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Fun can be mathematics Author(s): AUDREY KOPP and ROBERT HAMADA Source: The Arithmetic Teacher, Vol. 16, No. 7 (NOVEMBER 1969), pp. 575-577 Published by: National Council of Teachers of Mathematics Stable URL: http://www.jstor.org/stable/41186080 . Accessed: 21/06/2014 18:20 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Arithmetic Teacher. http://www.jstor.org This content downloaded from 185.44.78.113 on Sat, 21 Jun 2014 18:20:29 PM All use subject to JSTOR Terms and Conditions

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Fun can be mathematicsAuthor(s): AUDREY KOPP and ROBERT HAMADASource: The Arithmetic Teacher, Vol. 16, No. 7 (NOVEMBER 1969), pp. 575-577Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41186080 .

Accessed: 21/06/2014 18:20

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to The Arithmetic Teacher.

http://www.jstor.org

This content downloaded from 185.44.78.113 on Sat, 21 Jun 2014 18:20:29 PMAll use subject to JSTOR Terms and Conditions

Page 2: Fun can be mathematics

In the classroom Charlotte W. Junge

Fun can be mathematics AUDREY KOPP and ROBERT HAMADA Edison Model Mathematics Demonstration Center, Los Angeles City Schools, California

Juvery mathematics classroom can be a laboratory where students experiment with numerical ideas. Two-way communication between teacher and class by means of games can foster an atmosphere of eager participation in mathematical activities. The games suggested below have characteristics that can stimulate the student's mathemat- ical thinking by the use of number ideas and number sequences and patterns. Some of the exercises call for use of paper and pencil by students and either the chalk- board or the overhead projector for the teacher to show collection of data. Often each child may be asked for an oral re- sponse, thus allowing all to participate, as well as permitting the teacher to check if each student understands the rules of the game.

1. An easy way to establish the notion of patterns and sequences is to first ask the class to count by five. Tell the students to listen for a secret as each child in order gives an answer. Then again count by five but use 1, 2, 3, or 4 as the initial number. For example, count: 3, 8, 13, 18, 23. Again suggest that students listen for the secret. Try a third time. The teacher can judge if students know the secret by the promptness of their responses. If they have not caught on by the third round, the teacher may repeat each answer, enabling students to hear the terminal digits. By this

time, usually everyone knows the secret and you may have a perfect response on the fourth try.

The game can be used at the end of a class period in order to set the groundwork for the lessons to come on patterns and sequence. If only a few students discover the secret one day, try again the next.

2. Number sequences may be introduced by the teacher presenting the first numbers on the chalkboard:

1, 4, 7, -, -, -, •••

The class discovers more numbers in the sequence

1, 4, 7, 10, 13, 16, 19, •••

Then the teacher asks the class to verbalize the rule governing the sequence. If prac- tical, the class should be asked to write the rule in some form depending on the ability of the class: Add 3 to the previous number: 1 + (3-0), 1 + (3-1), 1 + (3-2), 1 + [3- (n - 1)] where n is the nth term in the sequence. Here is another example:

l, 4, 9, _,_,-, •••

After the class suggests the correct num- bers in sequence:

1, 4, 9, 16, 25, 36, •••

it is time to formulate rules again. Some classes may come up with the rule of squares; others may suggest adding the odd numbers in sequence. Teachers may find

Excellence in Mathematics Education - For AH 575

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Page 3: Fun can be mathematics

sequences or make up some that are com- mensurate with class levels. The patterns should not be difficult at first so that stu- dents can easily be acquainted with the idea of formulating a rule. After a few sequences the students can participate by offering sequences themselves. One stu- dent gives a series of numbers which the teacher records on the board. The student calls on other members of the class to complete the sequence and can either sup- ply the rule himself or ask another student to do so. Rules and sequences may be recorded by all students.

Occasionally not enough data is supplied to predict just one pattern.

1, 2, 4, -, -, -, •••

As many possibilities as can be found should be examined along with their rules.

1, 2, 4, 8, 16, 32, •••

1, 2, 4, 7, 11_, 16, •••

3. Student Participation Game Each child in the room is assigned a

number in sequence. Students may count off in order to make sure each knows his own number and those of the others around them. The teacher then asks students to stand according to a particular rule.

The relationship between ordinal and cardinal numbers may be illustrated by having every third student stand, or asking each student after the tenth student to stand. The game extends to practice with factors and primes. After the teacher makes a statement, the class members respond by standing, making it easy to check answers and locate missing ones. Sample directions:

The even numbers Multiples of three Factors of twelve Numbers with only two factors The prime numbers

Eventually, students will be able to make up appropriate directions themselves. Stu- dents often express ideas as they play this game. After a series of statements regard- ing factors, Carl asked, "Why do I have to stand all the time?" Dave replied im- mediately, "Because 1 is a factor of every

number." When Ernest was 1, he an- nounced that he would stand for every question that had to do with factors. He learned that he was wrong when the teacher asked all the prime factors to stand. Questions are easily found for each direc- tion. After the even numbers are stand- ing, for instance, one may ask 8 why she is standing. If she doesn't know, somebody will probably state that it's because 8 con- tains a factor of 2. The question of why 13 should stand in the prime numbers was answered by Mary as "13 has only 2 dif- ferent factors." Jerome, number 17, was tired of sitting so he suggested "the factors of 34 stand." Here is another example of a series of statements.

Factors of 12: Students assigned to numbers 1, 2, 3,

4, 6, and 12 should stand.

Factors of 18: Students assigned to numbers 1, 2, 3, 6,

9, and 18 should stand.

Common factors of 12 and 18: Only those students who stood up both

times previously should stand.

Greatest common factor of 12 and 18: The student with the largest number

remains standing. Another drill on fundamentals would have the teacher make a statement such as: 4 plus 3, multiplied by 2, minus 2, divided by 6.

Obviously, the student assigned to the num- ber 2 should stand up.

Instead of having pupils stand, each may be given a 3 X 5 card with his num- ber written on both sides in bold color. Cards may be held up for answers. After a while the numbers may be given out randomly, which may increase the difficulty of the game.

4. Ordered pairs lend themselves to games wherein students select and verbalize rules. Visual reproduction of a function machine is often helpful. Basic parts are Input, Output, and Function Rule. Many

576 The Arithmetic Teacher /November 1969

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Page 4: Fun can be mathematics

rules can be used, depending again on the level and needs of the class. Here are two sample sets of number pairs: (2, 4) (3, 6) (8, 16) (12, - ) (- , 100) (4, 9) (5, 11) (6, 13) (9, 19) (10, - ) (- , 51)

Students enjoy working out their own rules and trying them on the other members of the class. Thus the notion of function can be introduced at even a very elementary level.

Editor's Note. - You may wish to try this approach to finding the pattern for number sequences:

Term of the sequence: 1 4 7 10 13 16 19

Number of the term: 12 3 4 4 6 7

1 I I 1 1 I I (3Xl)-2 (3X2)- 2 (3X3)-2 (3X4)- 2 (3X5)-2 (3X6)-2 (3X7)-2

To find any term of the sequence, multiply the number of the term by 3 and subtract 2, or 3« - 2.

Term of the sequence: 1 4 9 16 25 36

Number of the term: 12 3 4 5 6

I I I I I I (1X1) (2X2) (3X3) (4X4) (5X5) (6X6)

To find any term of the sequence multiply the number of the term by itself, or n X n. Now try to develop the general form of the sequence for this sequence:

3 5 7 9 11 13

-Charlotte W. Junge

Solving story problems and liking it SALLY MATHISON North Junior High School, Portage, Michigan

JL/o you have difficulty motivating your students during story problem units? Here's an idea that makes the teaching of word problems a new story!

You will need a package of construc- tion papers (choose light colors because you are going to write on the paper), some glue, and some old magazines. Magazines are full of colorful pictures that can be

used to illustrate story problems. Keeping in mind your students' interests as well as the fact that they appreciate a sense of humor, use your imagination as you select your illustrations. How many pictures will you need? Mathematically speaking, if N = the number of students in your class, you will need N - 1 pictures. Cut out the pictures and paste them on the construc-

Excellence in Mathematics Education - For All 577

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