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

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5

6

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8

9

10

11

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13

14

15

2

0

2

4

6

8

10

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28

30

3

0

3

6

9

12

15

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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

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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.

 

Circumstance

The association of the student and his world

 

  • At the same time as the issue; where are they? who else is there? what is happening?
  • At the same time in another place; where are they?  who was there?  what is happening?
  • What is or was possible?
  • What is or was probable?
  • Power, will and opportunity
  • Who is interested?
  • What is the best choice?

The concerns of Grammar are people, places, things and times- the study of events.

The Logic of circumstance is concerned with causal relationships between events and ideas in the history of the world.

Rhetoric is the self aware inclusion of the student in circumstance, a relationship with the past that informs the present and indicates the future.

“The past is the present unrolled for inspection, the present is the past gathered for action.” -Will Durant

Omnibus

History:

Comparison

 

  • How is X the same as something/someone else
    • What makes it different from what it is similar to? (see the predicables)
    • What does it not share with the rest of its genus?
    • What was it in the past?
    • What might it be in the future?
    • What is its opposite in kind?
    • What is it analogous to?
    • Similitude:  The current situation is like one that we’ve seen before, i.e., the search for resemblances.
  • How is an X different from something/someone else
  • To what degree?
    • How does it compare to its normal version?
    • How does it differ from things that resemble it?
    • How is it different from precedents and parallel versions?
    • What is its range of variation?
    • Is its opposite or contrary better or worse
    • Is X better worse than Y?  more/less
    • How can it be evaluated?
    • What is the standard of evaluation?

Grammar includes the metaphor and simile.  Judgements of quality relate to beauty, goodness and truth.  Art is imitation.

Logic of comparison deals with the concrete and poesy, compared to the analytic mode of reason that subtracts from essence.   “my love is like a red red rose” “God is Love”.  Poetic reasoning is more compact and comprehensive than analytic.

Rhetoric of comparison is a formational discipline.  The ethical consideration determines what is best regarding choice.  The aesthetic consideration determines what is best regarding desirability.  Classical Christian Education is emphatically committed to the ‘best’.

 

Definition

Aliquis definiverit vinciet – the act of apprehension whereby a term is commonly understood

The first of the 5 Common Topics –   Aliquis definit vincit.

Associated with the first act of reason, or simple apprehension.   A perception proceeds to form a thought, which in turn is abstracted into an idea, and finally denoted by a term.  Terms are the subjects of definition — De-finitio – a ‘bounded’ idea.

  • Who or What is X?
  • What kind of thing is X?
    • Formal or logical cause  (Comprehension, Connotation)
    • Genus and Species
  • What is X not?
  • What are the parts of X?
  • To What group does X belong?
  • What examples of the thing exist?  (Extension, Denotation)
  • Etymological definitions, the history of the word

Grammar includes vocabulary and the practical and theoretical sciences.

Material Logic sets forth the art of definition, division, categories, predication and causes.

 

Iron Age

  • Early Iron Age
    • 1300-500 BC
    • Trojan War – Homer
    • Philistine invasion – Samuel and Kings
    • Fall of Israel/Judah
    • Roman Kingdom – Livy
  • Middle Iron Age
    • 500-250 AD
    • Greco/Persian War – Herodotus
    • Peloponnesian War – Thucydides
    • Roman Republic/Empire – Civil Wars – Livy, Suetonius, Tacitus
    • Return of Judah, destruction of the temple, diaspora – Esther, Nehemiah, Josephus
  • Late Iron Age
    • 250-500 AD
    • Western/Eastern Roman Empire
    • Fall of Rome
    • Gothic Invasions
    • Parthian Empire

Mathematics

Mathematics is the art of recognizing numerical patterns in God’s revelation. – Michaela Farrell. (In Lingua Latina: Mathe ̄matica est ars agnoscentis exempla ̄ra numero ̄rum in reve ̄la ̄tio ̄ne De ̄ı.)

  1. Arithmetic is that branch of mathematics dealing with the study of numbers directly, particularly the properties of the traditional operations between them – addition, subtraction, multiplication, and division. – Wikipedia.
  2.  Algebra is that branch of mathematics dealing with the manipulation of symbols to solve equations and inequalities.
  3. Statistics is that branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. – Wikipedia

 

Medieval

A time period used for the scheduling of Reading Lists and the understanding of historical grammar.  The period is broadly considered to be that time starting at the time of the Church, and lasting until the New Heaven and Earth.

In more common use, the period can be described as the time between St. Augustine and Martin Luther theologically, Aurelius and Renee Descartes epistemologically, between the fall of Rome and the fall of Constantinople politically.  The Medieval period occurs between the Ancient and Modern times.

The Medieval period is further divided into the early (400 – 1000 AD, high (1000 – 1200 AD, and late middle ages (1200 – 1500 AD) .