The Grammar of Arithmetic:
 Addition and Subtraction Facts up to 20. That is, the students should be able to add any two numbers, each of which is less than 20, and the same for subtraction.
 Multiplication Tables from 0 to 15. Students should be able to compute any numbers in the following table instantly, correctly, without thinking about it. Students should be able to count by x up to x · 15. Students should be able to compute 6 · 7 = 42 from memory, as well as 7 · 6 = 42 from memory. They should not be using the Commutative Property to compute either of these.
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 

0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
2 
0 
2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
26 
28 
30 
3 
0 
3 
6 
9 
12 
15 
18 
21 
24 
27 
30 
33 
36 
39 
42 
45 
4 
0 
4 
8 
12 
16 
20 
24 
28 
32 
36 
40 
44 
48 
52 
56 
60 
5 
0 
5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 
65 
70 
75 
6 
0 
6 
12 
18 
24 
30 
36 
42 
48 
54 
60 
66 
72 
78 
84 
90 
7 
0 
7 
14 
21 
28 
35 
42 
49 
56 
63 
70 
77 
84 
91 
98 
105 
8 
0 
8 
16 
24 
32 
40 
48 
56 
64 
72 
80 
88 
96 
104 
112 
120 
9 
0 
9 
18 
27 
36 
45 
54 
63 
72 
81 
90 
99 
108 
117 
126 
135 
10 
0 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
110 
120 
130 
140 
150 
11 
0 
11 
22 
33 
44 
55 
66 
77 
88 
99 
110 
121 
132 
143 
154 
165 
12 
0 
12 
24 
36 
48 
60 
72 
84 
96 
108 
120 
132 
144 
156 
168 
180 
13 
0 
13 
26 
39 
52 
65 
78 
91 
104 
117 
130 
143 
156 
169 
182 
195 
14 
0 
14 
28 
42 
56 
70 
84 
98 
112 
126 
140 
154 
168 
182 
196 
210 
15 
0 
15 
30 
45 
60 
75 
90 
105 
120 
135 
150 
165 
180 
195 
210 
225 
 Perfect Squares up to 15. That is, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
 A Prime Number is a number divisible only by two distinct numbers: 1 and itself. (Hence, 1 is not aprime number.)
 Prime Numbers up to 31. That is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
 Divisibility Rules for 2, 3, 5, 9. Even numbers are divisible by 2, numbers are divisible by 3 if their digit sum is divisible by 3, numbers are divisible by 5 if the last digit is 0 or 5, and numbers are divisible by 9 if their digit sum is divisible by 9.
 Order of Operations is PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
 Trachtenberg Rule for Multiplying by 11: Add the neighbor.
 Trachtenberg Rule for Multiplying by 12: Double the number and add the neighbor.
 Trachtenberg Rule for Multiplying by 6: Add 5 to the number if the number is odd; add nothing if it is even. Add “half” the neighbor.
 Trachtenberg Rule for Multiplying by 7: Double the number and add “half” the neighbor; add 5 if the number is odd.
 Trachtenberg Rule for Multiplying by 5: “Half” the neighbor, plus 5 if the number is odd.
 Trachtenberg Rule for Multiplying by 9:a. Subtract the righthand figure of the long number from ten. This gives the righthand figure of the answer.b. Taking each of the following figures in turn, up to the last one, subtract it from nine and add the neighbor.c. At the last step, when you are under the zero in front of the long number, subtract one from the neighbor and use that as the lefthand figure of the answer.
 Trachtenberg Rule for Multiplying by 8:a. First figure: subtract from ten and double.
b. Middle figures: subtract from nine and double what you get, then add the neighbor. c. Lefthand figure: subtract two from the lefthand figure of the long number.  Trachtenberg Rule for Multiplying by 4:
 Subtract the righthand digit of the given number from ten, and add five if that digit is odd.
 Subtract each digit of the given number in turn from nine, add five if the digit is odd, and add half the neighbor.
 Under the zero in front of the given number, write half the neighbor of this zero, less one.
 Trachtenberg Rule for Multiplying by 3:
 First figure: subtract from ten and double. Add five if the number is odd.
 Middle figures: subtract the number from nine and double what you get, then add half the neighbor. Add five if the number is odd.
 Lefthand figure: divide the lefthand figure of the long number in half; then subtract two.