1 
 divide whole objects into parts and identify and describe, through investigation, equalsized parts of the whole, using fractional names (e.g., halves; fourths or quarters).

2 
 determine, through investigation using concrete materials, the relationship between the number of fractional parts of a whole and the size of the fractional parts (e.g., a paper plate divided into fourths has larger parts than a
paper plate divided into eighths) (Sample problem: Use paper squares to show which is bigger, one half of a square or one fourth of a square.).

2 
 regroup fractional parts into wholes, using concrete materials (e.g., combine nine fourths to form two wholes and one fourth);

3 
 divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g., one half; three thirds; two fourths or two quarters), without using numbers in standard fractional notation.

4 
 represent fractions using concrete materials, words, and standard fractional notation, and explain the meaning of the denominator as the number of the fractional parts of a whole or a set, and the numerator as the number of
fractional parts being considered;

4 
 count forward by halves, thirds, fourths, and tenths to beyond one whole, using concrete materials and number lines (e.g., use fraction circles to count fourths: “One fourth, two fourths, three fourths, four fourths, five
fourths, six fourths, ...”);

5 
 represent, compare, and order fractional amounts with like denominators, including proper and improper fractions and mixed numbers, using a variety of tools (e.g., fraction circles, Cuisenaire rods, number lines) and using
standard fractional notation;

5 
 demonstrate and explain the concept of equivalent fractions, using concrete materials (e.g., use fraction strips to show that ^{3}⁄_{4}
is equal to ^{9}⁄_{12});

6 
 represent, compare, and order fractional amounts with unlike denominators, including proper and improper fractions and mixed numbers, using a variety of tools and using standard fractional notation;

7 
 use a variety of mental strategies to solve problems involving the addition and subtraction of fractions and decimals;

7 
 add and subtract fractions with simple like and unlike denominators, using a variety of tools and algorithms;

8 
 represent, compare, and order rational numbers;

8 
 use estimation when solving problems involving operations with whole numbers, decimals, percents, integers, and fractions, to help judge the reasonableness of a solution;

8 
 solve problems involving addition, subtraction, multiplication, and division with simple fractions.

9D 
 simplify numerical expressions involving integers and rational numbers, with and without the use of technology;

9D 
 solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion;

9D 
 solve for the unknown value in a proportion, using a variety of methods (e.g., concrete materials, algebraic reasoning, equivalent ratios, constant of proportionality) (Sample problem: Solve ^{x}⁄_{4} = ^{15}⁄_{20}.);

9P 
 solve problems requiring the expression of percents, fractions, and decimals in their equivalent forms

9P 
 simplify numerical expressions involving integers and rational numbers, with and without the use of technology;*
