Nuts & Bolts: Number Operations Continuum

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Number Operations Continuum

Addition and Subtraction (Combining and Comparing)

  • Develops computation strategies for combining and comparing based on number sense and number relationships.
  • Uses landmark numbers (multiples of 10 and 100) in comparing and combining quantities.
  • Examines how parts and the whole are related in addition and subtraction.
  • Solves addition problems with multiple addends.
  • Develops more than one way to solve a computation problem and uses one method to check another.
  • Solves compare and combine problems with strategies and records with standard addition and subtraction notation.
  • Makes comparisons of how things change over time.
  • Learns to weigh with a pan balance.
  • Explores number relationships in the context of time, money, and linear measure.
  • Uses important equivalencies of time, money, and linear measure.
  • Estimates solutions that can be adjusted to construct an exact solution.
  • Reads and writes numbers in the hundreds and thousands.
  • Develops strategies to combine and compare quantities in the hundreds and thousands.
  • Develops conjectures and predictions; evaluates data and evidence.
  • Collects, records, and graphs data.
  • Describes and interprets data.
  • Explores the mathematical characteristics of the calendar.
  • Develops strategies for problems that combine addition and subtraction.

Addition and Subtraction (Money, Miles, and Large Numbers)

  • Estimates sums, including total amounts of money.
  • Explores strategies for comparing and combining numbers, through hundreds and thousands.
  • Uses landmark numbers (multiples of 10 or 0.10 and 100 or 1.00) to compare and find differences between two quantities.
  • Uses standard addition and subtraction notation to record combining and comparing situations.
  • Uses the calculator to solve problems and interpret decimals on the calculator as amounts of money.
  • Estimates local distances in miles and tenths of miles: develops a sense of about how long a mile and 1/10 of a mile are.
  • Compares and combines decimal numbers and, later, quantities with decimal portions.
  • Sees the relationships of decimal parts to the whole.
  • Measures distances on maps using a scale.
  • Becomes familiar with common decimal and fraction equivalents.
  • Considers whether events are likely or unlikely to occur (Ten Minute Math).


Multiplication and Division (Things That Come in Groups)

  • Finds things that come in groups.
  • Uses multiplication to mean groups.
  • Recognizes that skip counting represents multiples of the same number and has a connection to multiplication.
  • Finds patterns in multiples of 2, 3, 4, 5, 6, 9, 10, 11, and 12 by using the 100 chart and the calculator.
  • Understands that number patterns can help in multiplication.
  • Recognizes that multiplication can be used to find the area of a rectangle.
  • Uses arrays to skip count; multiplies and divides with skip counting.
  • Finds factor pairs.
  • Understands relationships between multiplication and division.
  • Identifies whether word problems can be solved using division and/or multiplication.
  • Uses multiplication and/or division notation to write number sentences.
  • Uses patterns to solve multiplication and division problems.
  • Organizes and presents data in tables and line plots.
  • Sorts out complex problems that require both multiplication and addition.
  • Describes events as likely and unlikely (Ten Minute Math).


Multiplication and Division (Arrays and Shares)

  • Uses skip counting as a model for multiplication.
  • Sees multiplication as an accumulation of groups of a number.
  • Looks for the multiplication patterns of numbers (including patterns of multiples highlighted on the 100 chart).
  • Uses known multiplication relationships to solve harder relationships.
  • Uses an array as a model for multiplication.
  • Recognizes prime numbers as those that each have only one pair of factors and only one array.
  • Understands how division notation represents a variety of division situations (including sharing and partitioning situations).
  • Determines what to do with leftovers in division, depending on the situation.
  • Partitions numbers to multiply them more easily (e.g., 7x23 can be 7x10 plus 7x10 plus 7x3).
  • Learns about patterns that are useful for multiplying by multiples of 10.


Multiplication and Division (Packages and Groups)

  • Looks for and uses multiplication patterns of numbers (e.g., identifies multiples of 5 by seeing that the units digit is either a 5 or a 0).
  • Finds multiples and becomes familiar with the multiples of larger numbers (e.g., skip counting by 2-digit numbers like 25).
  • Identifies factors of larger numbers (including triple-digit numbers).
  • Uses familiar landmark numbers to solve problems (e.g., determining whether the solution is greater than 100, 200, 300, etc., or estimating 32x9 as 30x10 or 300).
  • Partitions large numbers to multiply them more easily (e.g., 24x8 is thought of as 20x8+4x8).
  • Solves double-digit multiplication problems (e.g., 32x21).
  • Understands how division notation can represent a variety of division situations, including sharing and grouping situations.
  • Creates a context that is representative of a division equation (e.g., represents 152÷4=38 with 152 apples divided into 38 packages of 4).
  • Uses multiplication and division relationships in order to solve problems.
  • Describes features of data; interprets and poses questions about data (Ten Minute Math).
  • Recognizes and describes characteristics of numbers and relationships among numbers (Ten Minute Math).


Computation and Estimation Strategies (Building on Numbers You Know)

  • Skip counts by 2-, 3-, and 4-digit numbers (including landmark numbers).
  • Relates skip counting to multiplication and division.
  • Finds and uses patterns in sequences of multiples.
  • Reads, writes, and orders large numbers, and approximates them to the nearest multiple of 100 or 1000.
  • Develops strategies for determining and comparing distances between numbers.
  • Uses random digits to approximate 4- or 5-digit numbers.
  • Develops, records, explains, and compares strategies for estimating and solving subtraction, multiplication, and division problems in more than one way.
  • Makes sense of remainders in a variety of contexts.
  • Interprets, records, and uses division and multiplication notation in a variety of situations.
  • Understands and explains the relationships among the four basic operations, and using those relationships to solve problems and model situations.
  • Develops real-life meaning for quantities in the thousands, ten thousands, and hundred thousands, and begins to acquire a sense of the size of 1,000,000.
  • Breaks difficult computation problems into manageable parts.
  • Uses a rectangular array model to represent factor pairs of numbers 10,000 and larger.
  • Visualizes ratios, makes predictions, and explores the relationship of a sample to its group (Ten Minute Math).
  • Develops concepts and language to communicate about shapes, patterns, and visual images (Ten Minute Math).

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