Grade 6 · printable course

Ratios, Relationships, and the Number System

Grade 6 · 404 pages · 78 lessons · PDF, print at home

This is the year your child learns to think in ratios instead of just counting differences: "for every 3 of these, 2 of those" instead of "3 more of these." That one idea, a multiplicative relationship between two quantities, gets used all year: to divide fractions, to make sense of negative numbers, to write algebra expressions, to compute area and volume, and finally to describe a set of data. By June they should be comfortable with signed numbers, one-step equations, decimals in all four operations, basic area/volume/surface area, and describing a data set with center and spread, not as nine separate topics, but as one idea applied nine times.

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What is inside

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.

Ratios, Relationships, and the Number System

78 lessons · try a lesson online free

Unit 1: Ratios and Rate Reasoning9 lessons
  1. 1.Ratio notation and ratio language for a described relationship
  2. 2.The distinction between multiplicative (ratio) and additive (difference) comparison
  3. 3.Ratio table with equivalent ratios generated by scaling
  4. 4.Double number line as a model of a ratio relationship
  5. 5.Unit rate comparison across differently formatted representations
  6. 6.Distinctions among ratio, rate, and unit rate
  7. 7.Percent as a rate per 100, applied to an unfamiliar context
  8. 8.Combining two distinct ratio relationships into a single new relationship
  9. 9.The general rule for recognizing equivalent ratios across arbitrary tables
Unit 2: Dividing Fractions9 lessons
  1. 10.Measurement (how-many-groups) meaning of division applied to whole ÷ unit fraction
  2. 11.Measurement meaning of division applied to fraction ÷ fraction
  3. 12.Common-denominator strategy for fraction division
  4. 13.Standard algorithm for fraction division
  5. 14.Relationship between divisor size and quotient size in fraction division
  6. 15.Measurement versus partitive division situations in word problems
  7. 16.Fraction division embedded in a ratio/rate context
  8. 17.Fraction division applied to a novel real-world context
  9. 18.Structural equivalence between bar-model and common-denominator justifications of fraction division
Unit 3: The Decimal System10 lessons
  1. 19.Place-value alignment in decimal addition and subtraction
  2. 20.The structural difference between decimal addition/subtraction and decimal multiplication
  3. 21.The standard multiplication algorithm applied to decimals
  4. 22.Decimal division as repeated grouping, connected to fraction division reasoning
  5. 23.The standard long-division algorithm with decimal dividends and divisors
  6. 24.The link between a denominator's prime factors (only 2s and 5s) and whether a decimal terminates
  7. 25.Terminating-decimal-to-fraction conversion via place value, including the reason the pre-simplified denominator is a power of ten
  8. 26.The algebraic method (setting x = the decimal, multiplying to shift the repeat, subtracting) for repeating-decimal-to-fraction conversion
  9. 27.Estimation strategies applied specifically to decimal operation results
  10. 28.Multi-step decimal word problems combining two or more of the four operations
Unit 4: Negative Numbers and the Coordinate Plane10 lessons
  1. 29.Signed numbers as representations of opposite-direction quantities
  2. 30.The extended number line below zero
  3. 31.The relationship between opposite numbers and zero
  4. 32.Ordering of positive and negative rational numbers
  5. 33.Absolute value as distance, independent of sign
  6. 34.Absolute value versus numerical order in context
  7. 35.The four-quadrant coordinate plane
  8. 36.Reflections of points across a coordinate axis
  9. 37.Horizontal and vertical distance between coordinate points
  10. 38.Four-quadrant coordinate systems as a representational tool
Unit 5: Expressions with Variables9 lessons
  1. 39.The structural parts of an algebraic expression (coefficient, variable, term, constant)
  2. 40.The correspondence between verbal quantity language and symbolic expressions
  3. 41.Substitution of signed values into an expression using order of operations
  4. 42.The structural meaning of exponential notation as repeated multiplication
  5. 43.Numerical expressions containing exponents evaluated with order of operations
  6. 44.The distributive property applied to expressions with a variable term and a constant term
  7. 45.Equivalence of algebraic expressions verified through both numeric testing and structural comparison
  8. 46.The logical distinction between numeric verification and general proof of expression equivalence
  9. 47.The non-equivalence of (x+y)^2 and x^2+y^2 as a case where surface-level distribution fails
Unit 6: Equations and Proportional Relationships10 lessons
  1. 48.One-step addition and subtraction equations solved using the balance model
  2. 49.One-step multiplication and division equations solved using the balance model
  3. 50.The balance model of equation equality
  4. 51.One-step equations translated from word-problem contexts
  5. 52.Reversal errors in translating comparison sentences to equations
  6. 53.Proportionality tested via constant ratio across a table
  7. 54.Independent and dependent variables in a two-quantity relationship
  8. 55.The constant of proportionality k in y = kx
  9. 56.Graphs of proportional relationships used for prediction beyond given data
  10. 57.Invariance of the graph of y = kx under changing real-world context
Unit 7: Area, Surface Area, and Volume12 lessons
  1. 58.Area of a triangle via decomposition into a rectangle/parallelogram
  2. 59.Conservation of area under decomposition and rearrangement
  3. 60.Area of a trapezoid via decomposition
  4. 61.Area of a polygon on the coordinate plane
  5. 62.Surface area of a composite solid built from familiar solids
  6. 63.Nets of right rectangular prisms
  7. 64.Surface area of prisms via nets
  8. 65.Net design against a surface-area constraint
  9. 66.Volume of a right rectangular prism with fractional edge lengths
  10. 67.Effect of fractional multiplication on volume magnitude
  11. 68.Distinguishing area, surface area, and volume in context
  12. 69.Surface area formula as an algebraic expression with a variable edge
Unit 8: Statistical Distributions9 lessons
  1. 70.The distinction between statistical and non-statistical questions
  2. 71.Dot plots as a graphical representation of a data set
  3. 72.The median as a measure of center
  4. 73.Mean absolute deviation as a measure of variability
  5. 74.Shape of a distribution as shown across two different graph types
  6. 75.The relationship between center, spread, and what counts as a 'typical' or 'consistent' value
  7. 76.The potential for a summary statistic to misrepresent a distribution
  8. 77.A complete statistical comparison of two distributions
  9. 78.The insufficiency of a single summary statistic to characterize a distribution

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