Below is a system to help students learn their addition facts by using their knowledge of counting, both forward and backward, and by needing to know only 7 simple rules which are learned by use, and 6 memorized sums (3+3=6, 4+4=8, 5+5=10, 6+6=12, 7+7=14, 3+7=10).

Students must thoroughly know the three key facts below to be able to use this system.

1. Count swiftly and easily to at least 20 and know these terms: units (ones) and tens, what they mean, and how they are used in writing our numbers.

2. Count backwards from 20: 20, 19, 18, etc.

3. Given a number between 1 and 20, the student should be able to quickly give the number before and after; i.e., given 6 they should give 5 and 7. They will also use two numbers before and two after; i.e., given 6 then two before is 4 and two after is 8. The two before and after numbers do not have to be memorized as they can be found by repeating one before or one after, i.e., given 6, 5 is before 6 and 4 is before 5; given 6, 7 is after 6 and 8 is after 7. Exercises with flash cards containing the numbers 1 through 10 are useful to firmly establish this knowledge.

Do not start the ten steps until the students thoroughly know the information required above. As noted below, the order of the Basics+ addition books will need to be changed for this system. Oral practice with flash cards is highly recommended and sums; mastered should be marked on the Addition Mastery Chart.

Step 1: Rule 1Addition of 1s: The answer is the number after the number added to one: 1+1=2, 1+2=3, 1+3=4, 1+4=5, etc. Practice in book AD1.

Step 2: Rule 2Addition of 2s: The answer increases by one the answer obtained when one is added to that number: 2+2=3+1=4 (2+1=3 and the number after 3 is 4); 2+3=4+1=5 (1+3=4 and the number after 4 is 5), etc. It may be presented as adding one and to that result adding one more. Use book AD2 for practice.

Step 3: Rule 3—Addition of 10s and 0s: For 10s the answer is obtained by adding a 1 to the tens column of the number being added to 10, except 10+10=20 which is the first sum to be memorized: 1+10=11, 2+10=12, 3+10=13, etc. For 0s the answer is the number added to the 0: 0+1=1, 0+2=2, 0+3=3, etc. Use book AD10 for practice but skip the review sections.

Step 4: Rule 4—Addition of 9s: The answer is obtained by using the number before the one you are adding to 9 and placing a one in the tens column: 9+3=12 (2 is before 3 and 1 in the tens column gives 12); 9+4=13 (3 is before 4 and 1 in the tens column gives 13); etc. Use book AD9 for practice but skip the review sections.

Do the next steps through 10 using book AD76 for practice and then at the completion of AD76, start at book AD3 and continue in sequence through the program completing the remaining books. The review exercises skipped should be done in the sequence of the regular program.

Step 5: Five new sums—Memorize the addition pairs which have the same addends, called doubling: 1+1=2*, 2+2=4*, 3+3=6, 4+4=8, 5+5=10, 6+6=12, 7+7=14, 8+8=16*, 9+9=18*, 10+10=20* (asterisk denotes ones already learned but reviewed here). Do lessons 1, 2, and 3 in book AD76.

Step 6: Rule 5—Adjacent numbers: When adding two numbers which when counting are adjacent, double the smaller number and use the next larger number following the result: 3+4=6+1=7 (3 doubled is 6 and the next larger number is 7); 4+5=8+1=9 (4 doubled is 8 and the next larger number is 9); 5+6=10+1=11 (5 doubled is 10 and the next larger number is 11); etc. Do lessons 4 through 7 and Review 1 in book AD76.

Step 7: Rule 6—Numbers separated by one: When adding two numbers which when counting are separated by one, use the number between them and double it: 3+5=8 (between 3 and 5 is 4, doubled is 8); 5+7=12 (between 5 and 7 is 6, doubled is 12); 6+8=14; etc. Do lessons 8 through 10 in book AD76.

Step 8: One new sum: Practice these addition pairs which add to 10 because they are very useful when doing long addition: 1+9=10*, 2+8=10*, 3+7=10, 4+6=10*, 5+5=10* (asterisk denotes combinations already learned but reviewed here). Practice is in book AD76, lessons 11 through 13, which are assigned at step 10.

Step 9: Three sums: These sums use the sum of 10’s. The following: 6+3=9, 8+3=11, and 7+4=11. Because it is known that 6+4=10, 6+3 is easy to determine as one less than 10 or 9. Because it is known that 8+2=10, 8+3 is easy to determine as one more than 10 or 11. Because it is known that 7+3=10, 7+4 is easy to determine as one more than 10 or 11. Do lessons 11 through 13 and Review 2 in book AD76.

Step 10: Last two sums: The sums 8+4 and 8+5 may be handled in several ways. The system for nines could be modified by counting backward two. So with 8+4 one would count back from 4 two digits to 2 and add ten to get 12. With 8+5 one would count back from 5 two digits to 3 and add ten to get 13.

Continue to practice, practice, practice with the goal of having all the addition tables committed to memory so that calculations are automatic. Do lessons 14 through 16 and Review 3 in book AD76 and then go to book AD3.