30 written chapters · 1 published teaching videos
Grade 8, one chapter at a time.
Connect algebraic expressions to lines, equations, and functions. Explore geometric relationships and data, using multiple representations to explain why a method works.
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Chapters and video availability
Each chapter follows an existing academy lesson. Numbers match the written study order; grade placement is a guide and school sequences may vary.
30 of 30 chapters · in study order
Chapter 1 · Algebra
Count factors to combine powers
Multiplication adds exponents of a common base, while division cancels matching nonzero factors.
Written lesson available · video pending
Study this chapterChapter 2 · Algebra
Distinguish a power of a power from a product
Raising a power to another integer power multiplies the exponents, and an outside power acts on every factor of a product.
Written lesson available · video pending
Study this chapterChapter 3 · Algebra
Extend exponent patterns through zero
For a nonzero base, the zero power is one and a negative integer exponent represents a reciprocal.
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Study this chapterChapter 4 · Algebra
Normalize very small numbers in scientific notation
Scientific notation writes a positive number as a coefficient from 1 up to but not including 10 times a power of ten.
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Study this chapterChapter 5 · Algebra
Align powers of ten before adding
Adding quantities in scientific notation requires expressing them with the same power of ten.
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Study this chapterChapter 6 · Algebra
Multiply and renormalize scientific notation
Products in scientific notation combine coefficients and powers separately, then restore the standard coefficient range.
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Study this chapterChapter 7 · Algebra
Distinguish a square root from solving a squared equation
The square-root symbol denotes the nonnegative root, while a positive squared equation has two real solutions.
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Study this chapterChapter 8 · Algebra
Use cube roots to recover a signed input
Cubing preserves a real number's sign, so a negative real number has one negative real cube root.
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Study this chapterChapter 9 · Algebra
Locate an irrational square root between decimals
Comparing squares gives reliable bounds on a positive square root without pretending a decimal approximation is exact.
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Study this chapterChapter 10 · Geometry
Recover a right triangle's missing leg
When the hypotenuse is known, the Pythagorean theorem finds a missing leg by subtraction of squares.
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Study this chapterChapter 11 · Geometry
Test whether three sides form a right triangle
The converse of the Pythagorean theorem recognizes a right triangle from a square-sum equality.
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Study this chapterChapter 12 · Geometry
Turn coordinate differences into a distance
The straight-line distance between two points is the hypotenuse formed by their horizontal and vertical differences.
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Study this chapterChapter 13 · Geometry
Translate a figure with one shared movement
A translation adds the same horizontal and vertical changes to every point, preserving shape, size, and orientation.
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Study this chapterChapter 14 · Geometry
Reflect points across coordinate axes
A reflection places each point the same perpendicular distance on the opposite side of a mirror line.
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Study this chapterChapter 15 · Geometry
Rotate a point through a quarter-turn
A quarter-turn about the origin exchanges coordinate roles with a sign determined by the rotation direction.
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Study this chapterChapter 16 · Geometry
Connect dilation with length and area factors
An origin-centred dilation multiplies coordinates by a scale factor; lengths scale by that factor and areas by its square.
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Study this chapterChapter 17 · Geometry
Establish congruence with a sequence of motions
Congruent figures can be matched by rigid motions that preserve every distance and angle.
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Study this chapterChapter 18 · Algebra
Explain constant slope with similar triangles
Right triangles drawn along one straight line have equal rise-to-run ratios because their corresponding angles match.
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Study this chapterChapter 19 · Algebra
Recover a linear rule from a table
A constant output change per input unit gives a linear rate, while substitution recovers the intercept.
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Study this chapterChapter 20 · Algebra
Solve an equation that needs distribution
Distributing and combining like terms can reveal the simpler equation hidden inside parentheses.
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Study this chapterChapter 21 · Algebra
Compare two expressions with the same unknown
When a variable appears on both sides, collecting its terms on one side reveals where two linear rules agree.
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Study this chapterChapter 22 · Algebra
Recognize equations with no solution or all real solutions
If variable terms cancel, the remaining statement tells whether an equation is impossible or true for every allowed input.
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Study this chapterChapter 23 · Algebra
Solve two linked conditions by substitution
A system's solution must satisfy both equations at once; substitution replaces a variable by an equal expression.
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Study this chapterChapter 24 · Algebra
Cancel one variable by adding equations
Adding valid equations produces another true equation and can eliminate a variable with opposite coefficients.
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Study this chapterChapter 25 · Algebra
Decide whether a relation is a function
A function assigns exactly one output to each input; different inputs are allowed to share an output.
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Study this chapterChapter 26 · Statistics
Read a scatter plot without claiming causation
A scatter plot can show direction, form, and unusual points, but an association alone does not identify a cause.
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Study this chapterChapter 27 · Statistics
Choose the denominator in a two-way table
A conditional relative frequency uses the size of the named group as its denominator, not automatically the whole sample.
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Study this chapterChapter 28 · Geometry
Build a cylinder from circular layers
A cylinder's volume is its circular base area multiplied by its perpendicular height.
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Study this chapterChapter 29 · Geometry
Compare a cone with its matching cylinder
A cone has one third of the volume of a cylinder with the same base area and perpendicular height.
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Study this chapterChapter 30 · Geometry
Connect sphere volume to cubic scaling
A sphere's volume is 4πr³/3, so changing its radius changes its volume by a cubic factor.
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