Grade 3 Operations and Algebraic Thinking
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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 2012 COMMON CORE STATE STANDARDS ALIGNED MODULES
2 THE NEWARK PUBLIC SCHOOLS Office of Mathematics MATH TASKS Operations and Algebraic Thinking - 3.OA.2, 3, 4 Represent and solve problems involving multiplication and division. Goal: Students develop an understanding of multiplication and division of whole numbers through activities and problems involving equal sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Essential Questions: How is multiplication related to division? What questions can be answered using multiplication and division? What are efficient methods for finding products and quotients? Prerequisites: Interpret products of whole numbers Multiplication Facts 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 3 3. OA.4 Determine the unknown whole number in multiplication or division equation relating three whole numbers. Lesson 2 3.OA.3 Use multiplication and division solve word problems involving equal groups, arrays & measurement quantities. Lesson 1 3. OA.2 Interpret whole-numbers quotients of whole numbers. Lesson 5 Golden Problem 3. OA.2-3. OA.4 Lesson 4 3. OA.4 Determine the unknown whole number in multiplication or division equation relating three whole numbers. Lesson Structure: Introductory Task Prerequisite Skills Focus Questions Guided Practice Homework Journal Question Page 2 of 28
3 Multiplication & Division 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 28
4 Manipulatives also play an important role in division. The first step that students take beyond basic division facts is division with remainders. Counters work well for modeling this concept. This model shows 19 counters divided into 3 groups, or There are 6 counters in each group. One counter is left over. The amount left over in division is the remainder = 6 R1 Base-ten blocks work well for modeling two- and three-digit quotients. The model below shows Divide the tens first. There is 1 ten in each group. Divide the ones next. There are 2 ones in each group and 1 one left over = 12 R1 Models become even more important when students learn regrouping in division. In this example, tens are regrouped as ones Divide the tens. There are 2 tens in each group and 1 ten left over. Regroup the 1 ten as 10 ones to divide. There are 16 ones. Divide the ones. There are 8 ones in each group. 2 tens + 8 ones = 28, so 56 2 = 28. Page 4 of 28
5 As you did with multiplication, provide students with many opportunities to model division before introducing the algorithm. Don't be discouraged if students progress slowly with this concept. Remember, this is the first time most students have ever encountered it. Take the time necessary to encourage students as they move through the process. Page 5 of 28
6 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 6 of 28
7 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 b) c = a (b c ) Partitive Division modeling Dividing a certain number of equal groups Identity Element 15 1 = 15 Identity element for division Distributive property of division over addition 84 2 We used 8 tens and 4 ones (80 + 4) 2 = (80 2) + (4 2) = = 44 Page 7 of 28
8 3.OA.2: Lesson 1 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 Introductory Task Guided Practice Collaborative Homework Assessment Sandra is having a party. She has arranged the chesseburgers on a tray for her and 8 friends. Help Sandra determine how many chesseburgers each person at the party will receive. Express the solution with a diagram, use the terms quotient, dividend, divisor and factor. Sandra also needs to write an equation for the problem. Can you help Sandra explain her problem? Focus Questions Journal Question How does an understanding of the meaning of multiplication and division affect the appropriate and accurate use of these operations? How is division similar to sharing? How do interim strategies connect to the standard algorithms? Page 8 of 28
9 3.OA.2: Lesson 1 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. Introductory Task Guided Practice Collaborative 1. Homework Assessment 2. Four children are sharing 48 dimes? How many dimes will each child get? Write a division equation for the pictures above = Ms. Powell had her class take a survey and found that 9 children owned 18 cats? How many cats does each child own? Diagram a model of the division equation above. Page 9 of 28
10 3.OA.2: Lesson 1 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. Introductory Task Guided Practice Collaborative Homework Assessment = 6. Diagram a model of the division equation above There are 56 bottle caps in 7 boxes. How many bottle caps are in each box? 9. Anne bought 27 granola bars. If the granola bars came in packs of 3, how many packs did Anne buy? 10. Page 10 of 28
11 3.OA.2: Lesson 1 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. Introductory Task Guided Practice Collaborative Homework Assessment Nine children are sharing 81 pennies? How many pennies will each child get? Write a division equation for the pictures above = Tommy bought 16 rubber ducks. If he wants to shre them among 4 friends, how many rubber ducks does each friend get? Diagram a model of the division equation above. Page 11 of 28
12 3.OA.2: Lesson 1 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 = 6. Diagram a model of the division equation above There are 64 apples in 8 baskets. How many apples are in each basket? 9. Anne bought 27 granola bars. If the granola bars came in packs of 3, how many packs did Anne buy? 10. Page 12 of 28
13 3.OA.3: Lesson 2 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. Helen gave each of her 7 grandchildren an equal amount of money. If she gave a total of $42, how much did each grandchild get? Show your work by using the following models equal groups, arrays models and measurement quantities to verify your answer is reasonable. Helen also needs to write an equation for the problem. Can you help Helen explain her problem? Focus Questions Question 1: How can an array model help solve a division problem? Question 2: What are the different ways to break wholes into parts? Journal Question Some people believe division is the opposite of multiplication is this a correct statement why or why not? Page 13 of 28
14 3.OA.3: Lesson 2 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. Show your work by using one of the following models equal groups, arrays, measurement quantities to verify your answer is reasonable. 1. Steven puts 18 cookies in 2 equal groups. How many cookies are in each group? 2. Nicole puts 54 cookies in 6 equal groups. How many cookies are in each group? 3. Paul puts 70 pens in 7 equal groups. How many pens are in each group? 4. There are 56 comic books being share among the students. Each box contains 7 comic books. How many students will receive comic books? Page 14 of 28
15 3.OA.3: Lesson 2 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. Show your work by using one of the following models equal groups, arrays, measurement quantities to verify your answer is reasonable. 5. Jon has 48 baseball cards. He wants to put 6 baseball cards in each box. How many boxes does he need? 6. Ted the lion loves steak. Kim, her trainer, has 64 steaks. If she gives Ted 8 steaks each day, how many days will the steaks last? 7. Mr. Cupcake baked 42 cupcakes. He placed 6 cupcakes in each box. How many boxes are there in all 8. There are 50 ice cream bars. If each box contains 10 ice cream bars. How many boxes are there in all? 9. Peter puts 45 bollons in 5 bags. Each bag has the same number of bollons. How many bollons are in each bag? 10. Coach Hoover has 24 soccer balls in his van. If each team needs 3 balls for a game of soccer. How many teams are there in all? Page 15 of 28
16 3.OA.3: Lesson 2 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. Show your work by using one of the following models equal groups, arrays, measurement quantities to verify your answer is reasonable. 1. Steven puts 21 cookies in 3 equal groups. How many cookies are in each group? 2. Sam the dolphin loves fish. Nicole, her trainer, has 42 fish. If she gives Sam 6 fish each day, how many days will the fish last? 3. Paul puts 100 pens in 5 equal groups. How many pens are in each group? 4. There are 64 comic books being share among the students. Each box contains 8 comic books. How many students will receive comic books? Page 16 of 28
17 3.OA.3: Lesson 2 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. Show your work by using one of the following models equal groups, arrays, measurement quantities to verify your answer is reasonable. 5. The bag has 92 hair clips, and Laura and her three friends want to share them equally. How many hair clips will each person receive? 6. Sam puts 80 flowers in 8 equal vases. How many flowers are in each group? 7. Mr. Cupcake baked 72 cupcakes. He placed 9 cupcakes in each box. How many boxes are there in all? 8. Max the monkey loves bananas. Molly, his trainer, has 24 bananas. If she gives Max 4 bananas each day, how many days will the bananas last? 9. Peter puts 35 bollons in 5 bags. Each bag has the same number of bollons. How many bollons are in each bag? 10. Coach Hoover has 80 jerseys in his van. If each team needs 8 jerseys for a game of soccer. How many teams are there in all? Page 17 of 28
18 3.OA.4: Lesson 3 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 Max the monkey loves bananas. Molly, his trainer, has 48 bananas. If she gives Max 8 bananas each day, how many days will the bananas last? Victoria needs to write an equation for the problem. Can you help Victoria explain her problem? Focus Questions Question 1: How do fractions relate to division of a whole? Question 2: How do you solve for an unknown? Journal Question How is equal grouping different from the array model? Page 18 of 28
19 3.OA.4: Lesson 3 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? = 48, 5 = Name Solve the problems below for the missing factor. 1. x 2 = 4 2.? x 7 = 49 Date 3. 8? = x 7 = n 5 = x = x = n 6 = n 6 = ? = 54 Page 19 of 28
20 3.OA.4: Lesson 3 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 Name Solve the problems below for the missing factor. 1. x 2 = 8 2.? x 6 = 36 Date 3. 8? = x 7 = n 5 = x = x = n 4 = n 6 = ? = 50 Page 20 of 28
21 3.OA.4: Lesson 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 Maria has 12 feet of ribbon and wants to wrap some gifts that need 3 feet of ribbon each. How many gifts can she wrap using the ribbon? Help Maria model a diagram below be sure to include terms. Maria also needs help with writing an equation for the problem. Can you help Maria explain her problem? Focus Questions Question 1: Describe a context in which a number of groups or a number of shares can be expressed as Question 2: Can the order of the factors be reversed in a multiplication problem? If so, is this always true? Why or why not? Journal Question What strategies help you solve an unknown fact? Page 21 of 28
22 3.OA.4: Lesson 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 Name Solve the problems and write a story problem for the diagram or equation below = Δ Page 22 of 28
23 3.OA.4: Lesson 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 Name Date Solve the problems and write an equations, story prolem for the diagram or equation below = = Δ Page 23 of 28
24 3.OA.4: Lesson 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 Name Date Solve the problems and write an equations, story prolem for the diagram or equation below = Δ = Δ 5. 8 Page 24 of 28
25 3.OA.4: Lesson 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 Name Date Solve the problems and write an equations, story prolem for the diagram or equation below = Δ Page 25 of 28
26 3.OA.2-3.OA.4 : Lesson 5 Students develop an understanding multiplication and division of whole numbers through activities and problems involving equal sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Mr. Smith s family is going to the carnival. Mr. Smith bought 72 carnival ride tickets in advance. He wants each person to have an equal amount of tickets. If there are six adults and three children, how many tickets will each person recieve? Express the solution with a diagram, use the terms quotient, dividend, divisor and factor as well as write an equation for the problem. Focus Questions Question 1: Can the order of the numbers be reversed in a division problem? If so, is this always true? Why or why not? Question 2: How can you explain the patterns observed in multiplication and division combinations/facts? Journal Questions How might multiplication help you solve a division problem? Page 26 of 28
27 LESSON 5 RUBRIC Score GOLDEN PROBLEM Description 3 Student has an understanding of what division is. Student correctly identifies 9 groups with 8 tickets per person. In addition, the student was then able to write an equation for the problem. Finally, the student was able to correctly label the equation with all the correct terms (quotient, dividend, divisor and factor). 2 Student has an understanding of what division is. Student correctly identifies 9 groups with 8 tickets per person. The student might do one of the following things incorrectly: (1) not identify the equation or (2) correctly label the equation with all the correct terminology or (3) make simple mathematical mistakes in the computation. 1 Student attempts to solve the problem, or solves it without utilizing division. There are many errors in the calculation. The problem lacks an appropriate explanation, or there is no explanation at all. The student does not select the correct terminology. 0 Does not address task, unresponsive, unrelated or inappropriate. Page 27 of 28
28 Third Grade CCSSM Fluencies Skills Multiply/divide within 100 (By end of year, know from memory all products of two one digit numbers) Add/subtract within Addition Worksheet Two Plus Two Digit Addition Version 2 Answer Key 2. Addition Worksheet Three Plus Two Digit Addition Version 2 Answer Key 3. Space Math Multiplication x 10 Version 3 Answer Key 4. Space Math Multiplication x 10 Version 4 Answer Key Page 28 of 28
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