Activities. A Geometric Look at Greatest Common Divisor

ACTIVITIES A GEOMETRIC LOOK AT GREATEST COMMON DIVISOR By MELFRIED OLSON, Western Illinois University, Macomb, IL 61455

Teacher's Guide

Introduction: One of the four standards specified at all grade levels in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) is mathematical connections. The cutTiculum standards suggest that the study of mathematics should include opportunities to make connections so that students can link conceptual and procedw-al knowledge, relate various representations of concepts or procedw-es to one another, recognize relationships among different topics in mathematics, use a mathematical idea to fw-ther their understanding of other mathematical ideas, and recognize equivalent representations of the same concept. It is important to afford opportunities for connections so that students will not view mathematics as a collection of isolated topics nor think of mathematics as computation only. The following activity examines an arithmetic concept, greatest common divisor, from a geometric standpoint. This activity could also be used to demonstate conversions between units of area. The method presented here engages students intellectually and challenges t hem to apply prior knowledge of numerical relationships to explore the concept of greatest common factor

from another slant. This activity uses t he area model for multiplication and division in another attempt to emphasize connections between arithmetic and geometric representations. Problem solving and reasoning in the exploration of number theory and patterns are also included.

Grade levels: 5-10 Materials: Copies of sheets 1-4 for each student, scissors, extra sheets of graph paper Objectives: To develop and enhance students' understanding of the idea of greatest common divisor Prerequisites: Before beginning this activity, the teacher should make sure that the students understand multiplication and division from an area-model perspective. Ideally, students should be able not only to model 3 x 7 in rectangular form as

11111111 but also to model the distributive property for 14 x 23 in rectangular form (see fig. 1).

Edited by Mm·y Kim Prichard, University of North Carolina at Charlotte, Charlotte, NC 28223 Nadine Bezuk, San Diego State University, San Diego, CA 92182 Mally .lloody, Oxford High School, Oxford, AL 36203 This section is designed to provide in reproducible formats mathematical activities appropriate for students in grades 7-12. This material may be reproduced by classroom teachers for use in their own classes. Readers who have developed successful classroom activities are encouraged to submit manuscripts, in a format similar to the "Activities" already published, to the editorial coordinator (or review. 0( particular interest are activities focusing 011 the Council's curriculum standards, its expanded concept of basic skills, problem solving and applications, and the u,ses of calculators and computers.

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20

3

10

used in the Euclidean algorithm to find the greatest common divisor of two numbers. For example, to find the greatest common divisor of 24 and 44 (problems 1-4, sheet 1) by using the Euclidean algorithm we calculate thus:

4 Fig. 1

Directions: This activity is probably best used over two or three days. This time can be shortened by not having all students cut each rectangle in problem 1 of sheet 2. Instead, the teacher can divide the class into three groups and have each group do two rectangles.

Sheet 1: Students should cut out the appropriate rectangle from sheet 4. Students may need some help in identifying the largest square that can be removed. The teacher should work through the first activity with the students to familiarize them with "removing the largest possible square from one of the rectangle's edges." Once this concept is clear the students should cut out the second square as directed. Students should be careful to keep the remaining rectangle. The information regarding the size of the last square removed will be recorded on the chart on sheet 1, as well as later on sheet 2. The purpose of the cutting is to find the special square related to each rectangle. This special square is the size of the last square removed, and the length of a side of the square is the greatest common divisor of the dimensions of the rectangle. As can be seen on sheet 3, each rectangle can be partitioned into squares all of which are the size of the last square removed. Sheet 2: Students must use their own grid paper for work on this sheet. They repeat the square-removal process from sheet 1 and record what they find. Students predict the size of the last square removed and check their prediction on other rectangles. This process yields the greatest common divisor of the dimensions of a rectangle because it geometrically pictures the steps

1 1. 24)44 24 20

One 24 x 24 square is removed; a 20 x 24 rectangle remains.

1 One 20 x 20 square is removed; 2. 20)24 a 4 x 20 rectangle remains. 20 4 3.

5 4)20 20 0

A 0 remainder gives 4 as the greatest common divisor of 44 and 24.

Similarly, using the Euclidean algorithm for finding the greatest common divisor of 18 and 51 (problems 5 and 6, sheet 1), we write this: 2 Two 18 x 18 squares are re1. 18)51 moved; a 15 x 18 rectangle 36 remains. 15 1 2. 15)18 15 3 5 3. 3)15 15 3

One 15 x 15 square is removed; a 3 x 15 rectangle remains.

A 0 remainder gives 3 as the greatest common divisor of 51 and 18.

Sheet 3: Students reverse the process of removing squares and try to draw in and visualize how many squares can be found. After drawing in squares, t he students should record the information in the chart in problem 3. The extension into least common multiple is probably expected but often not seen in the same manner as asked in question 4. That this strategy indeed does produce the least common multiple can be seen by one of the methods used to find the least common multiple of two numbers. For exam-

March 1991 - - -- - - - - - - - - - - - - - --

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203

pie, t he least common multiple of 20 and 42 is found by factoring both numbers: 20 42

= 2 ·10 = 2. 21

Therefore, the least common multiple of 20 and 42 is 2 · (10 · 21 ).

Answers: Sheet 1: l(a) 24 X 24, (b) no, (c) 20 x 24; 2(a ) 20 x 20, (b) no, (c) 4 x 20; 3(a) 4 x 4, (b) yes, 5, (c) No rectangle remains. 5(a ) F irst two 18 x 18 squares are removed leaving a 15 x 18 rectangle. (b) Next, one 15 x 15 square is removed leaving a 3 x 15 rectangle. (c) Finally, five 3 x 3 squares are removed so t hat no rectangle remains. 6. For the 24 x 44 rectangle: 4 x 4, 7; for the 18 x 51 rectangle: 3 x 3; 8 Sheet 2: 1. 20 x 42-2 x 2, 16 x 36--4 x 4, 21 X 35-7 X 7, 17 X 30-1 X 1, 12 X 42- 6 x 6, 18 x 45-9 x 9; 2. Among the observations are the following: The dimension of the last square removed is the greatest common divisor of the dimensions of the rectangle; the original rectangle can be divided into squares t hat are all t he same size; a last square can usually be found; for most rectangles, that is not a 1 x 1 square. 3(a) 7 X 7, (b) 8 X 8, (c) 1 X 1, (d ) 3 X 3, (e) 4 X 4; 4. See directions for sheet 2.

4. Students may need to be reminded of other uses, including "simplifying" fractions. Suppose we treat the length of the sides of a rectangle like the numerator and denominator of a fraction. When all the lines are drawn in the rectangle, as in figure d in problem 1, t he common fact or is revealed a long each side, with each side being "divided" by the greatest common divisor; t he "reduced" fraction can be seen by counting the number of divisions a long each side. This represen tation emphasizes dividing both numerator and denominator by the same amount. 5. The product of the number in column N and the length of a side in columnS gives t he least common multiple of the d imensions of the rectangle.

REFERENCES National Council of Teachers of Mathematics, Commission on Standards for School Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989. •

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Sheet 3 1. 66 squares

2. 102 squares of size 3 x 3 3. Dimensions Number of Size of Squares Squares of Rectangle (N ) (S ) 20 16 21 12 18 14 16 7 9 140 ab

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REMOVING SQUARES

SHEET 1

1. Cut out the 24 x 44 rectangle (rectangle A) on sheet 4. Remove the largest possible square ''from one of the rectangle's edges." (a) What size of square is removed?_ __ __ _ __ _ _ _ __ _ __ (b)

Can more than one of these squares be removed? _ _ _ __ __ __ If yes, how many can be removed?_ ______________

(c) What size of rectangle remains?________ _ _ _ _ _ __ _ 2. R~peat the process in step 1 with the 20 x 24 rectangle left after step 1c. (a)

What size of square is removed?______ _ _ _ _ _ _ _ __ _

(b)

Can more than one of these squares be removed?_ _ _ _ _ __ __ If yes, how many can be removed?_ _____________ _

(c) What size of rectangle remains?________ _ _ _ _ __ _ _ 3. Repeat the process in step 1 with the 4 x 20 rectangle after step 2c. (a) What size of square is removed?______________ __ (b)

Can more than one square be removed?_____________ If yes, how many can be removed?____ __ _ _ _ _ __ _ __

(c) What size of rectangle remains?________________ 4. Record in table 1 the size of the last square removed and the number of squares removed so that nothing remained. 5. Cut out one 18 x 51 rectangle (rectangle B) from sheet 4 and remove from it the largest possible square, as in step 1. Continue the process until the whole 18 x 51 rectangle has been depleted. Answer the following questions at each stage. (a) What size of square is removed?________________ (b)

Can more than one of these squares be removed?_ _ _ _ _ _ _ __ Ifso, how many?___________________________

(c) What size of rectangle is left?_ _ _ _ __ _ _ _ _ _ _ __ _ __ 6. When you get to the last step, record in the chart below the size of the last square removed and the number of squares removed so that nothing remained. Rectangle 24 18

X

X

Size of Last Square Removed

Number of Squares Removed

44 51 From the Mathematics Teacher, March 1991

SHEET 2

PREDICTING LAST SQUARE REMOVED

1. For the rectangles given, repeat the removing-squares process used on sheet 1. In the right-hand column of the chart, record the size of the last

square removed. The information for the 24 x 44 and 18 x 51 rectangles from sheet 1 is recorded for you.

Rectangle 24 18 20

X

16 21

X

17 12

X

X X

X

X

44 51 42

Size of Last Square Removed 4X4 3 X3

36 35 30 42

18 X 45 2. What observations can you make about the relationship between the size of the original rectangle and the size of the last square removed?

3. Given the dimensions of a rectangle, can you predict the size of the last square to be removed? Try your predictions on the following rectangles: (a) 14 x 35 _ _ __ (b)

(c)

16

X

40 _ _ __

7 x

24 _ __ _

9 X 21 - - - (e) 140 x 288 _ _ __

(d )

4. Why does this square-removal process give us information needed to find the greatest common divisor of the dimensions of a rectangle?_ _ _ _ __

From I he Mathematic• TroJ:h~r. March 1991

SHEET 3

HOW MANY SQUARES?

1. The results of the work on question 1 on sheet 1 can be represented in the following manner:

(d)

+

* i-

+ T

~ ~8= Itf-l

[t

~ Ft

4-t-f->

I++

rnmm

R=t=

~

By inserting heavy lines into figure c, we can obtain figure d, which contains squares of only size 4 x 4. How many 4 x 4 squares appear in figured? 2. Repeat the process shown above for the remaining 18 x 51 rectangle (rectangle C) from sheet 4 to find how many 3 x 3 squares are contained in the rectangle. 3. Together with the process shown above, use the information from questions 1 and 2 on sheet 2 to complete the following chart. Dimensions of Rectangle 20 X 42 16 X 36 21 X 35 12 X 42 18 X 45 14 X 35 16 X 40 7 X 24 9 X 21 140 X 288 *ab x ac

Number of Squares

Size of Squares

(N)

(S )

*Where b a nd c have no common factors

4. How does this concept of greatest common divisor relate to other uses you have made of the concept of greatest common divisor? 5. How can the numbers in columns N and S be used to find the least common multiple of the dimensions of the rectangle? From the Mathematics Trocher. March 1991

SHEET 4

GRIDS

A. I I

B.

c.

From the Mathematic• Teacher, March 1991