Period \(=\frac{2\pi}{|b|}\) (or \(\frac{360^\circ}{|b|}\) in degrees mode)
+
Vertical shift \(=d\)
+
+
+
Exam strategy
+
+
Check calculator mode (degree/radian) before solving.
+
State interval when listing trig equation solutions.
+
Draw quick triangle sketch for sine/cosine rule setup.
+
Keep exact radian forms when possible.
+
+
+
+
+
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
+
+
Wave modeling: sound, light, and AC signals.
+
Navigation: triangulation in maps and GPS support calculations.
+
Engineering: periodic motion and oscillation analysis.
+
+
+
References
+
+
Standard high-school advanced trigonometry syllabi (IB, A-Level, DSE extension).
+
Precalculus texts on trig identities, equations, and graphs.
+
+
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.
+
Period \(=\frac{2\pi}{|b|}\) (or \(\frac{360^\circ}{|b|}\) in degrees mode)
+
Vertical shift \(=d\)
+
+
+
Exam strategy
+
+
Check calculator mode (degree/radian) before solving.
+
State interval when listing trig equation solutions.
+
Draw quick triangle sketch for sine/cosine rule setup.
+
Keep exact radian forms when possible.
+
+
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
+
+
Wave modeling: sound, light, and AC signals.
+
Navigation: triangulation in maps and GPS support calculations.
+
Engineering: periodic motion and oscillation analysis.
+
+
+
References
+
+
Standard high-school advanced trigonometry syllabi (IB, A-Level, DSE extension).
+
Precalculus texts on trig identities, equations, and graphs.
+
+
+ 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
+
+
Angle at center is twice angle at circumference on same arc.
+
Angles in the same segment are equal.
+
Angle in a semicircle is \(90^\circ\).
+
Opposite angles of cyclic quadrilateral sum to \(180^\circ\).
+
Tangent is perpendicular to radius at point of contact.
+
Two tangents from same external point are equal in length.
+
Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.
+
+
+
Using theorems in proofs
+
+
Mark known equal angles/lengths clearly.
+
Name theorem used at each step.
+
Chain results to target statement.
+
+
+
Cyclic quadrilaterals
+
+ A quadrilateral is cyclic if all vertices lie on one circle.
+
+
+ Useful tests:
+
+
+
Opposite angles supplementary.
+
Exterior angle equals opposite interior angle.
+
+
+
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
+
+
Label all equal angles immediately.
+
Write theorem names, not just numbers.
+
Avoid assuming symmetry unless given or proved.
+
Finish with a clear conclusion sentence.
+
+
+
+
+
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
+
+
Design: arcs and tangency in mechanical/architectural drawings.
+
Optics: circular geometry in lenses and reflective paths.
+
CAD tools: tangent/chord constraints in modeling software.
+
+
+
References
+
+
Standard high-school geometry materials on circle theorems.
+
Classical Euclidean geometry 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
+
+
Angle at center is twice angle at circumference on same arc.
+
Angles in the same segment are equal.
+
Angle in a semicircle is \(90^\circ\).
+
Opposite angles of cyclic quadrilateral sum to \(180^\circ\).
+
Tangent is perpendicular to radius at point of contact.
+
Two tangents from same external point are equal in length.
+
Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.
+
+
+
Using theorems in proofs
+
+
Mark known equal angles/lengths clearly.
+
Name theorem used at each step.
+
Chain results to target statement.
+
+
+
Cyclic quadrilaterals
+
+ A quadrilateral is cyclic if all vertices lie on one circle.
+
+
+ Useful tests:
+
+
+
Opposite angles supplementary.
+
Exterior angle equals opposite interior angle.
+
+
+
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
+
+
Label all equal angles immediately.
+
Write theorem names, not just numbers.
+
Avoid assuming symmetry unless given or proved.
+
Finish with a clear conclusion sentence.
+
+
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
+
+
Design: arcs and tangency in mechanical/architectural drawings.
+
Optics: circular geometry in lenses and reflective paths.
+
CAD tools: tangent/chord constraints in modeling software.
+
+
+
References
+
+
Standard high-school geometry materials on circle theorems.
+ 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:
+
+
+
SSS (three sides equal)
+
SAS (two sides and included angle equal)
+
ASA / AAS (two angles and one corresponding side equal)
+
RHS (right triangle: hypotenuse and one side equal)
+
+
+
Similarity
+
+ Similar figures have the same shape but possibly different sizes.
+
+
+
Corresponding angles are equal.
+
Corresponding sides are in one constant ratio (scale factor).
+
+
+ Triangle similarity criteria:
+
+
+
AA (two angles equal)
+
SAS in proportion
+
SSS in proportion
+
+
+
Scale factor relationships
+
+ If linear scale factor from shape A to B is \(k\):
+
+
+
Lengths multiply by \(k\).
+
Areas multiply by \(k^2\).
+
Volumes multiply by \(k^3\).
+
+
+
Using similarity to solve problems
+
+
Identify corresponding vertices in correct order.
+
Write ratio equation using matching sides.
+
Solve unknown lengths/areas.
+
Check ratio consistency across all pairs.
+
+
+
Congruence and similarity in proofs
+
+ Common proof flow:
+
+
+
Show two triangles are congruent/similar by criteria.
+
Conclude equal angles/sides or proportional segments.
+
Use result to prove target statement.
+
+
+
Exam strategy
+
+
State criterion explicitly (e.g., “\(\triangle ABC\cong\triangle DEF\) by SAS”).
+
Match vertices in correct correspondence order.
+
Do not mix congruence and similarity conditions.
+
For ratio problems, keep fractions in consistent side order.
+
+
+
+
+
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
+
+
Scale maps/drawings: convert plan dimensions to real distances.
+
Shadow methods: estimate object heights using similar triangles.
+
Engineering design: preserve shape while resizing components.
+
+
+
References
+
+
Standard high-school geometry syllabi on triangle congruence and similarity.
+
Geometry textbooks on ratio, scale factor, and proof applications.
+
+
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:
+
+
+
SSS (three sides equal)
+
SAS (two sides and included angle equal)
+
ASA / AAS (two angles and one corresponding side equal)
+
RHS (right triangle: hypotenuse and one side equal)
+
+
+
Similarity
+
+ Similar figures have the same shape but possibly different sizes.
+
+
+
Corresponding angles are equal.
+
Corresponding sides are in one constant ratio (scale factor).
+
+
+ Triangle similarity criteria:
+
+
+
AA (two angles equal)
+
SAS in proportion
+
SSS in proportion
+
+
+
Scale factor relationships
+
+ If linear scale factor from shape A to B is \(k\):
+
+
+
Lengths multiply by \(k\).
+
Areas multiply by \(k^2\).
+
Volumes multiply by \(k^3\).
+
+
+
Using similarity to solve problems
+
+
Identify corresponding vertices in correct order.
+
Write ratio equation using matching sides.
+
Solve unknown lengths/areas.
+
Check ratio consistency across all pairs.
+
+
+
Congruence and similarity in proofs
+
+ Common proof flow:
+
+
+
Show two triangles are congruent/similar by criteria.
+
Conclude equal angles/sides or proportional segments.
+
Use result to prove target statement.
+
+
+
Exam strategy
+
+
State criterion explicitly (e.g., “\(\triangle ABC\cong\triangle DEF\) by SAS”).
+
Match vertices in correct correspondence order.
+
Do not mix congruence and similarity conditions.
+
For ratio problems, keep fractions in consistent side order.
+
+
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
+
+
Scale maps/drawings: convert plan dimensions to real distances.
+
Shadow methods: estimate object heights using similar triangles.
+
Engineering design: preserve shape while resizing components.
+
+
+
References
+
+
Standard high-school geometry syllabi on triangle congruence and similarity.
+
Geometry textbooks on ratio, scale factor, and proof applications.
+
+
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: \((x-3)^2+(y+2)^2=25\)
+
+
Center: \((3,-2)\)
+
Radius: \(5\)
+
+
+
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
+
+
A tangent touches a circle at exactly one point.
+
Radius to tangent point is perpendicular to the tangent.
+
+
+ If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined).
+
+
+
Point position relative to circle
+
+ For circle center \(C\), radius \(r\), and point \(P\):
+
+
+
\(CP<r\): inside
+
\(CP=r\): on circle
+
\(CP>r\): outside
+
+
+
Line-circle intersections
+
+ Substitute line equation into circle equation.
+
+
+
Two solutions: secant intersects twice.
+
One solution: tangent.
+
No real solution: no intersection.
+
+
+
Exam strategy
+
+
Write center and radius before computation.
+
Use exact values first; round at final step.
+
For tangents, use radius-perpendicular fact explicitly.
+ 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
+
+
Engineering: circular component alignment and tolerance checks.
+
Computer graphics: collision and boundary detection.
+
Robotics: turning path and sensor range modeling.
+
+
+
References
+
+
Standard high-school coordinate geometry materials on circles and tangents.
+
Analytic geometry references on conic sections (intro level).
+
+
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: \((x-3)^2+(y+2)^2=25\)
+
+
Center: \((3,-2)\)
+
Radius: \(5\)
+
+
+
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
+
+
A tangent touches a circle at exactly one point.
+
Radius to tangent point is perpendicular to the tangent.
+
+
+ If radius slope is \(m\), tangent slope is \(-\frac1m\) (when defined).
+
+
+
Point position relative to circle
+
+ For circle center \(C\), radius \(r\), and point \(P\):
+
+
+
\(CP<r\): inside
+
\(CP=r\): on circle
+
\(CP>r\): outside
+
+
+
Line-circle intersections
+
+ Substitute line equation into circle equation.
+
+
+
Two solutions: secant intersects twice.
+
One solution: tangent.
+
No real solution: no intersection.
+
+
+
Exam strategy
+
+
Write center and radius before computation.
+
Use exact values first; round at final step.
+
For tangents, use radius-perpendicular fact explicitly.
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
+
+
Engineering: circular component alignment and tolerance checks.
+
Computer graphics: collision and boundary detection.
+
Robotics: turning path and sensor range modeling.
+
+
+
References
+
+
Standard high-school coordinate geometry materials on circles and tangents.
+
Analytic geometry references on conic sections (intro level).
+
+
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:
+
+
+
Q1: \((+,+)\)
+
Q2: \((-,+)\)
+
Q3: \((-,-)\)
+
Q4: \((+,-)\)
+
+
+
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.
+
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 \(x_2\ne x_1\):
+
+
\[
+ m=\frac{y_2-y_1}{x_2-x_1}.
+ \]
+
+
\(m>0\): line rises left to right.
+
\(m<0\): line falls left to right.
+
\(m=0\): horizontal line.
+
Vertical line has undefined slope.
+
+
+
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
+
+
Parallel lines have equal slopes: \(m_1=m_2\).
+
Perpendicular lines satisfy \(m_1m_2=-1\) (when both slopes are defined).
+
+
+
Exam strategy
+
+
Write coordinates clearly before substituting formulas.
+
Keep exact form (roots/fractions) unless decimal is requested.
+
Check whether slope is undefined for vertical lines.
+
Use diagrams to avoid sign mistakes in differences.
+
+
+
+
+
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
+
+
Maps/GPS: locations represented using coordinate systems.
+
Computer graphics: points and lines placed on pixel coordinate planes.
+
Physics: trajectories and motion graphs depend on coordinates and slope.
+
+
+
References
+
+
Standard high-school coordinate geometry units.
+
Analytic geometry references on points, lines, distance, and midpoint.
+
+
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:
+
+
+
Q1: \((+,+)\)
+
Q2: \((-,+)\)
+
Q3: \((-,-)\)
+
Q4: \((+,-)\)
+
+
+
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.
+
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 \(x_2\ne x_1\):
+
+
\[
+ m=\frac{y_2-y_1}{x_2-x_1}.
+ \]
+
+
\(m>0\): line rises left to right.
+
\(m<0\): line falls left to right.
+
\(m=0\): horizontal line.
+
Vertical line has undefined slope.
+
+
+
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
+
+
Parallel lines have equal slopes: \(m_1=m_2\).
+
Perpendicular lines satisfy \(m_1m_2=-1\) (when both slopes are defined).
+
+
+
Exam strategy
+
+
Write coordinates clearly before substituting formulas.
+
Keep exact form (roots/fractions) unless decimal is requested.
+
Check whether slope is undefined for vertical lines.
+
Use diagrams to avoid sign mistakes in differences.
+
+
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
+
+
Maps/GPS: locations represented using coordinate systems.
+
Computer graphics: points and lines placed on pixel coordinate planes.
+
Physics: trajectories and motion graphs depend on coordinates and slope.
+
+
+
References
+
+
Standard high-school coordinate geometry units.
+
Analytic geometry references on points, lines, distance, and midpoint.
+
+
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
+
+
Given: information provided in the question.
+
To prove: target statement.
+
Because / therefore: links each conclusion to a reason.
+
+
+
Core angle facts
+
+
Angles on a straight line sum to \(180^\circ\).
+
Angles around a point sum to \(360^\circ\).
+
Vertically opposite angles are equal.
+
In parallel lines:
+
+
alternate interior angles are equal,
+
corresponding angles are equal,
+
co-interior angles sum to \(180^\circ\).
+
+
+
+
+
Triangle facts
+
+
Interior angles sum to \(180^\circ\).
+
Exterior angle equals sum of two opposite interior angles.
+ Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\).
+
+
+
Proof structure
+
+
Write what is given and what must be shown.
+
Mark known equal angles/lengths on diagram.
+
Use relevant theorems in short logical chain.
+
End with clear conclusion sentence.
+
+
+
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
+
+
Stating result without reason.
+
Using theorem conditions not satisfied by diagram.
+
Assuming diagram scale implies equality.
+
Skipping steps between statements.
+
+
+
Exam strategy
+
+
Use short statements with explicit reasons.
+
Reference standard theorem names precisely.
+
Keep symbols consistent with question labels.
+
Do not rely on appearance; rely on given facts and proven steps.
+
+
+
+
+
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 \(x\).
+
+
+
+
+
+ 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
+
+
Engineering drawings: design correctness depends on provable geometric constraints.
+
Architecture: angle and parallel-line reasoning ensures structural fit and symmetry.
+
Computer-aided design: geometric rule engines rely on deductive relationships.
+
+
+
References
+
+
Euclid, Elements (classical deductive geometry foundation).
+
Standard high-school geometry syllabi on angle and polygon proofs.
+
+
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
+
+
Given: information provided in the question.
+
To prove: target statement.
+
Because / therefore: links each conclusion to a reason.
+
+
+
Core angle facts
+
+
Angles on a straight line sum to \(180^\circ\).
+
Angles around a point sum to \(360^\circ\).
+
Vertically opposite angles are equal.
+
In parallel lines:
+
+
alternate interior angles are equal,
+
corresponding angles are equal,
+
co-interior angles sum to \(180^\circ\).
+
+
+
+
+
Triangle facts
+
+
Interior angles sum to \(180^\circ\).
+
Exterior angle equals sum of two opposite interior angles.
+ Sum of exterior angles (one per vertex, same direction) is always \(360^\circ\).
+
+
+
Proof structure
+
+
Write what is given and what must be shown.
+
Mark known equal angles/lengths on diagram.
+
Use relevant theorems in short logical chain.
+
End with clear conclusion sentence.
+
+
+
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
+
+
Stating result without reason.
+
Using theorem conditions not satisfied by diagram.
+
Assuming diagram scale implies equality.
+
Skipping steps between statements.
+
+
+
Exam strategy
+
+
Use short statements with explicit reasons.
+
Reference standard theorem names precisely.
+
Keep symbols consistent with question labels.
+
Do not rely on appearance; rely on given facts and proven steps.
+
+
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 \(x\).
+
+
+ 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
+
+
Choose ratio from known and required sides (SOH-CAH-TOA).
+
Set calculator mode to degrees unless stated otherwise.
+
Round only at final step unless instructed.
+
Include units and context in word-problem answers.
+
+
+
+
+
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 \(\cos\theta\).
+
+ 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
+
+
Choose ratio from known and required sides (SOH-CAH-TOA).
+
Set calculator mode to degrees unless stated otherwise.
+
Round only at final step unless instructed.
+
Include units and context in word-problem answers.
+
+
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 \(\cos\theta\).
+
Surveying: height and distance estimation without direct measurement.
+
Navigation: direction and angle-based positioning.
+
Engineering: force components and incline analysis.
+
+
+
References
+
+
Standard high-school trigonometry curricula (right-triangle ratios).
+
Geometry and trigonometry texts on SOH-CAH-TOA applications.
+
+
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
+
+
Fixed distance \(r\) from point \(A\): a circle center \(A\), radius \(r\).
+
Fixed distance from line \(l\): two parallel lines on each side of \(l\).
+
Equidistant from points \(A,B\): perpendicular bisector of \(AB\).
+
Equidistant from two intersecting lines: angle bisectors.
+
+
+
Constructing loci
+
+
Translate condition into known locus type.
+
Construct each required locus accurately.
+
Find intersection of loci for final solution points/region.
+
+
+
Loci and inequalities
+
+ Conditions like “distance at most 3 cm from line” describe a region, not a single line.
+
+
+ Boundary inclusion matters:
+
+
+
\(\le\): boundary included (solid)
+
\(<\): boundary excluded
+
+
+
Algebra links
+
+ Coordinate forms:
+
+
+
Circle: \((x-h)^2+(y-k)^2=r^2\)
+
Perpendicular bisector from equal-distance condition \(PA=PB\)
+
+
+
Modeling with loci
+
+ Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints.
+
+
+
Exam strategy
+
+
Underline keywords: equidistant, fixed distance, within, at least.
+
Draw boundaries first, then shade valid region.
+
State whether endpoints/boundaries are included.
+
Use intersections of loci to justify final points.
+
+
+
+
+
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
+
+
Cell/Wi-Fi planning: coverage zones around transmitters.
+
Urban design: buffer distances from roads or facilities.
+
Robotics: safe navigation boundaries and target zones.
+
+
+
References
+
+
Standard high-school geometry content on loci and construction.
+
Analytic geometry materials linking loci to equations.
+
+
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
+
+
Fixed distance \(r\) from point \(A\): a circle center \(A\), radius \(r\).
+
Fixed distance from line \(l\): two parallel lines on each side of \(l\).
+
Equidistant from points \(A,B\): perpendicular bisector of \(AB\).
+
Equidistant from two intersecting lines: angle bisectors.
+
+
+
Constructing loci
+
+
Translate condition into known locus type.
+
Construct each required locus accurately.
+
Find intersection of loci for final solution points/region.
+
+
+
Loci and inequalities
+
+ Conditions like “distance at most 3 cm from line” describe a region, not a single line.
+
+
+ Boundary inclusion matters:
+
+
+
\(\le\): boundary included (solid)
+
\(<\): boundary excluded
+
+
+
Algebra links
+
+ Coordinate forms:
+
+
+
Circle: \((x-h)^2+(y-k)^2=r^2\)
+
Perpendicular bisector from equal-distance condition \(PA=PB\)
+
+
+
Modeling with loci
+
+ Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints.
+
+
+
Exam strategy
+
+
Underline keywords: equidistant, fixed distance, within, at least.
+
Draw boundaries first, then shade valid region.
+
State whether endpoints/boundaries are included.
+
Use intersections of loci to justify final points.
+
+
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
+
+
Cell/Wi-Fi planning: coverage zones around transmitters.
+
Urban design: buffer distances from roads or facilities.
+
Robotics: safe navigation boundaries and target zones.
+
+
+
References
+
+
Standard high-school geometry content on loci and construction.
+
Analytic geometry materials linking loci to equations.
+
+
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.
+
Sketch and label all dimensions before formula use.
+
Choose exact \(\pi\) form first; round at final step if required.
+
Convert units early (e.g., cm to m).
+
For composite shapes, explain add/subtract logic briefly.
+
+
+
+
+
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.
+
Sketch and label all dimensions before formula use.
+
Choose exact \(\pi\) form first; round at final step if required.
+
Convert units early (e.g., cm to m).
+
For composite shapes, explain add/subtract logic briefly.
+
+
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.
+
+ 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 \(c\):
+
+
+
\(a^2+b^2=c^2\): right triangle.
+
\(a^2+b^2>c^2\): acute triangle.
+
\(a^2+b^2<c^2\): obtuse triangle.
+
+
+
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
+
+
Diagonal of rectangles/squares.
+
Ladders against walls.
+
Shortest straight-line distances on maps/grids.
+
Checking whether corners are right angles in construction.
+
+
+
Exam strategy
+
+
Identify the right angle first.
+
Label hypotenuse clearly before substitution.
+
Keep exact surd form unless decimal is requested.
+
Check answer size: hypotenuse must be longest.
+
+
+
+
+
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
+
+
Architecture: checking right angles and structural diagonals.
+
Navigation: shortest displacement calculations on grid maps.
+
Computer graphics: pixel distance and vector magnitude.
+
+
+
References
+
+
Standard high-school geometry curricula on Pythagoras and its converse.
+
Classical Euclidean geometry 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.
+
+ 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 \(c\):
+
+
+
\(a^2+b^2=c^2\): right triangle.
+
\(a^2+b^2>c^2\): acute triangle.
+
\(a^2+b^2<c^2\): obtuse triangle.
+
+
+
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
+
+
Diagonal of rectangles/squares.
+
Ladders against walls.
+
Shortest straight-line distances on maps/grids.
+
Checking whether corners are right angles in construction.
+
+
+
Exam strategy
+
+
Identify the right angle first.
+
Label hypotenuse clearly before substitution.
+
Keep exact surd form unless decimal is requested.
+
Check answer size: hypotenuse must be longest.
+
+
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
+
+
Architecture: checking right angles and structural diagonals.
+
Navigation: shortest displacement calculations on grid maps.
+
Computer graphics: pixel distance and vector magnitude.
+
+
+
References
+
+
Standard high-school geometry curricula on Pythagoras and its converse.
+
Classical Euclidean geometry 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
+
+
Bar chart: categorical comparisons.
+
Histogram: continuous grouped data (touching bars).
+
Line graph: change over time.
+
Pie chart: part-to-whole proportions.
+
Box plot: spread via quartiles and median.
+
+
+
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
+
+
Use mean when data has no extreme outliers.
+
Use median when distribution is skewed.
+
Use mode for most common category/value.
+
+
+
Exam strategy
+
+
Label axes and units clearly on charts.
+
Check scale intervals before reading values.
+
Order data before finding median and quartiles.
+
State if an answer is estimated from grouped data.
+
+
+
+
+
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.
+
+
+
+ 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 GMTRepresent 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 GMTModel 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 GMTWork 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 GMTModel direct, inverse, joint, and partial variation and solve combined variation problems.
@@ -75,13 +194,6 @@
Thu, 21 May 2026 05:05:38 GMTCount 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/
+ 2026-05-21T16:43:43.663Zhttps://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/2026-05-20T20:17:25.000Zhttps://openfreebooks.org/math/exponential-logarithmic/
- 2026-05-21T07:32:39.567Z
+ 2026-05-21T07:33:01.000Zhttps://openfreebooks.org/math/factor-remainder-theorems/
- 2026-05-21T07:32:39.697Z
+ 2026-05-21T07:33:01.000Zhttps://openfreebooks.org/math/functions-graphs/
- 2026-05-21T07:32:39.566Z
+ 2026-05-21T07:33:01.000Z
+
+
+ https://openfreebooks.org/math/introduction-trigonometry/
+ 2026-05-21T16:43:43.931Zhttps://openfreebooks.org/math/laws-of-indices/
@@ -74,7 +106,7 @@
https://openfreebooks.org/math/linear-equations-two-unknowns/
- 2026-05-21T05:05:38.000Z
+ 2026-05-21T07:33:01.000Zhttps://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/
+ 2026-05-21T16:43:43.799Zhttps://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/2026-05-20T20:17:25.000Zhttps://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/
+ 2026-05-21T16:43:43.394Zhttps://openfreebooks.org/math/variations/
- 2026-05-21T07:32:39.286Z
+ 2026-05-21T07:33:01.000Zhttps://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.