Cornelia C. Tierney
In elementary school, great importance is placed on memorization of number facts. In teaching fifth through eighth graders, I have assumed that most of my students had made a concerted effort to memorize facts in earlier grades. I have observed, however, that children who have had a similar amount of practice have a great range of recall. A few students complete tests of 100 multiplication or division facts perfectly in less than three minutes, whereas others are made miserable by the whole process. They skip many problems, look around the room to compare their progress with that of other students, and finally give up with a few correct answers. Although those who have memorized the facts do better than others at whole-number arithmetic, they do not always do well in work with fractions.
In addition to reinforcing knowledge of number facts, the following activities give students practice in making generalizations that improve problem-solving skills and provide a foundation for work with fractions. Reducing fractions and factoring quadratics require students to do what are essentially division problems with only the dividend given. Before they can find the quotient, they must provide the divisor. For this task, rote memorization of facts is not sufficient. Children with total recall of multiplication facts often do not see that a fraction can be "reduced" by any number but 2 or 10. For example, 6/9 is left untouched or changed to
by a creative student. Also, students do not recognize ways to factor numbers that are not among the answers to the facts they have memorized.
Before beginning a unit on fractions in the middle grades and before introducing factoring in algebra, I teach some number theory. The students "discover" divisibility rules by studying patterns in the multiplication tables. For example, they learn that multiples of 5 end in 5 or 0, whereas in multiples of 9, the sum of the digits is 9 or a multiple of 9. Although students who have memorized the tables increase their understanding, it is often those who have been unable to memorize the number facts who enjoy these activities the most. Although these students make errors with details, they see the larger picture and are able to make comparisons between tables and between parts of tables.
I use the same activities for all grades, except that some classes see more patterns and so I push them to look farther. A sixth-grade class I worked with spent an hour on only half of the first activity. They looked only at patterns in the 2s, 3s, 4s, and 6s tables. They found many tables within tables and patterns in diagonals. Third- and fourth-grade students can try some of these activities before doing regular work with the multiplication facts, and fifth, sixth, and seventh graders can try them before working with fractions.
Patterns in the multiplication tables
For all the tables, I use an overhead transparency of a standard 10 x 10 hundred chart with stencils cut from tagboard that just fit over the chart, allowing only one set of multiples to be seen (fig. 1). I use stencils for 2s, 3s, 4s, 5s, 7s, 8s, 9s, and prime numbers. I start by projecting the 2s table on the chalkboard and asking, "What do you see here?" (If the chalkboard is used as a screen, notes can be written on the chart, or certain numbers can be circled. If all the numbers are circled, an interesting pattern remains on the board when the projector is turned off, which the students come to recognize as "the shape" of that table.)
To allow all children to participate, this activity should at first be done quietly and independently. For a few minutes after each table is projected, the children look for patterns and write notes to themselves. They then share their discoveries, which a student or I write on the board. Often only a few children respond to the 2s table, but many see patterns in the 3s table. One child's idea leads another to see something new, so that students notice far more than they had originally written down.
After 2s and 3s, I ask students what they think will show when both the 2s and the 3s are used together. After looking at this table (the 6s), we examine the 5s through 9s table in order of increasing size. Although the 9s table, like the 3s, is rich with patterns, the 7s table is notable for its apparent dearth of patterns. The 10s table can be made in a fashion similar to the 6s. The 11s can be made using the 9s stencil turned around with two numbers blocked off. Last of all, I show the prime numbers and ask the students if they can figure out what this group of numbers is.
Other combinations can be tried. Which two stencils together cover the hundred chart and leave only the 15s tables exposed? What is the last number left exposed if the stencils are put on top of one another in order -- 2s, 3s, 4s, and so on?
To reinforce the overhead-projector activity or to introduce a table that is to be memorized, students can make their own charts of multiples by circling all the multiples of one number on a 10 x 10 hundred chart or by circling multiples of several numbers each in a different color.
Once children have done these activities, it is interesting for them to try the same activity on number charts with different dimensions. If, for example, one with a width of 12 instead of 10 is used, multiples of 3 fall in a vertical line, whereas multiples of 5 are on a diagonal (fig 2).
Sieve of Eratosthenes
On a hundred chart, students circle the 2 and then cross out all its multiples; using a different color each time, they repeat this step with 3 and then 5 (as a multiple of 2, 4 was crossed out earlier). Continuing in this way, students circle the prime numbers and cross out the composites. How high a factor do we need to test before we are sure that all composites are crossed out and only primes are left exposed? How high a factor if the chart is continued to 200?
To provide practice with the divisibility rules, I make some cards, about 5 cm x 15 cm, and along the bottom narrow edge of each, I write a three- or four-digit number that has one or more of the factors the students have investigated (fig. 3(a)). In groups of four to six with one serving as the leader, students can play "slap factors." They decide on one factor for that round and follow the form of slapjack. The leader turns over a card in a deck placed in the middle of the group, and, if the number on the card has the chosen factor, the first student to slap the card with his or her hand gets it and any cards underneath. The round continues until all cards have been turned over by the leader. The child with the most cards wins the round.
Before taking a correctly slapped card, the student must "prove" that it is contained in that table by explaining how it passes the divisibility-rule test or by naming the quotient of the number divided by the factor. A student who slaps a card that does not have the desired factor returns a captured card to the pile in the middle or sits out while the next three cards are played.
A deck of forty cards works well. Those pictured in figure 3(b) are multiples of at least some of the following numbers: 2, 3, 4, 5, 6, 9, 10.
After some work with fractions, students can play the same game, except that they use cards with fractions having three- or four-digit numerators and denominators (fig. 3(c)). Those fractions that can be "reduced" by the factor are slapped. Some possibilities are shown in figure 3(b).
Posters: Ending rules or sum-of-digit rules?
Groups of students can make posters to show which tables have ending rules and which have sum-of-digits rules: one that displays the 2s, 5s, and 10s tables and another that displays the 3s and 9s tables. These posters can be displayed in a public area with questions underneath for other students to think about (figs. 4 and 5). It is important to stress differences between ending rules and sum-of-digits rules. Students tend to generalize their early learning and conclude, for example, that 253 is in the 3s table because "it ends with a 3."
As a class activity children count aloud, reciting multiples of 3, for instance. Each child in turn names one number while the teacher records the list on the chalkboard, writing each number as it is spoken. Related tables -- 3s, 6s, and 9s, for instance -- can be recited and recorded in succession so that patterns can be compared. This activity uses both oral and visual patterns to reinforce students' learning of the tables.
In board games where a spinner or die determines the size of moves, remainders can be used instead. Division problems, with divisors chosen from the factors studied, are written on cards. Students draw a card from the top of a stack and move the number of spaces equal to the remainder (fig. 6). To prepare students for this game, give them numbers that are not divisible by a certain number and ask them how much must be subtracted to get the next lowest number that is divisible by that number. For example, we know that 349 is not divisible by 3 because the sum of its digits is 16, but subtracting 1 results in a digit sum of 15, so we know 348 is divisible by 3. If 349 is divided by 3, the remainder is 1.
Other number bases
Able students enjoy figuring out divisibility rules for numbers in bases other than base ten. The base number and its factors have ending rules. The number that is one less than the base number and the factors of that number have sum-of-digits rules.
Students with intermediate BASIC programming skills can write programs related to these topics. For example, here are programs some of my seventh- and eighth-grade students have written. The first program asks for an input number and lists all factor pairs:
10 REM ALL FACTOR PAIRS
20 INPUT "PICK A NUMBER AND THE COMPUTER WILL LIST ITS PAIRS OF FACTORS. ":X
30 FOR Y = 1 TO SQR (X)
40 IF X / Y = INT (X / Y) THEN PRINT Y "*" X / Y
50 NEXT Y
The following program lists all prime numbers up to 200:
2 PRINT "PRIME NUMBERS < 200"
5 PRINT" "2" ":
10 FOR X = 2 TO 200
20 FOR Y = 2 TO SQR (X)
40 IF X / Y = INT (X / Y) THEN GOTO 70
50 NEXT Y
60 PRINT X" ":
70 NEXT X
Marion Walter introduced me to some of these activities in a mathematics methods course in the sixties. Like our students, we were really examining these patterns for the first time. Although we were perceptive then, we certainly did not see all the patterns. It has been exciting to learn more each year from my students. A sixth grader recently led me to see for the first time that in the 3s chart, the fourth right-to-left diagonal is a continuation of the first, whereas the fifth is a continuation of the second. Once the student pointed out this fact, another saw that the 12s table appeared as a combination of the second and sixth left-to-right diagonals.
These activities encourage students to think both inductively and deductively -- to see number patterns and to recognize which numbers fit those patterns.
The Schools Council. Mathematics in Primary Schools: Curriculum Bulletin No. 1. 3d ed. London: Her Majesty's Stationery Office, 1969.
Walter, Marion I., and Stephen I. Brown. The Art of Problem Posing. Philadelphia: Franklin Institute Press, 1983.
A middle school mathematics teacher, Cornelia Tierney is presently a doctoral student at the Harvard University School of Education, Cambridge, MA 02138, and a consultant on learning disabilities and mathematics teaching.
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