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:
When given a fraction a/b (with denominators of 2, 3, 4, 6, 8),

• Use a number line and partition the interval between 0 and 1 into b equal parts, specified by the denominator.
• Use a number line and partition the interval between 0 and 1 into b equal parts and mark off a lengths of 1/b unit fractions.
• Model a fraction with a point on a number line and recognize the length of the fraction as the distance from the fraction point to 0.
• Extend the number to include fractions greater than one as a continuation of counting unit fractions.
• Given a fraction, draw a model to represent the fraction using a number line.
• Given a fraction and a number line with labeled points, identify the labeled point that represents the fraction.
• Given a point on a number line, identify the fraction modeled by the point.

Students know:

• How to use fraction strips as a model to connect to finding fractional parts on a number line.
• Fractions are numbers that can be represented on a number line.
• Fractions can be placed on the number line by marking off equal parts between two whole numbers.
• Fractions equal to 1 have the same numerator and same denominator.
• Fractions greater than 1 have a numerator that will be greater than the denominator.

Students are able to:

• Represent fractions on a number line.
• Locate fractions on a number line.
• Use a number line and partition an interval from 0 to 1 into equal parts as specified by the denominator of a fraction.
• Represent a non unit fraction on a number line by marking off unit fraction lengths as specified by the numerator from zero.
• Extend the number line to include fractions greater than one as a continuation of counting unit fractions.

Students understand that:

• A number line is a length model.
• Fractions are numbers that represent a quantity less than, equal to, or greater than 1 and can be placed on a number line.
• A number line can be partitioned to represent equal parts of a whole.

Learning Objectives:
M.3.14.1: Recognize fractions as lengths from zero to one.
M.3.14.2: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
M.3.14.3: Identify a number line.
M.3.14.4: Recognize whole numbers as lengths from zero to one.
M.3.14.5: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
M.3.14.6: Identify a number line.
M.3.14.7: Label the fractions on a pre-made number line diagram.
M.3.14.8: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2…, and represent whole-number sums and differences within 100 on a number diagram.
M.3.14.9: Recognize a number line diagram with equally spaced points.

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 problems involving division of whole numbers leading to quotients of a fraction or mixed number.

Example: Given that 3 cookies are shared equally with 6 people, find what fraction of the cookies each person receives. Each person receives 3/6 of a cookie or 1/2 of a cookie.
Example: Given that 3 cookies are shared equally with 2 people, find what fraction of the cookies each person receives. Each person receives 3/2 cookies or 1 1/2 cookies.
• Model and interpret a fraction as division.
• Use models, drawings, or equations to represent word problems.

Students know:

• Contextual situations for division.
• Strategies to equipartition.

Students are able to:

• Solve word problems involving division of whole numbers leading to quotients with fractions.
• Use fraction models, drawings, equations to represent word problems.
• Model and interpret a fraction as division.

Students understand that:

• a ÷ b is a division expression and can be written as a/b showing division of the numerator by the denominator (including cases where the value of a < b).

Learning Objectives:
M.5.11.1: Define a mixed number.
M.5.11.2: Generate equivalent fractions.
M.5.11.3: Recognize a fraction as a number on the number line; represent fractions on a number line diagram.

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.