Arithmetic

The Grammar of Arithmetic:

  1. 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.
  2. 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

  1. Perfect Squares up to 15. That is, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
  2. A Prime Number is a number divisible only by two distinct numbers: 1 and itself. (Hence, 1 is not aprime number.)
  3. Prime Numbers up to 31. That is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31.
  4. 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.
  5. Order of Operations is PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
  6. Trachtenberg Rule for Multiplying by 11: Add the neighbor.
  7. Trachtenberg Rule for Multiplying by 12: Double the number and add the neighbor.
  8. 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.
  9. Trachtenberg Rule for Multiplying by 7: Double the number and add “half” the neighbor; add 5 if the number is odd.
  10. Trachtenberg Rule for Multiplying by 5: “Half” the neighbor, plus 5 if the number is odd.
  11. 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.
  1. 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.
  2. Trachtenberg Rule for Multiplying by 4:
    1. Subtract the right-hand digit of the given number from ten, and add five if that digit is odd.
    2. Subtract each digit of the given number in turn from nine, add five if the digit is odd, and add half the neighbor.
    3. Under the zero in front of the given number, write half the neighbor of this zero, less one.
  3. Trachtenberg Rule for Multiplying by 3:
    1. First figure: subtract from ten and double. Add five if the number is odd.
    2. Middle figures: subtract the number from nine and double what you get, then add half the neighbor. Add five if the number is odd.
    3. 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

Authority

  • Are there personal testimonies?
  • What maxims or ancient wisdom applies?
  • What is assumed or supposed?
  • By what powers will you reason?
  • On what type of authority does the argument depend?
  • What law or rule applies to ____?
  • How recent are these statistics?  How was the data gathered?
  • Should we trust majority opinion on ____?
  • Is this universally true, or are there counter-examples? (Elenchus)
  • Who is a witness? ________
    • What is the testimony of the witness?
    • What did he see (event or character) to cause him to give this testimony?
    • Why does the witness think that?
    • If more than one witness, questions would be answered for each witness

Three Laws of Reason

A deductive or geometric certainty, also known as a demonstration, can be considered certainly true or certainly false, and therefore authoritative.  In this regard, Formal Logic provides authority.  This is the study of the Formal Logic of deduction.

Mathematical Certainty –

Number abstracted from things, and the concepts of quantity provide certainties and systems of certainty.  The grammar of multiplication tables or factors are complimented by the logic and laws applying to real and imaginary numbers.  Mathematics is traditionally assigned to the Quadrivium.

Testimony

The Axiom, or common knowledge is the strongest of the intuitive proofs from testimony.   Axioms and Proverbs provide the grammar of common authority.

The authority, or ethos of the testifier determines the force of persuasion in this type of proof.

Probability -“There are three kinds of lies: lies, damned lies, and statistics.” – Mark Twain — or — Benjamin Disraeli

Inductive probability is more authoritative when it is more probable, and correspondingly less authoritative when it is less probable.  Complete induction, and therefore certainty, is impossible.  This probability offered as an argument is called a proof.  The Formal Logic of Induction is studied in Statistics.  Francis Bacon originates the rigorous focus on this Logic with the Novum Organum ( a reference to Aristotle’s Organon – a staple of  Formal Logic)

Mill’s Five Canons of the General Methods of Science

These laws apply to the science of observation and experiment for the testing of hypothesis.

Laws 

Authority of possibility is governed by physical laws, and the laws of human conscience and prudence are authoritative concerning human choice, or ethics.