Content Standard(s):
Mathematics MA2019 (2019) Grade: 3 13. Demonstrate that a unit fraction represents one part of an area model or length model of a whole that has been equally partitioned; explain that a numerator greater than one indicates the number of unit pieces represented by the fraction.
Unpacked Content
Evidence Of Student Attainment:
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.
Teacher Vocabulary:
Unit fraction
Area model
Interval
Length (Linear) model
Partition
Numerator
Denominator
Part
Point
Whole Knowledge:
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. Skills:
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. Understanding:
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. Diverse Learning Needs:
Essential Skills:
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.
Prior Knowledge Skills:
Define halves, thirds, fourths, quarters, whole, parts (shares) and equal.
Distinguish between equal and non-qual parts.
Alabama Alternate Achievement Standards
AAS Standard: M.AAS.3.15 Compare fractions.
M.AAS.3.15a Use models to identify two equivalent fractions (limit to fourths and halves).
M.AAS.3.15b Recognize two equivalent fractions (limit to fourths and halves).
M.AAS.3.15c Use models of fourths and halves to make a whole.
Mathematics MA2019 (2019) Grade: 3 15. Explain equivalence and compare fractions by reasoning about their size using visual fraction models and number lines.
a. Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers.
b. Compare two fractions with the same numerator or with the same denominator by reasoning about their size (recognizing that fractions must refer to the same whole for the comparison to be valid). Record comparisons using < , >, or = and justify conclusions.
Unpacked Content
Evidence Of Student Attainment:
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.
Teacher Vocabulary:
Equivalence
Visual fraction model
Number line
Numerator
Denominator
Reasoning
Conclusions
Comparison
Point Knowledge:
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. Skills:
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. Understanding:
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. Diverse Learning Needs:
Essential Skills:
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.
Prior Knowledge Skills:
Label numerator, denominator, and fraction bar.
Match the number in the ones, tens, and hundreds position to a pictorial representation or manipulative of the value.
Distinguish between equal and non-qual parts.
Model partitioning circles and rectangles.
Identify two-dimensional shapes.
Sort two-dimensional shapes.
Name shapes.
Alabama Alternate Achievement Standards
AAS Standard: M.AAS.3.15 Compare fractions.
M.AAS.3.15a Use models to identify two equivalent fractions (limit to fourths and halves).
M.AAS.3.15b Recognize two equivalent fractions (limit to fourths and halves).
M.AAS.3.15c Use models of fourths and halves to make a whole.
Mathematics MA2019 (2019) Grade: 4 13. Using area and length fraction models, explain why one fraction is equivalent to another, taking into account that the number and size of the parts differ even though the two fractions themselves are the same size. a. Apply principles of fraction equivalence to recognize and generate equivalent fractions.
Example: ^{a} /_{b} is equivalent to ^{(n x a)} /_{(nĂ— b)} .
Unpacked Content
Evidence Of Student Attainment:
Students:
Use visual models to create equivalent fractions.
Explain the generalized pattern, a/b = (n x a) / (n x b).
Use the generalized pattern to create equivalent fractions.
Set models (parts of a group) are not models used in grade 4.
Teacher Vocabulary:
Fraction
Numerator
Denominator
Equivalent
Fraction model
Area model
-Length model Knowledge:
Students know:
Fractions can be equivalent even though the number of parts and size of the parts differ.
Two fractions are equivalent if they are at the same point on a number line or if they have the same area. Skills:
Students are able to:
Use area and length fraction models to explain why fractions are equivalent.
Recognize and generate equivalent fractions. Understanding:
Students understand that:
equivalent fractions are fractions that represent equal value. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.4.13.1: Identify fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts and size 1/b.
M.4.13.2: Identify a fraction as a number on the number line; represent fractions on a number line diagram.
M.4.13.3: Recognize a fraction as a number on the number line.
M.4.13.4: Represent fractions on a number line diagram.
M.4.13.5: Recognize fractions as numerals that may represent division problems.
M.4.13.6: Label numerator, denominator, and fraction bar.
M.4.13.7: Identify parts of a whole with two, three, or four equal parts.
M.4.13.8: Distinguish between equal and non-equal parts.
M.4.13.9: Define area, length, equivalent, fraction, numerator and denominator.
Prior Knowledge Skills:
Recognize fractions as lengths from zero to one.
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.
Identify a number line.
Recognize whole numbers as lengths from zero to one.
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.
Identify a number line.
Label the fractions on a pre-made number line diagram.
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.
Recognize a number line diagram with equally spaced points.
Compare length using non standard units to determine which is longer.
Alabama Alternate Achievement Standards
AAS Standard: M.AAS.4.13 Identify and compare models of a whole (1), one-half (1/2), one-third (1/3), and one fourth (1/4) using models, manipulatives, numbers lines, and a clock.
Mathematics MA2019 (2019) Grade: 4 15. Model and justify decompositions of fractions and explain addition and subtraction of fractions as joining or separating parts referring to the same whole.
a. Decompose a fraction as a sum of unit fractions and as a sum of fractions with the same denominator in more than one way using area models, length models, and equations.
b. Add and subtract fractions and mixed numbers with like denominators using fraction equivalence, properties of operations, and the relationship between addition and subtraction.
c. Solve word problems involving addition and subtraction of fractions and mixed numbers having like denominators, using drawings, visual fraction models, and equations to represent the problem.
Unpacked Content
Evidence Of Student Attainment:
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. Teacher Vocabulary:
Decomposition
Unit fraction
Area model
Length model
Equation
Mixed number
Visual fraction model
Whole
Sum
Difference
Recomposition Knowledge:
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. Skills:
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. Understanding:
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. Diverse Learning Needs:
Essential Skills:
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.
Prior Knowledge Skills:
Define fraction, numerator, and denominator.
Identify the parts of a fraction a/b as the quantity formed by a parts and size 1/b.
Label numerator, denominator, and fraction bar.
Identify parts of a whole with two, three, or four equal parts.
Distinguish between equal and non-qual parts.
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.
Recognize fractions as lengths from zero to one.
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.
Alabama Alternate Achievement Standards
AAS Standard: M.AAS.4.15 Model decomposing fractions having like denominators, using visual fraction models (limit to half and fourths).
Mathematics MA2019 (2019) Grade: 5 10. Add and subtract fractions and mixed numbers with unlike denominators, using fraction equivalence to calculate a sum or difference of fractions or mixed numbers with like denominators.
Unpacked Content
Evidence Of Student Attainment:
Students:
Use a variety of strategies and fraction equivalence to find sums and differences of fractions and mixed numbers with unlike denominators. Teacher Vocabulary:
Fraction
Denominator
Numerator
Visual Model
Sum
Difference
Equivalence
Unlike denominators
Unlike units Knowledge:
Students know:
Strategies to determine if two given fractions are equivalent.
How to use a visual model to illustrate fraction equivalency.
Contextual situations for addition and subtraction. Skills:
Students are able to:
Use fraction equivalence to add and subtract fractions and mixed numbers with unlike denominators. Understanding:
Students understand that:
Addition and subtraction of fractions and mixed numbers with unlike units,
Require strategies to find equivalent fractions in a common unit, and the sum or difference will be expressed in the common unit. Can be assessed for reasonableness of answers using estimation strategies. Diverse Learning Needs:
Essential Skills:
Learning Objectives: M.5.10.1: Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
M.5.10.2: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
M.5.10.3: Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
M.5.10.4: Recognize and generate simple equivalent fractions.
M.5.10.5: Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
M.5.10.6: Compare two fractions with the same numerator or the same denominator by reasoning about their size.
M.5.10.7: Recall basic addition, subtraction, multiplication, and division facts.
Prior Knowledge Skills:
Use fraction equivalence to add and subtract fractions and mixed numbers with unlike denominators.