Overview. Essential Questions. Grade 4 Mathematics, Quarter 1, Unit 1.4 Using Factors, Multiples, and Arrays

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1 Using Factors, Multiples, and Arrays Overview Number of instruction days: 9 11 (1 day = 90 minutes) Content to Be Learned Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range of is prime or composite. Solve multistep word problems involving multiplication and division of whole numbers. Interpret a multiplication problem as a comparison written times as many. Represent verbal statements of multiplication comparisons as multiplication equations. Find the area and perimeter of rectangles using factors and arrays. Use variables to represent unknown quantities in multistep word problems. Mathematical Practices to Be Integrated 7 Look for and make use of structure. Students will see that 7 X 8 equals the wellremembered (7 X 5) + (7 X 3), in preparation for learning about the distributive property. A number that has exactly two factors is a prime number. Students will use arrays to discover that products and area are related. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice whether calculations are repeated and look both for general methods and for shortcuts. Students work to solve a problem maintaining oversight of the process while attending to details. Essential Questions What is the difference between a prime and a composite number? How can an array help you solve a multiplication problem? How can you solve for an unknown quantity using multiplication and division? What strategies can you use to solve a multistep word problem? How can you find all the factor pairs of a whole number between 1 and 100? How are the products of two factors related to the area of a rectangle? Providence Public Schools D-21

2 Using Factors, Multiples, and Arrays (9 11 days) Standards Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems. 4.OA.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 1 1 See Glossary, Table 2. 4.OA.3 Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Gain familiarity with factors and multiples. 4.OA.4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. Common Core State Standards for Mathematical Practice 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see D-22 Providence Public Schools

3 Using Factors, Multiples, and Arrays (9 11 days) Grade 4 Mathematics, Quarter 1, Unit 1.4 complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In Grade 3, multiplication was a major cluster. Students represented and solved problems involving multiplication and division in situations involving equal groups, arrays, and measurement quantities. Students multiplied and divided within 100. By the end of Grade 3, students knew all the products of 2 one-digit numbers. Students solved problems involving the four operations. They explored using arrays to understand the concept of area. Students solved problems involving perimeter of polygons. Current Learning In Grade 4, factors and multiples are a supporting cluster. Students gain familiarity with factors and multiples. They identify numbers from as prime or composite. Area is addressed in this unit because it is multiplication, but the formula for area is not formally addressed until quarter 4. In Grade 4, multiplication is a major cluster. Students interpret a multiplication problem as a comparison written times as many. (See Table 2 on page 89 of the Common Core State Standards for Mathematics for common multiplication and division situations.) Students use the four operations with whole numbers to solve problems. Multi-digit multiplication is addressed in quarter 2. Future Learning In Grade 5, multiplication is a major cluster. Students will perform operations with multi-digit whole numbers and with decimals to the hundredths. Students will fluently multiply multi-digit whole numbers by the end of Grade 5. Additional Findings In Grades 3 5 students should focus on the meanings of and relationship between multiplication and division. It is important that students understand what each number in a multiplication and division expression represents. Modeling multiplication problems with pictures, diagrams, or concrete materials helps students learn what the factors and their products represent in various contexts (Principles and Standards for School Mathematics, p.151). Providence Public Schools D-23

4 Using Factors, Multiples, and Arrays (9 11 days) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Demonstrate how to find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range of is prime or composite. Explain the difference between a prime and composite number. Solve multi-step word problems involving multiplication and division of whole numbers. Interpret a multiplication problem as a comparison written times as many. Represent verbal statements of multiplication comparisons as multiplication equations. Find the area and perimeter of rectangles using factors and arrays. Use variables to represent unknown quantities in multi-step word problems. Instruction Learning Objectives Students will be able to: Find all factor pairs for a whole number in the range Determine whether a given whole number in the range of is prime or composite. Solve multi-step word problems involving multiplication and division of whole numbers. Interpret a multiplication problem as a comparison written times as many. Represent verbal statements of multiplication comparisons as multiplication equations. Find the area and perimeter of rectangles using factors and arrays. Use variables to represent unknown quantities in multi-step word problems. Demonstrate understanding of the concepts and skills in this unit. D-24 Providence Public Schools

5 Using Factors, Multiples, and Arrays (9 11 days) Grade 4 Mathematics, Quarter 1, Unit 1.4 Resources envision Math Grade 4, Pearson Education, Inc., 2009 Topic 3, Multiplication Meanings and Facts, Teacher Edition Topic 14, Area and Perimeter, Teacher Edition Section I, Supplemental Materials Lesson 14-7A, Solving Perimeter and Area Problems Resource Masters Student Edition Investigations in Numbers, Data, and Space, Grade 4, Pearson Education, Inc., 2008 Implementing Investigations in Grade 4- Implementation Guide Unit 1: Factors, Multiples, and Arrays, Teacher Edition Section I, Supplemental Materials Lesson 1-6A, Multiplicative Comparisons Unit 4: Size, Shape, and Symmetry, Teacher Edition Teacher Resources Binder Pearson Online Success Net, Investigations Site, Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for ample resources to refer to when planning your unit and individual lessons. Materials Place-value blocks, crayons, colored pencils, paper, pencils, two-color counters, markers, index cards, color tiles, centimeter grid paper Instructional Considerations Key Vocabulary multiple array associative property of multiplication commutative property of multiplication factor identity property of multiplication product zero property of multiplication area perimeter Providence Public Schools D-25

6 Using Factors, Multiples, and Arrays (9 11 days) Planning for Effective Instructional Design and Delivery In Lesson 3-4 (TE, page 62B) use nonlinguistic representations (physical models) to help students develop an understanding of the distributive property and multiplication facts. During the Interactive Learning, students use counters to create an array to show the distributive property of Students break up 12 into and write 4 (10 + 2), creating two smaller arrays of counters representing the problem (4 10) + (4 2). Model this process for the whole group. Lessons 3-4 through 3-6 refer to the distributive property and use array models and decomposition of numbers. Order of operations should be embedded and reinforced throughout this unit. Students will use a variety of strategies such as pictures and diagrams to solve real-world problems. Use concrete models, such as geoboards or grids, to build the concrete concept of area and perimeter. To meet individual needs, with Lesson 14-1, use the advanced activity (The Two Shape Challenge) on page 314D; students apply content they have learned by drawing two non-congruent hexagons on a grid and then comparing to identify similarities and differences in the areas and perimeters.(e.g., same perimeters/different areas, and same areas/different perimeters). Incorporate Ten Minute Math Activities, the Problem of the Day, and the Daily Spiral Review that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction. Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources, particularly the Algebra Connections and Teacher Notes section. D-26 Providence Public Schools

7 Using Factors, Multiples, and Arrays (9 11 days) Grade 4 Mathematics, Quarter 1, Unit 1.4 Notes Providence Public Schools D-27

8 Using Factors, Multiples, and Arrays (9 11 days) D-28 Providence Public Schools