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Rectangular Arrays Objectives To review rectangular arrays and the use of multiplication number models to represent such arrays. epresentations etoolkit Algorithms Practice EM

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Rectangular Arrays Objectives To review rectangular arrays and the use of multiplication number models to represent such arrays. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Find factors of a number. [Number and Numeration Goal 3] Write number sentences for rectangular arrays. [Operations and Computation Goal 7] Use the turn-around rule for multiplication. [Patterns, Functions, and Algebra Goal 4] Key Activities Students discuss rectangular arrays using examples in the Arrays Museum and ones they draw or make with counters. They write multiplication number models to represent rectangular arrays. Ongoing Assessment: Recognizing Student Achievement Use journal page. [Operations and Computation Goal 7] Key Vocabulary rectangular array number model Commutative Property of Multiplication turn-around rule (for multiplication) Materials Math Journal 1, p. Student Reference Book, p. 10 Study Link 1 1 Math Masters, p. 413 (optional) 18 counters Class Data Pad slate Recognizing Patterns in Extended Facts Math Journal 1, pp. 6 and 7 Students practice solving extended multiplication and division fact problems using Fact Triangles. Math Boxes Math Journal 1, p. 8 Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 8 Students practice and maintain skills through Study Link activities. READINESS Defining Rows and Columns Math Masters, p. 9 per partnership: 40 centimeter cubes 2 dice Students practice building arrays. ENRICHMENT Exploring Magic Square and Heterosquare Arrays Math Masters, p. 10 Students explore rectangular arrays by solving magic square and heterosquare array problems. ELL SUPPORT Describing Exhibits in the Arrays Museum Students practice new vocabulary by describing items in the Arrays Museum. Advance Preparation Post the Working with a Partner principles. See Teacher s Reference Manual, pages Prepare a display area for the Class Data Pad and Arrays Museum. Collect several arrays as examples. Refer to Teacher s Reference Manual, page 13. Teacher s Reference Manual, Grades 4 6 pp. 16, 79 83, , Lesson 21 Getting Started Mental Math and Reflexes Pose basic and extended division facts. Have students write the answers for each set of problems. At the end of each set, ask students to describe the patterns they see among the dividends, divisors, and quotients. Suggestions: ,000 1, , ,000 4, = = 8 60,000 70,000 = 8 Math Message Arrange 12 counters into as many different rectangular arrays as you can. Then choose and draw one of the arrays. Study Link 1 1 Follow-Up Discuss student s responses and their number pattern poems. Ask: If you were writing a poem about arithmetic, how would you finish this sentence: Arithmetic is? List the mathematics vocabulary that students use on the Class Data Pad. Emphasize how using mathematics vocabulary makes communicating their ideas to others easier and more efficient. NOTE Some students may benefit from doing the Readiness activity before you begin Part 1 of each lesson. See the Readiness activity in Part 3 for details. Whole Numbers Interactive whiteboard-ready epresentations are available at to help you teach the lesson. Factors of a Counting Number A rectangular array is an arrangement of objects into rows and columns that form a rectangle. All rows and columns must be filled. Each row has the same number of objects. Each column has the same number of objects. A multiplication number model can represent a rectangular array. This rectangular array has 1 red dots. It has 3 rows with dots in each row. 3 * 1 is a number model for this array. 3 and are counting-number factors of 1. 1 is the product of 3 and. 3 and are a factor pair for 1. Counting numbers can have more than one factor pair. 1 and 1 are another factor pair for 1 because 1 * 1 = 1. To test whether a counting number a is a factor of another counting number b, divide b by a. If the result is a counting number and the remainder is 0, then a is a factor of b. One way to find all the factors of a counting number is to find all the factor pairs for that number. List all the factors of each number Check your answers on page 433. The counting numbers are 1, 2, 3, and so on. Whenever you are asked to find the factors of a counting number: (1) each factor must be a counting number, and (2) the other number in its factor pair must also be a counting number. 4 is a factor of 12 because 12 / 4 gives 3 with a remainder of 0. 6 is not a factor of 14 because 14 / 6 gives 2 with a remainder of 2. Find all the factors of the number 24. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Student Reference Book, p. 10 Number Models Factor Pairs 24 1 * 24 1, * 12 2, * 8 3, * 6 4, 6 1 Teaching the Lesson Math Message Follow-Up Adjusting the Activity WHOLE-CLASS DISCUSSION Ask students to share the rectangular arrays they drew. Have one student describe the array and another draw the array from the description. To support English language learners, clarify the noun/adjective relationship between rectangle and rectangular. Mentally note students use and understanding of appropriate vocabulary (rows, columns, in each row, in each column). Array possibilities for 12: 1-by-12, 12-by-1, 2-by-6, 6-by-2, 3-by-4, and 4-by-3. Draw the following visual R O W C reference on the board: O L U M N A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Reviewing Arrays (Student Reference Book, p. 10; Math Masters, p. 413) WHOLE-CLASS Algebraic Thinking Display examples of rectangular arrays from the Arrays Museum. Stress these key elements: Each row has the same number of objects. Each column also has the same number of objects. Each array has a rectangular shape. ELL ELL 22 Unit 1 Number Theory Ask students to name the number of rows and columns in each example. During this unit, students should collect other examples of arrays to add to the Arrays Museum. Adjusting the Activity Consider reserving a section of the Arrays Museum for arrangements that are almost arrays but do not satisfy all the conditions for rectangular arrays. Examples include some calculator keypads; certain playing cards, such as the nine of diamonds; a double-3 domino; and a calendar month, such as the month of August. August S M T W T F S This is not an array: The fifth row has only 3 days. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Arrays were first introduced in Second Grade Everyday Mathematics. Have students focus on labeling arrays in terms of rows and columns and representing arrays with number models. Assign student groups to: 1. Read page 10 in the Student Reference Book. 2. Complete one of the Check Your Understanding problems. 3. Draw a rectangular array from one of their factor pairs and write the number model that represents the array. Circulate and assist. Review multiplication number models as a way of representing rectangular arrays. Have groups present their arrays and number models. Rectangular arrays can help students visualize factors and the Commutative Property of Multiplication. Ask students to take out and arrange 6 counters into an array. Show students responses on the board or transparency of Math Masters, page 413 until all four possibilities have been displayed and discussed. (See margin.) To avoid confusion when naming an r-by-c array, let r represent the number of rows and c the number of objects in each row (the number of columns). Point out that both the 3-row-by-2-column and the 2-row-by-3- column arrays have the same number of dots, but not the same number of rows and columns. Tell students that this models a property of multiplication. The order in which two numbers are multiplied makes no difference in their product: 2 3 = 6 and 3 2 = 6. 3-row-by-2-column array Number model: 3 2 = 6 1-row-by-6-column array Number model: 1 6 = 6 2-row-by-3-column array Number model: 2 3 = 6 6-row-by-1-column array Number model: 6 1 = 6 Lesson 23 Date Arrays A rectangular array is an arrangement of objects into rows and columns. Each row has the same number of objects, and each column has the same number of objects. A multiplication number model can be written to describe a rectangular array. The first factor is the number of rows in the array. The second factor is the number of columns. The product is the total number of objects. This is an array of 8 dots. It has 4 rows with 2 dots in each row. It has 2 columns with 4 dots in each column. 4 2 = 8 The number model is next to the array. This is another array of 8 dots. It has 2 rows with 4 dots in each row. 2 4 = 8 It has 4 columns with 2 dots in each column. Label this array by writing the number model next to it. 1. a. Take 10 counters. Make as many different rectangular arrays as you can using all 10 counters. b. Draw each array on the grid at the right by marking dots. c. Write the number model next to each array. 2. a. How many dots are in the array at the right? 18 dots b. Write a number model for the array. 3 * 6 = 18 c. Make as many other arrays as you can with the same number of dots that were used for the array in Part 2a. Draw each array on the grid at the right. Write a number model for each array. Math Journal, p = = = 18 2 = 10 2 = = = = = = 18 Students have used this property of multiplication in turn-around facts as shortcuts to learning new facts. Ask if students know what this property is called. Some students will respond that the property is the turn-around rule for multiplication. Some students might know to use the term Commutative Property of Multiplication, but do not insist that students use this term. Adjusting the Activity Teach students a physical representation of the Commutative Property of Multiplication to indicate turn-around facts. This gesture demonstrates the idea of switching the numbers and can be used to remind students when the turn-around rule is being applied. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L ELL EM3cuGMJ1_U01_ indd 1/11/11 11:29 AM Finding All Possible Rectangular Arrays for a Number (Math Journal 1, p. ) PARTNER PROBLEM SOLVING Date Multiplication and Division Extended Facts Read the information about extended multiplication and division facts on Student Reference Book, pages 18 and 21. If you know the basic multiplication and division facts, then you can solve extended fact problems such as and 1, mentally. Just as there are four related facts for each basic fact, there are also four related facts in an extended fact family. 2 3 = = = = = = = = Review the Working with a Partner principles. Ask students for additional suggestions to help make the classroom more pleasant when students are working with partners or in small groups. Ask partners to make all possible rectangular arrays using 8 counters. 1-by-8, 8-by-1, 2-by-4, 4-by-2 Partners then work on journal page. Circulate and assist. NOTE Some students might find it easier to work on a full sheet of dot paper for Problem 2. (Math Masters, p. 413) Ongoing Assessment: Recognizing Student Achievement Journal Page Use journal page to assess students ability to build arrays and identify factors that describe arrays. Students are making adequate progress if they correctly arrange and label the arrays for both 10 and 18 by using counters and/or drawing on the journal page. [Operations and Computation Goal 7] =? Think: 2 [3s] = 6. Then 20 [30s] is 100 times as much = Write the extended fact family represented by each of these Fact Triangles. a = 2, = 2,100 2, = 30 2, = b. 20 = 1, = 1,200 1, = 60 1, = 20 2, , Math Journal 1, p. 6 EM3cuGMJ1_U01_ indd 6 1/22/11 9:0 AM 24 Unit 1 Number Theory 2 Ongoing Learning & Practice Recognizing Patterns in Extended Facts (Math Journal 1, pp. 6 and 7) Students write the extended fact families represented by the numbers on multiplication and division Fact Triangles. They describe patterns in the number of zeros in the factors and products. Math Boxes (Math Journal 1, p. 8) PARTNER INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1-4. The skill in Problem previews Unit 2 content. Date Multiplication and Division Extended Facts cont. c. 3,200 = ,200 = = 3, = 3, Complete your own Fact Triangle with extended multiplication and division facts. = = = = 3. Look at the four sets of facts you wrote. a. Describe a pattern for finding the product when you multiply with extended facts. Sample answer: First find the basic fact. Then count the number of zeros in each factor, and attach that many zeros to the product. b. Describe a pattern for finding the quotient when you divide with extended facts. Sample answer: First find the basic fact. Then subtract the number of zeros in the divisor from the remaining zeros in the dividend. Attach that many zeros to the quotient. 4. Do your patterns in Problem 3 work for and for 2,000 40? If not, adjust your patterns as necessary. Answers vary. Math Journal 1, p. 7 EM3cuGMJ1_U01_ indd 7 3, Answers vary. 1/22/11 9:0 AM Study Link (Math Masters, p. 8) INDEPENDENT Home Connection Students build and draw rectangular arrays to represent numbers and write the associated number models. Study Link Master Name Date STUDY LINK More Array Play Date Math Boxes A rectangular array is an arrangement of objects in rows and columns. Each row has the same number of objects, and each column has the same number of objects. We can write a multiplication number model to describe a rectangular array. For each number below, use pennies or counters to make as many different arrays as possible. Draw each array on the grid with dots. Write the number model next to each array º º 1 18 º º = 12 1 º º Marcus drew 8 cards from a pile: 10, 8, 4,, 8, 6, 12, and 1. Find the following landmarks: a. Maximum b. Minimum c. Range d. Median 3. Make an array for each of these number sentences. a. 3 9 = Name five numbers between 0 and 1. Answers vary. 4. a. Write the largest number you can make using each of the digits 7, 1, 0, 2, and 9 just once. 97, º º 3 18 b. 6 7 = 42 b. Write the smallest number. (Do not start with 0.) 10,279 9 º º º º Draw a line from each spinner to the number that represents the shaded parts % Practice = = = = = 129 Math Masters, p. 8 8 Math Journal 1, p. 8 Lesson 2 Name Date 1 2 Teaching Master Rows and Columns A rectangular array is an arrangement of objects in rows and columns. Each row has the same number of objects, and each column has the same number of objects. Work with a partner to build arrays. For each array, take turns rolling dice. The first die is the number of rows. Write this number in the table under Rows. The second die is the number of cubes in each row. Write this number under Columns. Then use centimeter cubes to build the array on the dot grid. How many cubes are in the array? Write this number under Array Total on the dot grid table. 3 Differentiation Options READINESS Defining Rows and Columns (Math Masters, p. 9) PARTNER 1 30 Min Rows Columns Array Total Rows Columns Array Total To explore factoring numbers using a concrete model, have students build arrays and find the total number of counters for each array. Have students describe their arrays using the words row and column. Math Masters, p. 9 ENRICHMENT PARTNER Exploring Magic Square and Heterosquare Arrays (Math Masters, p. 10) 1 30 Min To further explore rectangular arrays, have students solve magic square and heterosquare array problems. Arrays are also used to organize numbers, numerical expressions, and symbols to represent rules. In a magic square, the rule is that the sum of each row, column, and diagonal is the same. In a heterosquare, these sums will be different. Partners complete these two types of arrays and make an array of either type using their own numbers. Have students display their arrays in the Arrays Museum. This activity also provides practice with adding, subtracting, and comparing whole numbers. Name Date 1 2 Magic Square and Heterosquare Arrays A rectangular array is an arrangement of objects in rows and columns. The objects in an array can be numbers or numerical expressions. The Multiplication/Division Facts Table on the inside front cover of your journal is an example of numbers arranged in an array. The objects can also be words or symbols that represent elements of a given situation. For example, a plan for after-school snacks could be arranged in a 1-by- array, using A for apple, B for banana, and so on. A magic square is an array of positive whole numbers. The sum of the numbers in each row, column, and diagonal will be the same. 1. Complete this magic square Teaching Master A heterosquare is like a magic square, except that the sum of the numbers in each row, column, and diagonal are different. A 3-by-3 array for a heterosquare will have an arrangement of the numbers Complete this heterosquare, and write the sum for each row, column, and the two diagonals Create a magic square or heterosquare for your partner to solve. Answers vary ELL SUPPORT SMALL-GROUP Describing Exhibits in the Arrays Museum 1 Min To provide language support for multiplication, have students look at the Arrays Museum. Ask them to describe the arrays in the museum using language from the lesson. They might describe the rows, columns, shape, and the contents of the arrays. Planning Ahead Remind students to collect examples of arrays for the Arrays Museum. The Arrays Museum will be used again in Lesson 1-3 and in subsequent lessons. Math Masters, p Unit 1 Number Theory

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