From 81c661472221bf540b25e6d4051dd685feab1811 Mon Sep 17 00:00:00 2001 From: Hananoshika Yomaru Date: Thu, 21 May 2026 09:54:30 -0700 Subject: [PATCH] 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. --- content/math/advanced-trigonometry/_index.md | 134 ++++++++++++++ content/math/advanced-trigonometry/core.html | 70 ++++++++ .../advanced-trigonometry/supplement.html | 52 ++++++ content/math/basic-probability/_index.md | 82 +++++++++ content/math/basic-probability/core.html | 43 +++++ .../math/basic-probability/supplement.html | 27 +++ content/math/circle-properties/_index.md | 114 ++++++++++++ content/math/circle-properties/core.html | 50 ++++++ .../math/circle-properties/supplement.html | 52 ++++++ content/math/congruence-similarity/_index.md | 154 ++++++++++++++++ content/math/congruence-similarity/core.html | 73 ++++++++ .../congruence-similarity/supplement.html | 69 ++++++++ content/math/coordinate-circles/_index.md | 136 ++++++++++++++ content/math/coordinate-circles/core.html | 72 ++++++++ .../math/coordinate-circles/supplement.html | 52 ++++++ .../coordinate-geometry-foundations/_index.md | 159 +++++++++++++++++ .../coordinate-geometry-foundations/core.html | 88 ++++++++++ .../supplement.html | 59 +++++++ content/math/deductive-geometry/_index.md | 161 +++++++++++++++++ content/math/deductive-geometry/core.html | 79 +++++++++ .../math/deductive-geometry/supplement.html | 70 ++++++++ .../math/introduction-trigonometry/_index.md | 132 ++++++++++++++ .../math/introduction-trigonometry/core.html | 61 +++++++ .../introduction-trigonometry/supplement.html | 59 +++++++ content/math/loci/_index.md | 119 +++++++++++++ content/math/loci/core.html | 55 ++++++ content/math/loci/supplement.html | 52 ++++++ content/math/measures-dispersion/_index.md | 84 +++++++++ content/math/measures-dispersion/core.html | 45 +++++ .../math/measures-dispersion/supplement.html | 27 +++ content/math/mensuration-3d/_index.md | 148 ++++++++++++++++ content/math/mensuration-3d/core.html | 77 ++++++++ content/math/mensuration-3d/supplement.html | 59 +++++++ .../normal-distribution-z-scores/_index.md | 85 +++++++++ .../normal-distribution-z-scores/core.html | 46 +++++ .../supplement.html | 27 +++ .../_index.md | 90 ++++++++++ .../core.html | 45 +++++ .../supplement.html | 33 ++++ content/math/pythagoras-theorem/_index.md | 144 +++++++++++++++ content/math/pythagoras-theorem/core.html | 72 ++++++++ .../math/pythagoras-theorem/supplement.html | 60 +++++++ .../statistical-charts-averages/_index.md | 97 ++++++++++ .../statistical-charts-averages/core.html | 52 ++++++ .../supplement.html | 33 ++++ content/math/statistical-literacy/_index.md | 93 ++++++++++ content/math/statistical-literacy/core.html | 48 +++++ .../math/statistical-literacy/supplement.html | 33 ++++ data/math-curriculum.json | 32 ++-- spec.md | 4 +- static/feed.xml | 166 +++++++++++++++--- static/sitemap.xml | 80 ++++++++- .../templates/partials/why-sections.html | 2 +- 53 files changed, 3903 insertions(+), 53 deletions(-) create mode 100644 content/math/advanced-trigonometry/_index.md create mode 100644 content/math/advanced-trigonometry/core.html create mode 100644 content/math/advanced-trigonometry/supplement.html create mode 100644 content/math/basic-probability/_index.md create mode 100644 content/math/basic-probability/core.html create mode 100644 content/math/basic-probability/supplement.html create mode 100644 content/math/circle-properties/_index.md create mode 100644 content/math/circle-properties/core.html create mode 100644 content/math/circle-properties/supplement.html create mode 100644 content/math/congruence-similarity/_index.md create mode 100644 content/math/congruence-similarity/core.html create mode 100644 content/math/congruence-similarity/supplement.html create mode 100644 content/math/coordinate-circles/_index.md create mode 100644 content/math/coordinate-circles/core.html create mode 100644 content/math/coordinate-circles/supplement.html create mode 100644 content/math/coordinate-geometry-foundations/_index.md create mode 100644 content/math/coordinate-geometry-foundations/core.html create mode 100644 content/math/coordinate-geometry-foundations/supplement.html create mode 100644 content/math/deductive-geometry/_index.md create mode 100644 content/math/deductive-geometry/core.html create mode 100644 content/math/deductive-geometry/supplement.html create mode 100644 content/math/introduction-trigonometry/_index.md create mode 100644 content/math/introduction-trigonometry/core.html create mode 100644 content/math/introduction-trigonometry/supplement.html create mode 100644 content/math/loci/_index.md create mode 100644 content/math/loci/core.html create mode 100644 content/math/loci/supplement.html create mode 100644 content/math/measures-dispersion/_index.md create mode 100644 content/math/measures-dispersion/core.html create mode 100644 content/math/measures-dispersion/supplement.html create mode 100644 content/math/mensuration-3d/_index.md create mode 100644 content/math/mensuration-3d/core.html create mode 100644 content/math/mensuration-3d/supplement.html create mode 100644 content/math/normal-distribution-z-scores/_index.md create mode 100644 content/math/normal-distribution-z-scores/core.html create mode 100644 content/math/normal-distribution-z-scores/supplement.html create mode 100644 content/math/permutations-combinations-probability/_index.md create mode 100644 content/math/permutations-combinations-probability/core.html create mode 100644 content/math/permutations-combinations-probability/supplement.html create mode 100644 content/math/pythagoras-theorem/_index.md create mode 100644 content/math/pythagoras-theorem/core.html create mode 100644 content/math/pythagoras-theorem/supplement.html create mode 100644 content/math/statistical-charts-averages/_index.md create mode 100644 content/math/statistical-charts-averages/core.html create mode 100644 content/math/statistical-charts-averages/supplement.html create mode 100644 content/math/statistical-literacy/_index.md create mode 100644 content/math/statistical-literacy/core.html create mode 100644 content/math/statistical-literacy/supplement.html diff --git a/content/math/advanced-trigonometry/_index.md b/content/math/advanced-trigonometry/_index.md new file mode 100644 index 0000000..65de94e --- /dev/null +++ b/content/math/advanced-trigonometry/_index.md @@ -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" ++++ + +
+

+ Advanced trigonometry extends right-triangle work to non-right triangles, radians, and trig + graphs used in periodic modeling. +

+ +

Radians and arc length

+

+ Convert degrees/radians: +

+

\[ + 180^\circ=\pi\ \text{rad}. + \]

+

+ Arc length and sector area: +

+

\[ + s=r\theta,\qquad A=\frac12r^2\theta + \]

+

with \(\theta\) in radians.

+ +

Sine and cosine rules

+

\[ + \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} + \]

+

\[ + c^2=a^2+b^2-2ab\cos C. + \]

+

+ Use sine rule for side-angle opposite pairs; cosine rule for SAS or SSS cases. +

+ +

Triangle area formula

+

\[ + \text{Area}=\frac12ab\sin C. + \]

+

+ Useful when two sides and included angle are known. +

+ +

Core identities

+

\[ + \sin^2x+\cos^2x=1,\qquad + \tan x=\frac{\sin x}{\cos x}. + \]

+ +

Solving basic trig equations

+

+ Solve in interval (e.g., \(0^\circ\le x<360^\circ\)) using unit-circle symmetry. +

+

Example: \(\sin x=\frac12\Rightarrow x=30^\circ,150^\circ\).

+ +

Trigonometric graphs

+

+ For \(y=a\sin(bx)+d\) (similarly cosine): +

+ + +

Exam strategy

+ +
+ +
+

History

+

+ Advanced trigonometric methods were developed for astronomy, navigation, and surveying. Sine and + cosine rule tables were essential long before modern calculators. +

+ +

Checkpoints

+
+ Convert \(150^\circ\) to radians. +
+

Answer: \(\frac{5\pi}{6}\).

+
+
+
+ Find arc length for \(r=8\), \(\theta=\frac{\pi}{3}\). +
+

Answer: \(s=r\theta=\frac{8\pi}{3}\).

+
+
+
+ In triangle, \(a=10,b=7,C=60^\circ\). Find \(c\). +
+

Answer: \(c^2=10^2+7^2-2(10)(7)\cos60^\circ=79\), so \(c=\sqrt{79}\).

+
+
+
+ Solve \(2\sin x=1\) for \(0^\circ\le x<360^\circ\). +
+

Answer: \(x=30^\circ,150^\circ\).

+
+
+
+ For \(y=3\cos(2x)-1\), state amplitude and period. +
+

Answer: Amplitude \(3\), period \(\pi\) radians (or \(180^\circ\)).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/advanced-trigonometry/core.html b/content/math/advanced-trigonometry/core.html new file mode 100644 index 0000000..d983850 --- /dev/null +++ b/content/math/advanced-trigonometry/core.html @@ -0,0 +1,70 @@ +
+

+ Advanced trigonometry extends right-triangle work to non-right triangles, radians, and trig + graphs used in periodic modeling. +

+ +

Radians and arc length

+

+ Convert degrees/radians: +

+

\[ + 180^\circ=\pi\ \text{rad}. + \]

+

+ Arc length and sector area: +

+

\[ + s=r\theta,\qquad A=\frac12r^2\theta + \]

+

with \(\theta\) in radians.

+ +

Sine and cosine rules

+

\[ + \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} + \]

+

\[ + c^2=a^2+b^2-2ab\cos C. + \]

+

+ Use sine rule for side-angle opposite pairs; cosine rule for SAS or SSS cases. +

+ +

Triangle area formula

+

\[ + \text{Area}=\frac12ab\sin C. + \]

+

+ Useful when two sides and included angle are known. +

+ +

Core identities

+

\[ + \sin^2x+\cos^2x=1,\qquad + \tan x=\frac{\sin x}{\cos x}. + \]

+ +

Solving basic trig equations

+

+ Solve in interval (e.g., \(0^\circ\le x<360^\circ\)) using unit-circle symmetry. +

+

Example: \(\sin x=\frac12\Rightarrow x=30^\circ,150^\circ\).

+ +

Trigonometric graphs

+

+ For \(y=a\sin(bx)+d\) (similarly cosine): +

+ + +

Exam strategy

+ +
diff --git a/content/math/advanced-trigonometry/supplement.html b/content/math/advanced-trigonometry/supplement.html new file mode 100644 index 0000000..4deb85f --- /dev/null +++ b/content/math/advanced-trigonometry/supplement.html @@ -0,0 +1,52 @@ +
+

History

+

+ Advanced trigonometric methods were developed for astronomy, navigation, and surveying. Sine and + cosine rule tables were essential long before modern calculators. +

+ +

Checkpoints

+
+ Convert \(150^\circ\) to radians. +
+

Answer: \(\frac{5\pi}{6}\).

+
+
+
+ Find arc length for \(r=8\), \(\theta=\frac{\pi}{3}\). +
+

Answer: \(s=r\theta=\frac{8\pi}{3}\).

+
+
+
+ In triangle, \(a=10,b=7,C=60^\circ\). Find \(c\). +
+

Answer: \(c^2=10^2+7^2-2(10)(7)\cos60^\circ=79\), so \(c=\sqrt{79}\).

+
+
+
+ Solve \(2\sin x=1\) for \(0^\circ\le x<360^\circ\). +
+

Answer: \(x=30^\circ,150^\circ\).

+
+
+
+ For \(y=3\cos(2x)-1\), state amplitude and period. +
+

Answer: Amplitude \(3\), period \(\pi\) radians (or \(180^\circ\)).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/basic-probability/_index.md b/content/math/basic-probability/_index.md new file mode 100644 index 0000000..32c64a1 --- /dev/null +++ b/content/math/basic-probability/_index.md @@ -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" ++++ + +
+

+ Probability measures how likely an event is, from 0 (impossible) to 1 (certain). +

+ +

Fundamentals

+

\[ + P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}. + \]

+

+ Also: \(0\le P(E)\le1\). +

+ +

Complement rule

+

\[ + P(E')=1-P(E). + \]

+ +

Combined events

+ + +

Experimental probability

+

\[ + P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}. + \]

+ +

Sample spaces and trees

+

+ Use sample-space tables or tree diagrams for multi-step outcomes. + Multiply along branches, add across valid branches. +

+ +

Exam strategy

+ +
+ +
+

Checkpoints

+
+ A fair die is rolled. Find \(P(\text{prime})\). +
+

Answer: \(\frac{3}{6}=\frac12\) (2,3,5).

+
+
+
+ A bag has 4 red and 6 blue balls. Find \(P(\text{not red})\). +
+

Answer: \(\frac{6}{10}=\frac35\).

+
+
+
+ For independent events, \(P(A)=0.3,P(B)=0.4\). Find \(P(A\cap B)\). +
+

Answer: \(0.12\).

+
+
+
+ An event occurs 18 times in 50 trials. Estimate probability. +
+

Answer: \(\frac{18}{50}=0.36\).

+
+
+
diff --git a/content/math/basic-probability/core.html b/content/math/basic-probability/core.html new file mode 100644 index 0000000..3dc05f9 --- /dev/null +++ b/content/math/basic-probability/core.html @@ -0,0 +1,43 @@ +
+

+ Probability measures how likely an event is, from 0 (impossible) to 1 (certain). +

+ +

Fundamentals

+

\[ + P(E)=\frac{\text{number of favorable outcomes}}{\text{number of equally likely outcomes}}. + \]

+

+ Also: \(0\le P(E)\le1\). +

+ +

Complement rule

+

\[ + P(E')=1-P(E). + \]

+ +

Combined events

+ + +

Experimental probability

+

\[ + P(E)\approx\frac{\text{observed frequency}}{\text{number of trials}}. + \]

+ +

Sample spaces and trees

+

+ Use sample-space tables or tree diagrams for multi-step outcomes. + Multiply along branches, add across valid branches. +

+ +

Exam strategy

+ +
diff --git a/content/math/basic-probability/supplement.html b/content/math/basic-probability/supplement.html new file mode 100644 index 0000000..4303a47 --- /dev/null +++ b/content/math/basic-probability/supplement.html @@ -0,0 +1,27 @@ +
+

Checkpoints

+
+ A fair die is rolled. Find \(P(\text{prime})\). +
+

Answer: \(\frac{3}{6}=\frac12\) (2,3,5).

+
+
+
+ A bag has 4 red and 6 blue balls. Find \(P(\text{not red})\). +
+

Answer: \(\frac{6}{10}=\frac35\).

+
+
+
+ For independent events, \(P(A)=0.3,P(B)=0.4\). Find \(P(A\cap B)\). +
+

Answer: \(0.12\).

+
+
+
+ An event occurs 18 times in 50 trials. Estimate probability. +
+

Answer: \(\frac{18}{50}=0.36\).

+
+
+
diff --git a/content/math/circle-properties/_index.md b/content/math/circle-properties/_index.md new file mode 100644 index 0000000..9956e65 --- /dev/null +++ b/content/math/circle-properties/_index.md @@ -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" ++++ + +
+

+ Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs + and exam reasoning. +

+ +

Key circle theorems

+ + +

Using theorems in proofs

+
    +
  1. Mark known equal angles/lengths clearly.
    2. +
  2. Name theorem used at each step.
    3. +
  3. Chain results to target statement.
    4. +
+ +

Cyclic quadrilaterals

+

+ A quadrilateral is cyclic if all vertices lie on one circle. +

+

+ Useful tests: +

+ + +

Tangent-chord applications

+

+ When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle + with an inscribed angle subtending that chord. +

+ +

Exam strategy

+ +
+ +
+

History

+

+ 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. +

+ +

Checkpoints

+
+ If central angle is \(100^\circ\), what is inscribed angle on same arc? +
+

Answer: \(50^\circ\).

+
+
+
+ In a cyclic quadrilateral, one angle is \(72^\circ\). Find opposite angle. +
+

Answer: \(108^\circ\).

+
+
+
+ What angle is formed by radius and tangent at point of contact? +
+

Answer: \(90^\circ\).

+
+
+
+ Two tangents from external point \(P\) touch at \(A,B\). Compare \(PA\) and \(PB\). +
+

Answer: \(PA=PB\).

+
+
+
+ Why does one theorem name per step improve proof quality? +
+

Answer: It makes logical justification explicit and verifiable.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/circle-properties/core.html b/content/math/circle-properties/core.html new file mode 100644 index 0000000..0dff817 --- /dev/null +++ b/content/math/circle-properties/core.html @@ -0,0 +1,50 @@ +
+

+ Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs + and exam reasoning. +

+ +

Key circle theorems

+ + +

Using theorems in proofs

+
    +
  1. Mark known equal angles/lengths clearly.
    2. +
  2. Name theorem used at each step.
    3. +
  3. Chain results to target statement.
    4. +
+ +

Cyclic quadrilaterals

+

+ A quadrilateral is cyclic if all vertices lie on one circle. +

+

+ Useful tests: +

+ + +

Tangent-chord applications

+

+ When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle + with an inscribed angle subtending that chord. +

+ +

Exam strategy

+ +
diff --git a/content/math/circle-properties/supplement.html b/content/math/circle-properties/supplement.html new file mode 100644 index 0000000..204207c --- /dev/null +++ b/content/math/circle-properties/supplement.html @@ -0,0 +1,52 @@ +
+

History

+

+ 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. +

+ +

Checkpoints

+
+ If central angle is \(100^\circ\), what is inscribed angle on same arc? +
+

Answer: \(50^\circ\).

+
+
+
+ In a cyclic quadrilateral, one angle is \(72^\circ\). Find opposite angle. +
+

Answer: \(108^\circ\).

+
+
+
+ What angle is formed by radius and tangent at point of contact? +
+

Answer: \(90^\circ\).

+
+
+
+ Two tangents from external point \(P\) touch at \(A,B\). Compare \(PA\) and \(PB\). +
+

Answer: \(PA=PB\).

+
+
+
+ Why does one theorem name per step improve proof quality? +
+

Answer: It makes logical justification explicit and verifiable.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/congruence-similarity/_index.md b/content/math/congruence-similarity/_index.md new file mode 100644 index 0000000..37cb6d1 --- /dev/null +++ b/content/math/congruence-similarity/_index.md @@ -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" ++++ + +
+

+ Congruence and similarity compare shapes by size and shape. They are central for triangle proofs, + scale drawings, and indirect measurement. +

+ +

Congruence

+

+ Congruent figures have exactly the same shape and size. Corresponding sides and angles are equal. +

+

+ For triangles, common criteria: +

+ + +

Similarity

+

+ Similar figures have the same shape but possibly different sizes. +

+ +

+ Triangle similarity criteria: +

+ + +

Scale factor relationships

+

+ If linear scale factor from shape A to B is \(k\): +

+ + +

Using similarity to solve problems

+
    +
  1. Identify corresponding vertices in correct order.
    2. +
  2. Write ratio equation using matching sides.
    3. +
  3. Solve unknown lengths/areas.
    4. +
  4. Check ratio consistency across all pairs.
    5. +
+ +

Congruence and similarity in proofs

+

+ Common proof flow: +

+ + +

Exam strategy

+ +
+ +
+

History

+

+ Similarity ideas appear in ancient surveying and architecture. Greek geometers formalized triangle + relationships, enabling rigorous methods for unknown lengths and indirect measurements. +

+ +

Derivation and reasoning

+

Why AA implies triangle similarity

+

+ 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. +

+ +

Checkpoints

+
+ + Triangles have side sets \((5,7,9)\) and \((5,7,9)\). Congruent or similar? + +
+

Answer: Congruent (SSS, identical side lengths).

+
+
+
+ + Two triangles have angles \(40^\circ,60^\circ,80^\circ\) and \(40^\circ,60^\circ,80^\circ\), but one is larger. What relation? + +
+

Answer: Similar (AA), not necessarily congruent.

+
+
+
+ + Similar triangles have scale factor \(k=3\). If a side is \(4\) in the small triangle, what is matching side in large triangle? + +
+

Answer: \(4\times3=12\).

+
+
+
+ + If two similar figures have area ratio \(25:9\), what is linear scale factor ratio? + +
+

Answer: \(5:3\) (square root of area ratio).

+
+
+
+ + Right triangles share one acute angle and each has a right angle. What criterion gives similarity? + +
+

Answer: AA similarity.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/congruence-similarity/core.html b/content/math/congruence-similarity/core.html new file mode 100644 index 0000000..2a1507a --- /dev/null +++ b/content/math/congruence-similarity/core.html @@ -0,0 +1,73 @@ +
+

+ Congruence and similarity compare shapes by size and shape. They are central for triangle proofs, + scale drawings, and indirect measurement. +

+ +

Congruence

+

+ Congruent figures have exactly the same shape and size. Corresponding sides and angles are equal. +

+

+ For triangles, common criteria: +

+ + +

Similarity

+

+ Similar figures have the same shape but possibly different sizes. +

+ +

+ Triangle similarity criteria: +

+ + +

Scale factor relationships

+

+ If linear scale factor from shape A to B is \(k\): +

+ + +

Using similarity to solve problems

+
    +
  1. Identify corresponding vertices in correct order.
    2. +
  2. Write ratio equation using matching sides.
    3. +
  3. Solve unknown lengths/areas.
    4. +
  4. Check ratio consistency across all pairs.
    5. +
+ +

Congruence and similarity in proofs

+

+ Common proof flow: +

+ + +

Exam strategy

+ +
diff --git a/content/math/congruence-similarity/supplement.html b/content/math/congruence-similarity/supplement.html new file mode 100644 index 0000000..3f127c0 --- /dev/null +++ b/content/math/congruence-similarity/supplement.html @@ -0,0 +1,69 @@ +
+

History

+

+ Similarity ideas appear in ancient surveying and architecture. Greek geometers formalized triangle + relationships, enabling rigorous methods for unknown lengths and indirect measurements. +

+ +

Derivation and reasoning

+

Why AA implies triangle similarity

+

+ 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. +

+ +

Checkpoints

+
+ + Triangles have side sets \((5,7,9)\) and \((5,7,9)\). Congruent or similar? + +
+

Answer: Congruent (SSS, identical side lengths).

+
+
+
+ + Two triangles have angles \(40^\circ,60^\circ,80^\circ\) and \(40^\circ,60^\circ,80^\circ\), but one is larger. What relation? + +
+

Answer: Similar (AA), not necessarily congruent.

+
+
+
+ + Similar triangles have scale factor \(k=3\). If a side is \(4\) in the small triangle, what is matching side in large triangle? + +
+

Answer: \(4\times3=12\).

+
+
+
+ + If two similar figures have area ratio \(25:9\), what is linear scale factor ratio? + +
+

Answer: \(5:3\) (square root of area ratio).

+
+
+
+ + Right triangles share one acute angle and each has a right angle. What criterion gives similarity? + +
+

Answer: AA similarity.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/coordinate-circles/_index.md b/content/math/coordinate-circles/_index.md new file mode 100644 index 0000000..5410562 --- /dev/null +++ b/content/math/coordinate-circles/_index.md @@ -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" ++++ + +
+

+ Coordinate geometry of circles combines algebra with geometric properties such as radius, center, + tangents, and intersections. +

+ +

Standard equation of a circle

+

+ A circle with center \((h,k)\) and radius \(r\) has equation: +

+

\[ + (x-h)^2+(y-k)^2=r^2. + \]

+

+ If center is origin, this simplifies to \(x^2+y^2=r^2\). +

+ +

Reading center and radius

+

Example: \((x-3)^2+(y+2)^2=25\)

+ + +

General form

+

+ A circle can also appear as: +

+

\[ + x^2+y^2+Dx+Ey+F=0. + \]

+

+ Complete the square in \(x\) and \(y\) to convert to standard form. +

+ +

Tangents and radii

+ +

+ If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined). +

+ +

Point position relative to circle

+

+ For circle center \(C\), radius \(r\), and point \(P\): +

+ + +

Line-circle intersections

+

+ Substitute line equation into circle equation. +

+ + +

Exam strategy

+ +
+ +
+

History

+

+ Coordinate methods made circle geometry algebraic: geometric questions about tangency and + intersection can be solved by equations. This approach grew from analytic geometry. +

+ +

Checkpoints

+
+ Find center and radius of \((x+4)^2+(y-1)^2=36\). +
+

Answer: Center \((-4,1)\), radius \(6\).

+
+
+
+ Write equation of circle with center \((2,-3)\), radius \(7\). +
+

Answer: \((x-2)^2+(y+3)^2=49\).

+
+
+
+ Is point \((5,1)\) on circle \((x-1)^2+(y-1)^2=16\)? +
+

Answer: Yes, \((5-1)^2+(1-1)^2=16\).

+
+
+
+ Convert \(x^2+y^2-6x+4y-12=0\) to standard form. +
+

Answer: \((x-3)^2+(y+2)^2=25\).

+
+
+
+ A line through tangent point has slope 2. What is slope of radius to that point? +
+

Answer: \(-\frac12\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/coordinate-circles/core.html b/content/math/coordinate-circles/core.html new file mode 100644 index 0000000..2c6b3ba --- /dev/null +++ b/content/math/coordinate-circles/core.html @@ -0,0 +1,72 @@ +
+

+ Coordinate geometry of circles combines algebra with geometric properties such as radius, center, + tangents, and intersections. +

+ +

Standard equation of a circle

+

+ A circle with center \((h,k)\) and radius \(r\) has equation: +

+

\[ + (x-h)^2+(y-k)^2=r^2. + \]

+

+ If center is origin, this simplifies to \(x^2+y^2=r^2\). +

+ +

Reading center and radius

+

Example: \((x-3)^2+(y+2)^2=25\)

+ + +

General form

+

+ A circle can also appear as: +

+

\[ + x^2+y^2+Dx+Ey+F=0. + \]

+

+ Complete the square in \(x\) and \(y\) to convert to standard form. +

+ +

Tangents and radii

+ +

+ If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined). +

+ +

Point position relative to circle

+

+ For circle center \(C\), radius \(r\), and point \(P\): +

+ + +

Line-circle intersections

+

+ Substitute line equation into circle equation. +

+ + +

Exam strategy

+ +
diff --git a/content/math/coordinate-circles/supplement.html b/content/math/coordinate-circles/supplement.html new file mode 100644 index 0000000..f36194b --- /dev/null +++ b/content/math/coordinate-circles/supplement.html @@ -0,0 +1,52 @@ +
+

History

+

+ Coordinate methods made circle geometry algebraic: geometric questions about tangency and + intersection can be solved by equations. This approach grew from analytic geometry. +

+ +

Checkpoints

+
+ Find center and radius of \((x+4)^2+(y-1)^2=36\). +
+

Answer: Center \((-4,1)\), radius \(6\).

+
+
+
+ Write equation of circle with center \((2,-3)\), radius \(7\). +
+

Answer: \((x-2)^2+(y+3)^2=49\).

+
+
+
+ Is point \((5,1)\) on circle \((x-1)^2+(y-1)^2=16\)? +
+

Answer: Yes, \((5-1)^2+(1-1)^2=16\).

+
+
+
+ Convert \(x^2+y^2-6x+4y-12=0\) to standard form. +
+

Answer: \((x-3)^2+(y+2)^2=25\).

+
+
+
+ A line through tangent point has slope 2. What is slope of radius to that point? +
+

Answer: \(-\frac12\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/coordinate-geometry-foundations/_index.md b/content/math/coordinate-geometry-foundations/_index.md new file mode 100644 index 0000000..1006099 --- /dev/null +++ b/content/math/coordinate-geometry-foundations/_index.md @@ -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" ++++ + +
+

+ Coordinate geometry connects algebra and geometry by representing points with ordered pairs + \((x,y)\) on the Cartesian plane. +

+ +

The Cartesian plane

+

+ Horizontal axis is \(x\)-axis, vertical axis is \(y\)-axis, and intersection is origin \((0,0)\). +

+

+ Quadrants: +

+ + +

Plotting and reading points

+

+ Ordered pair \((a,b)\) means move \(a\) units along \(x\), then \(b\) units along \(y\). +

+

Example: \((-3,2)\) lies in Quadrant II.

+ +

Distance between two points

+

+ For \(A(x_1,y_1)\), \(B(x_2,y_2)\): +

+

\[ + AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. + \]

+

+ This comes from Pythagoras' theorem on horizontal and vertical differences. +

+ +

Midpoint of a segment

+

\[ + M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). + \]

+

Example: Midpoint of \((2,5)\) and \((8,-1)\) is \((5,2)\).

+ +

Gradient (slope) of a line

+

+ For points \(A(x_1,y_1)\), \(B(x_2,y_2)\) with \(x_2\ne x_1\): +

+

\[ + m=\frac{y_2-y_1}{x_2-x_1}. + \]

+ + +

Equations of lines

+

+ Slope-intercept form: +

+

\[ + y=mx+c + \]

+

+ where \(m\) is slope and \(c\) is \(y\)-intercept. +

+

+ Point-slope form: +

+

\[ + y-y_1=m(x-x_1). + \]

+ +

Parallel and perpendicular lines

+ + +

Exam strategy

+ +
+ +
+

History

+

+ 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. +

+ +

Derivation and reasoning

+

Why the distance formula works

+

+ 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 \(\sqrt{(\Delta x)^2+(\Delta y)^2}\). +

+ +

Checkpoints

+
+ State the quadrant of \((-4,-7)\). +
+

Answer: Quadrant III.

+
+
+
+ Find distance between \((1,2)\) and \((7,10)\). +
+

Answer: \(\sqrt{6^2+8^2}=\sqrt{100}=10\).

+
+
+
+ Find midpoint of \((-3,5)\) and \((9,-1)\). +
+

Answer: \(\left(3,2\right)\).

+
+
+
+ Find slope through \((2,-1)\) and \((6,7)\). +
+

Answer: \(m=\frac{7-(-1)}{6-2}=\frac84=2\).

+
+
+
+ Line with slope \(3\) through \((1,4)\): find equation. +
+

Answer: \(y-4=3(x-1)\Rightarrow y=3x+1\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/coordinate-geometry-foundations/core.html b/content/math/coordinate-geometry-foundations/core.html new file mode 100644 index 0000000..5d6f716 --- /dev/null +++ b/content/math/coordinate-geometry-foundations/core.html @@ -0,0 +1,88 @@ +
+

+ Coordinate geometry connects algebra and geometry by representing points with ordered pairs + \((x,y)\) on the Cartesian plane. +

+ +

The Cartesian plane

+

+ Horizontal axis is \(x\)-axis, vertical axis is \(y\)-axis, and intersection is origin \((0,0)\). +

+

+ Quadrants: +

+ + +

Plotting and reading points

+

+ Ordered pair \((a,b)\) means move \(a\) units along \(x\), then \(b\) units along \(y\). +

+

Example: \((-3,2)\) lies in Quadrant II.

+ +

Distance between two points

+

+ For \(A(x_1,y_1)\), \(B(x_2,y_2)\): +

+

\[ + AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. + \]

+

+ This comes from Pythagoras' theorem on horizontal and vertical differences. +

+ +

Midpoint of a segment

+

\[ + M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right). + \]

+

Example: Midpoint of \((2,5)\) and \((8,-1)\) is \((5,2)\).

+ +

Gradient (slope) of a line

+

+ For points \(A(x_1,y_1)\), \(B(x_2,y_2)\) with \(x_2\ne x_1\): +

+

\[ + m=\frac{y_2-y_1}{x_2-x_1}. + \]

+ + +

Equations of lines

+

+ Slope-intercept form: +

+

\[ + y=mx+c + \]

+

+ where \(m\) is slope and \(c\) is \(y\)-intercept. +

+

+ Point-slope form: +

+

\[ + y-y_1=m(x-x_1). + \]

+ +

Parallel and perpendicular lines

+ + +

Exam strategy

+ +
diff --git a/content/math/coordinate-geometry-foundations/supplement.html b/content/math/coordinate-geometry-foundations/supplement.html new file mode 100644 index 0000000..d4c74fa --- /dev/null +++ b/content/math/coordinate-geometry-foundations/supplement.html @@ -0,0 +1,59 @@ +
+

History

+

+ 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. +

+ +

Derivation and reasoning

+

Why the distance formula works

+

+ 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 \(\sqrt{(\Delta x)^2+(\Delta y)^2}\). +

+ +

Checkpoints

+
+ State the quadrant of \((-4,-7)\). +
+

Answer: Quadrant III.

+
+
+
+ Find distance between \((1,2)\) and \((7,10)\). +
+

Answer: \(\sqrt{6^2+8^2}=\sqrt{100}=10\).

+
+
+
+ Find midpoint of \((-3,5)\) and \((9,-1)\). +
+

Answer: \(\left(3,2\right)\).

+
+
+
+ Find slope through \((2,-1)\) and \((6,7)\). +
+

Answer: \(m=\frac{7-(-1)}{6-2}=\frac84=2\).

+
+
+
+ Line with slope \(3\) through \((1,4)\): find equation. +
+

Answer: \(y-4=3(x-1)\Rightarrow y=3x+1\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/deductive-geometry/_index.md b/content/math/deductive-geometry/_index.md new file mode 100644 index 0000000..8644eba --- /dev/null +++ b/content/math/deductive-geometry/_index.md @@ -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" ++++ + +
+

+ 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. +

+ +

Reasoning language

+ + +

Core angle facts

+ + +

Triangle facts

+ + +

Polygon facts

+

+ Sum of interior angles of \(n\)-gon: +

+

\[ + (n-2)\times180^\circ. + \]

+

+ Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\). +

+ +

Proof structure

+
    +
  1. Write what is given and what must be shown.
    2. +
  2. Mark known equal angles/lengths on diagram.
    3. +
  3. Use relevant theorems in short logical chain.
    4. +
  4. End with clear conclusion sentence.
    5. +
+ +

Example proof pattern

+

+ If \(AB\parallel CD\) and line \(AD\) intersects them, prove one pair of alternate interior angles + is equal. +

+

+ Since \(AB\parallel CD\), alternate interior angles are equal (parallel-line theorem). Therefore + required pair is equal. +

+ +

Common proof errors

+ + +

Exam strategy

+ +
+ +
+

History

+

+ Deductive geometry traces to Euclid's Elements, where geometric knowledge was organized + into axioms, definitions, and logically derived propositions. This method shaped modern proof. +

+ +

Derivation and reasoning

+

Why proof matters

+

+ Measurement drawings can suggest answers, but proofs guarantee truth for all valid cases. Deductive + reasoning removes uncertainty caused by drawing scale or rounding. +

+ +

Checkpoints

+
+ + Two angles on a straight line are \(3x\) and \(x+40\). Find \(x\). + +
+

Answer: \(3x+(x+40)=180\Rightarrow 4x=140\Rightarrow x=35\).

+
+
+
+ + In a triangle, angles are \(x,2x,3x\). Find all three angles. + +
+

Answer: \(x+2x+3x=180\Rightarrow x=30\).

+

Angles: \(30^\circ,60^\circ,90^\circ\).

+
+
+
+ + If corresponding angles are equal for a transversal, what can you conclude? + +
+

Answer: The two lines are parallel (converse theorem).

+
+
+
+ + Find interior-angle sum of a decagon. + +
+

Answer: \((10-2)\times180=1440^\circ\).

+
+
+
+ + Explain one reason why “it looks equal” is not valid proof. + +
+

Answer: Diagram scale can be inaccurate; only theorem-based logic is reliable.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/deductive-geometry/core.html b/content/math/deductive-geometry/core.html new file mode 100644 index 0000000..82f0d93 --- /dev/null +++ b/content/math/deductive-geometry/core.html @@ -0,0 +1,79 @@ +
+

+ 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. +

+ +

Reasoning language

+ + +

Core angle facts

+ + +

Triangle facts

+ + +

Polygon facts

+

+ Sum of interior angles of \(n\)-gon: +

+

\[ + (n-2)\times180^\circ. + \]

+

+ Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\). +

+ +

Proof structure

+
    +
  1. Write what is given and what must be shown.
    2. +
  2. Mark known equal angles/lengths on diagram.
    3. +
  3. Use relevant theorems in short logical chain.
    4. +
  4. End with clear conclusion sentence.
    5. +
+ +

Example proof pattern

+

+ If \(AB\parallel CD\) and line \(AD\) intersects them, prove one pair of alternate interior angles + is equal. +

+

+ Since \(AB\parallel CD\), alternate interior angles are equal (parallel-line theorem). Therefore + required pair is equal. +

+ +

Common proof errors

+ + +

Exam strategy

+ +
diff --git a/content/math/deductive-geometry/supplement.html b/content/math/deductive-geometry/supplement.html new file mode 100644 index 0000000..f7c74e4 --- /dev/null +++ b/content/math/deductive-geometry/supplement.html @@ -0,0 +1,70 @@ +
+

History

+

+ Deductive geometry traces to Euclid's Elements, where geometric knowledge was organized + into axioms, definitions, and logically derived propositions. This method shaped modern proof. +

+ +

Derivation and reasoning

+

Why proof matters

+

+ Measurement drawings can suggest answers, but proofs guarantee truth for all valid cases. Deductive + reasoning removes uncertainty caused by drawing scale or rounding. +

+ +

Checkpoints

+
+ + Two angles on a straight line are \(3x\) and \(x+40\). Find \(x\). + +
+

Answer: \(3x+(x+40)=180\Rightarrow 4x=140\Rightarrow x=35\).

+
+
+
+ + In a triangle, angles are \(x,2x,3x\). Find all three angles. + +
+

Answer: \(x+2x+3x=180\Rightarrow x=30\).

+

Angles: \(30^\circ,60^\circ,90^\circ\).

+
+
+
+ + If corresponding angles are equal for a transversal, what can you conclude? + +
+

Answer: The two lines are parallel (converse theorem).

+
+
+
+ + Find interior-angle sum of a decagon. + +
+

Answer: \((10-2)\times180=1440^\circ\).

+
+
+
+ + Explain one reason why “it looks equal” is not valid proof. + +
+

Answer: Diagram scale can be inaccurate; only theorem-based logic is reliable.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/introduction-trigonometry/_index.md b/content/math/introduction-trigonometry/_index.md new file mode 100644 index 0000000..7dc9695 --- /dev/null +++ b/content/math/introduction-trigonometry/_index.md @@ -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" ++++ + +
+

+ Trigonometry relates angles and side lengths in right triangles. The three core ratios are sine, + cosine, and tangent. +

+ +

Right-triangle ratios

+

+ For angle \(\theta\) in a right triangle: +

+

\[ + \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad + \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad + \tan\theta=\frac{\text{opposite}}{\text{adjacent}}. + \]

+

+ Remember: opposite/adjacent depend on the chosen angle. +

+ +

Common exact values

+

+ For \(30^\circ,45^\circ,60^\circ\): +

+ + +

Finding missing sides

+

Example. Hypotenuse \(=10\), angle \(=37^\circ\). Find opposite side \(x\):

+

\[ + \sin37^\circ=\frac{x}{10}\Rightarrow x=10\sin37^\circ\approx 6.02. + \]

+ +

Finding missing angles

+

Example. Opposite \(=5\), adjacent \(=12\). Find \(\theta\):

+

\[ + \tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 22.6^\circ. + \]

+ +

Angles of elevation and depression

+

+ Angle of elevation looks upward from horizontal; angle of depression looks downward. Draw a clear + diagram and label horizontal lines before forming trig equations. +

+ +

Link with Pythagoras

+

+ In right triangles, trigonometry and Pythagoras work together: first find one side/angle, then use + the other method to complete the triangle. +

+ +

Exam strategy

+ +
+ +
+

History

+

+ 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. +

+ +

Derivation and reasoning

+

Why trig ratios depend only on angle

+

+ 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. +

+ +

Checkpoints

+
+ Find \(\sin30^\circ\) and \(\cos60^\circ\). +
+

Answer: Both are \(\frac12\).

+
+
+
+ In a right triangle, opposite to \(\theta\) is 9 and hypotenuse is 15. Find \(\sin\theta\). +
+

Answer: \(\sin\theta=\frac{9}{15}=\frac35\).

+
+
+
+ Adjacent is 7, hypotenuse is 25. Find \(\cos\theta\). +
+

Answer: \(\cos\theta=\frac{7}{25}\).

+
+
+
+ If \(\tan\theta=0.75\), find \(\theta\) (to 1 d.p.). +
+

Answer: \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\).

+
+
+
+ A ladder leans against a wall. Foot is 4 m from wall and ladder length is 6 m. Find angle with ground. +
+

Answer: \(\cos\theta=\frac{4}{6}=\frac23\Rightarrow \theta\approx 48.2^\circ\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/introduction-trigonometry/core.html b/content/math/introduction-trigonometry/core.html new file mode 100644 index 0000000..d002ed3 --- /dev/null +++ b/content/math/introduction-trigonometry/core.html @@ -0,0 +1,61 @@ +
+

+ Trigonometry relates angles and side lengths in right triangles. The three core ratios are sine, + cosine, and tangent. +

+ +

Right-triangle ratios

+

+ For angle \(\theta\) in a right triangle: +

+

\[ + \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad + \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad + \tan\theta=\frac{\text{opposite}}{\text{adjacent}}. + \]

+

+ Remember: opposite/adjacent depend on the chosen angle. +

+ +

Common exact values

+

+ For \(30^\circ,45^\circ,60^\circ\): +

+ + +

Finding missing sides

+

Example. Hypotenuse \(=10\), angle \(=37^\circ\). Find opposite side \(x\):

+

\[ + \sin37^\circ=\frac{x}{10}\Rightarrow x=10\sin37^\circ\approx 6.02. + \]

+ +

Finding missing angles

+

Example. Opposite \(=5\), adjacent \(=12\). Find \(\theta\):

+

\[ + \tan\theta=\frac{5}{12}\Rightarrow \theta=\tan^{-1}\!\left(\frac{5}{12}\right)\approx 22.6^\circ. + \]

+ +

Angles of elevation and depression

+

+ Angle of elevation looks upward from horizontal; angle of depression looks downward. Draw a clear + diagram and label horizontal lines before forming trig equations. +

+ +

Link with Pythagoras

+

+ In right triangles, trigonometry and Pythagoras work together: first find one side/angle, then use + the other method to complete the triangle. +

+ +

Exam strategy

+ +
diff --git a/content/math/introduction-trigonometry/supplement.html b/content/math/introduction-trigonometry/supplement.html new file mode 100644 index 0000000..ff9d5c3 --- /dev/null +++ b/content/math/introduction-trigonometry/supplement.html @@ -0,0 +1,59 @@ +
+

History

+

+ 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. +

+ +

Derivation and reasoning

+

Why trig ratios depend only on angle

+

+ 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. +

+ +

Checkpoints

+
+ Find \(\sin30^\circ\) and \(\cos60^\circ\). +
+

Answer: Both are \(\frac12\).

+
+
+
+ In a right triangle, opposite to \(\theta\) is 9 and hypotenuse is 15. Find \(\sin\theta\). +
+

Answer: \(\sin\theta=\frac{9}{15}=\frac35\).

+
+
+
+ Adjacent is 7, hypotenuse is 25. Find \(\cos\theta\). +
+

Answer: \(\cos\theta=\frac{7}{25}\).

+
+
+
+ If \(\tan\theta=0.75\), find \(\theta\) (to 1 d.p.). +
+

Answer: \(\theta=\tan^{-1}(0.75)\approx 36.9^\circ\).

+
+
+
+ A ladder leans against a wall. Foot is 4 m from wall and ladder length is 6 m. Find angle with ground. +
+

Answer: \(\cos\theta=\frac{4}{6}=\frac23\Rightarrow \theta\approx 48.2^\circ\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/loci/_index.md b/content/math/loci/_index.md new file mode 100644 index 0000000..add2aee --- /dev/null +++ b/content/math/loci/_index.md @@ -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" ++++ + +
+

+ A locus is the set of all points that satisfy a given condition. Loci connect + geometric constructions with equations and constraints. +

+ +

Basic loci patterns

+ + +

Constructing loci

+
    +
  1. Translate condition into known locus type.
    2. +
  2. Construct each required locus accurately.
    3. +
  3. Find intersection of loci for final solution points/region.
    4. +
+ +

Loci and inequalities

+

+ Conditions like “distance at most 3 cm from line” describe a region, not a single line. +

+

+ Boundary inclusion matters: +

+ + +

Algebra links

+

+ Coordinate forms: +

+ + +

Modeling with loci

+

+ Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints. +

+ +

Exam strategy

+ +
+ +
+

History

+

+ Locus ideas emerged from classical compass-and-straightedge geometry and later became a core part + of analytic geometry and optimization problems. +

+ +

Checkpoints

+
+ Describe the locus of points 5 cm from point \(A\). +
+

Answer: A circle center \(A\), radius 5 cm.

+
+
+
+ What is locus of points equidistant from \(A\) and \(B\)? +
+

Answer: Perpendicular bisector of segment \(AB\).

+
+
+
+ Points within 2 m of line \(l\): describe region. +
+

Answer: Strip between two parallels 2 m from \(l\), including boundaries if “within or equal to”.

+
+
+
+ Points equidistant from intersecting lines \(p,q\): what locus? +
+

Answer: The two angle bisectors of lines \(p,q\).

+
+
+
+ Why can a locus answer be a region instead of a curve? +
+

Answer: Inequality conditions describe many points satisfying a distance range.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/loci/core.html b/content/math/loci/core.html new file mode 100644 index 0000000..784e61a --- /dev/null +++ b/content/math/loci/core.html @@ -0,0 +1,55 @@ +
+

+ A locus is the set of all points that satisfy a given condition. Loci connect + geometric constructions with equations and constraints. +

+ +

Basic loci patterns

+ + +

Constructing loci

+
    +
  1. Translate condition into known locus type.
    2. +
  2. Construct each required locus accurately.
    3. +
  3. Find intersection of loci for final solution points/region.
    4. +
+ +

Loci and inequalities

+

+ Conditions like “distance at most 3 cm from line” describe a region, not a single line. +

+

+ Boundary inclusion matters: +

+ + +

Algebra links

+

+ Coordinate forms: +

+ + +

Modeling with loci

+

+ Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints. +

+ +

Exam strategy

+ +
diff --git a/content/math/loci/supplement.html b/content/math/loci/supplement.html new file mode 100644 index 0000000..0080372 --- /dev/null +++ b/content/math/loci/supplement.html @@ -0,0 +1,52 @@ +
+

History

+

+ Locus ideas emerged from classical compass-and-straightedge geometry and later became a core part + of analytic geometry and optimization problems. +

+ +

Checkpoints

+
+ Describe the locus of points 5 cm from point \(A\). +
+

Answer: A circle center \(A\), radius 5 cm.

+
+
+
+ What is locus of points equidistant from \(A\) and \(B\)? +
+

Answer: Perpendicular bisector of segment \(AB\).

+
+
+
+ Points within 2 m of line \(l\): describe region. +
+

Answer: Strip between two parallels 2 m from \(l\), including boundaries if “within or equal to”.

+
+
+
+ Points equidistant from intersecting lines \(p,q\): what locus? +
+

Answer: The two angle bisectors of lines \(p,q\).

+
+
+
+ Why can a locus answer be a region instead of a curve? +
+

Answer: Inequality conditions describe many points satisfying a distance range.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/measures-dispersion/_index.md b/content/math/measures-dispersion/_index.md new file mode 100644 index 0000000..0ea3c6d --- /dev/null +++ b/content/math/measures-dispersion/_index.md @@ -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" ++++ + +
+

+ Dispersion describes how spread out data is. Two datasets can have the same average but very + different variability. +

+ +

Range and interquartile range

+

\[ + \text{Range}=\text{max}-\text{min},\qquad + \text{IQR}=Q_3-Q_1. + \]

+

+ IQR is less affected by outliers than range. +

+ +

Variance and standard deviation

+

+ Variance measures average squared distance from mean. + Standard deviation is square root of variance and has same unit as data. +

+

\[ + \sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}} + \]

+ +

Comparing datasets

+ + +

Box plot interpretation

+

+ Box shows middle 50% (\(Q_1\) to \(Q_3\)); median line marks center. + Longer whisker/box indicates larger spread. +

+ +

Exam strategy

+ +
+ +
+

Checkpoints

+
+ Find range of 12, 9, 15, 20, 13. +
+

Answer: \(20-9=11\).

+
+
+
+ If \(Q_1=18\), \(Q_3=31\), find IQR. +
+

Answer: 13.

+
+
+
+ Dataset A and B have same mean. A has SD 2, B has SD 8. Which is more consistent? +
+

Answer: A (smaller SD).

+
+
+
+ Why is IQR preferred over range when outliers exist? +
+

Answer: IQR focuses on middle 50% and is less distorted by extreme values.

+
+
+
diff --git a/content/math/measures-dispersion/core.html b/content/math/measures-dispersion/core.html new file mode 100644 index 0000000..785d53d --- /dev/null +++ b/content/math/measures-dispersion/core.html @@ -0,0 +1,45 @@ +
+

+ Dispersion describes how spread out data is. Two datasets can have the same average but very + different variability. +

+ +

Range and interquartile range

+

\[ + \text{Range}=\text{max}-\text{min},\qquad + \text{IQR}=Q_3-Q_1. + \]

+

+ IQR is less affected by outliers than range. +

+ +

Variance and standard deviation

+

+ Variance measures average squared distance from mean. + Standard deviation is square root of variance and has same unit as data. +

+

\[ + \sigma=\sqrt{\frac{\sum (x-\bar{x})^2}{n}} + \]

+ +

Comparing datasets

+ + +

Box plot interpretation

+

+ Box shows middle 50% (\(Q_1\) to \(Q_3\)); median line marks center. + Longer whisker/box indicates larger spread. +

+ +

Exam strategy

+ +
diff --git a/content/math/measures-dispersion/supplement.html b/content/math/measures-dispersion/supplement.html new file mode 100644 index 0000000..ac30983 --- /dev/null +++ b/content/math/measures-dispersion/supplement.html @@ -0,0 +1,27 @@ +
+

Checkpoints

+
+ Find range of 12, 9, 15, 20, 13. +
+

Answer: \(20-9=11\).

+
+
+
+ If \(Q_1=18\), \(Q_3=31\), find IQR. +
+

Answer: 13.

+
+
+
+ Dataset A and B have same mean. A has SD 2, B has SD 8. Which is more consistent? +
+

Answer: A (smaller SD).

+
+
+
+ Why is IQR preferred over range when outliers exist? +
+

Answer: IQR focuses on middle 50% and is less distorted by extreme values.

+
+
+
diff --git a/content/math/mensuration-3d/_index.md b/content/math/mensuration-3d/_index.md new file mode 100644 index 0000000..cc0a328 --- /dev/null +++ b/content/math/mensuration-3d/_index.md @@ -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" ++++ + +
+

+ Mensuration in 3D focuses on volume (space inside solids) and surface area (total outer area). +

+ +

Units and interpretation

+ +

+ Always keep units consistent before substituting values. +

+ +

Prisms and cylinders

+

+ For any prism/cylinder: +

+

\[ + \text{Volume}=\text{base area}\times\text{height}. + \]

+

+ Cylinder formulas: +

+

\[ + V=\pi r^2h,\qquad + \text{TSA}=2\pi r^2+2\pi rh. + \]

+ +

Pyramids and cones

+

\[ + V_{\text{pyramid}}=\frac13(\text{base area})h,\qquad + V_{\text{cone}}=\frac13\pi r^2h. + \]

+

+ Cone curved surface area: +

+

\[ + \text{CSA}_{\text{cone}}=\pi rl + \]

+

+ where \(l\) is slant height. Total surface area adds base area \(\pi r^2\). +

+ +

Spheres

+

\[ + V=\frac43\pi r^3,\qquad + \text{Surface area}=4\pi r^2. + \]

+ +

Composite solids

+

+ Split complex shapes into known solids, then add/subtract volumes or areas appropriately. +

+ + +

Volume in context

+

+ Common contexts include capacity, material usage, and mass: +

+

\[ + \text{mass}=\text{density}\times\text{volume}. + \]

+ +

Exam strategy

+ +
+ +
+

History

+

+ Mensuration developed from practical needs in architecture, agriculture, and trade. Ancient + mathematicians studied area and volume formulas, with later refinement by Archimedes and others. +

+ +

Derivation and reasoning

+

Why pyramid and cone have one-third factor

+

+ 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. +

+ +

Checkpoints

+
+ Find volume of a cylinder with \(r=3\) cm, \(h=10\) cm. +
+

Answer: \(V=\pi r^2h=90\pi\) cm\(^3\).

+
+
+
+ Find total surface area of a closed cylinder with \(r=2\) cm, \(h=7\) cm. +
+

Answer: \(2\pi r^2+2\pi rh=8\pi+28\pi=36\pi\) cm\(^2\).

+
+
+
+ Find volume of a cone with \(r=6\) cm, \(h=9\) cm. +
+

Answer: \(\frac13\pi r^2h=\frac13\pi(36)(9)=108\pi\) cm\(^3\).

+
+
+
+ Sphere has radius 5 cm. Find its volume. +
+

Answer: \(V=\frac43\pi(5^3)=\frac{500}{3}\pi\) cm\(^3\).

+
+
+
+ A block is \(4\times3\times2\) m. Density is \(2500\) kg/m\(^3\). Find mass. +
+

Answer: Volume \(=24\) m\(^3\). Mass \(=2500\times24=60000\) kg.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/mensuration-3d/core.html b/content/math/mensuration-3d/core.html new file mode 100644 index 0000000..2d83268 --- /dev/null +++ b/content/math/mensuration-3d/core.html @@ -0,0 +1,77 @@ +
+

+ Mensuration in 3D focuses on volume (space inside solids) and surface area (total outer area). +

+ +

Units and interpretation

+ +

+ Always keep units consistent before substituting values. +

+ +

Prisms and cylinders

+

+ For any prism/cylinder: +

+

\[ + \text{Volume}=\text{base area}\times\text{height}. + \]

+

+ Cylinder formulas: +

+

\[ + V=\pi r^2h,\qquad + \text{TSA}=2\pi r^2+2\pi rh. + \]

+ +

Pyramids and cones

+

\[ + V_{\text{pyramid}}=\frac13(\text{base area})h,\qquad + V_{\text{cone}}=\frac13\pi r^2h. + \]

+

+ Cone curved surface area: +

+

\[ + \text{CSA}_{\text{cone}}=\pi rl + \]

+

+ where \(l\) is slant height. Total surface area adds base area \(\pi r^2\). +

+ +

Spheres

+

\[ + V=\frac43\pi r^3,\qquad + \text{Surface area}=4\pi r^2. + \]

+ +

Composite solids

+

+ Split complex shapes into known solids, then add/subtract volumes or areas appropriately. +

+ + +

Volume in context

+

+ Common contexts include capacity, material usage, and mass: +

+

\[ + \text{mass}=\text{density}\times\text{volume}. + \]

+ +

Exam strategy

+ +
diff --git a/content/math/mensuration-3d/supplement.html b/content/math/mensuration-3d/supplement.html new file mode 100644 index 0000000..146056a --- /dev/null +++ b/content/math/mensuration-3d/supplement.html @@ -0,0 +1,59 @@ +
+

History

+

+ Mensuration developed from practical needs in architecture, agriculture, and trade. Ancient + mathematicians studied area and volume formulas, with later refinement by Archimedes and others. +

+ +

Derivation and reasoning

+

Why pyramid and cone have one-third factor

+

+ 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. +

+ +

Checkpoints

+
+ Find volume of a cylinder with \(r=3\) cm, \(h=10\) cm. +
+

Answer: \(V=\pi r^2h=90\pi\) cm\(^3\).

+
+
+
+ Find total surface area of a closed cylinder with \(r=2\) cm, \(h=7\) cm. +
+

Answer: \(2\pi r^2+2\pi rh=8\pi+28\pi=36\pi\) cm\(^2\).

+
+
+
+ Find volume of a cone with \(r=6\) cm, \(h=9\) cm. +
+

Answer: \(\frac13\pi r^2h=\frac13\pi(36)(9)=108\pi\) cm\(^3\).

+
+
+
+ Sphere has radius 5 cm. Find its volume. +
+

Answer: \(V=\frac43\pi(5^3)=\frac{500}{3}\pi\) cm\(^3\).

+
+
+
+ A block is \(4\times3\times2\) m. Density is \(2500\) kg/m\(^3\). Find mass. +
+

Answer: Volume \(=24\) m\(^3\). Mass \(=2500\times24=60000\) kg.

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/normal-distribution-z-scores/_index.md b/content/math/normal-distribution-z-scores/_index.md new file mode 100644 index 0000000..2a650a6 --- /dev/null +++ b/content/math/normal-distribution-z-scores/_index.md @@ -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" ++++ + +
+

+ The normal distribution is a symmetric bell-shaped model for many real measurements. +

+ +

Normal model basics

+

+ A normal variable \(X\sim N(\mu,\sigma^2)\) has mean \(\mu\) and standard deviation \(\sigma\). +

+ + +

Standardizing with z-score

+

\[ + z=\frac{x-\mu}{\sigma}. + \]

+

+ This converts values to the standard normal \(Z\sim N(0,1)\). +

+ +

Finding probabilities

+
    +
  1. Convert boundary value(s) to z-score(s).
    2. +
  2. Use normal table/calculator for area.
    3. +
  3. Apply complement or subtraction if needed.
    4. +
+ +

Finding values from probabilities

+

+ If \(P(X\le x)=p\), find corresponding \(z_p\), then: +

+

\[ + x=\mu+z_p\sigma. + \]

+ +

Exam strategy

+ +
+ +
+

Checkpoints

+
+ If \(\mu=70,\sigma=10\), find z-score of \(x=85\). +
+

Answer: \(z=\frac{85-70}{10}=1.5\).

+
+
+
+ If \(z=-0.8\), \(\mu=50,\sigma=5\), find \(x\). +
+

Answer: \(x=50+(-0.8)(5)=46\).

+
+
+
+ In a normal model, what does \(P(Z>2)\) represent? +
+

Answer: Area to the right of z=2 (upper-tail probability).

+
+
+
+ Why do we standardize before using z-tables? +
+

Answer: Tables are for \(N(0,1)\), so we convert \(X\) to \(Z\).

+
+
+
diff --git a/content/math/normal-distribution-z-scores/core.html b/content/math/normal-distribution-z-scores/core.html new file mode 100644 index 0000000..554ceeb --- /dev/null +++ b/content/math/normal-distribution-z-scores/core.html @@ -0,0 +1,46 @@ +
+

+ The normal distribution is a symmetric bell-shaped model for many real measurements. +

+ +

Normal model basics

+

+ A normal variable \(X\sim N(\mu,\sigma^2)\) has mean \(\mu\) and standard deviation \(\sigma\). +

+ + +

Standardizing with z-score

+

\[ + z=\frac{x-\mu}{\sigma}. + \]

+

+ This converts values to the standard normal \(Z\sim N(0,1)\). +

+ +

Finding probabilities

+
    +
  1. Convert boundary value(s) to z-score(s).
    2. +
  2. Use normal table/calculator for area.
    3. +
  3. Apply complement or subtraction if needed.
    4. +
+ +

Finding values from probabilities

+

+ If \(P(X\le x)=p\), find corresponding \(z_p\), then: +

+

\[ + x=\mu+z_p\sigma. + \]

+ +

Exam strategy

+ +
diff --git a/content/math/normal-distribution-z-scores/supplement.html b/content/math/normal-distribution-z-scores/supplement.html new file mode 100644 index 0000000..a3319f7 --- /dev/null +++ b/content/math/normal-distribution-z-scores/supplement.html @@ -0,0 +1,27 @@ +
+

Checkpoints

+
+ If \(\mu=70,\sigma=10\), find z-score of \(x=85\). +
+

Answer: \(z=\frac{85-70}{10}=1.5\).

+
+
+
+ If \(z=-0.8\), \(\mu=50,\sigma=5\), find \(x\). +
+

Answer: \(x=50+(-0.8)(5)=46\).

+
+
+
+ In a normal model, what does \(P(Z>2)\) represent? +
+

Answer: Area to the right of z=2 (upper-tail probability).

+
+
+
+ Why do we standardize before using z-tables? +
+

Answer: Tables are for \(N(0,1)\), so we convert \(X\) to \(Z\).

+
+
+
diff --git a/content/math/permutations-combinations-probability/_index.md b/content/math/permutations-combinations-probability/_index.md new file mode 100644 index 0000000..eb4e7e6 --- /dev/null +++ b/content/math/permutations-combinations-probability/_index.md @@ -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" ++++ + +
+

+ Counting methods make probability efficient when listing all outcomes is impractical. +

+ +

Factorial

+

\[ + n!=n(n-1)(n-2)\cdots 1,\quad 0!=1. + \]

+ +

Permutations (order matters)

+

\[ + {}_nP_r=\frac{n!}{(n-r)!}. + \]

+

+ Use for arrangements where position/order is important. +

+ +

Combinations (order not important)

+

\[ + {}_nC_r=\frac{n!}{r!(n-r)!}. + \]

+

+ Use for selections/groups. +

+ +

Probability with counting

+

\[ + P(E)=\frac{\text{count of favorable outcomes}}{\text{count of all outcomes}}. + \]

+ +

With/without replacement

+ + +

Exam strategy

+ +
+ +
+

Checkpoints

+
+ Compute \({}_7P_3\). +
+

Answer: \(7\times6\times5=210\).

+
+
+
+ Compute \({}_8C_2\). +
+

Answer: \(\frac{8\cdot7}{2}=28\).

+
+
+
+ How many 3-letter arrangements from A,B,C,D (no repeats)? +
+

Answer: \({}_4P_3=24\).

+
+
+
+ How many ways to choose 3 students from 10? +
+

Answer: \({}_{10}C_3=120\).

+
+
+
+ A 5-card hand from 52 cards: probability all hearts. +
+

Answer: \(\frac{{}_{13}C_5}{{}_{52}C_5}\).

+
+
+
diff --git a/content/math/permutations-combinations-probability/core.html b/content/math/permutations-combinations-probability/core.html new file mode 100644 index 0000000..0a7e1fe --- /dev/null +++ b/content/math/permutations-combinations-probability/core.html @@ -0,0 +1,45 @@ +
+

+ Counting methods make probability efficient when listing all outcomes is impractical. +

+ +

Factorial

+

\[ + n!=n(n-1)(n-2)\cdots 1,\quad 0!=1. + \]

+ +

Permutations (order matters)

+

\[ + {}_nP_r=\frac{n!}{(n-r)!}. + \]

+

+ Use for arrangements where position/order is important. +

+ +

Combinations (order not important)

+

\[ + {}_nC_r=\frac{n!}{r!(n-r)!}. + \]

+

+ Use for selections/groups. +

+ +

Probability with counting

+

\[ + P(E)=\frac{\text{count of favorable outcomes}}{\text{count of all outcomes}}. + \]

+ +

With/without replacement

+ + +

Exam strategy

+ +
diff --git a/content/math/permutations-combinations-probability/supplement.html b/content/math/permutations-combinations-probability/supplement.html new file mode 100644 index 0000000..9eea946 --- /dev/null +++ b/content/math/permutations-combinations-probability/supplement.html @@ -0,0 +1,33 @@ +
+

Checkpoints

+
+ Compute \({}_7P_3\). +
+

Answer: \(7\times6\times5=210\).

+
+
+
+ Compute \({}_8C_2\). +
+

Answer: \(\frac{8\cdot7}{2}=28\).

+
+
+
+ How many 3-letter arrangements from A,B,C,D (no repeats)? +
+

Answer: \({}_4P_3=24\).

+
+
+
+ How many ways to choose 3 students from 10? +
+

Answer: \({}_{10}C_3=120\).

+
+
+
+ A 5-card hand from 52 cards: probability all hearts. +
+

Answer: \(\frac{{}_{13}C_5}{{}_{52}C_5}\).

+
+
+
diff --git a/content/math/pythagoras-theorem/_index.md b/content/math/pythagoras-theorem/_index.md new file mode 100644 index 0000000..f26eace --- /dev/null +++ b/content/math/pythagoras-theorem/_index.md @@ -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" ++++ + +
+

+ Pythagoras' theorem links the side lengths of a right-angled triangle. It is one of the most + useful tools in geometry and coordinate problems. +

+ +

Pythagoras' theorem

+

+ In a right triangle with legs \(a,b\) and hypotenuse \(c\): +

+

\[ + a^2+b^2=c^2. + \]

+

+ The hypotenuse is always the side opposite the right angle and is the longest side. +

+ +

Finding a missing side

+

Example 1. If \(a=6\), \(b=8\), then:

+

\[ + c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10. + \]

+

Example 2. If \(c=13\), \(a=5\), then:

+

\[ + b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12. + \]

+ +

Converse of Pythagoras

+

+ If side lengths satisfy \(a^2+b^2=c^2\), the triangle is right-angled. +

+

+ Example: \(7,24,25\) gives \(7^2+24^2=49+576=625=25^2\), so right-angled. +

+ +

Classifying triangle type

+

+ For largest side \(c\): +

+ + +

Using in coordinate geometry

+

+ Distance between \(A(x_1,y_1)\) and \(B(x_2,y_2)\): +

+

\[ + AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. + \]

+

+ This is Pythagoras applied to horizontal and vertical changes. +

+ +

Typical applications

+ + +

Exam strategy

+ +
+ +
+

History

+

+ 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. +

+ +

Derivation and reasoning

+

Area-based proof idea

+

+ A classic proof rearranges four identical right triangles inside a square in two different ways. + Equating uncovered areas leads to \(a^2+b^2=c^2\). +

+ +

Checkpoints

+
+ Find hypotenuse if legs are 9 and 12. +
+

Answer: \(c=\sqrt{9^2+12^2}=\sqrt{225}=15\).

+
+
+
+ A right triangle has hypotenuse 17 and one leg 8. Find the other leg. +
+

Answer: \(\sqrt{17^2-8^2}=\sqrt{225}=15\).

+
+
+
+ Are sides 10, 24, 26 right-angled? +
+

Answer: Yes, \(10^2+24^2=100+576=676=26^2\).

+
+
+
+ Distance between \((-2,1)\) and \((4,9)\). +
+

Answer: \(\sqrt{6^2+8^2}=10\).

+
+
+
+ Rectangle has side lengths 7 and 24. Find diagonal. +
+

Answer: \(\sqrt{7^2+24^2}=25\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/pythagoras-theorem/core.html b/content/math/pythagoras-theorem/core.html new file mode 100644 index 0000000..3829759 --- /dev/null +++ b/content/math/pythagoras-theorem/core.html @@ -0,0 +1,72 @@ +
+

+ Pythagoras' theorem links the side lengths of a right-angled triangle. It is one of the most + useful tools in geometry and coordinate problems. +

+ +

Pythagoras' theorem

+

+ In a right triangle with legs \(a,b\) and hypotenuse \(c\): +

+

\[ + a^2+b^2=c^2. + \]

+

+ The hypotenuse is always the side opposite the right angle and is the longest side. +

+ +

Finding a missing side

+

Example 1. If \(a=6\), \(b=8\), then:

+

\[ + c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10. + \]

+

Example 2. If \(c=13\), \(a=5\), then:

+

\[ + b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12. + \]

+ +

Converse of Pythagoras

+

+ If side lengths satisfy \(a^2+b^2=c^2\), the triangle is right-angled. +

+

+ Example: \(7,24,25\) gives \(7^2+24^2=49+576=625=25^2\), so right-angled. +

+ +

Classifying triangle type

+

+ For largest side \(c\): +

+ + +

Using in coordinate geometry

+

+ Distance between \(A(x_1,y_1)\) and \(B(x_2,y_2)\): +

+

\[ + AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. + \]

+

+ This is Pythagoras applied to horizontal and vertical changes. +

+ +

Typical applications

+ + +

Exam strategy

+ +
diff --git a/content/math/pythagoras-theorem/supplement.html b/content/math/pythagoras-theorem/supplement.html new file mode 100644 index 0000000..881f771 --- /dev/null +++ b/content/math/pythagoras-theorem/supplement.html @@ -0,0 +1,60 @@ +
+

History

+

+ 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. +

+ +

Derivation and reasoning

+

Area-based proof idea

+

+ A classic proof rearranges four identical right triangles inside a square in two different ways. + Equating uncovered areas leads to \(a^2+b^2=c^2\). +

+ +

Checkpoints

+
+ Find hypotenuse if legs are 9 and 12. +
+

Answer: \(c=\sqrt{9^2+12^2}=\sqrt{225}=15\).

+
+
+
+ A right triangle has hypotenuse 17 and one leg 8. Find the other leg. +
+

Answer: \(\sqrt{17^2-8^2}=\sqrt{225}=15\).

+
+
+
+ Are sides 10, 24, 26 right-angled? +
+

Answer: Yes, \(10^2+24^2=100+576=676=26^2\).

+
+
+
+ Distance between \((-2,1)\) and \((4,9)\). +
+

Answer: \(\sqrt{6^2+8^2}=10\).

+
+
+
+ Rectangle has side lengths 7 and 24. Find diagonal. +
+

Answer: \(\sqrt{7^2+24^2}=25\).

+
+
+ +

Applications

+ + +

References

+ +
diff --git a/content/math/statistical-charts-averages/_index.md b/content/math/statistical-charts-averages/_index.md new file mode 100644 index 0000000..9c6233e --- /dev/null +++ b/content/math/statistical-charts-averages/_index.md @@ -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" ++++ + +
+

+ Data handling starts with organizing data clearly, choosing suitable charts, and summarizing a + dataset with averages. +

+ +

Frequency tables

+

+ A frequency table records each value (or class interval) and how often it appears. + For grouped data, use class midpoints for estimation. +

+ +

Common charts

+ + +

Mean, median, and mode

+

\[ + \text{Mean}=\frac{\text{sum of values}}{\text{number of values}}. + \]

+

+ Median is middle value when ordered. Mode is most frequent value. +

+ +

Mean from frequency table

+

\[ + \bar{x}=\frac{\sum fx}{\sum f}. + \]

+

+ where \(x\) is value (or midpoint), \(f\) is frequency. +

+ +

Choosing an average

+ + +

Exam strategy

+ +
+ +
+

Checkpoints

+
+ Find mean of 4, 7, 7, 10, 12. +
+

Answer: \(\frac{40}{5}=8\).

+
+
+
+ Find median of 5, 2, 9, 4, 11, 6. +
+

Answer: Order: 2,4,5,6,9,11. Median \(=\frac{5+6}{2}=5.5\).

+
+
+
+ Find mode of 3, 5, 5, 5, 8, 8, 9. +
+

Answer: 5.

+
+
+
+ For values 1,2,3 with frequencies 4,5,1, find mean. +
+

Answer: \(\bar{x}=\frac{1(4)+2(5)+3(1)}{10}=\frac{17}{10}=1.7\).

+
+
+
+ Which average is best for house prices with extreme luxury homes? +
+

Answer: Median (less affected by outliers).

+
+
+
diff --git a/content/math/statistical-charts-averages/core.html b/content/math/statistical-charts-averages/core.html new file mode 100644 index 0000000..d113560 --- /dev/null +++ b/content/math/statistical-charts-averages/core.html @@ -0,0 +1,52 @@ +
+

+ Data handling starts with organizing data clearly, choosing suitable charts, and summarizing a + dataset with averages. +

+ +

Frequency tables

+

+ A frequency table records each value (or class interval) and how often it appears. + For grouped data, use class midpoints for estimation. +

+ +

Common charts

+ + +

Mean, median, and mode

+

\[ + \text{Mean}=\frac{\text{sum of values}}{\text{number of values}}. + \]

+

+ Median is middle value when ordered. Mode is most frequent value. +

+ +

Mean from frequency table

+

\[ + \bar{x}=\frac{\sum fx}{\sum f}. + \]

+

+ where \(x\) is value (or midpoint), \(f\) is frequency. +

+ +

Choosing an average

+ + +

Exam strategy

+ +
diff --git a/content/math/statistical-charts-averages/supplement.html b/content/math/statistical-charts-averages/supplement.html new file mode 100644 index 0000000..4c6a224 --- /dev/null +++ b/content/math/statistical-charts-averages/supplement.html @@ -0,0 +1,33 @@ +
+

Checkpoints

+
+ Find mean of 4, 7, 7, 10, 12. +
+

Answer: \(\frac{40}{5}=8\).

+
+
+
+ Find median of 5, 2, 9, 4, 11, 6. +
+

Answer: Order: 2,4,5,6,9,11. Median \(=\frac{5+6}{2}=5.5\).

+
+
+
+ Find mode of 3, 5, 5, 5, 8, 8, 9. +
+

Answer: 5.

+
+
+
+ For values 1,2,3 with frequencies 4,5,1, find mean. +
+

Answer: \(\bar{x}=\frac{1(4)+2(5)+3(1)}{10}=\frac{17}{10}=1.7\).

+
+
+
+ Which average is best for house prices with extreme luxury homes? +
+

Answer: Median (less affected by outliers).

+
+
+
diff --git a/content/math/statistical-literacy/_index.md b/content/math/statistical-literacy/_index.md new file mode 100644 index 0000000..b885fb4 --- /dev/null +++ b/content/math/statistical-literacy/_index.md @@ -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" ++++ + +
+

+ Statistical literacy is the ability to read claims critically, question data quality, and avoid + misleading conclusions. +

+ +

Questions to ask about a claim

+ + +

Sampling and bias

+

+ Biased samples produce unreliable conclusions. Prefer random, sufficiently large, and + representative samples. +

+ +

Correlation vs causation

+

+ Correlation means variables move together. It does not prove one causes the other. + Confounders may explain both. +

+ +

Misleading charts

+ + +

Interpreting risk language

+

+ Distinguish absolute change from relative change. + “Risk doubled” may still represent a very small absolute increase. +

+ +

Exam strategy

+ +
+ +
+

Checkpoints

+
+ A survey about school meals only asks students in one class. Name one issue. +
+

Answer: Sampling bias; sample may not represent the whole school.

+
+
+
+ Two variables are strongly correlated. Can we conclude causation? +
+

Answer: No, correlation alone is insufficient.

+
+
+
+ A bar chart axis starts at 95 instead of 0. Why can this mislead? +
+

Answer: Differences look much larger than they really are.

+
+
+
+ “Risk increased from 1% to 2%.” State absolute and relative increase. +
+

Answer: Absolute +1 percentage point; relative increase 100% (doubled).

+
+
+
+ Name one way to improve reliability of a poll. +
+

Answer: Use a larger random sample from the target population.

+
+
+
diff --git a/content/math/statistical-literacy/core.html b/content/math/statistical-literacy/core.html new file mode 100644 index 0000000..cedca42 --- /dev/null +++ b/content/math/statistical-literacy/core.html @@ -0,0 +1,48 @@ +
+

+ Statistical literacy is the ability to read claims critically, question data quality, and avoid + misleading conclusions. +

+ +

Questions to ask about a claim

+ + +

Sampling and bias

+

+ Biased samples produce unreliable conclusions. Prefer random, sufficiently large, and + representative samples. +

+ +

Correlation vs causation

+

+ Correlation means variables move together. It does not prove one causes the other. + Confounders may explain both. +

+ +

Misleading charts

+ + +

Interpreting risk language

+

+ Distinguish absolute change from relative change. + “Risk doubled” may still represent a very small absolute increase. +

+ +

Exam strategy

+ +
diff --git a/content/math/statistical-literacy/supplement.html b/content/math/statistical-literacy/supplement.html new file mode 100644 index 0000000..b022a70 --- /dev/null +++ b/content/math/statistical-literacy/supplement.html @@ -0,0 +1,33 @@ +
+

Checkpoints

+
+ A survey about school meals only asks students in one class. Name one issue. +
+

Answer: Sampling bias; sample may not represent the whole school.

+
+
+
+ Two variables are strongly correlated. Can we conclude causation? +
+

Answer: No, correlation alone is insufficient.

+
+
+
+ A bar chart axis starts at 95 instead of 0. Why can this mislead? +
+

Answer: Differences look much larger than they really are.

+
+
+
+ “Risk increased from 1% to 2%.” State absolute and relative increase. +
+

Answer: Absolute +1 percentage point; relative increase 100% (doubled).

+
+
+
+ Name one way to improve reliability of a poll. +
+

Answer: Use a larger random sample from the target population.

+
+
+
diff --git a/data/math-curriculum.json b/data/math-curriculum.json index 2a29c5a..fa9cedf 100644 --- a/data/math-curriculum.json +++ b/data/math-curriculum.json @@ -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", diff --git a/spec.md b/spec.md index 47593ae..24984bf 100644 --- a/spec.md +++ b/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 diff --git a/static/feed.xml b/static/feed.xml index 5103b49..81944c6 100644 --- a/static/feed.xml +++ b/static/feed.xml @@ -4,54 +4,173 @@ Open Free Books https://openfreebooks.org/ Recent updates to Open Free Books pages and live chapters. - Thu, 21 May 2026 07:32:39 GMT + Thu, 21 May 2026 16:43:43 GMT - Linear programming - https://openfreebooks.org/math/linear-programming/ - https://openfreebooks.org/math/linear-programming/ - Thu, 21 May 2026 07:32:39 GMT - Graph linear inequalities, identify feasible regions, and optimise objective functions in two variables. + Loci + https://openfreebooks.org/math/loci/ + https://openfreebooks.org/math/loci/ + Thu, 21 May 2026 16:43:43 GMT + Describe and construct loci in the plane, linking geometric constraints to algebraic equations. - Factor & remainder theorems - https://openfreebooks.org/math/factor-remainder-theorems/ - https://openfreebooks.org/math/factor-remainder-theorems/ - Thu, 21 May 2026 07:32:39 GMT - Use the factor and remainder theorems to factorise higher-degree polynomials and solve related equations. + Introduction to trigonometry + https://openfreebooks.org/math/introduction-trigonometry/ + https://openfreebooks.org/math/introduction-trigonometry/ + Thu, 21 May 2026 16:43:43 GMT + Define sine, cosine, and tangent ratios and solve right-angled triangle problems. - Exponential & logarithmic functions - https://openfreebooks.org/math/exponential-logarithmic/ - https://openfreebooks.org/math/exponential-logarithmic/ - Thu, 21 May 2026 07:32:39 GMT - Use exponential and logarithmic models, apply log laws, and solve growth and decay equations. + Permutations, combinations & probability + https://openfreebooks.org/math/permutations-combinations-probability/ + https://openfreebooks.org/math/permutations-combinations-probability/ + Thu, 21 May 2026 16:43:43 GMT + Count outcomes with permutations and combinations, then compute probabilities in finite sample spaces. + + + Mensuration in 3D + https://openfreebooks.org/math/mensuration-3d/ + https://openfreebooks.org/math/mensuration-3d/ + Thu, 21 May 2026 16:43:43 GMT + Find volumes and surface areas of prisms, cylinders, pyramids, cones, and spheres. + + + Normal distribution & z-scores + https://openfreebooks.org/math/normal-distribution-z-scores/ + https://openfreebooks.org/math/normal-distribution-z-scores/ + Thu, 21 May 2026 16:43:43 GMT + Model continuous data with the normal curve, standardise with z-scores, and read probabilities from tables. + + + Coordinate geometry foundations + https://openfreebooks.org/math/coordinate-geometry-foundations/ + https://openfreebooks.org/math/coordinate-geometry-foundations/ + Thu, 21 May 2026 16:43:43 GMT + Plot points, find distances and midpoints, and work with slopes of lines in the Cartesian plane. + + + Measures of dispersion + https://openfreebooks.org/math/measures-dispersion/ + https://openfreebooks.org/math/measures-dispersion/ + Thu, 21 May 2026 16:43:43 GMT + Interpret range, interquartile range, variance, and standard deviation for comparing data sets. + + + Advanced trigonometry + https://openfreebooks.org/math/advanced-trigonometry/ + https://openfreebooks.org/math/advanced-trigonometry/ + Thu, 21 May 2026 16:43:43 GMT + Use sine and cosine rules, radians, and trigonometric graphs in 2D and 3D applications. + + + Properties of circles + https://openfreebooks.org/math/circle-properties/ + https://openfreebooks.org/math/circle-properties/ + Thu, 21 May 2026 16:43:43 GMT + Apply chord, tangent, and angle theorems in circles to prove results and solve geometry problems. + + + Deductive geometry + https://openfreebooks.org/math/deductive-geometry/ + https://openfreebooks.org/math/deductive-geometry/ + Thu, 21 May 2026 16:43:43 GMT + Prove angle and side relationships for intersecting lines, triangles, and polygons. + + + Coordinate geometry of circles + https://openfreebooks.org/math/coordinate-circles/ + https://openfreebooks.org/math/coordinate-circles/ + Thu, 21 May 2026 16:43:43 GMT + Find equations of circles, tangents, and intersections using coordinate methods and distance. + + + Basic probability + https://openfreebooks.org/math/basic-probability/ + https://openfreebooks.org/math/basic-probability/ + Thu, 21 May 2026 16:43:43 GMT + Calculate theoretical and experimental probabilities for simple events and complements. + + + Charts & averages + https://openfreebooks.org/math/statistical-charts-averages/ + https://openfreebooks.org/math/statistical-charts-averages/ + Thu, 21 May 2026 16:43:43 GMT + Organise data with tables and charts and calculate mean, median, and mode. + + + Statistical literacy + https://openfreebooks.org/math/statistical-literacy/ + https://openfreebooks.org/math/statistical-literacy/ + Thu, 21 May 2026 16:43:43 GMT + Evaluate claims in media and research: sampling bias, correlation versus causation, and misleading charts. + + + Congruence & similarity + https://openfreebooks.org/math/congruence-similarity/ + https://openfreebooks.org/math/congruence-similarity/ + Thu, 21 May 2026 16:43:43 GMT + Apply congruence and similarity criteria to justify properties of triangles and scale figures. + + + Pythagoras' theorem + https://openfreebooks.org/math/pythagoras-theorem/ + https://openfreebooks.org/math/pythagoras-theorem/ + Thu, 21 May 2026 16:43:43 GMT + Use Pythagoras' theorem and its converse in 2D problems and coordinate settings. Complex numbers https://openfreebooks.org/math/complex-numbers/ https://openfreebooks.org/math/complex-numbers/ - Thu, 21 May 2026 07:32:39 GMT + Thu, 21 May 2026 07:33:01 GMT Represent complex numbers in Cartesian form and perform operations including conjugates and division. + + Exponential & logarithmic functions + https://openfreebooks.org/math/exponential-logarithmic/ + https://openfreebooks.org/math/exponential-logarithmic/ + Thu, 21 May 2026 07:33:01 GMT + Use exponential and logarithmic models, apply log laws, and solve growth and decay equations. + + + Factor & remainder theorems + https://openfreebooks.org/math/factor-remainder-theorems/ + https://openfreebooks.org/math/factor-remainder-theorems/ + Thu, 21 May 2026 07:33:01 GMT + Use the factor and remainder theorems to factorise higher-degree polynomials and solve related equations. + Functions & graphs https://openfreebooks.org/math/functions-graphs/ https://openfreebooks.org/math/functions-graphs/ - Thu, 21 May 2026 07:32:39 GMT + Thu, 21 May 2026 07:33:01 GMT Model input-output rules, work with domain and range, and sketch key function graphs and transformations. + + Linear equations in two unknowns + https://openfreebooks.org/math/linear-equations-two-unknowns/ + https://openfreebooks.org/math/linear-equations-two-unknowns/ + Thu, 21 May 2026 07:33:01 GMT + Solve pairs of linear equations by substitution, elimination, and graphs; classify unique, parallel, and coincident cases. + + + Linear programming + https://openfreebooks.org/math/linear-programming/ + https://openfreebooks.org/math/linear-programming/ + Thu, 21 May 2026 07:33:01 GMT + Graph linear inequalities, identify feasible regions, and optimise objective functions in two variables. + Sequences & series https://openfreebooks.org/math/sequences-series/ https://openfreebooks.org/math/sequences-series/ - Thu, 21 May 2026 07:32:39 GMT + Thu, 21 May 2026 07:33:01 GMT Work with arithmetic and geometric sequences, sigma notation, and finite summation formulas. Variations https://openfreebooks.org/math/variations/ https://openfreebooks.org/math/variations/ - Thu, 21 May 2026 07:32:39 GMT + Thu, 21 May 2026 07:33:01 GMT Model direct, inverse, joint, and partial variation and solve combined variation problems. @@ -75,13 +194,6 @@ Thu, 21 May 2026 05:05:38 GMT Count objects, spot patterns, add, multiply, and divide; then meet place value and big numbers from hundreds to billions. - - Linear equations in two unknowns - https://openfreebooks.org/math/linear-equations-two-unknowns/ - https://openfreebooks.org/math/linear-equations-two-unknowns/ - Thu, 21 May 2026 05:05:38 GMT - Solve pairs of linear equations by substitution, elimination, and graphs; classify unique, parallel, and coincident cases. - Written calculation https://openfreebooks.org/math/written-calculation/ diff --git a/static/sitemap.xml b/static/sitemap.xml index 78f4c1a..4a83df9 100644 --- a/static/sitemap.xml +++ b/static/sitemap.xml @@ -36,6 +36,10 @@ https://openfreebooks.org/math/ 2026-05-19T10:55:50.000Z + + https://openfreebooks.org/math/advanced-trigonometry/ + 2026-05-21T16:43:43.537Z + https://openfreebooks.org/math/algebraic-fractions-formulas/ 2026-05-20T20:17:25.000Z @@ -44,29 +48,57 @@ https://openfreebooks.org/math/basic-math-notation/ 2026-05-21T01:07:39.000Z + + https://openfreebooks.org/math/basic-probability/ + 2026-05-21T16:43:43.535Z + + + https://openfreebooks.org/math/circle-properties/ + 2026-05-21T16:43:43.537Z + https://openfreebooks.org/math/complex-numbers/ - 2026-05-21T07:32:39.566Z + 2026-05-21T07:33:01.000Z + + + https://openfreebooks.org/math/congruence-similarity/ + 2026-05-21T16:43:43.137Z + + + https://openfreebooks.org/math/coordinate-circles/ + 2026-05-21T16:43:43.536Z + + + https://openfreebooks.org/math/coordinate-geometry-foundations/ + 2026-05-21T16:43:43.663Z https://openfreebooks.org/math/counting-and-numbers/ 2026-05-21T05:05:38.000Z + + https://openfreebooks.org/math/deductive-geometry/ + 2026-05-21T16:43:43.537Z + https://openfreebooks.org/math/directed-numbers-rates/ 2026-05-20T20:17:25.000Z https://openfreebooks.org/math/exponential-logarithmic/ - 2026-05-21T07:32:39.567Z + 2026-05-21T07:33:01.000Z https://openfreebooks.org/math/factor-remainder-theorems/ - 2026-05-21T07:32:39.697Z + 2026-05-21T07:33:01.000Z https://openfreebooks.org/math/functions-graphs/ - 2026-05-21T07:32:39.566Z + 2026-05-21T07:33:01.000Z + + + https://openfreebooks.org/math/introduction-trigonometry/ + 2026-05-21T16:43:43.931Z https://openfreebooks.org/math/laws-of-indices/ @@ -74,7 +106,7 @@ https://openfreebooks.org/math/linear-equations-two-unknowns/ - 2026-05-21T05:05:38.000Z + 2026-05-21T07:33:01.000Z https://openfreebooks.org/math/linear-inequalities-one-unknown/ @@ -82,7 +114,27 @@ https://openfreebooks.org/math/linear-programming/ - 2026-05-21T07:32:39.831Z + 2026-05-21T07:33:01.000Z + + + https://openfreebooks.org/math/loci/ + 2026-05-21T16:43:43.932Z + + + https://openfreebooks.org/math/measures-dispersion/ + 2026-05-21T16:43:43.662Z + + + https://openfreebooks.org/math/mensuration-3d/ + 2026-05-21T16:43:43.798Z + + + https://openfreebooks.org/math/normal-distribution-z-scores/ + 2026-05-21T16:43:43.798Z + + + https://openfreebooks.org/math/permutations-combinations-probability/ + 2026-05-21T16:43:43.799Z https://openfreebooks.org/math/polynomials-expansion-factorisation/ @@ -92,17 +144,29 @@ https://openfreebooks.org/math/properties-of-operations/ 2026-05-20T20:17:25.000Z + + https://openfreebooks.org/math/pythagoras-theorem/ + 2026-05-21T16:43:43.137Z + https://openfreebooks.org/math/quadratic-equations/ 2026-05-20T20:17:25.000Z https://openfreebooks.org/math/sequences-series/ - 2026-05-21T07:32:39.286Z + 2026-05-21T07:33:01.000Z + + + https://openfreebooks.org/math/statistical-charts-averages/ + 2026-05-21T16:43:43.394Z + + + https://openfreebooks.org/math/statistical-literacy/ + 2026-05-21T16:43:43.394Z https://openfreebooks.org/math/variations/ - 2026-05-21T07:32:39.286Z + 2026-05-21T07:33:01.000Z https://openfreebooks.org/math/written-calculation/ diff --git a/themes/openfreebooks/templates/partials/why-sections.html b/themes/openfreebooks/templates/partials/why-sections.html index 198e1bd..5de5940 100644 --- a/themes/openfreebooks/templates/partials/why-sections.html +++ b/themes/openfreebooks/templates/partials/why-sections.html @@ -91,7 +91,7 @@

Planet

-

Open textbooks are gentler on the Earth

+

OFB are gentler on the Earth

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.