Grade 7 · printable course

Mathematics 7: Proportional Reasoning and Rational Number Systems

Grade 7 · 425 pages · 76 lessons · PDF, print at home

A full-year grade 7 mathematics course built around proportional relationships as the organizing spine, extended to the full rational number system, algebraic expressions and equations, and geometric measurement. Students move from unit rates and tables to equations of proportional lines, then extend arithmetic to signed rational numbers, then use both to solve multi-step equations and inequalities. The second half applies this algebra to scale drawings, angle relationships, circles, and surface area/volume, closing with an introduction to statistical sampling and probability. Aligned to CCSS Mathematics, Grade 7.

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Every unit and lesson in this book, in order. The same lessons are taught online at lettersanddevotion.com, so a family can move between paper and screen.

Mathematics 7: Proportional Reasoning and Rational Number Systems

76 lessons · try a lesson online free

Unit 1: Proportional Relationships9 lessons
  1. 1.Unit rate calculation with fractional quantities
  2. 2.Constant of proportionality k derived from a table
  3. 3.Proportional versus non-proportional relationships identified across representations
  4. 4.The role of the origin in distinguishing proportional from linear-but-non-proportional graphs
  5. 5.The equation y = kx as a representation of a proportional relationship
  6. 6.The point (1, r) as the unit rate on a proportional graph
  7. 7.Multi-step percent problems involving tax, tip, markup, and discount
  8. 8.The non-commutativity of sequential percent changes applied to an unfamiliar order-of-operations scenario
  9. 9.Proportionality analysis applied to a pricing structure with a fixed cost, in a context not used during instruction
Unit 2: Operations with Rational Numbers9 lessons
  1. 10.Additive inverse as a distance-preserving reflection across zero on the number line
  2. 11.Addition of rational numbers as directed movement on the number line
  3. 12.The equivalence of subtraction and addition of the inverse (p - q = p + (-q))
  4. 13.The sign rule for multiplication of rational numbers, derived from a repeated-addition pattern
  5. 14.Multiplication and division of signed rational numbers, including fractions and decimals
  6. 15.Fraction-to-decimal conversion via long division, including repeating decimals
  7. 16.Multi-step rational-number word problems requiring selection of operations without contextual cues
  8. 17.Justification of a signed-number operation rule using a self-constructed number-line diagram
  9. 18.The distinction between absolute distance between two rational numbers and signed change from one to another
Unit 3: Expressions and Multi-Step Equations11 lessons
  1. 19.Like terms in a linear expression with rational, including negative, coefficients
  2. 20.The distributive property applied with a negative multiplier
  3. 21.Equivalence between an expanded and a factored form of the same linear expression
  4. 22.Two-step linear equations with rational coefficients, form px + q = r
  5. 23.Multi-step linear equations of form p(x + q) = r and equations with variables on both sides
  6. 24.Translation of a word problem into a px + q = r equation
  7. 25.Arithmetic versus algebraic solution methods for the same word problem
  8. 26.One-variable linear inequalities requiring a sign reversal
  9. 27.The solution set of a linear inequality represented on a number line
  10. 28.Selecting between an equation and an inequality model for a novel real-world constraint problem
  11. 29.Application of the equation-as-balance principle to an unfamiliar quantitative domain
Unit 4: Scale Drawings and Geometric Constructions10 lessons
  1. 30.Scale factor as a constant of proportionality applied to a single length
  2. 31.The relationship between a linear scale factor and the resulting area, computed via actual lengths rather than by squaring the scale factor directly
  3. 32.The multiplicative (not additive) relationship between linear scale factor and area scale factor
  4. 33.The defining property of a scaled copy: every corresponding length pair shares the same constant ratio
  5. 34.Combining two scale ratios (original-to-actual and actual-to-new) to reproduce a figure at a new scale
  6. 35.Deriving a scale factor between two representations with no scale given, using measured corresponding distances
  7. 36.The triangle inequality as the condition for SSS construction to succeed
  8. 37.The conditions (SSS, SAS, ASA) under which a triangle is uniquely determined, versus conditions that permit more than one triangle or none
  9. 38.Generalizing the existence/uniqueness rule for triangles beyond the SSS/SAS/ASA cases explicitly taught
  10. 39.Discriminating between scale-factor problems and triangle-construction problems based on structural features of the prompt, not its position in the unit
Unit 5: Angle Relationships and Area9 lessons
  1. 40.Complementary and supplementary angle equations
  2. 41.Vertical and adjacent angle relationships in a crossing-lines figure
  3. 42.Multi-step reasoning across combined angle relationships (vertical, adjacent, supplementary) in a single figure
  4. 43.The invariance of the triangle angle sum
  5. 44.Circle area and circumference formulas
  6. 45.Decomposition strategies for composite figures involving circles
  7. 46.Surface area of polyhedra built from triangles and quadrilaterals
  8. 47.Transfer of angle-relationship reasoning to unfamiliar contexts
  9. 48.Generalized decomposition strategy for surface area of scaled composite solids
Unit 6: Circles, Surface Area, and Volume10 lessons
  1. 49.Circumference and area formulas for a circle (C = πd or 2πr; A = πr²)
  2. 50.Pi as a fixed ratio of circumference to diameter
  3. 51.Surface area of prisms via nets
  4. 52.Surface area of pyramids via nets
  5. 53.Volume of a rectangular prism with fractional edge lengths
  6. 54.The independence of surface area and volume as measures of a solid
  7. 55.Solving V = lwh for an unknown dimension
  8. 56.Solving a surface-area equation for an unknown dimension
  9. 57.The relationship between scale factor and volume ratio for similar solids
  10. 58.Multi-step application combining surface area, volume, and dimension-solving on composite or unfamiliar solids
Unit 7: Introduction to Probability9 lessons
  1. 59.Theoretical probability of a single event expressed as fraction, decimal, and percent
  2. 60.The divergence between theoretical and experimental probability under small sample sizes
  3. 61.A probability model with outcomes assigned probabilities summing to one
  4. 62.Agreement between a theoretical probability model and observed relative frequency
  5. 63.The sample space of a two-stage compound event
  6. 64.Probability of a compound event as a fraction of the sample space
  7. 65.The structural difference between combining independent stage probabilities correctly versus combining them by addition
  8. 66.A simulation designed to estimate the probability of a novel compound event
  9. 67.The relationship between sample size and confidence in a probability model
Unit 8: Sampling and Statistical Inference9 lessons
  1. 68.Random vs. biased sampling methods and who gets over/under-represented
  2. 69.The terms population, sample, random sample
  3. 70.Point estimates of a population proportion or mean from a single random sample
  4. 71.Sampling variability across repeated same-size samples
  5. 72.Generating repeated random samples to bound a population estimate
  6. 73.Difference in means expressed as a multiple of MAD, for distributions with similar variability
  7. 74.Informal visual and numerical judgment of whether two groups differ meaningfully
  8. 75.A complete original sampling plan: population, method, size, measurement
  9. 76.Critique of a stated survey claim's sampling method

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