Grade 5 · printable course

Place Value, Parts, and Space: Grade 5 Mathematics

Grade 5 · 317 pages · 72 lessons · PDF, print at home

A full-year grade 5 mathematics course built on three connected stories: how our place-value system extends to decimals and governs all arithmetic, how fractions behave as amounts that can be combined and split, and how measurement and space (volume, the coordinate plane, patterns) let us model the world. Every new procedure is introduced with an area model, number line, or fraction bar before any symbolic algorithm. Units are sequenced so each new topic depends on a named skill from the one before, with cumulative retrieval built into every unit after the first. Aligned to CCSS grade 5 mathematics.

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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.

Place Value, Parts, and Space: Grade 5 Mathematics

72 lessons · try a lesson online free

Unit 1: Place Value to the Thousandths8 lessons
  1. 1.The ten-times relationship between adjacent place value positions
  2. 2.Base-ten block representation of decimals to the thousandths place
  3. 3.The zero-pattern produced by multiplying by powers of ten
  4. 4.The decimal-point shift caused by multiplying or dividing by a power of ten written in exponent form
  5. 5.The three representations of a decimal (word, standard, expanded form)
  6. 6.Place-value-based comparison of two decimals with unequal digit counts
  7. 7.Rounding a decimal to a specified place using place value reasoning
  8. 8.Ranking real-world decimal measurements by place value in an unfamiliar context
Unit 2: Decimal Arithmetic9 lessons
  1. 9.Decimal addition and subtraction with aligned place value
  2. 10.The rule that decimal point alignment (not rightmost-digit alignment) preserves place value in addition/subtraction
  3. 11.Decimal-by-whole-number multiplication represented as an area model
  4. 12.The effect of multiplying by a factor less than one on the size of the product
  5. 13.Estimation as a check on decimal point placement in multiplication
  6. 14.Decimal division by a whole number represented as equal-group hops on a number line
  7. 15.The relationship between divisor size and quotient size in decimal division
  8. 16.Decimal arithmetic applied across all four operations in a combined real-world context
  9. 17.Categories of decimal computation errors tied to specific place-value breakdowns
Unit 3: Multi-Digit Division10 lessons
  1. 18.Estimation of quotients using rounded, place-value-friendly divisors and dividends
  2. 19.The area model for division with a two-digit divisor
  3. 20.The relationship between partial quotients in an area model and the standard algorithm's digits
  4. 21.The standard division algorithm with two-digit divisors
  5. 22.The adjust-and-retry step in multi-digit division
  6. 23.Remainder interpretation rules matched to real-world contexts
  7. 24.Justification of remainder-handling choice for a specific context
  8. 25.Numerical expressions with parentheses, brackets, or braces including a division operation
  9. 26.Translation of a word-problem calculation into a numerical expression with grouping symbols
  10. 27.Reasonableness-checking of a computed quotient against an estimate
Unit 4: Adding and Subtracting Fractions8 lessons
  1. 28.Why unlike-size fraction pieces cannot be directly added
  2. 29.Common denominators found via equivalence models
  3. 30.Addition and subtraction of unlike-denominator fractions
  4. 31.Regrouping a whole number into fraction pieces during mixed-number subtraction
  5. 32.Benchmark fractions 0, 1/2, and 1 as reference points for magnitude
  6. 33.Benchmark estimation of a fraction sum prior to exact computation
  7. 34.Operation selection in unlike-denominator fraction word problems
  8. 35.Applicability of unlike-denominator fraction addition/subtraction to an unfamiliar multi-quantity context
Unit 5: Multiplying Fractions10 lessons
  1. 36.Whole number times a fraction as repeated parts (repeated addition of the same fraction)
  2. 37.Area model for fraction times fraction
  3. 38.The numeric equation that matches an already-completed area model
  4. 39.Multiplication as scaling: product size relative to a factor
  5. 40.The fraction multiplication algorithm (numerator times numerator over denominator times denominator)
  6. 41.Multiplying mixed numbers via improper fraction conversion
  7. 42.Why multiplying two proper fractions produces a smaller result than either starting fraction
  8. 43.Real-world area problems involving a fraction of a fraction
  9. 44.Scaling reasoning and fraction multiplication applied to an unfamiliar real-world context
  10. 45.Contrast between the area model for multiplication and the fraction-bar model for addition
Unit 6: Dividing Unit Fractions9 lessons
  1. 46.Whole number divided by a unit fraction, modeled as counting how many pieces fit
  2. 47.Counting the total number of unit-fraction pieces shown in an already-drawn model
  3. 48.The numeric expression that matches a whole ÷ unit fraction model
  4. 49.Unit fraction divided by a whole number, modeled as splitting one piece into smaller equal pieces
  5. 50.Why dividing a whole number by a fraction less than one increases the quotient
  6. 51.The structural difference between whole ÷ unit fraction and unit fraction ÷ whole number
  7. 52.Selecting the correct fraction-division model for a given word problem context
  8. 53.Distinguishing fraction multiplication stories from fraction division stories
  9. 54.Applying the whole ÷ unit fraction model to an unfamiliar measurement context
Unit 7: Volume: Packing and Multiplying9 lessons
  1. 55.Volume of a rectangular prism found by counting unit cubes
  2. 56.The equivalence between counting unit cubes and multiplying three edge lengths
  3. 57.Volume formula for a rectangular prism
  4. 58.Unit cubes needed to complete a single layer of a rectangular prism
  5. 59.Volume formula applied to real-world rectangular containers
  6. 60.Volume of a composite figure made of two rectangular prisms
  7. 61.Missing edge length of a rectangular prism found by division
  8. 62.Problem type classification among packing, multiplying, composite, and missing-edge volume tasks
  9. 63.Reconciling a counted volume with a multiplied volume for a novel composite figure
Unit 8: The Coordinate Plane and Numerical Patterns9 lessons
  1. 64.The parts of a first-quadrant coordinate grid (x-axis, y-axis, origin)
  2. 65.The ordered-pair plotting procedure (x-distance then y-distance)
  3. 66.The non-commutativity of ordered-pair order (why (2,5) and (5,2) name different points)
  4. 67.Coordinate values interpreted as quantities in a real-world situation (e.g., minutes and miles)
  5. 68.A numerical sequence generated by iterating a stated rule (e.g., add 3 each time)
  6. 69.Two corresponding numerical sequences generated from two rules involving fraction or decimal operations
  7. 70.The multiplicative or additive relationship between two corresponding numerical sequences, as shown by their graphed points
  8. 71.An unstated multiplicative relationship between two sequences, inferred solely from a scatter of graphed points
  9. 72.The difference between a coordinate pair's numeric value and its meaning within a specific real-world situation

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