Rectangles and Factors - NC Mathematicsmaccss.ncdpi.wikispaces.net/file/view/3-5+Fall+RESA+Handouts.pdf · The rectangles are different in that they show two distinct arrays,

DPI 3-5 Mathematics Fall REAS, 2011 Page 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.


Materials needed:

one inch color tiles

cm grid paper

Task: (this task could be 1-3 days)

• Make as many different rectangles as you can using 12 square-inch 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

2 6 2 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.


Now….. Let’s work together to make a class table from 1-25

• 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


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 1-by-1 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.


Number Factors Number of

Rectangles Prime or Composite

1 1 1 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


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 5 × 7.

3.OA.2 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 56 ÷ 8.

3.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.) 3 × 5 ×

2 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) = 40 + 16 = 56. (Distributive property.)

3.OA.6. Understand division as an unknown-factor 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 1–100. Recognize that a whole

number is a multiple of each of its factors. Determine whether a given whole number in the range

1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the

range 1–100 is prime or composite.


This task came from these resources:

1. Math Matters Grade K-8 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 4th

Grade, Unit 4: Product and Factors

A TIMS Curriculum from University at Chicago

4. Grades 3-4 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!


Rectangles and Factors 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25


cm grid paper



Factor Game

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6

1 2 3 4 5 6

7 8 9 10 12 14

7 8 9 10 12 14

15 16 18 20 21 24

15 16 18 20 21 24

25 27 28 30 32 35

25 27 28 30 32 35

36 40 42 45 48 49

36 40 42 45 48 49

54 56 63 64 72 81

54 56 63 64 72 81

1 2 3 4 5 6

1 2 3 4 5 6

7 8 9 10 12 14

7 8 9 10 12 14

15 16 18 20 21 24

15 16 18 20 21 24

25 27 28 30 32 35

25 27 28 30 32 35

36 40 42 45 48 49

36 40 42 45 48 49

54 56 63 64 72 81

54 56 63 64 72 81


Table 2 Common multiplication and division situations.7

7The first examples in each cell are examples of discrete things. These are easier for students and

should be given before the measurement examples.


NCTM Illuminations http://illuminations.nctm.org

The Factor Game Factor Game: http://illuminations.nctm.org/LessonDetail.aspx?ID=L620

Factor Game

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

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

http://illuminations.nctm.org/

http://illuminations.nctm.org/LessonDetail.aspx?ID=L620


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 1 + 2 + 3 + 4 + 6 = 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. 1-16.

http://www.math.msu.edu/cmp


Product Game Product Game: http://illuminations.nctm.org/ActivityDetail.aspx?ID=29

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 1-9

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 2.

3. Player 1 moves either marker to another number, and the new product is colored blue for

Player 1.

4. 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. 17-25.

http://illuminations.nctm.org/ActivityDetail.aspx?ID=29

http://www.math.msu.edu/cmp/


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 problem-

solving 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 base-ten 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)

Documents

Rectangles and Factors - NC Mathematicsmaccss.ncdpi.wikispaces.net/file/view/3-5+Fall+RESA+Handouts.pdf · The rectangles are different in that they show two distinct arrays,