Grade 8 · Level 75 · Transition to abstraction
MATH 75
The taxonomy's house example arrives.
Meridian Note
The Pythagorean Theorem — the taxonomy's own house example — enters here and must exit at KS-7: missing sides found in problems that never announce the theorem. Functions enter at KS-3–4. Linear equations reach KS-7, positioned for KS-8 analysis in Algebra I.
The Fix Registry
Skills of This Course
Every skill is pinned to the meridian by fix notation — where along the scale, and at what depth. The highlighted state is this course's target: the state an honest instrument must certify before the skill counts as covered here.
| Skill | Fixes | Target at Level 75 |
|---|---|---|
| One-variable equations | 60-KS468-KS781-KS8 | KS-7Applied |
| Linear functions | 75-KS481-KS788-KS9 | KS-4Recalled |
| Pythagorean Theorem | 75-KS785-KS890-KS9 | KS-7Applied |
| Systems of linear equations | 75-KS681-KS8 | KS-6Understood |
| Roots & irrational numbers | 75-KS685-KS8 | KS-6Understood |
| Transformational geometry | 75-KS585-KS7 | KS-5Retained |
Course of Instruction
Units
Unit 1.Linear Equations: Solving with Structure
8 practice itemsAn equation is a claim: the expression on the left names the same number as the expression on the right. Solving means finding every value of the variable that makes the claim true. This year the equations get more interesting — the variable shows up on both sides, hides inside parentheses, and sometimes the honest answer is "no value works" or "every value works." You will handle all of it with one idea you already trust: keep the balance.
Multiple choiceFill in the blankMultiple choice (misconception distractors)Short answerProblem solving
Unit 2.Functions: One Input, One Output
8 practice itemsA function is a rule with a promise: give it an input and it returns **exactly one** output. Not two, not "it depends" — exactly one, every time. That promise is what makes functions the most useful objects in mathematics: they let you predict. This unit introduces the function idea and then looks closely at the friendliest family of all — linear functions, the rules that change at a perfectly steady rate.
MatchingMultiple choiceFill in the blankShort answerOral probe
Unit 3.Systems of Equations: Two Conditions at Once
7 practice itemsOne equation describes one condition. But real questions often impose two conditions at the same time — a family bought twelve tickets AND paid seventy-five dollars. A **system of equations** is exactly that: two claims that must both be true at once. The solution is the pair of numbers that satisfies both — and this unit gives you two reliable ways to find it, plus the reason a system can sometimes have no solution at all.
Multiple choiceFill in the blankMultiple choice (misconception distractors)Problem solvingShort answer
Unit 4.Roots and Irrational Numbers
6 practice itemsSquaring a number is easy. This unit asks the reverse question — what number, squared, gives 50? — and that question opens a door to a genuinely new kind of number. You will learn to compute roots, to trap a messy root between numbers you know, and to meet the discovery that shocked the ancient Greeks: some lengths cannot be written as any fraction at all.
MatchingMultiple choiceFill in the blankMultiple choice (misconception distractors)Short answer
Unit 5.The Pythagorean Theorem
8 practice itemsThis unit contains the most famous theorem in mathematics — a fact about right triangles that builders, navigators, and engineers have leaned on for over 2,500 years. You will learn what the theorem claims, why it is true (not just that it is), how to run it in both directions, and — the real goal this year — how to spot the right triangles hiding inside problems that never mention triangles at all.
Multiple choiceFill in the blankMultiple choice (misconception distractors)Short answerProblem solving
Unit 6.Transformational Geometry: Slides, Flips, Turns, and Scaling
7 practice itemsGeometry gets moving. A **transformation** takes every point of a figure and sends it somewhere new, all according to one rule. Three transformations — the slide, the flip, and the turn — move a figure without changing its size or shape at all, and a fourth, the dilation, changes size on purpose. The coordinate plane turns all four into precise rules you can compute with, which is exactly what this unit teaches.
MatchingMultiple choiceFill in the blankMultiple choice (misconception distractors)Short answer