Rectangles and Factors


 Cecil Tyler
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1 Rectangles and Factors Background information: Unlike addition, in multiplication, each factor has a different purpose; one identifies the number in each group and the other, the number of groups. Therefore we don t just want to introduce multiplication as repeated addition but also explore an array or rectangle. Properties of arithmetic provide the conceptual underpinnings for computational strategies and foundation for algebraic thinking. It is important to introduce these properties at the basic fact level so that students can apply them easily as multiplication examples become more complex. The array or area model of multiplication is particularly powerful for modeling the commutative and distributive properties of multiplication. By turning the rectangle or looking at it sideways, students can see that the total stays the same, that a x b = b x a. The distributive property of multiplication over addition states that a(b + c) = ab + ac. It is this property that allows us to find 9 x 6 or 9 x (3 + 3) by realizing that this is the same as (9 x 3) = (9 x 3) or 2(9 x 3). So if students know the product when multiplying by three, they can double it to multiply by six. Though not helpful to learning of basic facts, multiplication is also associative, that is (a x b) x c = a x(b x c). This property is useful when multiplying three numbers, for example, 7 x 2 x 5. We can choose to start with 2 x 5, rather than 7 x 2. Square numbers, like doubles in addition, should be emphasized. Once a student knows a fact such as 7 x 7 = 49, he or she can use this fact to find 8 x 7, by adding one more 7 to 49 to get 56. Having such anchors is essential, as unlike addition and subtraction facts, most basic multiplication facts cannot be determined quickly by counting on or back. Think about divisibility rules! A prime number is a positive integer that has two and only two unique positive factors. The two most pervasive misconceptions that students demonstrate when discussing prime and composite numbers are that one is a prime number that all prime numbers are odd. It is important to be very explicit that one is a special number, as it has only one unique factor, and that two is the only even prime number. DPI 35 Mathematics Fall REAS, 2011 Page 1
2 Materials needed: one inch color tiles cm grid paper Task: (this task could be 13 days) Make as many different rectangles as you can using 12 squareinch color tiles. Complete a table like the one below to record your rectangles. Rectangle Possible with 12 Tiles Number of Rows Number in each Row Multiplication Sentence x 6 = 12 What did you notice? Were the rectangles the same? Were the rectangles different? How would you describe your rectangle? Does that description fit someone else's rectangle? Possible Classroom Discussion: Does the orientation of the rectangles matter? Is a 2 x 6 rectangle the same as a 6 x 2 rectangle? The rectangles are different in that they show two distinct arrays, but they cover the same area: a 2 x 6 rectangle can be rotated to represent a 6 x 2 rectangle. Third grade (If fourth grade) For this activity we are interested in the dimensions of the rectangles (or the factors of the numbers), not in their vertical or horizontal orientation. What is important to note is that 2 and 6 are both factors of 12. Now let s find all the rectangles you can make with 18 tiles. Record your rectangles on grid paper. DPI 35 Mathematics Fall REAS, 2011 Page 2
3 Now.. Let s work together to make a class table from 125 Which numbers have rectangles with 2 rows? List them from smallest to largest. Which numbers have rectangles with 3 rows? List them from smallest to largest. Which numbers on the chart are multiples of 4? (have rectangles with 4 rows) Which numbers on the chart are multiples of 5? How many different rectangles can you make with 5 tiles How many with 7 tiles? List the prime numbers between 1 and 25 Are all odd numbers prime? Explain. Look at the number nine, what do you notice? What other numbers have rectangles that are squares? What is the next largest square after 25? Things to think about: A number is a multiple of 2 if it equals 2 times another whole number. If you can make a rectangle with 2 rows for a number then it is a multiple of 2. Numbers that are multiples of two (2, 4, 6, 8, etc ) are called even numbers. Numbers that are not multiplies of 2 (1, 3, 5, 7 etc ) are called odd numbers. When you skip counts you say the multiples of a number. For example, skip counting by 3 gives the multiples of 3. The multiples of 3 are 3, 6, 9, 12 and so on. They are all the numbers that have rectangles with 3 rows. Numbers that are larger than one and have only one rectangle have a special name. They are called prime numbers. For example, 5, 7, are prime numbers. Prime numbers are defined as numbers that are larger than one and have only on rectangle. One is Not prime, since a prime number is conventionally taken to be number with exactly two factors (itself and one), one is not considered to be prime. Note that two is prime since its only factors are itself and one. Two is the only even prime number. During the discussion of prime numbers, the question sometimes arises of what to call numbers that are not prime numbers, the ones that have more than one rectangular array. Tell students that these are called composite numbers. The definition of a composite number is one DPI 35 Mathematics Fall REAS, 2011 Page 3
4 that has more than two distinct factors. All numbers that are not prime and are not one are composite numbers. Possible Classroom Discussion What did you find out about the number of factors? The number 1 has one factor (itself) and forms one rectangle (a 1by1 square); it is classified by mathematicians as a special number and is neither prime nor composite. Many numbers have only two factors and make just one rectangle: 2, 3, 5, 7, 1, 13, 17, 19, 23. These numbers are the prime numbers. Prime numbers are composite numbers and have more than two factors. All the other numbers are composite numbers and have more than two factors; composite numbers can be represented by at least two unique rectangular arrays. Examine the rectangles representing the numbers 1, 4, 9, 16, and 25. Did you notice that in each case one of the rectangles that can be found is also a square? Ancient Greek mathematicians thought of number relationships in geometric terms and called numbers like this square numbers, because of the rectangular arrays they can be represented by a square. The square numbers have an odd number of factors, whereas the other numbers examined have an even number of factors. Numbers that are not square always have factor pairs. For example 12, the factor pairs are 1 and 12, 2, and 6, and 3 and 4. But square numbers always have one factor that has no partner other than itself. For example 9, 1 and 9 are factor pair, but 3 is its own partner because 3 x 3 = 9. The factor of a square number that has no partner 3 for the square number 9 isn t listed twice. Therefore, the factors of 9 are 1, 3, and 9, and the factors of 16 are 1, 2, 4, 8, and 16 an odd number of factors. DPI 35 Mathematics Fall REAS, 2011 Page 4
5 Number Factors Number of Rectangles Prime or Composite Neither 2 1, 2 2 Prime 3 1, 3 2 Prime 4 1, 2, 4 3 Composite 5 1, 5 2 Prime 6 1, 2, 3, 6 4 Composite 7 1, 7 2 Prime 8 1, 2, 4, 8 4 Composite 9 1, 3, 9 3 Composite 10 1, 2, 5, 10 4 Composite 11 1, 11 2 Prime 12 1, 2, 3, 4, 6, 12 6 Composite 13 1, 13 2 Prime 14 1, 2, 7, 14 4 Composite 15 1, 3, 5, 15 4 Composite 16 1, 2, 4, 8, 16 5 Composite 17 1, 17 2 Prime 18 1, 2, 3, 6, 9, 18 6 Composite 19 1, 19 2 Prime 20 1, 2, 4, 5, 10, 20 6 Composite 21 1, 3, 7, 21 4 Composite 22 1, 2, 11, 22 4 Composite 23 1, 23 2 Prime 24 1, 2, 3, 4, 6, 8, 12, 24 8 Composite 25 1, 5, 25 3 Composite DPI 35 Mathematics Fall REAS, 2011 Page 5
6 Possible Extension: If my rectangle has a total of 18 square tiles and 3 rows of tiles How many tiles are in each row? Write a number sentence for this rectangle. Is 34 a multiple of 2? Explain why or why not. Is 3 a factor of 35? Explain why or why not. Name ten numbers that have 3 as a factor. Common Core State Standards Third Grade: Represent and solve problems involving multiplication and division. 3.OA.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 OA.2 Interpret wholenumber 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 OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8? = 48, 5 = 3, 6 6 =?. Understand properties of multiplication and the relationship between multiplication and division. 3.OA.5. Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 4 = 24 is known, then 4 6 = 24 is also known. (Commutative property of multiplication.) can be found by 3 5 = 15, then 15 2 = 30, or by 5 2 = 10, then 3 10 = 30. (Associative property of multiplication.) Knowing that 8 5 = 40 and 8 2 = 16, one can find 8 7 as 8 (5 + 2) = (8 5) + (8 2) = = 56. (Distributive property.) 3.OA.6. Understand division as an unknownfactor problem. For example, find 32 8 by finding the number that makes 32 when multiplied by 8. Fourth Grade: Gain familiarity with factors and multiples. 4.OA.4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given onedigit number. Determine whether a given whole number in the range is prime or composite. DPI 35 Mathematics Fall REAS, 2011 Page 6
7 This task came from these resources: 1. Math Matters Grade K8 Understanding the Math You Teach by Suzanne Chapin and Art Johnson 2. A collection of Math Lessons from Grades 3 through 6 by Marilyn Burns 3. Trailblazer 4 th Grade, Unit 4: Product and Factors A TIMS Curriculum from University at Chicago 4. Grades 34 Zeroing in on Number and Operations Key Ideas and Common Misconceptions by Linda Dacey and Anne Collins Possible starting task: Many people have a number that they think is interesting. Choose a whole number between 1 and 100 that you think is special. Record your number. Explain why you chose that number. List three or four mathematical facts about your number. List three or four connections you can make between your number and your world. Revisit the task above after the Rectangle and Factor lesson, what mathematical facts about the selected number do students think about now! DPI 35 Mathematics Fall REAS, 2011 Page 7
8 Rectangles and Factors DPI 35 Mathematics Fall REAS, 2011 Page 8
9 cm grid paper DPI 35 Mathematics Fall REAS, 2011 Page 9
10 DPI 35 Mathematics Fall REAS, 2011 Page 10
11 Factor Game DPI 35 Mathematics Fall REAS, 2011 Page 11
12 Table 2 Common multiplication and division situations. 7 7 The first examples in each cell are examples of discrete things. These are easier for students and should be given before the measurement examples. DPI 35 Mathematics Fall REAS, 2011 Page 12
13 NCTM Illuminations The Factor Game Factor Game: Factor Game Player A chooses a number on the game board by clicking on it. The square will be colored blue, as shown for 12. Player A receives 12 points for this choice. Player B then clicks on all the proper factors of Player A s number. The proper factors of a number are all the factors of that number, except the number itself. For example, the proper factors of 12 are 1, 2, 3, 4, and 6. Although 12 is also a factor of 12, it is not DPI 35 Mathematics Fall REAS, 2011 Page 13
14 considered a proper factor. All of the proper factors that Player B selects will be colored green, as shown to the left. Player B will receive = 16 points for selecting all of the proper factors. Players reverse roles. On the next turn, Player B colors a new number and gets that many points, and Player A colors all the factors of the number that are not already colored and receives the sum of those numbers in points. The players take turns choosing numbers and coloring factors. If a player chooses a number with no uncolored factors remaining, that player loses a turn and does not get the points for the number selected. The game ends when there are no numbers remaining with uncolored factors. The player with the greater total when the game ends is the winner. The Factor Game applet was adapted with permission and guidance from "Prime Time: Factors and Multiples," Connected Mathematics Project, G. Lappan, J. Fey, W. Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996), pp DPI 35 Mathematics Fall REAS, 2011 Page 14
15 Product Game Product Game: Object of the Game: To get four squares in a row vertically, horizontally, or diagonally. 1. To begin the game, Player 1 moves a marker to a number in the factor list of numbers 19 along the bottom of the game screen. 2. Player 2 then moves the other marker to any number in the factor list (including the number marked by Player 1). The product of the two marked numbers is determined, and that product is colored red for Player Player 1 moves either marker to another number, and the new product is colored blue for Player Players take turns moving a marker, and each product is marked red or blue, depending on which player made the product. However, if a product is already colored, the player does not get a square for that turn. 5. Play continues until one player wins, or until all squares have been colored. This Product Game Investigation was adapted with permission and guidance from Prime Time: Factors and Multiples, Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications (1996), pp DPI 35 Mathematics Fall REAS, 2011 Page 15
16 What does Computational Fluency mean? Developing fluency requires a balance and connection between conceptual understanding and computational proficiency. On the one hand, computational methods that are over practiced without understanding are often forgotten or remembered incorrectly On the other hand; understanding without fluency can inhibit the problemsolving process. (PSSM, Page 35 How do students demonstrate Computational Fluency? Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the baseten number system properties of multiplication and division (and of course, of the other operations, as well), and the number relationships. (PSSM, page 152) Is there such a thing as effective drill? There is little doubt that strategy development and general number sense (number relationship and operation meanings) are the best contributors to fact mastery. Drill in the absence of these factors has repeatedly been demonstrated as ineffective. However, the positive value of drill should not be completely ignored. Drill of nearly any mental activity strengths memory and retrieval capabilities. (Van de Walle) What about Timed Tests? Teachers who use timed test believe that the test help children learn basic facts. This makes no instructional sense. Children who perform well under time pressure display their skills. Children who have difficulty with skills, or who work more slowly, run the risk of reinforcing wrong learning under pressure. In addition, children can become fearful and negative toward their math learning. (Burns 2000, p.157) DPI 35 Mathematics Fall REAS, 2011 Page 16
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