Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Overview In this unit students will explore the relationship between multiplication and division. The goal is that students recognize that any division problem can be solved using multiplication. Students will also be able to solve real-world problems involving joining equal groups, and separating equal groups using multiplication and division. This unit is a continuation of Unit 2 which focused on the relationship between addition and multiplication, including building arrays and repeated addition. Patterns in addition develop the foundation for multiplication, while multiplication is critical to the future progress of mathematical students. Emphasized Mathematical Practices MP. 1. Make sense of problems and persevere in solving them. MP. 4 Model with mathematics. MP. 6 Attend to precision. MP. 7 Look for and make use of structure. MP. 8 Look for and express regularity in repeated reasoning. 1 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Priority Standards (Taught, assessed, and re-taught if necessary.) Operations and Algebraic Thinking 3.OA. Represent and solve problems involving multiplication and division. 3.OA.1. Interpret products of whole numbers, e.g., interpret 5 X 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 X 7. 3.OA.2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 / 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56/ 8. 3.OA.3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 3.OA.4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 X ? = 48, 5 = � / 3, 6 X 6 = ?. 2 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 3.OA.6. Understand division as an unknown-factor problem. For example, find 32 / 8 by finding the number that makes 32 when multiplied by 8. Multiply and divide within 100. 3.OA.7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 X 5 = 40, one knows 40 / 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. Supporting Standards These are the standards that will be taught in conjunction with the Essential Standards - they will not be commonly assessed (although some requirements from some may appear on rubrics. Solve problems involving the four operations, and identify and explain patterns in arithmetic. 3.OA.8. Solve two-step word problems using the four operations. 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. 3 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) 3.OA.9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. Measurement and Data 3.MD. Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. 3.MD.1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. 3.MD.2. Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. Represent and interpret data. 3.MD.3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve oneand 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. 3.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. Number and Operations in Base Ten 3.NBT. 3.NBT.3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) 4 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Enduring Understandings and Essential Questions for Content Standards Enduring Understandings are those concepts we want students to remember ten years from now. They are the important concepts underlying the content. The goal is that after instruction, students should be able to independently answer the Essential Question with a grade-appropriate version of the Enduring Understanding. Activities should be designed to allow the student to discover the Enduring Understanding Essential Questions are questions based on the Enduring Understandings that we use to guide or drive instruction and assessment. Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. ● How are multiplication and division related? ● What patterns of multiplication can we discover by studying a times table chart? ● What strategies are there to help learn and memorize multiplication facts? The same number sentence (e.g. 12-4 = 8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. ● What strategies can help you solve real world multiplication and division problems? ● How can you use multiplication facts to solve unknown factor problems? 5 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Chapter 1 of 3 Approximate Length: 10 days Multiplication KEY Pre-requisite knowledge or skills that should assessed prior to teaching this chapter: Basic Understanding of Multiplication Facts (see Unit 2) Understanding of Multiplication Tools : Calculators and Multiplication Fact Grids Enduring Understandings and Essential Questions addressed in this chapter: Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. The same number sentence (e.g. 12-4 = 8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. Emphasized Mathematical Practices: MP. 1. Make sense of problems and persevere in solving them. MP. 4 Model with mathematics. MP. 6 Attend to precision. MP. 7 Look for and make use of structure. MP. 8 Look for and express regularity in repeated reasoning. 6 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Learning Objectives Criteria for Success “What is good enough?” Essential Questions Type(s) of assessments recommended (SR/SCR/PT) Interpret and describe a given multiplication problem as the total number of objects in a group by the number of groups. (OA.1) Given a multiplication problem using digits 1-10, students will be able to use a multiplication strategy to solve the problem. What patterns of multiplication can we discover by studying a times table chart? What strategies are there to help learn and memorize multiplication facts? Short Constructed Response Apply commutative property of operations as a strategy to multiply. (OA.5) Students will demonstrate knowledge of the commutative property by modeling a series of fact families and how they relate to each other. What strategies are there to help learn and memorize multiplication facts? Short Response Short Constructed Response Use multiplication to solve real world situations using equal groups, arrays, and measurement quantities; i.e. drawings, equations, variables, and symbols. (OA.3) Given a word problem, students will be able to identify the quantities needed to solve a problem, and apply a multiplication strategy to solve the problem, including eliminating nonessential parts. How can you apply multiplication knowledge to real life situations? Short Constructed Response How can you determine what information is essential to solving the problem? 7 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Resources: * EDM Skills Links (BlueBook) pages 29, 38, 39, 41, 42, 48, 53, 58, 60, 68, 75, 80, 91, 94, 101 * EDM Lessons: 4.1, 4.2, 4.5, 4.6, 4.7, 7.1, 7.2, 7.3, 7.5, 7.6, 7.8, 9.2, 9.3, 9.4, 9.5, 9.9, 9.11, 9.12 9, * EDM Games and Projects: Baseball Multiplication, Multiplication Bingo, Top-It: Multiplication, Beat the Computer: Multiplication Chapter 2 of 3 Approximate Length: 10 days Division KEY Pre-requisite knowledge or skills that should assessed prior to teaching this chapter: Basic Understanding of Multiplication Facts (see Unit 2) Understanding of Multiplication Tools : Calculators and Multiplication Fact Grids Enduring Understandings and Essential Questions addressed in this chapter: Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. (For example: The same number sentence (e.g. 12-4 = 8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation.) 8 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Emphasized Mathematical Practices: MP. 1. Make sense of problems and persevere in solving them. MP. 4 Model with mathematics. MP. 6 Attend to precision. MP. 7 Look for and make use of structure. MP. 8 Look for and express regularity in repeated reasoning. Learning Objectives Criteria for Success “What is good enough?” Essential Questions Type(s) of assessments recommended (SR/SCR/PT) Interpret whole-number quotients with no remainders. (3.OA.2) Students will demonstrate division concepts by modeling, arrays, and grouping. How are multiplication and Selected Response division related? Short Constructed Response How can you use multiplication facts to solve unknown factor problems? Apply properties of operations as strategies to multiply and divide. (Commutative, Associative, and Distributive property.)(3.OA.5) Given a number sentence, students will apply appropriate properties to find a solution or rewrite the sentence How do the properties help us solve multiplication and division problems? Short Constructed Response Extended Response Rich Task Understand division as an unknown-factor Using unknowns and variables, students will be able to solve an unknown in How can you use your knowledge of multiplication and division to solve an Selected Response 9 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) problem. (3.OA.6) different positions in a given equation. unknown factor? Use division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities. (3.OA.3) Given a word problem, students will be able to identify the quantities needed to solve a problem, and apply a division strategy to solve the problem, including eliminating non- essential parts. How can you apply knowledge of division to real life situations? Short Constructed Response Extended Response How can you determine what information is essential to solving the problem? Resources: EDM Skills Links (BlueBook) pages 39, 53, 58, 68, 73, 91, 101. EDM Lessons: 4.3, 4.4, 4.6, 7.6, 9.7, 9.8 EDM Games and Projects: Baseball Multiplication, Multiplication Bingo, Top-It: Multiplication, Beat the Computer: Multiplication Chapter 3 of 3 Approximate Length: 10 days Real-Word Problems (Mult/Div) KEY Pre-requisite knowledge or skills that should assessed prior to teaching this chapter: Knowledge and skill sets established in Chapter 1 and 2, including multiplication and division strategies, modeling with arrays, drawings, or number models. 10 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Enduring Understandings and Essential Questions addressed in this chapter: Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. The same number sentence (e.g. 12-4 = 8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. How are multiplication and division related? What strategies can help you solve real world multiplication and division problems? Emphasized Mathematical Practices: MP. 1. Make sense of problems and persevere in solving them. MP. 6 Attend to precision. MP. 7 Look for and make use of structure 11 Hueneme Elementary School District Grade 3 Math Unit Outline of Study Unit 3 – Relationships Between Multiplication and Division (Approximate Length: 30 days) Learning Objectives Criteria for Success “What is good enough?” Essential Questions Type(s) of assessments recommended (SR/SCR/PT) Students will fluently and accurately multiply and divide from 1 to 100. (3.OA.7) Given basic division and multiplication problems, students will be able to use a strategy to solve the problem. How can you use patterning, skip counting, basic facts, missing factors, decomposing and square numbers to understand multiplication and division. Selected Response Short Constructed Response Solve two-step word problems using variables and assess the reasonableness using estimation, rounding, and mental computation. (3.OA.8) Students will demonstrate a reasonableness of answers using estimation and rounding. How can you use estimation to adjust your answer for better accuracy and precision. Short Constructed Response Extended Response Rich Task Students will apply concepts of multiplication and division in real life applications. Students will explain their thinking, show their work by using at least one representation, and verify that their answer is reasonable. What strategies can help you solve real world multiplication and division problems? Performance Task Resources: EDM Skills Links (BlueBook) pages 32, 33, 75, 80, 95, 99, 100. EDM Lessons: 4.5, 4.8, 4.9, 7.3, 7.7, 7.9, 9.1, 9.7, 9.8, 9.9, 9.10 EDM Games and Projects: Baseball Multiplication, Multiplication Bingo, Top-It: Multiplication, Beat the Computer: Multiplication, CA Project 8 12