The founder of the ‘The Lyceum’ and tutor of Alexander the Great. Aristotle differs with his teacher Plato concerning the reality and comprehensibility of the ‘Ideal’. Radical idealism, the notion that the universal concepts of things are of greater force than the perception of the particular things themselves, is countered with a tempered view that the particulars are necessary to our knowledge of the ‘Ideal’. It may even be said that the real ideas arise from the entelechy of things themselves. (see the four causes.)

**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.

- 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 right-hand figure of the long number from ten. This gives the right-hand 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 left-hand 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. Left-hand figure: subtract two from the left-hand figure of the long number.
- Trachtenberg Rule for Multiplying by 4:
- Subtract the right-hand 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.
- Left-hand figure: divide the left-hand figure of the long number in half; then subtract two.

*Thanks to Dr. Adrian Keister for this list