ALEX Classroom Resource

Students:
When given any fraction in form a/b,

• Create an area model to represent the fraction.
• Use a number line to represent the fraction.
• Explain the relationship between the fraction and the model including the corresponding number of unit fractions.
  Example: 3/4 is composed of 3 units of 1/4 or 3/4 is the same as 1/4 + 1/4 + 1/4.
• Identify a point to represent the fraction when given on a number line labeled with multiple points.

Note: Set models (parts of a group) are not models used in grade 3.

Students know:

• Fractional parts of a whole must be of equal size but not necessarily equal shape.
• Denominators represent the number of equal size parts that make a whole.
• The more equal pieces in the whole, the smaller the size of the pieces.
• The numerator represents the number of equal pieces in the whole that are being counted or considered.

Students are able to:

• Use an area model and length model to show a unit fraction as one part of an equally partitioned whole.
• Explain that given a fraction with a numerator greater than one, the numerator indicates the number of unit fraction pieces represented by the fraction.
  Example: 3/4 is the same as 3 units of 1/4 size, or three 1/4 pieces, 3 copies of 1/4, or 3 iterations of 1/4.
• Identify and describe the fractional name given a visual fraction model.
• Identify and demonstrate fractional parts of a whole that are the same size but not the same shape using concrete materials.

Students understand that:

• Given the same size whole, the larger the denominator, indicating the number of equal parts in the whole, the smaller the size of the pieces because there are more pieces in the whole.
• Fractions are numbers that represent a quantity less than, equal to, or greater than 1.
• Fractions represent equal partitions of a whole.

Learning Objectives:
M.3.13.1: Define fraction, numerator, and denominator.
M.3.13.2: Identify the parts of a fraction.
M.3.13.3: Label numerator, denominator, and fraction bar.
M.3.13.4: Identify parts of a whole with two, three, or four equal parts.
M.3.13.5: Distinguish between equal and non-equal parts.
M.3.13.6: Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of.

Students:

• Use a variety of area models and length models to identify equivalent fractions.
• Use a variety of area models and length models to illustrate equivalent fractions.
• Use visual representations to find fractions equal to 1.
• Illustrate and explain fractions equivalent to whole numbers (limited to 0 through 5).
• Compare two fractions by reasoning about their size and use <, >, or = to record the comparison.
• Compare two fractions using visual fraction models.
• Use symbols <, >, or = to record the comparison between two fractions.

Note: Tasks in grade 3 are limited to fractions with denominators 2, 3, 4, 6, or 8.

Students know:

• Fractions with different names can be equal.
• Two fractions are equivalent if they are the same size, cover the same area, or are at the same point on a number line.
• Unit fraction counting continues beyond 1 and whole numbers can be written as fractions.
• Use a variety of area models and length models to show that a whole number can be expressed as a fraction and to show that fractions can be equivalent to whole numbers.
• Comparing two fractions is only reasonable if they refer to the same whole.
• The meaning of comparison symbols <, >, = .
• Reason about the size of a fraction to help compare fractions.
• Use a variety of area and length models to represent two fractions that are the same size but have different names.
• Use a fraction model to explain how equivalent fractions can be found.
• Use a variety of area models and length models to demonstrate that any fraction that has the same nonzero numerator and denominator is equivalent to 1.
• Use models to show that the numerator of a fraction indicates the number of parts, so if the denominators of two fractions are the same, the fraction with the greater numerator is the greater fraction.
• Use models to show that the denominator of a fraction indicates the size of equal parts a whole is partitioned into, and that the greater the denominator, the smaller the parts. -Determine when two fractions can not be compared because they do not refer to the same size whole.

Students are able to:

• Explain equivalence of two fractions using visual models and reasoning about their size.
• Compare two fractions with same numerators or with same denominators using visual models and reasoning about their size.
• Express whole numbers as fractions.
• Identify fractions equivalent to whole numbers.
• Record comparisons of two fractions using <, >, or = and justify conclusion.
• Explain that the whole must be the same for the comparing of fractions to be valid.

Students understand that:

• A fraction is a quantity which can be illustrated with a length model or an area model.
• Two fractions can be the same size but have different fraction names.
• A fraction can be equivalent to a whole number.
• Any fraction that has the same nonzero numerator and denominator is equivalent to 1.
• The numerator of a fraction indicates the number of parts, so if the denominators of two fractions are the same, the fraction with the greater number of parts is the greater fraction.
• The denominator of a fraction indicates the size of equal parts in a whole, so the greater the denominator, the smaller the size of the parts in a whole.

Learning Objectives:
M.3.15.1: Define equivalent.
M.3.15.2: Recognize pictorial representations of equivalent fractions.
M.3.15.3: Recognize different interpretations of fractions, including parts of a set or a collection, points on a number line, numbers that lie between two consecutive whole numbers, and lengths of segments on a ruler.
M.3.15.4: Recognize that equal shares of identical wholes need not have the same shape.
M.3.15.5: Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.
M.3.15.6: Label a fraction with multiple representations.
M.3.15.7: Recognize that a whole can be partitioned into differing equal parts (halves, fourths, eighths, etc.).
M.3.15.8: Partition circles and rectangles into two and four equal shares; and describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of.
M.3.15.9: Recognize different interpretations of fractions, including parts of a set or a collection, points on a number line, numbers that lie between two consecutive whole numbers, and lengths of segments on a ruler.
M.3.15.10: Label a pictorial representation.
M.3.15.11: Recognize that a fraction is a part of a whole.
M.3.15.12: Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of.

Students:

• When given any fraction or mixed number, apply unit fraction understanding to decompose the given fraction or mixed number into the sum of smaller fractions, including unit fractions.
• When given a problem solving situation involving addition and subtraction of fractions or mixed numbers with like denominators, explain and justify solutions using unit fractions, visual models, and equations involving a single unknown.

Students know:

• Situation contexts for addition and subtraction problems.
• A variety of strategies and models to represent addition and subtraction situations.
• The fraction a/b is equivalent to the unit fraction 1/b being iterated or "copied" the number of times indicated by the numerator, a.
• A fraction can represent a whole number or fraction greater than 1 and can be illustrated by decomposing the fraction.
  Example: 6/3 = 3/3 + 3/3 = 2 and 5/3 = 3/3 + 2/3 = 1 2/3.

Students are able to:

• Decompose fractions as a sum of unit fractions.
• Model decomposition of fractions as a sum of unit fractions.
• Add and subtract fractions with like denominators using properties of operations and the relationship between addition and subtraction.
• Solve word problems involving addition and subtraction using visual models, drawings, and equations to represent the problem.

Students understand that:

• A unit fraction (1/b) names the size of the unit with respect to the whole and that the denominator tells the number of parts the whole is partitioned, and the numerator indicates the number of parts referenced.
• A variety of models and strategies can be used to represent and solve word situations involving addition and subtraction.
• The operations of addition and subtraction are performed with quantities expressed in like units, and the sum or difference retains the same unit.

Learning Objectives:
M.4.15.1: Recognize that a whole can be partitioned into differing equal parts (halves, fourths, eighths, etc.).
M.4.15.2: Identify numerator and denominator.
M.4.15.3: Recall basic addition and subtraction facts.
M.4.15.4: Demonstrate an understanding of fractional parts.
M.4.15.5: Recall basic addition and subtraction facts.
M.4.15.6: Define mixed numbers.
M.4.15.7: Recall basic addition and subtraction facts.
M.4.15.8: Demonstrate an understanding of fractional parts.
M.4.15.9: Solve basic word problems using whole numbers.
M.4.15.10: Express parts of a whole as a fraction.
M.4.15.11: Write number sentences for word problems.
M.4.15.12: Identify key terms in word problems.
M.4.15.13: Recall basic addition and subtraction facts.

Students:

• Solve real-world problems involving division of a unit fraction by a non-zero whole number, or division of a whole number by a unit fraction.
• Justify solutions using visual models, drawings, and equations to represent the problem context.
• Explain quotients using the relationship between multiplication and division.
• Create a story context for a unit fraction divided by a whole number and use models to illustrate the quotient.
• Create a story context for a whole number divided by a unit fraction and use models to illustrate the quotient.

Students know:

• Contextual situations involving division with whole numbers and unit fractions.
• Strategies for representing a division problem with a visual model.

Students are able to:

• Use previous understandings of operations to
• Divide unit fractions by a whole number and whole numbers by unit fractions.
• Use visual models to illustrate quotients.
• Create story contexts for division.
• Use the relationship between multiplication and division to explain quotients.

Students understand that:

• A variety of contextual situations are represented with division of a whole number by a fraction or a fraction by a whole number.
• Quotients resulting from division of a whole number by a fraction or a fraction by a whole number can be illustrated and justified with a visual model.
• The relationship between multiplication and division can be used to justify quotients resulting from division of a whole number by a fraction or a fraction by a whole number.

Learning Objectives:
M.5.15.1: Define quotient.
M.5.15.2: Multiply a fraction by a whole number.
M.5.15.3: Recognize key terms to solve word problems.
Examples: times, every, at this rate, each, per, equal/equally, in all, total.
M.5.15.4: Recall basic multiplication and division facts.
M.5.15.5: Express whole numbers as fractions.
M.5.15.6: Recognize fractions that are equivalent to whole numbers.
M.5.15.7: Recall basic multiplication and division facts.
M.5.15.8: Solve word problems involving multiplication of a fraction by a whole number.
M.5.15.9: Recognize key terms to solve word problems.
M.5.15.10: Recall basic multiplication and division facts.