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UNIT 5 STANDARDS:
Develop understanding of fractions as numbers
MGSE3.NF.1 Understand a fraction 1 𝑏 as the quantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a fraction 𝑎 𝑏 as the quantity formed by a parts of size 1 𝑏 . For example, 3 4 means there are three 1 4 parts, so 3 4 = 1 4 + 1 4 + 1 4 .
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1 𝑏 on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1 𝑏 . Recognize that a unit fraction 1 𝑏 is located 1 𝑏 whole unit from 0 on the number line.
b. Represent a non-unit fraction 𝑎 𝑏 on a number line diagram by marking off a lengths of 1 𝑏 (unit fractions) from 0. Recognize that the resulting interval has size 𝑎 𝑏 and that its endpoint locates the non-unit fraction 𝑎 𝑏 on the number line.
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models. Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and 8, e.g., 1 2 = 2 4 , 4 6 = 2 3 . Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 6 2 (3 wholes is equal to six halves); recognize that 3 1 = 3; locate 4 4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
MGSE3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
OVERVIEW:
In this unit, students will:
● Develop an understanding of fractions, beginning with unit fractions.
● View fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole.
● Understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one.
● Solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.
● Recognize that the numerator is the top number (term) of a fraction and that it represents the number of equal-sized parts of a set or whole; recognize that the denominator is the bottom number (term) of a fraction and that it represents the total number of equal-sized parts or the total number of objects of the set
● Explain the concept that the larger the denominator, the smaller the size of the piece
● Compare common fractions with like denominators and tell why one fraction is greater than, less than, or equal to the other
● Represent halves, thirds, fourths, sixths, and eighths using various fraction models.
BIG IDEAS
In first grade and second grades, students discuss partitioning and equal shares. Students will have partitioned circles and rectangles into two, three, and four equal shares. This is the first time students are understanding/representing fractions through the use of a number line, and developing deep understanding of fractional parts, sizes, and relationships between fractions. This is a foundational building block of fractions, which will be extended in future grades. Students should have ample experiences using the words, halves, thirds, fourths, and quarters, and the phrases half of, third of, fourth of, and quarter of. Students should also work with the idea of the whole, which is composed of two halves, four fourths or four quarters, etc.
Example: How can you and a friend share equally (partition) this piece of paper so that you both have the same amount of paper to paint a picture?
● Fractional parts are equal shares of a whole or a whole set.
● The more equal sized pieces that form a whole, the smaller the pieces of the whole become.
● When the numerator and denominator are the same number, the fraction equals one whole.
● When the wholes are the same size, the smaller the denominator, the larger the pieces.
● The fraction name (half, third, etc) indicates the number of equal parts in the whole.
ESSENTIAL QUESTIONS
● How are fractions used in problem-solving situations?
● How can I compare fractions?
● What are the important features of a unit fraction?
● What relationships can I discover about fractions?
CONCEPTS/SKILLS TO MAINTAIN
Third-grade students will have prior knowledge/experience related to the concepts and skills identified in this unit.
● In first grade, students are expected to partition circles and rectangles into two or four equal shares, and use the words, halves, half of, a fourth of, and quarter of.
● In second grade, students are expected to partition circles and rectangles into two, three, or four equal shares, and use the words, halves, thirds, half of, a third of, fourth of, quarter of.
● Students should also understand that decomposing into more equal shares equals smaller shares, and that equal shares of identical wholes need not have the same shape.
Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.
Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.
Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.
Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.
Fluent students:
• flexibly use a combination of deep understanding, number sense, and memorization.
• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.
• are able to articulate their reasoning.
• find solutions through a number of different paths.
UNIT 5 STANDARDS:
Develop understanding of fractions as numbers
MGSE3.NF.1 Understand a fraction 1 𝑏 as the quantity formed by 1 part when a whole is partitioned into b equal parts (unit fraction); understand a fraction 𝑎 𝑏 as the quantity formed by a parts of size 1 𝑏 . For example, 3 4 means there are three 1 4 parts, so 3 4 = 1 4 + 1 4 + 1 4 .
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
a. Represent a fraction 1 𝑏 on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1 𝑏 . Recognize that a unit fraction 1 𝑏 is located 1 𝑏 whole unit from 0 on the number line.
b. Represent a non-unit fraction 𝑎 𝑏 on a number line diagram by marking off a lengths of 1 𝑏 (unit fractions) from 0. Recognize that the resulting interval has size 𝑎 𝑏 and that its endpoint locates the non-unit fraction 𝑎 𝑏 on the number line.
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models. Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and 8, e.g., 1 2 = 2 4 , 4 6 = 2 3 . Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 6 2 (3 wholes is equal to six halves); recognize that 3 1 = 3; locate 4 4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
MGSE3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.
MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.
OVERVIEW:
In this unit, students will:
● Develop an understanding of fractions, beginning with unit fractions.
● View fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole.
● Understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one.
● Solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.
● Recognize that the numerator is the top number (term) of a fraction and that it represents the number of equal-sized parts of a set or whole; recognize that the denominator is the bottom number (term) of a fraction and that it represents the total number of equal-sized parts or the total number of objects of the set
● Explain the concept that the larger the denominator, the smaller the size of the piece
● Compare common fractions with like denominators and tell why one fraction is greater than, less than, or equal to the other
● Represent halves, thirds, fourths, sixths, and eighths using various fraction models.
BIG IDEAS
In first grade and second grades, students discuss partitioning and equal shares. Students will have partitioned circles and rectangles into two, three, and four equal shares. This is the first time students are understanding/representing fractions through the use of a number line, and developing deep understanding of fractional parts, sizes, and relationships between fractions. This is a foundational building block of fractions, which will be extended in future grades. Students should have ample experiences using the words, halves, thirds, fourths, and quarters, and the phrases half of, third of, fourth of, and quarter of. Students should also work with the idea of the whole, which is composed of two halves, four fourths or four quarters, etc.
Example: How can you and a friend share equally (partition) this piece of paper so that you both have the same amount of paper to paint a picture?
● Fractional parts are equal shares of a whole or a whole set.
● The more equal sized pieces that form a whole, the smaller the pieces of the whole become.
● When the numerator and denominator are the same number, the fraction equals one whole.
● When the wholes are the same size, the smaller the denominator, the larger the pieces.
● The fraction name (half, third, etc) indicates the number of equal parts in the whole.
ESSENTIAL QUESTIONS
● How are fractions used in problem-solving situations?
● How can I compare fractions?
● What are the important features of a unit fraction?
● What relationships can I discover about fractions?
CONCEPTS/SKILLS TO MAINTAIN
Third-grade students will have prior knowledge/experience related to the concepts and skills identified in this unit.
● In first grade, students are expected to partition circles and rectangles into two or four equal shares, and use the words, halves, half of, a fourth of, and quarter of.
● In second grade, students are expected to partition circles and rectangles into two, three, or four equal shares, and use the words, halves, thirds, half of, a third of, fourth of, quarter of.
● Students should also understand that decomposing into more equal shares equals smaller shares, and that equal shares of identical wholes need not have the same shape.
Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.
Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.
Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.
Number Sense: Students consider the context of a problem, look at the numbers in a problem, make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.
Fluent students:
• flexibly use a combination of deep understanding, number sense, and memorization.
• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.
• are able to articulate their reasoning.
• find solutions through a number of different paths.