# Grade 3 3.OA.1. Operations and Algebraic Thinking 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES

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1 THE NEWARK PUBLIC SCHOOLS THE NEWARK PUBLIC SCHOOLS THE OFFICE OF MATHEMATICS THE OFFICE OF MATHEMATICS Grade 3 Operations and Algebraic Thinking 3.OA COMMON CORE STATE STANDARDS ALIGNED MODULES

2 THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH TASKS Operations and Algebraic Thinking - 3.OA.1 Represent and solve problems involving multiplication and division. Goal: Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models. Essential Questions: How is area related to multiplication? How is the area of a rectangle determined? For what purpose do you calculate area? Prerequisites: skip counting area model expanded notation commutative property of multiplication associative property of multiplication distributive property of multiplication Embedded Mathematical Practices MP.1 Make sense of problems and persevere in solving them MP.2 Reason abstractly and quantitatively MP.3 Construct viable arguments and critique the reasoning of others MP.4 Model with mathematics MP.5 Use appropriate tools strategically MP.6 Attend to precision MP.7 Look for and make use of structure MP.8 Look for and express regularity in repeated reasoning Lesson 5 Golden Problem 3. OA.1 Lesson 4 3. OA.1 Interpret products of whole numbers using numbers and equations Lesson 3 3. OA.1 Interpret products of whole numbers using words Lesson 2 3. OA.1 Interpret products of whole numbers using objects Lesson 1 3. OA.1 Interpret products of whole numbers using pictures Lesson Structure: Introductory Task Prerequisite Skills Focus Questions Guided Practice Homework Journal Question Page 2 of 26

3 Multiplication Concepts Multiplication can be defined in terms of repeated addition. For example, 3 6 can be viewed as More generally, for any positive integer n, n b can be represented as n b = b + b + + b, where the sum on the right consists of n addends. A rectangular array provides a visual model for multiplication. For example, the product 3 6 can be represented as By displaying 18 dots as 3 rows with 6 dots in each row, this array provides a visual representation of 3 6 as An equivalent area model can be made in which the dots of the array are replaced by unit squares. Besides representing 3 6 as an array of 18 unit squares, this model also shows that the area of a rectangle with a height of 3 units and a base of 6 units is 3 6 square units, or 18 square units. Multiplication is a binary operation that operates on a pair of numbers to produce another number. Given a pair of numbers a and b called factors, multiplication assigns them a value a b = c, called their product. Multiplication has certain fundamental properties that are of great importance in arithmetic. The Commutative Property of Multiplication states that changing the order in which two numbers are multiplied does not change the product. That is, for all numbers a and b, a b = b a. The array model can be used to make this plausible. For example, because 3 6 = 6 3, an array with 3 rows and 6 dots in each row has the same number of dots as an array with 6 rows and 3 dots in each row. Another important property of multiplication is the Identity Property of Multiplication. It states that the product of any number and 1 is that number. That is, for all numbers a, a 1 = 1 a = a. The Zero Property of Multiplication states that when a number is multiplied by zero, the product is zero. That is, for all numbers a, a 0 = 0 a = 0. Page 3 of 26

4 Teaching Tips Teaching Tip 1 Digit Name vs. Digit Value Stress place value in multiplication by distinguishing between the name of the digit and the value it stands for. The 2 in 24 stands for 2 10 = 20, not 2. Base-10 blocks and area model diagrams emphasize the value that each digit stands for because they use expanded notation to build the answer. Teaching Tip 2 Drawing Rectangles for an Area Model The area model is an alternative and efficient way to multiply. Encourage students to draw rectangles, even though the rectangles may not be drawn to scale. If students need to use base-10 blocks as a transitional step, change the numbers in the problems to match the quantity of blocks that are available. Teaching Tip 3 Using an Area Model to Record Multiplication Is it okay to permit students to use the area model as a recording method for multiplication? Yes. An area model not only helps to explain why the standard algorithm commonly taught in the United States for multiplication works, it is an efficient recording alternative. Some students (especially visual learners and those who have difficulty keeping numbers lined up in multiplication problems) may prefer it. Furthermore, this method has certain benefits. It illuminates important mathematical concepts (such as the distributive property), allows for computational flexibility (expanded notations allow students to use derived facts), and reinforces the concept of area. Finally, when students take algebra, they are likely to see the area model when they learn to multiply and factor polynomials. Page 4 of 26

5 Multiple Representations Framework In the identity 3(4 + 5) = 3(4) + 3(5), the 3 is distributed over the 4 and the 5. Distributive Property a(b + c ) = ab + ac and (b + c )a = ba + ca Commutative Properties of Multiplication a b = b a 3 4 = 4 3 (3 4) 5 = 12 5 = 60 or 3 (4 5) = 3 20 = 60 Associative Properties of Multiplication (a c = a (b ) Area Model Array Model Interpret products of whole numbers 5 7 as the total number of objects in 5 groups of 7 objects each Page 5 of 26

6 3.OA.1: Lesson 1 Interpret products of whole numbers, e.g., interpret 5 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 Devon loves math and he has just learned about multiplication. Devon is having a party. He has arranged the hamburgers on a tray for his friends. Help Devon use multiplication terms to express the diagram below by using the terms product and factors. Devon also needs to write an equation for the problem. Can you help Devon explain his multiplication problem? Focus Questions Question 1: How does an understanding of multiplication affect the appropriate and accurate use of operations? Question 2: How do interim strategies connect to the standard algorithms? Journal Question Write a story problem to go along with 8 x 10 =. Page 6 of 26

7 3.OA.1: Lesson 1 (Using Pictures) Page 7 of 26

8 3.OA.1: Lesson 1 (Using Pictures) Page 8 of 26

9 3.OA.1: Lesson 1 (Using Pictures) Page 9 of 26

10 3.OA.1: Lesson 1 (Using Pictures) Page 10 of 26

11 3.OA.1: Lesson 2 objects each. For example, describe a context in which a total number of objects can be expressed as 5 7. (Using Objects) Judy is having a party. She has arranged the 5 bags with 9 pieces of candy in each for her friends. Help Judy use multiplication terms to express the diagram below. Be sure to include terms such as product and factors. Judy also needs help with writing an equation for the problem. Can you help Judy explain her multiplication problem? Focus Questions Question 1: What are the groups? Question 2: What are the objects? What is the number sentence associated with this? Journal Question How can multiplication be represented by groups and objects? Page 11 of 26

12 3.OA.1: Lesson 2 (Using Objects) Page 12 of 26

13 3.OA.1: Lesson 2 (Using Objects) Page 13 of 26

14 3.OA.1: Lesson 2 (Using Objects) Page 14 of 26

15 3.OA.1: Lesson 2 (Using Objects) Page 15 of 26

16 3.OA.1: Lesson 3 (Using Words) Tom has seven boxes of cookies. Each box contains eight cookies. How many cookies did Tom buy in all? Create a diagram to help Tom use multiplication terms to model this situation below. Be sure to include terms such as product and factors. Tom also needs help with writing an equation for the problem. Can you help Tom explain his multiplication problem? Focus Questions Question 1: What are some interim strategies students can use to develop multiplication sense? Question 2: How do interim strategies connect to the standard algorithms? Journal Question How can we use modeling to help us solve multiplication problems? Page 16 of 26

17 3.OA.1: Lesson 3 (Using Words) Draw a diagram or write an equation to help you solve the problems below. 1. There are three eggs in each box. How many eggs 2. Each child has five oranges. If there are seven are in two boxes? children, how many oranges are there in total? 3. Each child has two marbles. If there are ten children, how many marbles are there in total? 4. There are nine bottle caps in each box. How many bottle caps are in six boxes? 5. Paula has six boxes of peanuts. Each box holds ten peanuts. How many peanuts does Paula have? 6. Eric has three boxes of pencils. Each box holds nine pencils. How many pencils does Eric have? 7. There are six marbles in each box. How many marbles are in three boxes? 8. Each child has nine tickets. If there are nine children, how many tickets are there in total? 9. There are nine eggs in each box. How many eggs are in two boxes? 10. Erica has six boxes of stickers. Each box holds nine stickers. How many pencils does Eric have? Page 17 of 26

18 3.OA.1: Lesson 3 (Using Words) Name Date Draw a diagram or write an equation to help you solve the problems below. 1. There are six eggs in each box. How many eggs 2. Each child has seven oranges. If there are seven are in four boxes? children, how many oranges are there in total? 3. Each child has nine marbles. If there are seven children, how many marbles are there in total? 4. There are nine bottle caps in each box. How many bottle caps are in four boxes? 5. Paula has seven boxes of peanuts. Each box holds eight peanuts. How many peanuts does Paula have? 6. Eric has seven boxes of pencils. Each box holds five pencils. How many pencils does Eric have? 7. There are six marbles in each box. How many marbles are in six boxes? 8. Each child has seven tickets. If there are nine children, how many tickets are there in total? 9. There are seven eggs in each box. How many eggs are in three boxes? 10. Erica has six boxes of stickers. Each box holds two stickers. How many pencils does Eric have? Page 18 of 26

19 3.OA.1: Lesson 4 (Using numbers and equations) Mr. Herreara lost the word problem that matches the diagram below. Help Mr. Herreara create a word problem that matches the diagram. Be sure to include terms such as product and factors in your word problem. Mr. Herreara also needs help with writing an equation for the problem can you help him? Focus Questions Question 1: What are some interim strategies students can use to develop multiplication sense? Question 2: How do interim strategies connect to the standard algorithms? Journal Question Does the mathematical meaning change if the array is rotated 90 degrees? Page 19 of 26

20 3.OA.1: Lesson 4 (Using numbers and equations) Match the word problem or equations with the diagram and explain why you chose your answer. 1. There are 7 eggs in each box. How many eggs are A. in 5 boxes? 2. There are 9 bottle caps in each box. How many bottle caps are in 4 boxes? B. 3. Paula has 7 boxes of peanuts. Each box holds 8 peanuts. How many peanuts does Paula have? C. 4. Each child has 5 tickets. If there are 7 children, how many tickets are there in total? D. 5. There are 4 eggs in each box. How many eggs are in 9 boxes? E. 6. Each child has 7 stickers. If there are 7 children, how many stickers are there in total? F. Page 20 of 26

21 3.OA.1: Lesson 4 (Using numbers and equations) Match the word problem or equations with the diagram and explain your answer. 7. There are 3 candies in each box. How many G. candies are in 7 boxes? 8. There are 6 caps in each box. How many caps are in 4 boxes? H. 9. Paul has 7 boxes of apples. Each box holds 10 apples. How many apples does Paul have? I. 10. Each child has 7 balls. If there are 3 children, how many balls are there in total? J. 11. There are 4 dolls in each box. How many dolls are in 6 boxes? K. 12. Each child has 7 books. If there are 10 children how many books are there in total? L. Page 21 of 26

22 3.OA.1: Lesson 4 (Using numbers and equations) Name Date Match the word problem or equations with the diagram and explain your answer. 1. There are 9 cars in each box. How many cars are in A. 8 boxes? 2. There are 4 baseballs in each box. How many baseballs are in 5 boxes? B. 3. Tom has 7 boxes. Each box holds 8 movies. How many movies does Tom have? C. 4. Each child has 5 Skittles bags. If there are 4 children, how many Skittles bags are there in total? D. 5. There are 9 Snickers in each box. How many Snickers are in 8 boxes? E. 6. Each child has 8 Swedish-Fish. If there are 7 children, how many Swedish-Fish are there in total? F. Page 22 of 26

23 3.OA.1: Lesson 4 (Using numbers and equations) Name Match the word problem or equations with the diagram and explain your answer. 7. There are 4 Air-Heads in each box. How many G. Air-Heads are in 7 boxes? Date 8. There are 7 toys in each box. How many toys are in 4 boxes? H. 9. Nicole has 7 boxes of balloons. Each box holds 8 balloons. How many balloons does Nicole have? I. 10. Each child has 5 cupcakes. If there are 9 children, how many cupcakes are there in total? J. 11. There are 9 coins in each box. How many coins are in 5 boxes? K. 12. Each child has 7 gifts. If there are 8 children, how many gifts are there in total? L. Page 23 of 26

24 3.OA.1: Lesson 5 (Using pictures, objects, words, numbers and equations) Mr. Titen s children are going to the carnival. Each child was given 7 tickets. If there are 5 children, how many tickets are there in total? Which diagram below describes the problem above? Explain how you know and explain why the other diagram does not describe the problem. Be sure to include terms such as product and factors in your answer as well as write an equation for the problem. Diagram A Diagram B Focus Questions Question 1: What are some interim strategies students can use to develop multiplication sense? Question 2: How do interim strategies connect to the standard algorithms? Journal Question How would you explain the difference between these two diagrams to a second grader? Page 24 of 26

25 LESSON 5 RUBRIC Score GOLDEN PROBLEM Description 3 Student demonstrates a clear understanding of multiplication. The student can correctly identify that there are 35 tickets in total. Furthermore, the student selects the correct number model diagram (B) that represents the multiplication problem that is required. In addition the students will be able to explain why the other diagram does not represent the number model for the problem. The student also utilizes the appropriate mathematical language, i.e. (product, factors, etc.) 2 Student demonstrates a basic understanding of multiplication. The student can correctly identify that there are 35 tickets in total. The student selects the correct number model diagram (B) that represents the multiplication problem that is required, but is unable to explain why the other diagram does not represent the number model for the problem. The student s explanation and reasoning are not detailed and clear. 1 Student attempts to solve the problem, or solves it without utilizing multiplication. There are many errors in calculation. The problem lacks an appropriate explanation, or there is no explanation at all. The student does not select the correct diagram model, and/or cannot explain the reasoning behind that choice. 0 Does not address task, unresponsive, unrelated or inappropriate. Page 25 of 26

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