Unit 3

Fraction, Equivalence, Ordering, and Operations

Math

Unit Length and Description:

40 days

Students recognize and generate equivalent fractions.  They work with visual models using these to find patterns to develop understanding. Students use their knowledge and understanding of equivalence and ordering of fractions to create line plots to show a data set of objects and solve simple word problems.

Students extend their understanding of composing and decomposing unit fractions to understanding how to add and subtract fractions with like denominators and composing and decomposing non-unit fractions.  They understand that fractional units behave just like other units, for example, 3 fifths + 1 fifth = 4 fifths.  Students begin with visual models such as the area model, fraction strips, and number lines, then progress to making generalizations for addition and subtraction of fractions with like denominators.

Students begin multiplication of a fraction by a whole number using visual representations. Students connect the meaning of multiplication of whole numbers to multiplication of a fraction by a whole number for example, 5 x 1/3 means 5 groups of 1/3.

Students use area models, fraction strips, number lines, and tape diagrams to solve word problems involving addition and subtraction of fractions with like denominators, and multiplication of a fraction by a whole number.

Standards:

 Major Cluster: Number and Operations-Fractions Extend understanding of fraction equivalence and ordering. 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. 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. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 4.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? Supporting Cluster:  Measurement and Data Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit 4.MD.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. Represent and interpret data 4.MD.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. Standards for Mathematical Practices 1.   Make sense of problems and persevere in solving them. 2.   Reason abstractly and quantitatively. 3.   Construct viable arguments and critique the reasoning of others. 4.   Model with mathematics. 5.   Use appropriate tools strategically. 6.   Attend to precision. 7.   Look for and make use of structure. 8.   Look for and express regularity in repeated reasoning.

Enduring Understandings:

·        Fractions can be composed and decomposed from unit fractions.

·        Mixed numbers and improper fractions can be used interchangeably.

·        Fractions can be represented visually and in written form.

·        Fractions with differing parts can be the same size.

·        Fractions of the same whole can be compared.

·        Fractional numbers and mixed numbers can be added, subtracted, and multiplied.

Essential Questions:

·        What does a fraction represent?

·        What is a mixed number and how can it be represented?

·        How can common numerators or denominators be created?

·        How can equivalent fractions be identified?

·        How can fractions with different numerators and different denominators be compared?

·        How can fractions and mixed numbers be used interchangeably?

·        How do we apply our understanding of fractions in everyday life?

·        What is the relationship between a mixed number and a fraction?