Getting to the Core Grade 5 Unit of Study Multiplication and Division of Fractions 1 Table of Contents 5th Grade Math Multiplication & Division of Fractions Pages Lessons and Activities 3–8 Unit Overview 9–11 Prerequisite Skills Test 12–16 Preparing the Learner Lesson A: Preparing a Fraction Bar Toolkit 17–21 Preparing the Learner Lesson B: Launching Mathematical Discourse 22–26 Preparing the Learner Lesson C: Learning the Language of Contrast 27–31 Lesson 1: Quotients of Whole Numbers 32–41 Lesson 2: Multiplying Whole Numbers and Fractions 42–49 Lesson 3: Multiplying Fractions with Whole Numbers 50–58 Lesson 4: Multiplying Fractions with Fractions 59–65 Lesson 5: Multiplying Fractions by Fractions 66–76 Lesson 6: Comparing Size of Products 77–86 Lesson 7: Multiplying Fractions with Whole Numbers and Fractions 87–93 Lesson 8: Division of Fractions 94–101 Lesson 9: Dividing Unit Fractions by Whole Numbers 102–108 Lesson 10: Dividing Unit Fractions by Whole Numbers and Whole Numbers by Unit Fractions 109–114 Lesson 11: Dividing Unit Fractions by Whole Numbers 115–124 Lesson 12: Culminating Task and Unit Assessment 118–122 Multiple Choice Test with Answer Key 123–124 Performance-Based Task with Rubric 125–129 Additional Menu Activities and Check Off Sheets 2 Santa Ana Unified School District Common Core Unit Planner-Mathematics Unit Title: Multiplying and Dividing Fractions Grade Level: 5th Grade Time Frame: 3 weeks Big Idea The properties of multiplication and division of whole numbers apply also to the multiplication and division of (Enduring fractions. Understandings): Essential • How are fractions related to division? Questions: • How can the area of a rectangle with fractional sides be represented? • How can a visual model help to show multiplication of a fraction by a whole number? • How does multiplying by a fraction or by a mixed number affect the size of the product? • How can multiplication of fractions and mixed numbers be used in real life situations? • How can division of fractions be used in real life situations? 21st Century Learning and Innovation: Skills: Critical Thinking & Problem Solving Communication & Collaboration Creativity & Innovation Information, Media and Technology: Online Tools Software Hardware Essential Tier II: Tier III: Academic Contrast Multiply Unit fraction Language: However Divide Improper fraction Although Simplest form Mixed number Nevertheless Mixed number Equivalent fraction Moreover Denominator Reciprocal In addition Numerator Similarly What pre-assessment will be given? How will pre-assessment guide instruction? Prerequisite Skills Test Students missing two or more in any section will need intervention through the Preparing the Learner lessons. 3 Instructional Activities: (What learning experiences will students engage in? How will you use these learning experiences to drive responsive teaching?) Preparing the Learner Lesson A Preparing the Learner Lesson B Preparing the Learner Lesson C Preparing a Fraction Bar Toolkit Launching Mathematical Discourse Learning the Language of Contrast CCS 5.4.b Multiply fractions by whole numbers and CCS 5.3 Interpret Fractions as CCS 5.4.a Multiply fractions by whole by other fractions: Division: numbers and by other fractions: Find the area of a rectangle with fractional side Fractions are defined as Apply and extend previous lengths by tiling it with unit squares of the division of the numerator by understandings of multiplication to appropriate unit fraction side lengths, and show that the denominator. multiply a fraction or whole number by a the area is the same as would be found by fraction. multiplying the side lengths. Multiply fractional side When multiplying a fraction times a lengths to find areas of rectangles, and represent whole, the parts of the fraction are fraction products as rectangular areas. partitioned among the whole number. The area of a rectangle with fractional lengths can be found by multiplying the length times the width, just as with whole numbers. CCS5 .7.a,b,c Real world problems with division CCS 5.5.a,b Scaling: Performance Task Assessment or CCS 5. 6 Real world problems of fractions and whole numbers: Comparing the size of a product to the with multiplication of fractions Apply and extend previous understandings of size of one factor on the basis of the size and mixed numbers: of the other factor, without performing division to divide unit fractions by whole Visual and numeric models of the indicated multiplication. numbers and whole numbers by unit fractions: Multiplying by a fraction reduces the multiplication of fractions and Interpret division of a unit fraction by a non- size of the product, while multiplying mixed numbers are used to zero whole number, and compute such by a mixed number increases the size of solve problems in daily life. quotients. Interpret division of a whole number by a unit the product. fraction, and compute such quotients. Division of fractions is used to solve problems in daily life. Performance Task 4 Standards Assessment of Standards Common Core Learning Standards Taught and Assessed What assessments will be utilized for this What does the unit? (F = formative, S = summative) assessment tell us? Common Core Mathematics Content Standards: F: Problem solving journal Ongoing evidence of Number and Operations–Fractions F: Visual representation of thinking students’ Apply and extend previous understandings of multiplication and F: Performance Task : Lesson 1-4 Review Tasks understanding of the division to multiply and divide fractions. F: Lesson 7 Performance Task concepts presented 3. Interpret a fraction as division of the numerator by the denominator (a/b = Diagnostic a ÷ b). Solve word problems involving division of whole numbers leading to information for answers in the form of fractions, mixed numbers, e.g., by using visual intervention or fraction models or equations to represent the problem. acceleration 4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. S: Performance Task: Culminating Task Student a. Interpret the product (a/b) × q as a parts of a partition of q into b equal S: End of Unit Assessment comprehension of parts; equivalently, as the result of a sequence of operations a × q ÷ b. For S: Benchmark Tests unit concepts and the example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a big idea: story context for this equation. Do the same with (2/3) × (4/5) = 8/15.(In Other Evidence: The properties of general, (a/b) × (c/d) = ac/bd.) Teacher observations multiplication and b. Find the area of a rectangle with fractional side lengths by tiling it with division of whole unit squares of the appropriate unit fraction side lengths, and show that the numbers apply also area is the same as would be found by multiplying the side lengths. Multiply to the multiplication fractional side lengths to find areas of rectangles, and represent fraction and division of products as rectangular areas. fractions. 5. Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n b) to the effect of multiplying a/b by 1. 6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship 5 between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? Bundled Language Standard(s): F: Teacher evaluation of student use of Do students use the 3. Use knowledge of language and its conventions when writing, speaking, appropriate mathematical academic language appropriate academic reading, or listening. during partner, small group, and class discussions. language when speaking in class 6. Acquire and use accurately grade-appropriate general academic and discussions and domain-specific words and phrases, including those that signal contrast, S: Use of accurate mathematical terms and presentations and addition, and other logical relationships (e.g., however, although, appropriate relationship language in culminating when writing in their nevertheless, similarly, moreover, in addition). written word problem and its solution. daily math journals? Bundled Speaking and Listening Standard(s): Teacher Evaluation of student speaking and When talking about 1. Engage effectively in a range of collaborative discussions (one-on-one, in listening: mathematics in pairs groups, and teacher-led) with diverse partners on grade 5 topics and texts, and groups, do building on others’ ideas and expressing their own clearly.