Below is a system to help students learn their multiplication facts by using their knowledge of doubling and addition. The Basics+ MU86 workbook is designed to help students learn the ten steps of this system, which is based on eight simple rules and six multiplication facts (3´3=9, 4´4=16, 6´6=36, 7´7=49, 8´8=64, 3´7=21). This system assumes that the student is not required to memorize the multiplication facts using the 8/6 system, but it is necessary eventually to memorize these facts. The exercise in the Basics+ books is designed to help the students memorize the tables; and it also helps students think mathematically. In addition, oral practice with flash cards is highly recommended.
Students must thoroughly know the following key facts to be able to use this system.
1. Know how to double numbers quickly and accurately.
2. Be able to perform basic addition quickly and accurately. Do the following steps using the Basics+ books specified. Some books will be out of sequence so some review exercises will be skipped until a more appropriate time. Use the flash cards to help with learning, sorting out the appropriate cards to use with each lesson. As each product is learned, mark the Multiplication Mastery Chart to record the student’s progress.
Ten Steps to Multiplication
Step 1: Rule 1—Any number times 1 is the number (1´1=1, 1´2=2, 1´3=3, etc.). Do the multiplication of 1’s in book MU1.
Step 2: Rule 2—Doubling is adding a number to itself or any number times 2 is the number doubled (2´1=2, 1+1=2; 2´2=4, 2+2=4; 2´3=6, 3+3=6, etc.). Do the multiplication of 2’s in book MU1.
Step 3: Rule 3—Any number times 10 is the number with a zero added to the ones column (10´1=10, 10´2=20, 10´3=30, etc.). Do the multiplication of 10’s section of book 9. Any number times 0 is 0 (0´1=0, 0´2=0, etc.). Do the multiplication of 0’s section in book MU9.
Step 4: Rule 4—The result of a number times 9 is obtained by reducing the number by 1; and that number is the tens. The units digit is the number which when added to the tens digit gives a total of 9 (9´8=72, 8 reduced by one is 7 and 7+2=9 so the answer is 72, 9´7=63, 7 reduced by one is 6 and 6+3=9 so the answer is 63). Do book MU8 except any review problems not covered yet.
Step 5: Rule 5—Any number times 5 gives either a 0 or 5 in the ones column: odd numbers result in 5s and even numbers in 0s. To multiply even numbers, halve the multiple and this gives the tens (5´6=30, 6 halved is 3 and 6 is an even number so the ones column is zero). For odd numbers, reduce the odd number by one and halve the result to obtain the tens digit and use 5 for the ones (7´5=35, 7 reduced by one is 6, halved is 3 so the answer is 35). Do lessons 1 and 2 in book MU86 for practice in halving and lessons 3 and 4 for practice identifying odd and even numbers. Then book MU4 except any review problems not covered yet.
Step 6: Rule 6—Any number times 4 is the result in the two’s table doubled (4´6=24 which is 2´6=12 doubled or 24; 4´8=32 which is 2´8=16 doubled or 32). If the two’s answer is not known, then the number times 4 may be doubled-doubled (4´6=24, 6 double 12 doubled is 24). To be efficient the student needs to have learned the twos by now. Do book MU3.
Step 7: Five new products—Memorize the multiplication pairs which have the same multiples, called squaring: 1´1=1*, 2´2=4*, 3´3=9, 4´4=16, 5´5=25*, 6´6=36, 7´7=49, 8´8=64, 9´9=81*, 10´10=100* (asterisk denotes facts already learned but reviewed here). Do lessons 5, 6, 7 and Review 1 in book MU86.
Step 8: Rule 7—Adjacent numbers: When two numbers are multiplied which when counting are adjacent, square the smaller number and add the smaller number to the result: 3´4=9+3=12 (3 squared is 9 and 3 added to 9 is 12); 7´8=49+7=56 (7 squared is 49 and 7 added to 49 is 56); 5´6=25+5=30 (5 squared is 25 and 5 added to 25 is 30); etc. Do lessons 8 through 11 and Review 2 in book MU86.
Step 9: Rule 8—Numbers separated by one: When two numbers are multiplied which when counting are separated by one, square the number between them and reduce it by one: 3´5=15 (between 3 and 5 is 4, squared is 16 and 16 reduced by 1 is 15); 6´8=48 (between 6 and 8 is 7, squared is 49 and 49 reduced by 1 is 48); etc. Do lessons 12 through 14 in book MU86.
Step 10: All the combinations have been covered except three. Two can be reasoned or all three can be memorized. The answer which must be memorized is 3´7=21. The other two combinations are 3´6=18 and 3´8=24. The product of 3´6 is the 3’s squared (3´3) and the answer doubled (3´3=9 and 9 doubled is 18); 3´8 is the 4´3=12 answer doubled (4´3=12 and 12 doubled is 24). Do lessons 15 & 16 in book MU86 for review. Do books MU2, MU5, MU6, MU7 including the review exercises, and complete the review problems skipped in other books.
The complete tables should be mastered now, and continued practice will improve the recall of these facts. Continue to practice with the flash cards. Complete book MU9 and do book MU10 which is the review of multiplication.