Numbers & Measurement
Build confidence with signed numbers, decimals, fractions, ratios, percents, measurement, metric conversions, and rates.
Starting point: Start here for an arithmetic refresher and the measurement skills used in science.
Purpose
The purpose of this curriculum is to build mathematical fluency for scientific reasoning. Rather than learning formulas in isolation, the learner develops the ability to measure, calculate, recognize mathematical relationships, solve problems, and interpret scientific data.
Drawing on earlier work with negative numbers, decimals, algebra, coordinate geometry, special triangles, and trigonometry, the series is organized into four sequential courses, with five reference infographics supporting the topics. It suits independent study or structured instruction, and it makes no assumption that the learner is a child.
Materials. Scientific calculator with degree/radian modes, graph paper or graphing software, ruler, protractor, simple measuring tools, notebook, and the five accompanying online reference infographics.
Choose your course
Start with Course 1 and continue in order, or choose a later course if you already have the preparation described below. Every course costs $250.
Algebra & Scientific Notation
Use powers and roots, work with scientific notation, simplify expressions, solve equations, and rearrange scientific formulas.
Starting point: Take Course 1 first, or bring equivalent skills with signed arithmetic, fractions, ratios, and unit conversions.
Graphs & Scientific Models
Plot coordinates, interpret slope and functions, explore linear and nonlinear models, and use units and data to explain a scientific relationship.
Starting point: Take Course 2 first, or bring equivalent skills with equations, formulas, exponents, and square roots.
Geometry & Trigonometry
Work with right triangles, sine and cosine, radians, the unit circle, waves, the Laws of Sines and Cosines, and a practical science capstone.
Starting point: Take Course 3 first, or bring equivalent skills with algebra, coordinate graphs, functions, and units.
How each week works
Each week contains two structured lessons, independent problem practice, a worked scientific application, and a brief assessment. The series begins with arithmetic fluency and culminates in multistep scientific problems involving algebra, graphs, and triangles.
Courses 1, 2, and 3 each run for 4 weeks (8 lessons); Course 4 runs for 8 weeks (16 lessons). Together, they cover 20 weeks and 40 lessons. Each week includes two 60- to 75-minute lessons, plus three short independent practice sessions. All pacing is adjustable after the diagnostic assessment.
The lesson cycle
- Recall (10 min) short cumulative arithmetic and notation drill.
- Demonstrate (15 min) model a worked example and narrate each mathematical decision.
- Practice (20–25 min) solve supported then independent problems; emphasize signs, substitutions, and showing steps.
- Apply (15–20 min) solve a measurement, graph, chemistry, physics, astronomy, or other science problem.
- Exit check (5 min) one procedural item plus one conceptual 'why' item.
Error-checking routine: Set up → solve one step → check sign → inspect units → estimate → verify.
Week-by-week curriculum
Open a course for its weekly lessons, science applications, and evidence of mastery. Week numbers follow the full 20-week sequence; Courses 2, 3, and 4 can also be taken separately with the right preparation.
Course 1 · $250 Numbers & Measurement 4 weeks · 8 lessons · Sequence weeks 1–4
Preparation: Start here for an arithmetic refresher and the measurement skills used in science.
Reference infographics: Numbers, Signs & Operations →
Week 1 Integers, negatives, and absolute value
- Lesson A
- Classify whole, integer, rational, and real numbers; place signed numbers on a number line; compare magnitude and direction.
- Lesson B
- Add and subtract signed integers; understand subtraction as adding the opposite; calculate absolute value as distance from zero.
- Science application
- Interpret temperatures, elevation changes, and measurements above/below a reference level.
- Evidence of mastery
- Explain why −7 − (−3) = −4, and verify with a number line.
Week 2 Multiplication, division, decimals, and order of operations
- Lesson A
- Multiply and divide signed values; distinguish (−3)² from −3²; solve multistep expressions.
- Lesson B
- Place-value alignment, decimal operations, estimation, and precision of measurements.
- Science application
- Signed rates of change and arithmetic with laboratory measurements.
- Evidence of mastery
- Complete mixed operations with signs correctly and justify magnitude by estimation.
Week 3 Fractions, mixed numbers, ratios, and proportions
- Lesson A
- Equivalent fractions, simplification, common denominators, fraction operations, and mixed/improper conversions.
- Lesson B
- Unit rates, ratios, proportions, percent conversion, and percent change.
- Science application
- Sample proportions, map scales, and rate calculations.
- Evidence of mastery
- Solve proportional relationships and show cancellation of common factors.
Week 4 Measurement, conversions, and Course 1 synthesis
- Lesson A
- Metric prefixes, powers of 10, unit-conversion chains, and interpreting units in multiplication/division.
- Lesson B
- Mixed arithmetic science problems; Course 1 assessment and corrections.
- Science application
- Convert length, time, area, and speed using explicit unit factors.
- Evidence of mastery
- Produce 250 cm from 2.5 m using 2.5 m × (100 cm / 1 m), not an unexplained numerical rule.
Course 2 · $250 Algebra & Scientific Notation 4 weeks · 8 lessons · Sequence weeks 5–8
Preparation: Take Course 1 first, or bring equivalent skills with signed arithmetic, fractions, ratios, and unit conversions.
Reference infographics: Powers, Roots & Algebra →
Week 5 Exponents and square roots
- Lesson A
- Repeated multiplication, exponent laws, zero and negative exponents; powers of ten.
- Lesson B
- Perfect squares, principal square root, absolute value in √(x²), and simplifying radicals.
- Science application
- Area scaling, inverse-square relationships (conceptual), and squared quantities.
- Evidence of mastery
- Identify when a proposed square-root simplification is invalid.
Week 6 Scientific notation and scale
- Lesson A
- Write numbers in a × 10ⁿ form; multiply and divide values with powers of ten.
- Lesson B
- Work with large/small scientific quantities, significant figures (introduction), and estimation.
- Science application
- Molecular scales, astronomical distances, and population counts.
- Evidence of mastery
- Correctly compare 4.5 × 10⁻⁴ and 4.5 × 10⁻³.
Week 7 Expressions and equations
- Lesson A
- Coefficients, variables, constants, distributive property, and combining like terms with signed coefficients.
- Lesson B
- One- and two-step linear equations, equations with variables on both sides, and checking by substitution.
- Science application
- Solve an unknown measurement in a simple scientific model.
- Evidence of mastery
- Justify each inverse operation and correctly classify an equation as one, none, or infinitely many solutions.
Week 8 Rearranging scientific formulas
- Lesson A
- Isolate any one variable in v = d/t, F = ma, and ρ = m/V; maintain units and nonzero-denominator assumptions.
- Lesson B
- Inequalities and interval meaning; Course 2 assessment.
- Science application
- Calculate time from speed/distance and mass from density/volume.
- Evidence of mastery
- Derive t = d/v when v ≠ 0 and check by substitution.
Course 3 · $250 Graphs & Scientific Models 4 weeks · 8 lessons · Sequence weeks 9–12
Preparation: Take Course 2 first, or bring equivalent skills with equations, formulas, exponents, and square roots.
Reference infographics: Graphs, Functions & Units →
Week 9 Cartesian coordinates and slope
- Lesson A
- Plot ordered pairs in four quadrants, identify intercepts, and interpret scales.
- Lesson B
- Slope as rise/run and Δy/Δx; positive, negative, zero, and undefined slope.
- Science application
- Estimate average velocity from a position-time graph.
- Evidence of mastery
- Compute slope with a consistent subtraction order.
Week 10 Lines and coordinate geometry
- Lesson A
- Slope-intercept, point-slope, and standard forms; move between forms.
- Lesson B
- Distance and midpoint formulas; graph a linear relation with units.
- Science application
- Interpolate values and estimate displacement between positions.
- Evidence of mastery
- Show equivalence between forms and check two points on the same line.
Week 11 Functions and scientific patterns
- Lesson A
- Identify input/output, domain/range, graph transformations, linear versus quadratic models.
- Lesson B
- Introductory exponential change; recognize periodic data; optional introduction to logarithmic scales.
- Science application
- Compare constant-speed, accelerating, growth/decay, and cyclic patterns.
- Evidence of mastery
- Explain which model is plausible for a given set of observations, without implying that correlation proves causation.
Week 12 Dimensional analysis and data reasoning
- Lesson A
- Compound units, scientific equations, dimensional consistency, density, and rate calculations.
- Lesson B
- Plot a small data set, fit/estimate a reasonable line, discuss measurement error and significant figures; Course 3 assessment.
- Science application
- Interpret an experiment measuring travel distance over time.
- Evidence of mastery
- Label both graph axes, attach units, and interpret slope in context.
Course 4 · $250 Geometry & Trigonometry 8 weeks · 16 lessons · Sequence weeks 13–20
Preparation: Take Course 3 first, or bring equivalent skills with algebra, coordinate graphs, functions, and units.
Reference infographics: Trigonometry & the Unit Circle → · Solve Any Triangle →
Week 13 Triangle geometry and Pythagorean reasoning
- Lesson A
- Angles, triangle sum, triangle inequality, right-triangle vocabulary, and the Pythagorean theorem.
- Lesson B
- Exact ratios for 45°–45°–90° and 30°–60°–90° triangles; square-root simplification.
- Science application
- Geometric components of a rectangular field or inclined structure.
- Evidence of mastery
- Derive a diagonal from perpendicular components.
Week 14 Sine, cosine, tangent, and inverse trig
- Lesson A
- Label opposite/adjacent/hypotenuse relative to a specified angle; calculate sin, cos, tan.
- Lesson B
- Inverse trig for a missing angle; angle of elevation/depression; choose the appropriate trig function.
- Science application
- Determine unknown heights or distances from a measured angle.
- Evidence of mastery
- Distinguish a ratio problem from an inverse-ratio problem and set calculator mode correctly.
Week 15 Radians and unit-circle values
- Lesson A
- Degrees ↔ radians, full rotation, quadrants, reference angles, exact unit-circle coordinates (cos θ, sin θ).
- Lesson B
- Evaluate sine and cosine at reference and quadrant angles; identify sign patterns and tangent where defined.
- Science application
- Angular position and rotational motion.
- Evidence of mastery
- Explain why sin 150° = 1/2 and cos 150° = −√3/2.
Week 16 Trigonometric identities and periodic models
- Lesson A
- Unit circle identity sin²θ + cos²θ = 1; use tan θ = sin θ/cos θ when cos θ ≠ 0.
- Lesson B
- Sketch sine and cosine waves; interpret amplitude, period, and phase conceptually; trigonometry progress check.
- Science application
- Oscillation, wave motion, and recurring signals.
- Evidence of mastery
- Connect an angle to its circle coordinates and corresponding periodic graph values.
Week 17 Right-triangle choice and Law of Sines
- Lesson A
- Decision process for right triangles; combine the Pythagorean theorem, trig ratios, and inverse trig.
- Lesson B
- Law of Sines for AAS and ASA configurations, after finding the third angle.
- Science application
- Measured angle-and-baseline surveys.
- Evidence of mastery
- Identify the matching side-angle pairs (a, A), (b, B), (c, C).
Week 18 Law of Cosines and the SSA ambiguous case
- Lesson A
- Use Law of Cosines for SAS or SSS and then complete the triangle.
- Lesson B
- Use Law of Sines with SSA and test all possible valid angle configurations (0, 1, or 2 solutions); check domain and 180° sum.
- Science application
- Solve distance measurements from two known lengths and an included angle.
- Evidence of mastery
- Determine why one inverse-sine result may not represent the only valid triangle.
Week 19 Triangle area and applications
- Lesson A
- Area = ½bh; area = ½ab sin C; Heron's formula given SSS.
- Lesson B
- Apply triangle decomposition, vector components, and geometry in physical contexts.
- Science application
- Plan-area estimates, forces on an incline, and surveying.
- Evidence of mastery
- Match area formula to available data and maintain square units.
Week 20 Cumulative science capstone
- Lesson A
- Integrative problem solving: signed measurements, decimal/fraction calculations, conversions, exponents, graph slope, and triangles.
- Lesson B
- Capstone presentation/explanation and cumulative assessment; reflect on corrected misconceptions.
- Investigation option
- Measure a baseline and angles, solve a triangle, estimate the target distance/height, compare with an independent measurement, and graph repeated readings. Explain the assumptions, units, and uncertainty.
- Evidence of mastery
- Reproducible calculations, clearly labeled diagrams, correct method choice, and an evidence-based interpretation.
Learning outcomes
By the end of all four courses, a learner should be able to:
- Perform operations with integers, signed fractions, and decimals, with accurate use of parentheses, absolute value, and order of operations.
- Calculate ratios, proportions, percents, rates, and unit conversions, and assess the reasonableness of scientific quantities.
- Apply integer and fractional exponents, square roots, radicals, and scientific notation.
- Simplify expressions, solve linear equations, rearrange scientific formulas, and verify solutions through substitution.
- Plot and interpret coordinates, slope, intercepts, and basic linear, quadratic, exponential, and periodic relationships.
- Use the Pythagorean theorem, special right-triangle ratios, the six trigonometric functions when appropriate, inverse trigonometry, degrees, radians, and the unit circle.
- Identify whether a triangle is right, acute, or obtuse; select the appropriate method from right-triangle methods, SSS, SAS, ASA, AAS, or SSA; and address the SSA ambiguous case.
- Apply the Law of Sines, Law of Cosines, and appropriate triangle area formulas in scientifically grounded problems.
- Document steps, units, precision, checks, and interpretations clearly enough that another person can reproduce the result.
Assessments and mastery
- Starting diagnostic. A brief skills check at the start of each course confirms readiness and identifies review needs; diagnostic only, not graded.
- Each week. 12–20 problems balanced between new content and cumulative review; at least two interpretation/explanation problems.
- Progress checks. Brief weekly checks and a trigonometry check at sequence Week 16, with error analysis and a retest after corrections.
- Courses 1, 2, and 3 completion. Assessments at sequence weeks 4, 8, and 12 consolidate each course’s skills.
- Course 4 completion (sequence Week 20). Cumulative quantitative assessment and science application capstone.
- Suggested mastery threshold. at least 85% on each course’s targeted skills, plus correct unit handling and a valid independent explanation of the central method. This is an instructional recommendation, not a standardized external benchmark.
Triangle-solving reference
| Given information | Starting technique | Watch for |
|---|---|---|
| Right triangle: 2 side lengths | Pythagorean theorem; trig to find angles | Hypotenuse is the longest side |
| Right triangle: 1 side + 1 acute angle | sin / cos / tan; angle sum | Opposite and adjacent depend on chosen angle |
| Non-right triangle: SSS | Law of Cosines | Verify triangle inequality |
| Non-right triangle: SAS | Law of Cosines | Included angle is between known sides |
| Non-right triangle: ASA | Third angle, then Law of Sines | All angles sum to 180° |
| Non-right triangle: AAS | Third angle, then Law of Sines | Match opposite sides and angles |
| Non-right triangle: SSA | Law of Sines and ambiguous-case check | Zero, one, or two valid triangles |
| Only AAA | Not enough to determine unique side lengths | Similar triangles may differ in scale |
Main formulas
- Right triangle: a² + b² = c² (c is hypotenuse).
- Sine rule: a/sin A = b/sin B = c/sin C.
- Cosine rule: c² = a² + b² − 2ab cos C (with corresponding permutations).
- Trigonometric area: K = ½ab sin C.
- Heron: s = (a + b + c)/2; K = √[s(s − a)(s − b)(s − c)].
- Unit circle: (x, y) = (cos θ, sin θ); sin²θ + cos²θ = 1.
After the four-course series
For deeper chemistry, physics, statistics, and research readiness, continue with: logarithms (pH and decibels); uncertainty propagation; probability and descriptive statistics; algebraic systems; vector notation and components; polynomial and rational functions; and introductory limits/calculus.
Accuracy note: Trigonometric formulas must use a consistent calculator angle mode; scientific reporting must not imply more measurement precision than the inputs support.