Putting Research into Practice

Research Best Practices Putting Research into Practice From Our Curriculum Research Project: Multiplication Methods Dr. Karen C. Fuson, Math Expressions Author We show three methods for multidigit multipli cation that we have found to be understood by students and that capture important mathematical ideas. Many less-advanced students prefer the area method because they sometimes get confused about which numbers to multiply by which in the numerical methods. In the area method, students find and write the products in the rectangle and add them up. In the algebraic method, students multiply horizontally. Most students choose the Expanded Notation Method because they feel more comfortable seeing the tens and ones in each number and writing out the products they are finding. This method can be done from the right or from the left. Students often find it easier to start from the left because they get the greatest product first and then they can align all of the other products under this number. From Current Research: The Influence of Instruction [An] observation is that proficiency with multidigit computation is more heavily influenced by instruction than is single-digit computation. Many features of multidigit procedures (for example, the base 10 elements and how they are repre sented by place-value notation) are not part of children s everyday experience and need to be learned in the classroom. In fact, many students are likely to need help learning efficient forms of multidigit procedures. Developing Proficiency with Whole Numbers Adding It Up: Helping Children Learn Mathematics, Washington, DC: National Academy Press, 001, p. 187. 115T UNIT Overview

Math Drawings Support Thinking and Math Talk A central result of our research project was the power of having students make math drawings and relate them to mathematical notation. Math drawings are simplified drawings that show quantities and relationships in simple ways. The Visual Math-Talk Community then is an on-going process of building and relating these math drawing visuals and language. Students can begin working from their strength (visual or language), and the continuing connections enable them to develop the other aspect. Fuson, Karen; Alter, Todd; Roedel, Sheri; Zaccariello, Janet. New England Mathematics Journal "Building a Nurturing Visual Math- Talk Teaching-Learning Community to Support Learning by English Language Learners and Students from Backgrounds of Poverty," New England Mathematics Journal, ATME, May 009, p. 6 UNIT R e s e a r c h Other Useful References: Multiplication Number and Operations Standard for Grades 3 5 Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 000, pp. 148 180. Ma, Liping. Knowing and Teaching Elementary Mathematics, pp. 8 54. Mahwah, NJ: Lawrence Eribaum Associates, Inc., 1999. Van de Walle, John A. Models for Multiplication and Division. Elementary and Middle School Mathematics: Teaching Developmentally, (Fourth Edition). New York: Longman, 001, pp. 10 1. UNIT Overview 115U

Getting Ready to Teach Unit Using the Common Core Standards for Mathematical Practice The Common Core State Standards for Mathematical Content indicate what concepts, skills, and problem solving students should learn. The Common Core State Standards for Mathematical Practice indicate how students should demonstrate understanding. These Mathematical Practices are embedded directly into the Student and Teacher Editions for each unit in Math Expressions. As you use the teaching suggestions, you will automatically implement a teaching style that encourages students to demonstrate a thorough understanding of concepts, skills, and problems. In this program, Math Talk suggestions are a vehicle used to encourage discussion that supports all eight Mathematical Practices. See examples in Mathematical Practice 6. Mathematical Practice 1 Make sense of problems and persevere in solving them. Students analyze and make conjectures about how to solve a problem. They plan, monitor, and check their solutions. They determine if their answers are reasonable and can justify their reasoning. Teacher Edition: examples from Unit MP.1 Make Sense of Problems Use a Different Method Write the following problem on the board: Ms. Delgado is spreading stone dust as a base for a walkway. The walkway is 5 feet by 43 feet. What is the area of the walkway Ms. Delgado needs to cover with stone dust? 15 square feet Read the problem. Tell students that we can use the same area model that was used for the Place Value Sections Method and the Expanded Notation Method to represent this problem. MP.1 Make Sense of Problems Justify Reasoning Discuss whether or not it is safe for Joshua and Sue to use their rounded estimate to buy their tiles. The model shows that rounding 7 up to 30 adds 40 3 = 10 more tiles, and rounding 4 down to 40 removes 7 = 54 tiles. Since 10 > 54, rounding is safe in this case. Lesson 14 ACTIVITY Lesson 7 ACTIVITY Mathematical Practice 1 is integrated into Unit in the following ways: Make Sense of Problems Analyze Relationships Analyze the Problem 115V UNIT Overview Solve a Simpler Problem Look for a Pattern Check Answers Use a Different Method Justify Reasoning Reasonable Answers

Mathematical Practice Reason abstractly and quantitatively. Students make sense of quantities and their relationships in problem situations. They can connect diagrams and equations for a given situation. Quantitative reasoning entails attending to the meaning of quantities. In this unit, this involves relating models and drawings to multiplication situations and products. Teacher Edition: examples from Unit MP. Reason Abstractly and Quantitatively Connect Diagrams and Equations Use Problems 8 15 on Student Book page 48 to help students make connections How might you find the product for Problem 9, using addition rather than multiplication? Why? Add 9 + 9 + 9. 9 + 9 + 9 is equal to 3 9. How would the rectangle for Problem 10 be different from the rectangle for Problem 11? The rectangle for Problem 10 would have only a tens part, not a ones part. How could you use your answer from Problem 14 to quickly find the answer to Problem 15? Add 8 to the answer to Problem 14. Lesson 4 ACTIVITY Mathematical Practice is integrated into Unit in the following ways: Reason Quantitatively Reason Abstractly and Quantitatively Connect Diagrams and Equations MP. Reason Abstractly and Quantitatively Connect Diagrams and Equations On the board, write 9 8 =. Then have students share any solution methods they know that have not been discussed on previous days. They should be able to relate their method to the area model. Connect Symbols and Words Connect Symbols and Models Lesson 9 ACTIVITY 1 UNIT MATH BACKGROUND UNIT Overview 115W

Mathematical Practice 3 Construct viable arguments and critique the reasoning of others. Students use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students are also able to distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Students can listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Math Talk is a conversation tool by which students formulate ideas and analyze responses and engage in discourse. See also MP.6 Attend to Precision. Teacher Edition: examples from Unit MP.3 Construct a Viable Argument Compare Methods Direct students attention to the steps for the Shortcut Method with New Groups Below (Method F). Ask students to compare the two recordings of the Shortcut Method. How are Methods E and F alike? How are they different? Both methods write the new group of tens in the tens column to be added to the other tens, but where the new groups are recorded is different. Lesson 9 ACTIVITY What s the Error? PA I R S MP.3, MP.6 Construct Viable Arguments/Critique Reasoning of Others Puzzled Penguin In pairs, students should read the letter from Puzzled Penguin on Student Book page 174 and discuss whether the estimate is appropriate for the situation. Students should recognize that Puzzled Penguin rounded both factors down to make an underestimate. Puzzled Penguin can correct the mistake by using an overestimate. Lesson 14 ACTIVITY 3 Mathematical Practice 3 is integrated into Unit in the following ways: Construct a Viable Argument Compare Models Compare Representations Compare Methods 115X UNIT Overview

Mathematical Practice 4 Model with mathematics. Students can apply the mathematics they know to solve problems that arise in everyday life. This might be as simple as writing an equation to solve a problem. Students might draw diagrams to lead them to a solution for a problem. Students apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation. They are able to identify important quantities in a practical situation and represent the relationships of such quantities using such tools as diagrams, tables, graphs, and formulas. Teacher Edition: examples from Unit MP.4, MP.5 Use Appropriate Tools/ Model with Mathematics Draw a Diagram Write the following problem on the board: A school bus company has space in their lot to park 6 rows of 9 school buses. How many school buses can the bus company park in their lot? Ask students to read the word problem and draw an area model to represent the situation on their MathBoards. Explain to students that the area model for 6 9 can also be used to demonstrate another method for multiplying by a multidigit number. Lesson 6 ACTIVITY Mathematical Practice 4 is integrated into Unit in the following ways: Model with Mathematics Draw a Diagram MP.4 Model with Mathematics Write an Expression For Exercise 6, ask student volunteers to show how they solved the problem. Students should begin by writing a multiplication expression they can use to solve the problem. Students should be able to explain how the area model is related to the situation given in the problem. Make a Model Write an Expression Lesson 7 ACTIVITY UNIT MATH BACKGROUND UNIT Overview 115Y

Mathematical Practice 5 Use appropriate tools strategically. Students consider the available tools and models when solving mathematical problems. Students make sound decisions about when each of these tools might be helpful. These tools might include paper and pencil, a straightedge, a ruler, or the MathBoard. Students recognize both the insight to be gained from using the tool and the tool s limitations. When making mathematical models, they are able to identify quantities in a practical situation and represent relationships using modeling tools such as diagrams, grid paper, tables, graphs, and equations. Modeling numbers in problems and in computations is a central focus in Math Expressions lessons. Students learn how to develop models to solve numerical problems and to model problem situations. Students continually use both kinds of modeling throughout the program. Teacher Edition: examples from Unit MP.5 Use Appropriate Tools Model Mathematics Use two different colors to draw the 4-by-5 array shown. Explain the two ways to find the total number of dots in the array. MP.5 Use Appropriate Tools Draw a Diagram Invite volunteers to draw rectangles showing 4 37 on the board as everyone else sketches on their MathBoards. 37 = 00 + 30 + 7 4 4 00 = 800 4 30 = 10 4 = 8 7 4 Method 1: First, add the number of the first color of columns and the number of the second color of columns to get the total number of columns. Then, multiply the total number of columns by the number of rows. (3 + ) 4 or 4 (3 + ). Method : First, multiply to find the number of dots in each array. Then, add the results. 4 3 + 4 Ask students to find the area of each section and then add to find the total area. Lesson 10 ACTIVITY Lesson 7 ACTIVITY 1 Mathematical Practice 5 is integrated into Unit in the following ways: Use Appropriate Tools MathBoard Modeling Model Mathematics Draw a Diagram MathBoard 115Z UNIT Overview

Mathematical Practice 6 Attend to precision. Students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. They are careful about specifying units of measure to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, expressing numerical answers with a degree of precision appropriate for the problem context. Students give carefully formulated explanations to each other. Teacher Edition: Examples from Unit Lesson 11 ACTIVITY 1 MATH TALK in ACTION MP.6 Attend to Precision Explain Solutions Use Solve and Discuss to find the product of 68 84. Have four volunteers go to the board to solve the problem using: Place Value Sections Method, Expanded Notation Method, Algebraic Notation Method, Shortcut Method Lesson 14 ACTIVITY 3 ACTIVITY 1 ou can use this dialogue to compare Y the three numerical methods of multiplication. What do the three methods have in common? Steve: Somewhere in each method, you have to multiply 4 times 00, 4 times 30, and 4 times 7. Mirella: In all of the methods, you have to add 800 + 10 + 8 to get your final answer. Ahmed: Even though one of the methods is called The Expanded Notation Method, you really do expanded form in all of the methods. You have to break apart 37 into 00 + 30 + 7 for all three methods. Lesson 10 ACTIVITY Mathematical Practice 6 is integrated into Unit in the following ways: Attend to Precision Explain a Method Explain a Solution Verify a Solution Puzzled Penguin Describe Methods Explain a Representation UNIT Overview 115AA MATH BACKGRO UN D Lesson 9 When students finish this section, have volunteers show their work and explain. Encourage questioning by the other students to clarify different strategies. M AT H TA L K UNIT MP.6 Attend to Precision Describe Methods Send five students to the board. Each student should neatly copy onto the board one of the numerical methods shown. Ask students to describe each step of their method. Then have the class identify the changes in each succeeding method.

Mathematical Practice 7 Look for structure. Students analyze problems to discern a pattern or structure. They draw conclusions about the structure of the relationships they have identified. Teacher Edition: examples from Unit MP.7 Look for Structure Identify Relationships Discuss solutions relating each step inside the rectangle to the products added outside the rectangle. Be sure that listeners understand the steps for finding the product of 4 37 by using the Place Value Sections Method. First multiply 4 00 = 800, then multiply 4 30 = 10, then multiply 4 7 = 8. 800 + 10 + 8 = 948. Lesson 10 ACTIVITY MP.7 Look for Structure Students should discuss how the two methods at the top of Student Book page 7 are alike and different. They should notice that the partial products shown on the left are recorded more concisely in the method on the right. Point out to students that the small above 81 represents the tens from the first partial product, 3 7 = 1. Lesson 13 ACTIVITY Mathematical Practice 7 is integrated into Unit in the following ways: Look for Structure Identify Relationships 115BB UNIT Overview

Mathematical Practice 8 Look for and express regularity in repeated reasoning. Students use repeated reasoning as they analyze patterns, relationships, and calculations to generalize methods, rules, and shortcuts. As they work to solve a problem, students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Teacher Edition: Examples from Unit MP.8 Use Repeated Reasoning Generalize Discuss the different solutions. Encourage student volunteers to explain how to do the steps mentally without writing them out. Students can suggest other examples and describe how the methods would be used. Lesson 15 ACTIVITY 1 MATH BACKGROUN D Lesson UNIT MP.8 Use Repeated Reasoning Generalize Lead students to generalize that 10 times any number gives you that number of tens and 10 times any number of tens give you that number of hundreds. This is the underlying concept upon which our place value system is built. ACTIVITY 1 Mathematical Practice 8 is integrated into Unit in the following ways: Use Repeated Reasoning Generalize Student EDITION: Lesson 19, pages 85 86-19 Name Date -19 Class Activity Name Date Class Activity 4-1_MNLESE84475_8A final -9-1 LKell Math and Games Big City Building 4-1_MNLESE84475_7A final This is a game called Big City Building. The goal of the -9-1 game is to design and build a successful city within a LKell The table shows the cost of different features on the Big City Building game. Below is Scott s design, so far, for his city in Big City Building. budget. To win the game, the city must have all of the features of a real-life city such as apartments, schools, parks, and shops, so its residents will be happy. Item Tree Shrub Lake Road Apartment building Any other building Cost $83 per tree $54 per shrub $198 per square unit $88 per square unit $9,179 per building $89 per square unit Currently, Scott has $156,34 in Big City Building money to create his city. 1. Each city in Big City Building requires a fire station, a police station, and a post office. These each cost $,657 in taxes per year to maintain. How much does it cost to maintain the fire station, the police station, and the post office building for one year? 3. Scott buys 4 trees to put in the park. The trees cost $83 each. How much money does Scott pay for the trees? $3,486 $7,971 4. Each apartment building contains 59 apartment units. Scott has 4 apartment buildings in his city. How many apartment units does Scott s city have?. In Big City Building, the roads are standard two-lane roads. The total width of the road is 9 meters. If each block is 8 meters long, what is the area of the road of one city block in square meters? 36 apartment units 5. If Scott s city is 7 units long and 19 units wide, what is the area of Scott s city in square units? 738 square meters Houghton Mifflin Harcourt Publishing Company Houghton Mifflin Harcourt Publishing Company Focus on Mathematical Practices Unit includes a special lesson that involves solving real world problems and incorporates all eight Mathematical Practices. In this lesson, students use what they know about multiplication to solve problems about simulation games. 513 square units UNIT LESSON 19 4_MNLESE84475_U0L19.indd 85 Focus on Mathematical Practices 85 1/04/1 3:45 PM 86 UNIT LESSON 19 4_MNLESE84475_U0L19.indd 86 Focus on Mathematical Practices 04/04/1 7:1 PM

Getting Ready to Teach Unit Learning Path in the Common Core Standards In previous grades, students learned basic facts and used these facts along with patterns, place value, and modeling to multiply one-digit numbers by multiples of 10. Unit broadens and deepens students experiences with multiplication to include multiplying numbers through thousands by one-digit numbers and finding the product of two two-digit numbers. In this unit, students model the concepts of arrays, single-digit multiplication, place value, and area. The activities in this unit help students gain a conceptual understanding of multidigit multiplication. Students are expected to apply their understanding of multidigit multiplication to numeric calculations and real world problem solving situations, including multistep problems. Math Expressions VOCABULARY As you teach the unit, emphasize understanding of these terms. Place Value Sections Method Expanded Notation Method Algebraic Notation Method Shortcut Method See the Teacher Glossary Help Students Avoid Common Errors Math Expressions gives students opportunities to analyze and correct errors, explaining why the reasoning was flawed. In this unit, we use Puzzled Penguin to show typical errors that students make. Students enjoy explaining Puzzled Penguin s error and teaching Puzzled Penguin the correct way to multiply whole numbers. The following common errors are presented to the students as letters from Puzzled Penguin and as problems in the Teacher Edition that were solved incorrectly by Puzzled Penguin. Lesson : Using incorrect operations when factoring Lesson 11: When using new groups in multiplication, multiplying by the new groups rather than adding Lesson 14: Incorrectly choosing to overestimate or underestimate to solve a real world problem Lesson 17: Incorrectly rounding to estimate a product In addition to Puzzled Penguin, there are other suggestions listed in the Teacher Edition to help you watch for situations that may lead to common errors. As a part of the Unit Test Teacher Edition pages, you will find a common error and prescription listed for each test item. 115DD UNIT Overview

Multiplying with Multiples of 10, 100, and 1000 Lessons 1 3 Array and Area Models In Grade 3, students modeled multiplication using array and area models. The models below show that a rectangle is made up of pushed-together squares of an array. All of these models show that 3 = 6. The models also illustrate that a rectangular model can be used to show equal groups as well as area. When working with the rectangular model with the small squares, students can visualize equal groups. When working with the rectangular model with no squares, students can visualize area. 3 columns 3 3 cm rows cm cm 3 3 cm Using these models is useful for students because it helps them utilize an idea with which they are familiar and expand it to represent more unfamiliar and complex concepts. This unit builds on the concept of area as pushed-together arrays of squares to help students understand multidigit multiplication. Multiplication and Place Value The exploration of multiplication begins by connecting students understanding of place value to the concept of multiplication. Lessons on place value in Unit 1 provided visual support for the students understanding that a 3 in the hundreds place does not stand for 3 ones, but actually represents 3 hundreds. It also helps prepare the way for the understanding of place value needed to conceptualize multidigit multiplication. UNIT MATH BACKGROUND To multiply multiples of tens, students use dot drawings. When using dot drawings to model place value concepts, students counted the dots. In this unit, students find the number of unit spaces between the dots in order to find total areas of rectangles. The drawing below shows that 3 = 6 because there are 6 unit spaces enclosed in the drawing. 3 3 3 3 UNIT Overview 115EE

Multiplying a One-Digit Number by a Multiple of 10, 100, or 1000 In the multiplication models presented in the unit, little plus signs (+) every 10 spaces horizontally and vertically make it easier to count the units. The drawing below shows representations for the multiplication 30. 1 + 1 30 = 10 + 10 + 10 10 = 0 10 = 0 10 = 0 10 30 1 30 = 30 1 30 = 30 30 + 10 + 10 1 + 1 from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN Multiplying by Multiples of 10, 100, and 1000 One component of understanding general methods for multiplication is understanding how to compute products of one-digit numbers and multiples of 10, 100, and 1000. 30 = 10 + 10 + 10 1 1 10 = 10 1 10 = 10 1 10 = 10 1 1 1 10 = 10 1 10 = 10 1 10 = 10 1 10 + 10 + 10 The drawings connect a visual representation to: 30 = ( 1) (3 10) = ( 3) (1 10) = 6 10 = 60 Multiplying Two Multiples of 10, 100, or 1,000 This drawing shows a representation for the multiplication 0 30. The drawing has all the squares visible in the first section to emphasize that the rectangular models can be used for both equal groups and area problems. 30 = 10 + 10 + 10 0 = 10 10 10 = 100 10 10 = 100 10 10 = 100 10 + + 10 10 10 = 100 10 10 = 100 10 10 = 100 10 10 + 10 + 10 The drawing connects a visual representation to: 0 30 = ( 10) (3 10) = ( 3) (10 10) = 6 100 115FF UNIT Overview = 600

Patterns in Products In this unit, students also learn how to move from the pictorial to the abstract as they look at patterns of products that involve multiples of 10, 100, and 1,000. Students examine groups of related multiplications such as: from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN A B C D 6 3 6 1 3 1 18 1 18 6 30 6 1 3 10 18 10 60 30 6 10 3 10 18 100 180 1,800 They rewrite each product so one factor is a power of ten. Then they see that the product is the product of the non-zero digits (in this case, 6 3), followed by the number of zeros in the power of 10. Area models help students understand why they can count the zeros when multiplying by tens and tens groups. These patterns can also help students understand how to estimate products by rounding and multiplying to check the accuracy of their calculations. Understanding the connection between place value and multiplication forms the foundation for the multiplication work that students will experience throughout the unit. It provides the conceptual foundation necessary to understand the different multiplication methods students will learn. Patterns We can calculate 6 700 by calculating 6 7 and then shifting the result to the left two places (by placing two zeros at the end to show that these are hundreds) because 6 groups of 7 hundred is 6 7 hundreds, which is 4 hundreds, or 4,00. Students can use this place value reasoning, which can also be supported with diagrams of arrays or areas, as they develop and practice using the patterns in relationships among products such as 6 7, 6 70, 6 700, and 6 7000. UNIT MATH BACKGROUND UNIT Overview 115GG

Lessons Multiplying by One-Digit Numbers 4 6 7 9 10 16 8 17 Using the Area Model As students begin to explore multiplying any two-digit number by a one-digit number, they expand the use of the area model they previously used when multiplying by multiples of 10. The first area models they use are dot drawings. In the area model below, the length represents 6, or 0 + 6. The width represents 4. The area of the first rectangle is the product of 4 0. The second product is 4 6. The areas, taken together, encompass the area of the whole rectangle, 4 6. 0 + 4 0 = 80 4 6 4 6 = 4 Thus, the model is a visual representation of: 4 6 = 4 (0 + 6) = 4 0 + 4 6 = 80 + 4 = 104 As students gain a higher level of abstraction, they draw models without dots. 35 = 30 9 + 5 9 As with the dot drawing, this model shows that: 9 35 = 9 (30 + 5) = 9 30 + 9 5 = 70 + 45 = 315 The area model is powerful because it can be generalized to any multidigit multiplication problem. 115HH UNIT Overview from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN Modeling Multiplication As with addition and subtraction, students should use methods they understand and can explain. Visual representations such as area and array diagrams that students draw and connect to equations and other written numerical work are useful for this purpose.

Multiplication Methods This unit presents students with a variety of numerical multiplication methods. Each method can be represented by an area model. Additionally, in all of the methods, the value of the digit in each place of one number is multiplied by the value of the digit in each place of the other number. However, in each method, the manner in which the multiplication steps are recorded varies. Place Value Sections Method Students draw a rectangle and add the areas of all the sections. The product equations are recorded inside the appropriate section of the rectangle. Then the products are added outside the rectangle. Notice that in this example, the first product, 70, is a result of multiplying 30 by 9. The second product, 45, is a result of multiplying 5 by 9. Students will come to know these products as partial products. 35 = 30 9 9 30 = 70 9 5 = 45 9 + 5 70 + 45 315 Expanded Notation Method In the Expanded Notation Method, the multiplication steps are recorded outside the rectangle. Arcs are used to ensure that every digit in one number is multiplied by every digit in the other number. Although students are introduced to using arcs in the one-digit multiplication lessons, it becomes a valuable strategy as students move into multidigit multiplication in which there are two or more digits in each factor. 9 = 0 + 9 6 6 6 6 9 6 0 9 = 0 = = = + 9 6 10 54 174 Distributive Property The connection is made between the Distributive Property and the multiplication methods that students have learned. Students realize that if a (b + c) = ab + ac and a two-digit number can be written as tens + ones, then it is possible to write, for example 5 43 as 5 (40 + 3), and thus find the sum of 5 40 and 5 3 to find the product. from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN The Distributive Property Another part of understanding general base-ten methods for multidigit multiplication is understanding the role played by the distributive property. This allows numbers to be decomposed into base-ten units, products of the units to be computed, then combined. By decomposing the factors into like base-ten units and applying the distributive property, multiplication computations are reduced to single-digit multiplications and products of numbers with multiples of 10, of 100, and of 1000. Students can connect diagrams of areas or arrays to numerical work to develop understanding of general base-ten multiplication methods. UNIT MATH BACKGROUND UNIT Overview 115II

Algebraic Notation Method Students extend their understanding of the Distributive Property to using equations to record the multiplication steps. Notice that in this method, arcs are also used to keep track of the number of steps as well as to relate the numerical steps to the model. 43 = 40 + 3 5 5 43 = 5 (40 + 3) = 5 40 + 5 3 = 00 + 15 = 15 Shortcut Method This is the common algorithm taught in most U.S. schools. Not all students are expected to gain proficiency using this method in this unit. Rather, it is preferred that students are capable of understanding multiplication using any of the methods presented. Just as in addition and subtraction, the new groups can be written above or below the problems. Shortcut Method with New Groups Above Shortcut Method with New Groups Below Method E: Step 1 Step Method F: Step 1 Step 7 7 8 _ 9 8 _ 9 5 8 _ 9 7 8 9 _ 7 5 115JJ UNIT Overview

Three- and Four-Digit Factors Multiplication in this unit is broadened to include multiplying one-digit numbers by three- and four-digit numbers. As students explore these problems, they gain the ability to generalize the methods that they learned. They observe that these same methods can be applied to three- and four-digit numbers. The following examples show the application of modeling and the Shortcut Method for recording multiplication to one-digit by threeand four-digit numbers. Multiply 4 37 4 37 = 00 + 30 + 7 4 00 = 800 4 30 = 10 4 7 = 8 4 800 10 + 8 948 1 37 _ 4 948 Multiply 9 6,435 6,435 = 6,000 + 400 + 30 + 5 9 400 9 30 9 5 9 9 6,000 = 54,000 9 = 3,600 = 70 = 45 54,000 3,600 70 + 45 57,915 3 3 4 6,435 9 57,915 UNIT MATH BACKGROUND UNIT Overview 115KK

Multiplying Two Two-Digit Numbers Lessons 1 13 15 18 Connecting Two-Digit Multiplication with One-digit Multiplication As students continue to explore multiplication, they move to multiplying two two-digit numbers. Although the problems in these lessons become more complex, the work that students did with one-digit factors forms the foundation for the understandings they need to proceed to greater numbers. The model and the methods already presented are powerful because they can be generalized to multiplication with any number of digits. Thus, as students progress through the unit, they continue to expand the application of the models and procedures they have already learned. Modeling and Methods The following examples show how the models and methods for which students have already developed fluency are extended to multiplication with greater numbers. The use of the arcs is expanded. Using color to code the arcs benefits students because it allows them to more carefully track and organize the steps in the multiplication. from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON NUMBER AND OPERATIONS IN BASE TEN Two-Digit by Two-Digit Multiplication Computing products of two two-digit numbers requires using the distributive property several times when the factors are decomposed into base-ten units. For example, 36 94 = (30 + 6) (90 + 4) = (30 + 6) 90 + (30 + 6) 4 = (30 90) + (6 90) + (30 4) + (6 4) Place Value Sections Method Expanded Notation Method Algebraic Notation Method 40 60 =,400 40 7 = 80 3 60 = 180 3 7 = + 1,881 67 60 + 7 43 = 40 + 3 40 60 =,400 40 7 = 80 3 60 = 180 3 7 = + 1,881 43 67 = (40 + 3) (60 + 7) =,400 + 80 + 180 + 1 =,881 115LL UNIT Overview

Shortcut Method Notice that the new groups are written above the numbers. Often when multiplying two-digit numbers, it is necessary to write more than one set of new groups. Since there is usually more room above the problem than below, recording the new groups in this fashion is more efficient. New Groups Above Step 1 Step Step 3 Step 4 Step 5 67 43 _ 1 67 _ 43 01 67 _ 43 67 _ 43 67 _ 43 01 8 01 68 01 + 68,881 Estimating Products Lessons 5 14 In Lessons 5 and 14 students learn how to estimate products. Students learn that by rounding the factors and multiplying them, an approximate answer for a product can be attained. Once the factors have been rounded, students can apply their understanding of multiplying with multiples of ten. A rounded estimate helps students decide if their answers are reasonable. Students estimate products with one- and two-digit factors. Problem Solving Lesson 11 Problem Solving Plan In Math Expressions a research-based problemsolving approach that focuses on problem types is used. Interpret the problem. Represent the situation. Solve the problem. Check that the answer makes sense. Hidden Questions In some problems, students must find the answer to a hidden question and use that answer to answer the question of the problem. We use the term hidden questions to make the conceptual point that students may need to answer these questions even if they do not appear in the problem. from THE PROGRESSIONS FOR THE COMMON CORE STATE STANDARDS ON OPERATIONS AND ALGEBRAIC THINKING Word Problems Fourth graders extend problem solving to multistep word problems using the four operations posed with whole numbers. The same limitations discussed for two-step problems concerning representing such problems using equations apply here. Some problems might easily be represented with a single equation, and others will be more sensibly represented by more than one equation or a diagram and one or more equations. UNIT MATH BACKGROUND UNIT Overview 115MM

Too Much Information/Too Little Information Throughout the unit, real world situations are used as the context for problem solving situations, including problems that involve two or more different operations. Students are also expected to analyze problems to determine whether they have too little or too much information. Focus on Mathematical Practices Lesson 19 The standards for Mathematical Practice are included in every lesson of this unit. However, there is an additional lesson that focuses on all eight Mathematical Practices. In this lesson, students use what they know about multiplying whole numbers to play a game in which they design and build a successful city while adhering to a specific budget. Multiplication and Division Basic Facts Fluency At this grade level students should be able to recall multiplication and division facts. If some students are still struggling with basic multiplication and division, you can use the diagnostic quizzes in the Teacher s Resource Book (M57 and M58) to assess their needs. Follow-up practice sheets are also provided. These practice sheets are structured so students can focus on a small group of multiplication and division facts on one sheet. There are also blank multiplication tables and scrambled multiplication tables to help students to develop instant recall. 115NN UNIT Overview