 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Compute \({}_7P_3\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(7\times6\times5=210\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Compute \({}_8C_2\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{8\cdot7}{2}=28\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">How many 3-letter arrangements from A,B,C,D (no repeats)?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \({}_4P_3=24\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">How many ways to choose 3 students from 10?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \({}_{10}C_3=120\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A 5-card hand from 52 cards: probability all hearts.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{{}_{13}C_5}{{}_{52}C_5}\).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
144
content/math/pythagoras-theorem/_index.md
Normal file
144
content/math/pythagoras-theorem/_index.md
Normal file
|
|
@ -0,0 +1,144 @@
|
|||
+++
|
||||
title = "Pythagoras' theorem"
|
||||
description = "Use Pythagoras' theorem and its converse in 2D problems and coordinate settings."
|
||||
template = "chapter.html"
|
||||
weight = 23
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "pythagoras-theorem"
|
||||
strand = "Measures, Shape & Space"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Pythagoras' theorem links the side lengths of a right-angled triangle. It is one of the most
|
||||
useful tools in geometry and coordinate problems.
|
||||
</p>
|
||||
|
||||
<h2 id="theorem" class="book-prose__heading">Pythagoras' theorem</h2>
|
||||
<p>
|
||||
In a right triangle with legs \(a,b\) and hypotenuse \(c\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
a^2+b^2=c^2.
|
||||
\]</p>
|
||||
<p>
|
||||
The hypotenuse is always the side opposite the right angle and is the longest side.
|
||||
</p>
|
||||
|
||||
<h2 id="finding-missing-side" class="book-prose__heading">Finding a missing side</h2>
|
||||
<p><strong>Example 1.</strong> If \(a=6\), \(b=8\), then:</p>
|
||||
<p class="book-formula">\[
|
||||
c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10.
|
||||
\]</p>
|
||||
<p><strong>Example 2.</strong> If \(c=13\), \(a=5\), then:</p>
|
||||
<p class="book-formula">\[
|
||||
b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12.
|
||||
\]</p>
|
||||
|
||||
<h2 id="converse" class="book-prose__heading">Converse of Pythagoras</h2>
|
||||
<p>
|
||||
If side lengths satisfy \(a^2+b^2=c^2\), the triangle is right-angled.
|
||||
</p>
|
||||
<p>
|
||||
Example: \(7,24,25\) gives \(7^2+24^2=49+576=625=25^2\), so right-angled.
|
||||
</p>
|
||||
|
||||
<h2 id="classify-triangle" class="book-prose__heading">Classifying triangle type</h2>
|
||||
<p>
|
||||
For largest side \(c\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(a^2+b^2=c^2\): right triangle.</li>
|
||||
<li>\(a^2+b^2>c^2\): acute triangle.</li>
|
||||
<li>\(a^2+b^2<c^2\): obtuse triangle.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="coordinate-use" class="book-prose__heading">Using in coordinate geometry</h2>
|
||||
<p>
|
||||
Distance between \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.
|
||||
\]</p>
|
||||
<p>
|
||||
This is Pythagoras applied to horizontal and vertical changes.
|
||||
</p>
|
||||
|
||||
<h2 id="word-problems" class="book-prose__heading">Typical applications</h2>
|
||||
<ul>
|
||||
<li>Diagonal of rectangles/squares.</li>
|
||||
<li>Ladders against walls.</li>
|
||||
<li>Shortest straight-line distances on maps/grids.</li>
|
||||
<li>Checking whether corners are right angles in construction.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Identify the right angle first.</li>
|
||||
<li>Label hypotenuse clearly before substitution.</li>
|
||||
<li>Keep exact surd form unless decimal is requested.</li>
|
||||
<li>Check answer size: hypotenuse must be longest.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
The theorem is named after Pythagoras, but knowledge of right-triangle relationships existed in
|
||||
earlier Babylonian and Indian mathematics. It became central in Euclidean geometry and later in
|
||||
coordinate methods.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="area-idea" class="book-prose__heading">Area-based proof idea</h3>
|
||||
<p>
|
||||
A classic proof rearranges four identical right triangles inside a square in two different ways.
|
||||
Equating uncovered areas leads to \(a^2+b^2=c^2\).
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find hypotenuse if legs are 9 and 12.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(c=\sqrt{9^2+12^2}=\sqrt{225}=15\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A right triangle has hypotenuse 17 and one leg 8. Find the other leg.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{17^2-8^2}=\sqrt{225}=15\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Are sides 10, 24, 26 right-angled?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Yes, \(10^2+24^2=100+576=676=26^2\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Distance between \((-2,1)\) and \((4,9)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{6^2+8^2}=10\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Rectangle has side lengths 7 and 24. Find diagonal.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{7^2+24^2}=25\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Architecture:</strong> checking right angles and structural diagonals.</li>
|
||||
<li><strong>Navigation:</strong> shortest displacement calculations on grid maps.</li>
|
||||
<li><strong>Computer graphics:</strong> pixel distance and vector magnitude.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry curricula on Pythagoras and its converse.</li>
|
||||
<li>Classical Euclidean geometry references.</li>
|
||||
</ul>
|
||||
</div>
|
||||
72
content/math/pythagoras-theorem/core.html
Normal file
72
content/math/pythagoras-theorem/core.html
Normal file
|
|
@ -0,0 +1,72 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Pythagoras' theorem links the side lengths of a right-angled triangle. It is one of the most
|
||||
useful tools in geometry and coordinate problems.
|
||||
</p>
|
||||
|
||||
<h2 id="theorem" class="book-prose__heading">Pythagoras' theorem</h2>
|
||||
<p>
|
||||
In a right triangle with legs \(a,b\) and hypotenuse \(c\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
a^2+b^2=c^2.
|
||||
\]</p>
|
||||
<p>
|
||||
The hypotenuse is always the side opposite the right angle and is the longest side.
|
||||
</p>
|
||||
|
||||
<h2 id="finding-missing-side" class="book-prose__heading">Finding a missing side</h2>
|
||||
<p><strong>Example 1.</strong> If \(a=6\), \(b=8\), then:</p>
|
||||
<p class="book-formula">\[
|
||||
c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10.
|
||||
\]</p>
|
||||
<p><strong>Example 2.</strong> If \(c=13\), \(a=5\), then:</p>
|
||||
<p class="book-formula">\[
|
||||
b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12.
|
||||
\]</p>
|
||||
|
||||
<h2 id="converse" class="book-prose__heading">Converse of Pythagoras</h2>
|
||||
<p>
|
||||
If side lengths satisfy \(a^2+b^2=c^2\), the triangle is right-angled.
|
||||
</p>
|
||||
<p>
|
||||
Example: \(7,24,25\) gives \(7^2+24^2=49+576=625=25^2\), so right-angled.
|
||||
</p>
|
||||
|
||||
<h2 id="classify-triangle" class="book-prose__heading">Classifying triangle type</h2>
|
||||
<p>
|
||||
For largest side \(c\):
|
||||
</p>
|
||||
<ul>
|
||||
<li>\(a^2+b^2=c^2\): right triangle.</li>
|
||||
<li>\(a^2+b^2>c^2\): acute triangle.</li>
|
||||
<li>\(a^2+b^2<c^2\): obtuse triangle.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="coordinate-use" class="book-prose__heading">Using in coordinate geometry</h2>
|
||||
<p>
|
||||
Distance between \(A(x_1,y_1)\) and \(B(x_2,y_2)\):
|
||||
</p>
|
||||
<p class="book-formula">\[
|
||||
AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.
|
||||
\]</p>
|
||||
<p>
|
||||
This is Pythagoras applied to horizontal and vertical changes.
|
||||
</p>
|
||||
|
||||
<h2 id="word-problems" class="book-prose__heading">Typical applications</h2>
|
||||
<ul>
|
||||
<li>Diagonal of rectangles/squares.</li>
|
||||
<li>Ladders against walls.</li>
|
||||
<li>Shortest straight-line distances on maps/grids.</li>
|
||||
<li>Checking whether corners are right angles in construction.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Identify the right angle first.</li>
|
||||
<li>Label hypotenuse clearly before substitution.</li>
|
||||
<li>Keep exact surd form unless decimal is requested.</li>
|
||||
<li>Check answer size: hypotenuse must be longest.</li>
|
||||
</ul>
|
||||
</div>
|
||||
60
content/math/pythagoras-theorem/supplement.html
Normal file
60
content/math/pythagoras-theorem/supplement.html
Normal file
|
|
@ -0,0 +1,60 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="history" class="book-prose__heading">History</h2>
|
||||
<p>
|
||||
The theorem is named after Pythagoras, but knowledge of right-triangle relationships existed in
|
||||
earlier Babylonian and Indian mathematics. It became central in Euclidean geometry and later in
|
||||
coordinate methods.
|
||||
</p>
|
||||
|
||||
<h2 id="derivation" class="book-prose__heading">Derivation and reasoning</h2>
|
||||
<h3 id="area-idea" class="book-prose__heading">Area-based proof idea</h3>
|
||||
<p>
|
||||
A classic proof rearranges four identical right triangles inside a square in two different ways.
|
||||
Equating uncovered areas leads to \(a^2+b^2=c^2\).
|
||||
</p>
|
||||
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find hypotenuse if legs are 9 and 12.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(c=\sqrt{9^2+12^2}=\sqrt{225}=15\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A right triangle has hypotenuse 17 and one leg 8. Find the other leg.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{17^2-8^2}=\sqrt{225}=15\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Are sides 10, 24, 26 right-angled?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Yes, \(10^2+24^2=100+576=676=26^2\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Distance between \((-2,1)\) and \((4,9)\).</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{6^2+8^2}=10\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Rectangle has side lengths 7 and 24. Find diagonal.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\sqrt{7^2+24^2}=25\).</p>
|
||||
</div>
|
||||
</details>
|
||||
|
||||
<h2 id="applications" class="book-prose__heading">Applications</h2>
|
||||
<ul>
|
||||
<li><strong>Architecture:</strong> checking right angles and structural diagonals.</li>
|
||||
<li><strong>Navigation:</strong> shortest displacement calculations on grid maps.</li>
|
||||
<li><strong>Computer graphics:</strong> pixel distance and vector magnitude.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="references" class="book-prose__heading">References</h2>
|
||||
<ul>
|
||||
<li>Standard high-school geometry curricula on Pythagoras and its converse.</li>
|
||||
<li>Classical Euclidean geometry references.</li>
|
||||
</ul>
|
||||
</div>
|
||||
97
content/math/statistical-charts-averages/_index.md
Normal file
97
content/math/statistical-charts-averages/_index.md
Normal file
|
|
@ -0,0 +1,97 @@
|
|||
+++
|
||||
title = "Charts & averages"
|
||||
description = "Organise data with tables and charts and calculate mean, median, and mode."
|
||||
template = "chapter.html"
|
||||
weight = 30
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "statistical-charts-averages"
|
||||
strand = "Data Handling"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Data handling starts with organizing data clearly, choosing suitable charts, and summarizing a
|
||||
dataset with averages.
|
||||
</p>
|
||||
|
||||
<h2 id="data-tables" class="book-prose__heading">Frequency tables</h2>
|
||||
<p>
|
||||
A frequency table records each value (or class interval) and how often it appears.
|
||||
For grouped data, use class midpoints for estimation.
|
||||
</p>
|
||||
|
||||
<h2 id="common-charts" class="book-prose__heading">Common charts</h2>
|
||||
<ul>
|
||||
<li><strong>Bar chart:</strong> categorical comparisons.</li>
|
||||
<li><strong>Histogram:</strong> continuous grouped data (touching bars).</li>
|
||||
<li><strong>Line graph:</strong> change over time.</li>
|
||||
<li><strong>Pie chart:</strong> part-to-whole proportions.</li>
|
||||
<li><strong>Box plot:</strong> spread via quartiles and median.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="mean-median-mode" class="book-prose__heading">Mean, median, and mode</h2>
|
||||
<p class="book-formula">\[
|
||||
\text{Mean}=\frac{\text{sum of values}}{\text{number of values}}.
|
||||
\]</p>
|
||||
<p>
|
||||
<strong>Median</strong> is middle value when ordered. <strong>Mode</strong> is most frequent value.
|
||||
</p>
|
||||
|
||||
<h2 id="weighted-mean" class="book-prose__heading">Mean from frequency table</h2>
|
||||
<p class="book-formula">\[
|
||||
\bar{x}=\frac{\sum fx}{\sum f}.
|
||||
\]</p>
|
||||
<p>
|
||||
where \(x\) is value (or midpoint), \(f\) is frequency.
|
||||
</p>
|
||||
|
||||
<h2 id="which-average" class="book-prose__heading">Choosing an average</h2>
|
||||
<ul>
|
||||
<li>Use <strong>mean</strong> when data has no extreme outliers.</li>
|
||||
<li>Use <strong>median</strong> when distribution is skewed.</li>
|
||||
<li>Use <strong>mode</strong> for most common category/value.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Label axes and units clearly on charts.</li>
|
||||
<li>Check scale intervals before reading values.</li>
|
||||
<li>Order data before finding median and quartiles.</li>
|
||||
<li>State if an answer is estimated from grouped data.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find mean of 4, 7, 7, 10, 12.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{40}{5}=8\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find median of 5, 2, 9, 4, 11, 6.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Order: 2,4,5,6,9,11. Median \(=\frac{5+6}{2}=5.5\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find mode of 3, 5, 5, 5, 8, 8, 9.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> 5.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">For values 1,2,3 with frequencies 4,5,1, find mean.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\bar{x}=\frac{1(4)+2(5)+3(1)}{10}=\frac{17}{10}=1.7\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Which average is best for house prices with extreme luxury homes?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Median (less affected by outliers).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
52
content/math/statistical-charts-averages/core.html
Normal file
52
content/math/statistical-charts-averages/core.html
Normal file
|
|
@ -0,0 +1,52 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Data handling starts with organizing data clearly, choosing suitable charts, and summarizing a
|
||||
dataset with averages.
|
||||
</p>
|
||||
|
||||
<h2 id="data-tables" class="book-prose__heading">Frequency tables</h2>
|
||||
<p>
|
||||
A frequency table records each value (or class interval) and how often it appears.
|
||||
For grouped data, use class midpoints for estimation.
|
||||
</p>
|
||||
|
||||
<h2 id="common-charts" class="book-prose__heading">Common charts</h2>
|
||||
<ul>
|
||||
<li><strong>Bar chart:</strong> categorical comparisons.</li>
|
||||
<li><strong>Histogram:</strong> continuous grouped data (touching bars).</li>
|
||||
<li><strong>Line graph:</strong> change over time.</li>
|
||||
<li><strong>Pie chart:</strong> part-to-whole proportions.</li>
|
||||
<li><strong>Box plot:</strong> spread via quartiles and median.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="mean-median-mode" class="book-prose__heading">Mean, median, and mode</h2>
|
||||
<p class="book-formula">\[
|
||||
\text{Mean}=\frac{\text{sum of values}}{\text{number of values}}.
|
||||
\]</p>
|
||||
<p>
|
||||
<strong>Median</strong> is middle value when ordered. <strong>Mode</strong> is most frequent value.
|
||||
</p>
|
||||
|
||||
<h2 id="weighted-mean" class="book-prose__heading">Mean from frequency table</h2>
|
||||
<p class="book-formula">\[
|
||||
\bar{x}=\frac{\sum fx}{\sum f}.
|
||||
\]</p>
|
||||
<p>
|
||||
where \(x\) is value (or midpoint), \(f\) is frequency.
|
||||
</p>
|
||||
|
||||
<h2 id="which-average" class="book-prose__heading">Choosing an average</h2>
|
||||
<ul>
|
||||
<li>Use <strong>mean</strong> when data has no extreme outliers.</li>
|
||||
<li>Use <strong>median</strong> when distribution is skewed.</li>
|
||||
<li>Use <strong>mode</strong> for most common category/value.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Label axes and units clearly on charts.</li>
|
||||
<li>Check scale intervals before reading values.</li>
|
||||
<li>Order data before finding median and quartiles.</li>
|
||||
<li>State if an answer is estimated from grouped data.</li>
|
||||
</ul>
|
||||
</div>
|
||||
33
content/math/statistical-charts-averages/supplement.html
Normal file
33
content/math/statistical-charts-averages/supplement.html
Normal file
|
|
@ -0,0 +1,33 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find mean of 4, 7, 7, 10, 12.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\frac{40}{5}=8\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find median of 5, 2, 9, 4, 11, 6.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Order: 2,4,5,6,9,11. Median \(=\frac{5+6}{2}=5.5\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Find mode of 3, 5, 5, 5, 8, 8, 9.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> 5.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">For values 1,2,3 with frequencies 4,5,1, find mean.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> \(\bar{x}=\frac{1(4)+2(5)+3(1)}{10}=\frac{17}{10}=1.7\).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Which average is best for house prices with extreme luxury homes?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Median (less affected by outliers).</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
93
content/math/statistical-literacy/_index.md
Normal file
93
content/math/statistical-literacy/_index.md
Normal file
|
|
@ -0,0 +1,93 @@
|
|||
+++
|
||||
title = "Statistical literacy"
|
||||
description = "Evaluate claims in media and research: sampling bias, correlation versus causation, and misleading charts."
|
||||
template = "chapter.html"
|
||||
weight = 35
|
||||
[extra]
|
||||
subject = "math"
|
||||
chapter_id = "statistical-literacy"
|
||||
strand = "Data Handling"
|
||||
+++
|
||||
|
||||
<div class="book-prose">
|
||||
<p>
|
||||
Statistical literacy is the ability to read claims critically, question data quality, and avoid
|
||||
misleading conclusions.
|
||||
</p>
|
||||
|
||||
<h2 id="good-questions" class="book-prose__heading">Questions to ask about a claim</h2>
|
||||
<ul>
|
||||
<li>Who collected the data and why?</li>
|
||||
<li>How large and representative is the sample?</li>
|
||||
<li>How was the data measured?</li>
|
||||
<li>Are conclusions consistent with evidence?</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="sampling-bias" class="book-prose__heading">Sampling and bias</h2>
|
||||
<p>
|
||||
Biased samples produce unreliable conclusions. Prefer random, sufficiently large, and
|
||||
representative samples.
|
||||
</p>
|
||||
|
||||
<h2 id="correlation-causation" class="book-prose__heading">Correlation vs causation</h2>
|
||||
<p>
|
||||
Correlation means variables move together. It does <strong>not</strong> prove one causes the other.
|
||||
Confounders may explain both.
|
||||
</p>
|
||||
|
||||
<h2 id="misleading-charts" class="book-prose__heading">Misleading charts</h2>
|
||||
<ul>
|
||||
<li>Truncated axes exaggerating differences.</li>
|
||||
<li>Inconsistent scales.</li>
|
||||
<li>Cherry-picked time ranges.</li>
|
||||
<li>3D chart distortion.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="risk-language" class="book-prose__heading">Interpreting risk language</h2>
|
||||
<p>
|
||||
Distinguish <strong>absolute change</strong> from <strong>relative change</strong>.
|
||||
“Risk doubled” may still represent a very small absolute increase.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Quote numerical evidence when critiquing claims.</li>
|
||||
<li>Mention possible bias/confounding clearly.</li>
|
||||
<li>Use “suggests” unless causation is justified by design.</li>
|
||||
<li>Comment on uncertainty and data limitations.</li>
|
||||
</ul>
|
||||
</div>
|
||||
|
||||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A survey about school meals only asks students in one class. Name one issue.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Sampling bias; sample may not represent the whole school.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Two variables are strongly correlated. Can we conclude causation?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> No, correlation alone is insufficient.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A bar chart axis starts at 95 instead of 0. Why can this mislead?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Differences look much larger than they really are.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">“Risk increased from 1% to 2%.” State absolute and relative increase.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Absolute +1 percentage point; relative increase 100% (doubled).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Name one way to improve reliability of a poll.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Use a larger random sample from the target population.</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
48
content/math/statistical-literacy/core.html
Normal file
48
content/math/statistical-literacy/core.html
Normal file
|
|
@ -0,0 +1,48 @@
|
|||
<div class="book-prose">
|
||||
<p>
|
||||
Statistical literacy is the ability to read claims critically, question data quality, and avoid
|
||||
misleading conclusions.
|
||||
</p>
|
||||
|
||||
<h2 id="good-questions" class="book-prose__heading">Questions to ask about a claim</h2>
|
||||
<ul>
|
||||
<li>Who collected the data and why?</li>
|
||||
<li>How large and representative is the sample?</li>
|
||||
<li>How was the data measured?</li>
|
||||
<li>Are conclusions consistent with evidence?</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="sampling-bias" class="book-prose__heading">Sampling and bias</h2>
|
||||
<p>
|
||||
Biased samples produce unreliable conclusions. Prefer random, sufficiently large, and
|
||||
representative samples.
|
||||
</p>
|
||||
|
||||
<h2 id="correlation-causation" class="book-prose__heading">Correlation vs causation</h2>
|
||||
<p>
|
||||
Correlation means variables move together. It does <strong>not</strong> prove one causes the other.
|
||||
Confounders may explain both.
|
||||
</p>
|
||||
|
||||
<h2 id="misleading-charts" class="book-prose__heading">Misleading charts</h2>
|
||||
<ul>
|
||||
<li>Truncated axes exaggerating differences.</li>
|
||||
<li>Inconsistent scales.</li>
|
||||
<li>Cherry-picked time ranges.</li>
|
||||
<li>3D chart distortion.</li>
|
||||
</ul>
|
||||
|
||||
<h2 id="risk-language" class="book-prose__heading">Interpreting risk language</h2>
|
||||
<p>
|
||||
Distinguish <strong>absolute change</strong> from <strong>relative change</strong>.
|
||||
“Risk doubled” may still represent a very small absolute increase.
|
||||
</p>
|
||||
|
||||
<h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
|
||||
<ul>
|
||||
<li>Quote numerical evidence when critiquing claims.</li>
|
||||
<li>Mention possible bias/confounding clearly.</li>
|
||||
<li>Use “suggests” unless causation is justified by design.</li>
|
||||
<li>Comment on uncertainty and data limitations.</li>
|
||||
</ul>
|
||||
</div>
|
||||
33
content/math/statistical-literacy/supplement.html
Normal file
33
content/math/statistical-literacy/supplement.html
Normal file
|
|
@ -0,0 +1,33 @@
|
|||
<div class="book-prose">
|
||||
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A survey about school meals only asks students in one class. Name one issue.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Sampling bias; sample may not represent the whole school.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Two variables are strongly correlated. Can we conclude causation?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> No, correlation alone is insufficient.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">A bar chart axis starts at 95 instead of 0. Why can this mislead?</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Differences look much larger than they really are.</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">“Risk increased from 1% to 2%.” State absolute and relative increase.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Absolute +1 percentage point; relative increase 100% (doubled).</p>
|
||||
</div>
|
||||
</details>
|
||||
<details class="book-question">
|
||||
<summary class="book-question__prompt">Name one way to improve reliability of a poll.</summary>
|
||||
<div class="book-question__solution">
|
||||
<p><strong>Answer:</strong> Use a larger random sample from the target population.</p>
|
||||
</div>
|
||||
</details>
|
||||
</div>
|
||||
|
|
@ -638,7 +638,7 @@
|
|||
"slug": "coordinate-geometry-foundations",
|
||||
"title": "Coordinate geometry foundations",
|
||||
"description": "Plot points, find distances and midpoints, and work with slopes of lines in the Cartesian plane.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -655,7 +655,7 @@
|
|||
"slug": "deductive-geometry",
|
||||
"title": "Deductive geometry",
|
||||
"description": "Prove angle and side relationships for intersecting lines, triangles, and polygons.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -671,7 +671,7 @@
|
|||
"slug": "congruence-similarity",
|
||||
"title": "Congruence & similarity",
|
||||
"description": "Apply congruence and similarity criteria to justify properties of triangles and scale figures.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -687,7 +687,7 @@
|
|||
"slug": "pythagoras-theorem",
|
||||
"title": "Pythagoras' theorem",
|
||||
"description": "Use Pythagoras' theorem and its converse in 2D problems and coordinate settings.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -704,7 +704,7 @@
|
|||
"slug": "introduction-trigonometry",
|
||||
"title": "Introduction to trigonometry",
|
||||
"description": "Define sine, cosine, and tangent ratios and solve right-angled triangle problems.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -721,7 +721,7 @@
|
|||
"slug": "mensuration-3d",
|
||||
"title": "Mensuration in 3D",
|
||||
"description": "Find volumes and surface areas of prisms, cylinders, pyramids, cones, and spheres.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -737,7 +737,7 @@
|
|||
"slug": "coordinate-circles",
|
||||
"title": "Coordinate geometry of circles",
|
||||
"description": "Find equations of circles, tangents, and intersections using coordinate methods and distance.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "related",
|
||||
|
|
@ -752,7 +752,7 @@
|
|||
"slug": "advanced-trigonometry",
|
||||
"title": "Advanced trigonometry",
|
||||
"description": "Use sine and cosine rules, radians, and trigonometric graphs in 2D and 3D applications.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"tier": "non-foundation",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "extension",
|
||||
|
|
@ -770,7 +770,7 @@
|
|||
"slug": "circle-properties",
|
||||
"title": "Properties of circles",
|
||||
"description": "Apply chord, tangent, and angle theorems in circles to prove results and solve geometry problems.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"tier": "non-foundation",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "extension",
|
||||
|
|
@ -786,7 +786,7 @@
|
|||
"slug": "loci",
|
||||
"title": "Loci",
|
||||
"description": "Describe and construct loci in the plane, linking geometric constraints to algebraic equations.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"tier": "non-foundation",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "extension",
|
||||
|
|
@ -806,7 +806,7 @@
|
|||
"slug": "statistical-charts-averages",
|
||||
"title": "Charts & averages",
|
||||
"description": "Organise data with tables and charts and calculate mean, median, and mode.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -823,7 +823,7 @@
|
|||
"slug": "measures-dispersion",
|
||||
"title": "Measures of dispersion",
|
||||
"description": "Interpret range, interquartile range, variance, and standard deviation for comparing data sets.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -840,7 +840,7 @@
|
|||
"slug": "basic-probability",
|
||||
"title": "Basic probability",
|
||||
"description": "Calculate theoretical and experimental probabilities for simple events and complements.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IGCSE": "core",
|
||||
|
|
@ -857,7 +857,7 @@
|
|||
"slug": "permutations-combinations-probability",
|
||||
"title": "Permutations, combinations & probability",
|
||||
"description": "Count outcomes with permutations and combinations, then compute probabilities in finite sample spaces.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"tier": "non-foundation",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "extension",
|
||||
|
|
@ -873,7 +873,7 @@
|
|||
"slug": "normal-distribution-z-scores",
|
||||
"title": "Normal distribution & z-scores",
|
||||
"description": "Model continuous data with the normal curve, standardise with z-scores, and read probabilities from tables.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"tier": "non-foundation",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "extension",
|
||||
|
|
@ -889,7 +889,7 @@
|
|||
"slug": "statistical-literacy",
|
||||
"title": "Statistical literacy",
|
||||
"description": "Evaluate claims in media and research: sampling bias, correlation versus causation, and misleading charts.",
|
||||
"status": "planned",
|
||||
"status": "live",
|
||||
"curriculumCoverage": {
|
||||
"DSE": "core",
|
||||
"IB": "related",
|
||||
|
|
|
|||
4
spec.md
4
spec.md
|
|
@ -18,7 +18,9 @@ we need to have a question bank feature. since this is purely static sites, ques
|
|||
|
||||
question bank not only show answer but also steps, explanation, calcultion.
|
||||
|
||||
best practice for creating chapters: correctness > completeness > order and structure > accessibility > optimization
|
||||
best practice for creating chapters: correctness (knowledge wise) > completeness > order and structure > accessibility > optimization > correctness of everything else (credits, reference etc.)
|
||||
|
||||
a textbook should only start when someone owns it and capable of completing it.
|
||||
|
||||
## Chapters requirements
|
||||
|
||||
|
|
|
|||
166
static/feed.xml
166
static/feed.xml
|
|
@ -4,54 +4,173 @@
|
|||
<title>Open Free Books</title>
|
||||
<link>https://openfreebooks.org/</link>
|
||||
<description>Recent updates to Open Free Books pages and live chapters.</description>
|
||||
<lastBuildDate>Thu, 21 May 2026 07:32:39 GMT</lastBuildDate>
|
||||
<lastBuildDate>Thu, 21 May 2026 16:43:43 GMT</lastBuildDate>
|
||||
<item>
|
||||
<title>Linear programming</title>
|
||||
<link>https://openfreebooks.org/math/linear-programming/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/linear-programming/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<description>Graph linear inequalities, identify feasible regions, and optimise objective functions in two variables.</description>
|
||||
<title>Loci</title>
|
||||
<link>https://openfreebooks.org/math/loci/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/loci/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Describe and construct loci in the plane, linking geometric constraints to algebraic equations.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Factor & remainder theorems</title>
|
||||
<link>https://openfreebooks.org/math/factor-remainder-theorems/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/factor-remainder-theorems/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<description>Use the factor and remainder theorems to factorise higher-degree polynomials and solve related equations.</description>
|
||||
<title>Introduction to trigonometry</title>
|
||||
<link>https://openfreebooks.org/math/introduction-trigonometry/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/introduction-trigonometry/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Define sine, cosine, and tangent ratios and solve right-angled triangle problems.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Exponential & logarithmic functions</title>
|
||||
<link>https://openfreebooks.org/math/exponential-logarithmic/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/exponential-logarithmic/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<description>Use exponential and logarithmic models, apply log laws, and solve growth and decay equations.</description>
|
||||
<title>Permutations, combinations & probability</title>
|
||||
<link>https://openfreebooks.org/math/permutations-combinations-probability/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/permutations-combinations-probability/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Count outcomes with permutations and combinations, then compute probabilities in finite sample spaces.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Mensuration in 3D</title>
|
||||
<link>https://openfreebooks.org/math/mensuration-3d/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/mensuration-3d/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Find volumes and surface areas of prisms, cylinders, pyramids, cones, and spheres.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Normal distribution & z-scores</title>
|
||||
<link>https://openfreebooks.org/math/normal-distribution-z-scores/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/normal-distribution-z-scores/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Model continuous data with the normal curve, standardise with z-scores, and read probabilities from tables.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Coordinate geometry foundations</title>
|
||||
<link>https://openfreebooks.org/math/coordinate-geometry-foundations/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/coordinate-geometry-foundations/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Plot points, find distances and midpoints, and work with slopes of lines in the Cartesian plane.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Measures of dispersion</title>
|
||||
<link>https://openfreebooks.org/math/measures-dispersion/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/measures-dispersion/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Interpret range, interquartile range, variance, and standard deviation for comparing data sets.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Advanced trigonometry</title>
|
||||
<link>https://openfreebooks.org/math/advanced-trigonometry/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/advanced-trigonometry/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Use sine and cosine rules, radians, and trigonometric graphs in 2D and 3D applications.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Properties of circles</title>
|
||||
<link>https://openfreebooks.org/math/circle-properties/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/circle-properties/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Apply chord, tangent, and angle theorems in circles to prove results and solve geometry problems.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Deductive geometry</title>
|
||||
<link>https://openfreebooks.org/math/deductive-geometry/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/deductive-geometry/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Prove angle and side relationships for intersecting lines, triangles, and polygons.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Coordinate geometry of circles</title>
|
||||
<link>https://openfreebooks.org/math/coordinate-circles/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/coordinate-circles/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Find equations of circles, tangents, and intersections using coordinate methods and distance.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Basic probability</title>
|
||||
<link>https://openfreebooks.org/math/basic-probability/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/basic-probability/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Calculate theoretical and experimental probabilities for simple events and complements.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Charts & averages</title>
|
||||
<link>https://openfreebooks.org/math/statistical-charts-averages/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/statistical-charts-averages/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Organise data with tables and charts and calculate mean, median, and mode.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Statistical literacy</title>
|
||||
<link>https://openfreebooks.org/math/statistical-literacy/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/statistical-literacy/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Evaluate claims in media and research: sampling bias, correlation versus causation, and misleading charts.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Congruence & similarity</title>
|
||||
<link>https://openfreebooks.org/math/congruence-similarity/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/congruence-similarity/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Apply congruence and similarity criteria to justify properties of triangles and scale figures.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Pythagoras' theorem</title>
|
||||
<link>https://openfreebooks.org/math/pythagoras-theorem/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/pythagoras-theorem/</guid>
|
||||
<pubDate>Thu, 21 May 2026 16:43:43 GMT</pubDate>
|
||||
<description>Use Pythagoras' theorem and its converse in 2D problems and coordinate settings.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Complex numbers</title>
|
||||
<link>https://openfreebooks.org/math/complex-numbers/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/complex-numbers/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Represent complex numbers in Cartesian form and perform operations including conjugates and division.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Exponential & logarithmic functions</title>
|
||||
<link>https://openfreebooks.org/math/exponential-logarithmic/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/exponential-logarithmic/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Use exponential and logarithmic models, apply log laws, and solve growth and decay equations.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Factor & remainder theorems</title>
|
||||
<link>https://openfreebooks.org/math/factor-remainder-theorems/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/factor-remainder-theorems/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Use the factor and remainder theorems to factorise higher-degree polynomials and solve related equations.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Functions & graphs</title>
|
||||
<link>https://openfreebooks.org/math/functions-graphs/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/functions-graphs/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Model input-output rules, work with domain and range, and sketch key function graphs and transformations.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Linear equations in two unknowns</title>
|
||||
<link>https://openfreebooks.org/math/linear-equations-two-unknowns/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/linear-equations-two-unknowns/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Solve pairs of linear equations by substitution, elimination, and graphs; classify unique, parallel, and coincident cases.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Linear programming</title>
|
||||
<link>https://openfreebooks.org/math/linear-programming/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/linear-programming/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Graph linear inequalities, identify feasible regions, and optimise objective functions in two variables.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Sequences & series</title>
|
||||
<link>https://openfreebooks.org/math/sequences-series/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/sequences-series/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Work with arithmetic and geometric sequences, sigma notation, and finite summation formulas.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Variations</title>
|
||||
<link>https://openfreebooks.org/math/variations/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/variations/</guid>
|
||||
<pubDate>Thu, 21 May 2026 07:32:39 GMT</pubDate>
|
||||
<pubDate>Thu, 21 May 2026 07:33:01 GMT</pubDate>
|
||||
<description>Model direct, inverse, joint, and partial variation and solve combined variation problems.</description>
|
||||
</item>
|
||||
<item>
|
||||
|
|
@ -75,13 +194,6 @@
|
|||
<pubDate>Thu, 21 May 2026 05:05:38 GMT</pubDate>
|
||||
<description>Count objects, spot patterns, add, multiply, and divide; then meet place value and big numbers from hundreds to billions.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Linear equations in two unknowns</title>
|
||||
<link>https://openfreebooks.org/math/linear-equations-two-unknowns/</link>
|
||||
<guid isPermaLink="true">https://openfreebooks.org/math/linear-equations-two-unknowns/</guid>
|
||||
<pubDate>Thu, 21 May 2026 05:05:38 GMT</pubDate>
|
||||
<description>Solve pairs of linear equations by substitution, elimination, and graphs; classify unique, parallel, and coincident cases.</description>
|
||||
</item>
|
||||
<item>
|
||||
<title>Written calculation</title>
|
||||
<link>https://openfreebooks.org/math/written-calculation/</link>
|
||||
|
|
|
|||
|
|
@ -36,6 +36,10 @@
|
|||
<loc>https://openfreebooks.org/math/</loc>
|
||||
<lastmod>2026-05-19T10:55:50.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/advanced-trigonometry/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.537Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/algebraic-fractions-formulas/</loc>
|
||||
<lastmod>2026-05-20T20:17:25.000Z</lastmod>
|
||||
|
|
@ -44,29 +48,57 @@
|
|||
<loc>https://openfreebooks.org/math/basic-math-notation/</loc>
|
||||
<lastmod>2026-05-21T01:07:39.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/basic-probability/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.535Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/circle-properties/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.537Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/complex-numbers/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.566Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/congruence-similarity/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.137Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/coordinate-circles/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.536Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/coordinate-geometry-foundations/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.663Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/counting-and-numbers/</loc>
|
||||
<lastmod>2026-05-21T05:05:38.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/deductive-geometry/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.537Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/directed-numbers-rates/</loc>
|
||||
<lastmod>2026-05-20T20:17:25.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/exponential-logarithmic/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.567Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/factor-remainder-theorems/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.697Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/functions-graphs/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.566Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/introduction-trigonometry/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.931Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/laws-of-indices/</loc>
|
||||
|
|
@ -74,7 +106,7 @@
|
|||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/linear-equations-two-unknowns/</loc>
|
||||
<lastmod>2026-05-21T05:05:38.000Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/linear-inequalities-one-unknown/</loc>
|
||||
|
|
@ -82,7 +114,27 @@
|
|||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/linear-programming/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.831Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/loci/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.932Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/measures-dispersion/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.662Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/mensuration-3d/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.798Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/normal-distribution-z-scores/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.798Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/permutations-combinations-probability/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.799Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/polynomials-expansion-factorisation/</loc>
|
||||
|
|
@ -92,17 +144,29 @@
|
|||
<loc>https://openfreebooks.org/math/properties-of-operations/</loc>
|
||||
<lastmod>2026-05-20T20:17:25.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/pythagoras-theorem/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.137Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/quadratic-equations/</loc>
|
||||
<lastmod>2026-05-20T20:17:25.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/sequences-series/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.286Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/statistical-charts-averages/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.394Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/statistical-literacy/</loc>
|
||||
<lastmod>2026-05-21T16:43:43.394Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/variations/</loc>
|
||||
<lastmod>2026-05-21T07:32:39.286Z</lastmod>
|
||||
<lastmod>2026-05-21T07:33:01.000Z</lastmod>
|
||||
</url>
|
||||
<url>
|
||||
<loc>https://openfreebooks.org/math/written-calculation/</loc>
|
||||
|
|
|
|||
|
|
@ -91,7 +91,7 @@
|
|||
</div>
|
||||
<div class="why-copy" data-scroll-reveal-card>
|
||||
<p class="section-kicker">Planet</p>
|
||||
<h2 class="section-title">Open textbooks are gentler on the Earth</h2>
|
||||
<h2 class="section-title">OFB are gentler on the Earth</h2>
|
||||
<p>
|
||||
Everybody learns the same basics. Everybody prompts AI the same thing. Core curriculum changes slowly.
|
||||
Newton’s laws, the structure of a cell, and how fractions work do not need a new AI run every September.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue