Vertical Articulation of Individual
Grade Expectations for Mathematics
Vermont‘s Framework of Standards and Learning Opportunities 

Standard M:1

Conceptual understanding of numeracy and development of number sense.

Introduction: 

Standard M:1 refers to the base ten, place value number system used in the US to identify whole numbers, common fractions, decimal fractions
and percentages. This standard is based on the idea that there are three components of a number as defined in the triangle of meaning; e.g.
 the numeral (5), the quantity (*****) and the spoken word (“five”).

 Kindergarten

MK: 1   Demonstrates conceptual understanding of rational numbers with respect to whole numbers by connecting oral  number words and
  numerals (up to and including two-digit numbers to 50) to the quantities they represent using physical models and representations
  and shows correct sequence of cardinal numbers.

 First grade

 M1: 1   Demonstrates conceptual understanding of rational numbers with respect to whole numbers from 0 to 100 using place value
            (a grouping system wherein a digit’s place in a number denotes its value; e.g., in 34, 3 represents 3 tens, or 30); by applying
            the concepts of equivalency in composing or decomposing numbers (e.g., 12 = 7 + 5); and in expanded notation
            (e.g., 41 = 4 tens + 1 one or 41 = 40 + 1) using models, explanations, or other representations. Shows correct sequence
            of ordinal and cardinal numbers and compares cardinal numbers and positive fractional numbers (benchmark fractions:
            a/2, a/3,or a/4 where a is a whole number greater than 0 and less than or equal to the denominator) as part/whole
            relationships of benchmark fractions with models, diagrams, or written or verbal/scribed response.

 Second Grade

 M2: 1  Demonstrates conceptual understanding of rational numbers with respect to:  whole numbers from 0 to 199 using place value,
  by applying the concepts of equivalency in composing or decomposing numbers (e.g., 34 = 17 + 17; 34 = 29 + 5); and in
  expanded notation (e.g., 141 = 1 hundred + 4 tens + 1 one or 141 = 100 + 40 + 1) using models, explanations, or other
  representations; and  positive fractional numbers (benchmark fractions: a/2, a/3, or a/4, where a is a whole number greater
  than 0 and less than or equal to the denominator) as a part to whole relationship in area and set models where the
  denominator is equal to the number of parts in the whole using models, explanations, or other representations. M(N&O)–2–1

 Third Grade

 M3: 1  Demonstrates conceptual understanding of rational numbers with respect to:  whole numbers from 0 to 999 through
           equivalency, composition, decomposition, or place value using models, explanations, or other representations; and
           positive fractional numbers (benchmark fractions: a/2, a/3, a/4, a/6, or a/8, where a is a whole number greater than 0
           and less than or equal to the denominator) as a part to whole relationship in area and set models where the number of
           parts in the whole is equal to the denominator; and decimals (within a context of money) as a part of 100 using models,
           explanations, or other representations. M(N&O)–3–1

 Fourth Grade

 M4: 1  Demonstrates conceptual understanding of rational numbers with respect to: whole numbers from 0 to 999,999 through
           equivalency, composition, decomposition, or place value using models, explanations, or other representations; and positive
           fractional numbers (benchmark fractions: a/2, a/3, a/4, a/5, a/6, a/8, or a/10, where a is a whole number greater than 0 and
           less than or equal to the denominator) as a part to whole relationship in area, set, or linear models where the number of parts
           in the whole are equal to, and a multiple or factor of the denominator; and decimals as hundredths within the context of money,
           or tenths within the context of metric measurements (e.g., 2.3 cm) using models, explanations, or other representations. M(N&O)–4–1

 Fifth Grade

 M5: 1  Demonstrates conceptual understanding of rational numbers with respect to: whole numbers from 0 to 9,999,999 through
           equivalency, composition, decomposition, or place value using models, explanations, or other representations; positive fractional
           numbers (proper, mixed number, and improper) (halves, fourths, eighths, thirds, sixths, twelfths, fifths, or powers of ten [10, 100, 1000]),
           decimals (to  thousandths), or benchmark percents (10%,25%, 50%, 75% or 100%) as a part to whole relationship in area, set,
           or linear models using models, explanations, or other representations.*M(N&O)–5–1

   *Specifications for area, set, and linear models for grades 5–8: Fractions: The number of parts in the whole is equal to the denominator,
 a multiple of the denominator, or a factor of the denominator. Percents: The number of parts in the whole is equal to 100, a multiple
 of 100, or a factor of 100 (for grade 5); the number of parts in the whole is a multiple or a factor of the numeric value representing the
 whole (for grades 6–8). Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the
 fractional equivalent of the decimal, a multiple of the denominator of the fractional equivalent of the decimal, or a factor of the
 denominator of the fractional equivalent of the decimal.

Sixth Grade

M6: 1 Demonstrates conceptual understanding of rational numbers with respect to ratios (comparison of two whole numbers by division a/b, a : b, 
       and a ÷ b , where b  ≠ 0); and rates (e.g., a out of b, 25%) using models, explanations, or other representations.*Demonstrates conceptual
        understanding of proportional reasoning, and fluently moves between equivalent representations of commonly used fractions and decimals. M
        (N&O)–6–1

       *Specifications for area, set, and linear models for grades 5–8: Fractions: The number of parts in the whole is equal to the denominator, a 
       multiple of the denominator, or a factor of the denominator. Percents: The number of parts in the whole is equal to 100, a multiple of 100, or a
       factor of 100 (for grade 5); the number of parts in the whole is a multiple or a factor of the numeric value representing the whole (for grades 6–8).
       Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a
       multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal.

 Seventh Grade

M7: 1 Demonstrates conceptual understanding of rational numbers with respect to percents as a means of comparing the same or different parts of
         the whole when the wholes vary in magnitude (e.g., 8 girls in a classroom of 16 students compared to 8 girls in a classroom of 20 students, or
         20% of 400 compared to 50% of 100); and percents as a way of expressing multiples of a number (e.g., 200% of 50) using models,
         explanations, or other representations.*M(N&O)–7–1

         *Specifications for area, set, and linear models for grades 5–8: Fractions: The number of parts in the whole is equal to the denominator, a
          multiple of the denominator, or a factor of the denominator. Percents: The number of parts in the whole is equal to 100, a multiple of 100, or a
          factor of 100 (for grade 5); the number of parts in the whole is a multiple or a factor of the numeric value representing the whole (for grades 6–8)        
          Decimals (including powers of ten): The number of parts in the whole is equal to the denominator of the fractional equivalent of the decimal, a
         multiple of the denominator of the fractional equivalent of the decimal, or a factor of the denominator of the fractional equivalent of the decimal.

Eighth Grade

M8: 1 Demonstrates conceptual understanding of rational numbers with respect to percents as a way of describing change (percent increase and
         decrease) using explanations, models, or other representations.

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