Whole Numbers and Primes

Phase 1 aims for the following: 

Show work format: This is required for evaluation of arithmetic and later algebraic expressions in a manner that aids and records the steps in that evaluation, so that a student and others may follow and check the steps. Requiring the format helps or forces students to respect the operation order or priority: BEDMAS. 

Remark: Students may have confidence in their mastery of arithmetic when they learn to do calculations carefully in a repeatable, reproducible and readable manner for the sake of verification or correction.  Further skill and confidence may come, when a student does not resist,  from explanation of why arithmetic methods for fractions and work. 

Remark: Imagine A = an arithmetic expression . The problem of evaluation A should have a solution of the form

A = the arithmetic expression
    = another arithmetic expression 
    = ... 
    = simplified results

Subexpressions should be replaced by their values in place, so that the written work shows a sequence of such replacements. Require the presence and vertical alignment of equal signs in the format.

I. Operations on Whole Numbers (44 steps)

Whole Numbers are used to count.  Counting principles imply operations on whole numbers. 

A. Physical Interpretation and Origins of Arithmetic Operations and the Decimal Representation of Whole Numbers (Extrinsic Development)

1. Counting and Tracking Objects - From tally marks and grouping to decimals
2. Addition and Subtraction - Counting Viewpoint
3. Link between sum and summands  If A = B + C then B = A - C and C = A - B
4. Repeated Addition and Multiplication - Counting equipollent groups.
5. Division by a large number by a smaller whole number, and forming groups whose size is given by the smaller whole number.  Illustrate with lengths that are integral multiples of another length. 
6. Counting Principles:  the total tally is independent of how objects are ordered and/or grouped for counting - groups must be disjoint to ensure no object is counted twice.
7. Optional: Different ways to count squares in rectangles formed by squares and Different ways to count cubes in boxes formed by cubes implies multiplication of whole numbers is commutative and associative. 
8. Optional: Different ways to count unit lengths in line segments partitioned into two or more subsegments (multiples of unit length, non-overlapping) implies addition of whole numbers is commutative and associative. 

B. Decimal Place Value

1. Reading one to three digit numbers aloud
2. North American and/or UK meaning for use of terms billion, trillion, quadrillion, quintillion, sextillion, septillion, 
3. Reading one to 30 digit numbers aloud in groups of 3 following North American use of terms million to septillion
4. (Alternative to 3): Reading one to 30 digit numbers aloud in groups of 3 following UK use of terms million to septillion

C. Arithmetic Operations Using Decimal Representation

1. Column (Place Value) Methods for Addition
   - prerequisite addition table - link to counting in terms of units, then groups of ten, groups of 100 and so on.
2. Multiplying powers of ten: 1, 10, 100 and 1000.  Try to explain why 10 one hundreds is the same as 100 tens. 
3. Column (Place Value) Methods for Multiplication
   - requires times table
4. Column (Place Value) Methods for Comparison
5. Column (Place Value) Methods for Subtraction of smaller from larger numbers:
   
   Method 1: Missing Addend (Carries but no borrows) - Simplest Method:|
   
   • No Carries
   • With Carries  

   Foregoing Link between sum and summands  If A = B + C then B = A - C and C = A - B
   
   Method 2: subtraction with conversions (borrows) - Standard Method

   • No Conversions
   • Conversion from next column - When is it possible
   • Conversion from multiple columns.  Include a progression of examples, for example:  (i) 43 -18 (ii)  456 - 268  (iii) 6432 - 3545  (iv) 643 -318 (vi)  8456 - 5268  (vii) 96432 - 43545  (viii) 436 -  ....
     where borrows are in lowest value places, then in middle locations, in highest value places; and where two or more borrows are needed in a single subtraction.
6. How many times does a smaller number go into a larger number - solution by dot representation of the divisor and dividend
7. Division by a single digit: Short Method
8. Division by one or two digit numbers: Long Division Method
9. Show or imply with examples that the number of unit squares in an A by B unit rectangle is A x B

D. Prime Numbers and Prime Number Decomposition

1. Define (explain) what is a Prime Number
2. Define (explain) what is a composite number
3. Explain why are whole numbers 2, 3, 5 and 7 prime?
4. Decimal Based Rules for recognizing multiples of 2,3 and 5
5. List all multiples of 7 less than 121 and identify which ones are not multiples of 2, 3 and 5
6. State and Use Theorem I: A whole number < 121  is composite if and only if is a multiple of the first four primes 2, 3, 5 and 7.
7. State and Use Theorem I': A whole number < 121  is prime if and only if is not divisible by each of the first four primes 2, 3, 5 and 7.
8. For each whole number < 121 recognize it as prime or give its prime number decomposition.  Use a tree for this. 
9. Use the List Method to introduce and find the least common multiple LCM of a pair of whole numbers
10. Use prime decomposition to calculate the least common multiple of a pair or triplet of whole numbers starting from the whole numbers themselves or from a prime number decomposition - latter may be given for numbers > 121.
11. Use prime decomposition and a tree diagram to generate all divisors and factor pairs of whole numbers < 121
12. Use prime decomposition to calculate the genera common multiple of a pair or triplet of whole numbers starting from the whole numbers themselves or from a prime number decomposition - latter may be given for numbers > 121.
13. Use calculators to calculate square and cube roots. 
14. Express the square root of m as p times the square root of a whole number q where q = 1 or the q prime decomposition given by the product of primes to the first  power. Here sqrt (m) =  p sqrt(q) where q = 1 or q has  prime decomposition given by the product of primes to the first power.  Each whole number can be expressed as a perfect square times another whole whose prime decomposition contains no duplicate primes.
15. Express the cube root of m as p times the cube root of a whole number q where q = 1 or the q prime decomposition given by the product of primes to the first or second power. Each whole number can be expressed as a perfect cube times another whole whose prime decomposition contains no prime to the cube or higher power. 

E.  Working with Signed Whole Numbers

1. Show how to add and subtract Collinear Vectors (displacements) along a straight line.   Show why addition commutes. 
2. Define additive inverse.  
3. For collinear vectors, show how subtraction of a vector gives the same result as adding its additive inverse.
4. Show how signed numbers may multiply displacement vectors - act as multipliers.  Show how -1 times a vector gives its additive inverse.
5. Show how to add and subtract multiples of a single vector and how that leads to another multiple  - Extrinsic motivation for definition of sum of signed numbers - saying how add them the multipliers defines the sum.
6. Show how to subtract multiples of a single vector and how that leads to another multiple - Extrinsic motivation for definition of difference of signed numbers -  saying how calculate the difference (subtract) defines the operation. 
7. Show how to divide a longer vector by a shorter collinear vector. 
8. Show how to divide a multiple of a vector by a shorter multiple of the vector.  Latter gives extrinsic viewpoint of division of integers.

II. Operations on Fractions (29 steps)

A. Whole Numbers and Fractions are used to Measure

Aim: Using Fractions to describe quantities and show how physical operations on discrete sets and continuous quantities determine arithmetic operations on fractions.

1. Unit Fractions of 12, 20 and 60:  Dependent on Divisors
2. Unit Fractions of lengths and areas - Division into parts identical in value.
3. Simple Fractions  as Whole Number Multiples of Unit Fractions - Counting Unit Fractions
4. Fraction of Fractions - Discrete and Continuous Cases
5. Why a fraction of a fraction is a fraction - product of fractions
6. Fractions with the same value: Equivalent Fractions - Discrete and Continuous Examples
7. Generation of Equivalent Fractions by Raising and Lowering Terms
8. Simplifying Fractions by Lowering Terms - Efficient Use of common factors and prime number decomposition of numerators and denominators
9. Product of Fractions - Simplification and Efficient Ways to Simplify
10. Show or imply with examples that the number of unit squares in an A/M by B/N unit rectangle is A x B / M x N or AB  times MN-ths.   First use a unit square to show the area of a 1/3 times 1/4 unit subrectangle is 1/(3 x 4) of a unit square. Then show a 7/3 times 5/4 unit region is  7 x 5 of those subrectangles 1/(3 x4)  of a unit square in measure. That and few more examples may suggest and/or confirm the general pattern.
11. Measurement with Whole Numbers and Fractions
12. How many times does one length or area go into another - Physical Viewpoint without and with fractions
13. Reciprocals - how many times does a fraction go into one?
14. How many times does one length or area go into another - Calculation method - describing with fractions and operations on fractions. 
15. Division - how many times does one fraction go into another? Saying how to compute the answer defines division. 
16. Comparison of pairs and triplets of lengths  - Physical and measurement with fraction viewpoint - raising terms to get a common denominator.  How to use the Least Common Denominator (LCM) as a least common denominator. 
17. Adding and Subtracting Simple Fractions with like denominators gives a fraction
18. Addition and Subtraction Lengths - Physical Viewpoint
19. Addition and Subtraction of Lengths - Calculation Method - describing with fractions and operations on fractions.  Like Denominator Case - Counting Unit Fractions
20. Addition and Subtraction of Lengths - Calculation Method - describing with fractions and operations on fractions.  Unlike Denominator Case - raising terms to get a common denominator. 
21. Efficient ways to Add and Subtract Fractions - Why Least Common Denominator is Preferred. Why use of least common denominator usually gives least amount of figuring. There are exceptions.
22. A whole numbers plus a proper fraction is said to be a mixed number.  Show how Improper Fractions have the same value as a Whole Numbers Plus a Proper Fractions, and vice versa.
23. Compound Fractions - Alternate Notation for Division.
24. Format for evaluating arithmetic expressions and showing work - proving mastery.

B.  Working with Signed Fractions

1. Show how signed Fractions, proper or improper, may multiply displacement vectors - act as multipliers. 
2. Show how to add and subtract multiples of a single vector and how that leads to another multiple  - Extrinsic motivation for definition of sum of signed fractions - saying how add them the multipliers defines the sum.
3. Show how to subtract multiples of a single vector and how that leads to another multiple - Extrinsic motivation for definition of difference of signed fractions -  saying how calculate the difference (subtract) defines the operation. 
4. Show how to divide a longer vector by a shorter collinear vector. 
5. Show how to divide a multiple of a vector by a shorter multiple of the vector.  Latter gives extrinsic viewpoint of division of fractions.

(26 steps)

III.  Decimal Representation of Fractions  

1. A Fraction is decimal if and only if its denominator equal to a power of ten.  Student should be able to identify decimal and non-decimal fractions.
2. Show if the denominator of a fraction is a power of 2 ( 2, 4, 8, 16, 32, etc) then it is equivalent to a decimal fraction
3. Show if the denominator of a fraction is a power of 5 ( 5, 25, 125, 625 etc) then it is equivalent to a decimal fraction
4. Show if the denominator of a fraction equals 2^m5ⁿ then it is equivalent to a decimal fraction where the denominator is a  2^m-n times 10ⁿ if m > n and the denominator is a power of   5^m-n times 10^m if m < n.  If m = n, the denominator   2^m5ⁿ  =  10ⁿ .
5. Show simplification of a proper decimal fraction leads to denominator with prime decomposition given by a product 2^m5ⁿ of twos and fives, and no other factors.

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6. Introduce Decimal Place Value Representation of Proper and Improper Decimal Fractions. 
7. Decimal Place Value Representation of Improper Decimal Fractions and their equivalence to a whole number plus a decimal fraction.
8. Column Methods for Addition of Proper and Improper Decimal Fractions
9. Comparison of Proper Decimal Fraction - raising terms if need-be, to common denominator
10. Comparison of Improper Decimal Fraction - raising terms if need-be, to common denominator
11. Lexicographic Method for Comparison of Numbers
12. Column Methods for Subtraction of a smaller Decimal Fractions from a larger may based on the column methods for subtraction of whole numbers via conversion or a missing addend approach.
13. Checking Results:  Verification modulo 9.  Look up Rules of 9. Odds of an error in check.

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14. Each mixed decimal number A can be expressed as fraction in which the numerator is N is not a multiple of 10, and the denominator D = 10^k.  Show how the decimal representation of A equals the decimal representation of N with the decimal point k places from the left.  Examples where N has k, fewer than k and more than k are needed to illustrate this relationship. The decimal representation of A has a nonzero digit in the k-th place after the decimal point.
15. Introduce or justify column methods for multiplication of decimals fractions with the aid of the previous item. Consider or illustrate the cases where both factors have non-zero digits after the decimal point, where one has non-zero digits after the decimal point and the other ends in zeroes before an explicit or implicit decimal point, and where both ends in zeroes before an explicit or implicit decimal points.  In essence, the product of a pair of whole numbers, the numerators, is calculated using decimal methods for obtaining products of whole numbers, and then the powers of ten in the "denominators" are added to determined the location of a decimal point.
16. The short and long division methods may be modified to give a place-value based, computational method for dividing one decimal by a whole number.  Answers can be obtained in the form of a whole number plus remainder, or in the form of terminating or periodic, non-terminating decimal expansion.
17. The short and long division methods may be further modified to give a place-value based, computational method for dividing one decimal by another decimal which has k decimals after the the decimal point and a nonzero digit in the k-th position, by shifting the decimal points k spaces in both dividend and divisor, so that the divisor  at least becomes a whole number.  This shift is justified by writing a compound fraction with the dividend in decimal fraction form A/10^k in the  denominator and the divisor in decimal fraction B/10^m  form as numerator, and then applying the rules for evaluation of a compound fraction to obtain  [A/10^k][10^m/B] and then to obtain the decimal representation of  the latter from the exact or approximate calculation of the decimal representation of A/B.  The last steps may require a knowledge or discussion of powers of ten and their properties.
18. Measurement and Decimals.  Rulers with unit lengths and fractions there-of may be employed to give or approximate the lengths of line segments.  In particular, lengths that are non-decimal may be approximated by rulers with decimal divisions (tenths, hundredths, thousandths) of a unit length. That implies the possibility of decimal approximation of all lengths and all fractional multiples of a unit within some resolution limit.  The latter in turn gives rise the notion of significant digits.
    
19. Each fraction may be written in the form A/B where A and B are whole numbers with at least one not being a multiple of ten. When B is a product of 2s and 5s, and no other primes, the fraction has a finite decimal expansion. The long division methods for dividing A by B may be continued until there is a zero remainder.  In all other cases, long division results in a periodic, non-terminating fraction where the period is less than B. That is as soon as the current remainder is less than B (Or more precisely the  current remainder times a power of 10 is a whole number less than B) then all further remainders, the next B remainders in particular,  will have that form. Since there are only B-1 instances of that form, that B-1 numbers from 1 to B-1, the Pigeon whole principle implies a at least one of the remainders must appear twice. Whence there will be a period and least possible period.  
    
20. Theorem: To each periodic non-terminating decimals is the decimal expansion of a fraction A/B where the denominator B is not a product of 2 and 5s. 
    
    One proof follows from identifying the periodic, non-terminating decimal with a geometric series and then calculating its limiting value A/B.  Another proof follows from defining arithmetic operations on decimals - the definition of addition, subtraction and multiplication by powers of ten suffice - and then doing a calculation.
    
21. Completeness: In geometry, we may form lengths that are mixed number multiples of a given length.  In geometry, we may calculate areas in terms of products of lengths and through that define products of mixed numbers.  But we cannot assume all lengths are mixed number multiples of a given length nor can we assume that all areas are mixed number multiples of a unit area.  That being said, geometrically, we may interpret an infinite decimal multiple of a unit length as the limit of a sequence of finite decimal multiples. If we suppose that infinite decimal expansion multiples of a length exist due to limit consideration, then that extends of the notion of what is a number.  Within that notion,  the Pythagorean theorem implies the hypotenuse of an isosceles right triangle should have a length the sqrt(2) times the length of a one of the legs of the triangle, but the properties of whole numbers implies that sqrt(2) is not a mixed number.  That being said, we can calculate a series of decimal approximation to it, an infinite decimal expansion, which we may regard as defining and representing sqrt(2) exactly, which we can use to approximate sqrt(2) when decimal results are required. THIS ITEM NEEDS TO BE REWRITTEN IN ALL OR PART.  Some error control analysis may be needed to show that the decimal expansion of the product is essentially unique (Modulo the 0.999 - 9 recurring phenomenon)
    
    Geometric Completeness Assumption: For each decimal, finite or infinite, there exists a line segment whose length relative to a unit length is given by that decimal.
    
    Remark:  Lengths given by square roots of whole numbers 2, 3, 4, 5, and so on, relative to a unit length, can be constructed with the aid of the successive applications of the Pythagorean theorem.  The square roots of prime numbers are not equal to fractions (are irrational) and so have infinite, non-terminating, non-repeating  decimal expansions. 
    
    Remark: The issue of infinite decimal expansions may be glossed over. For example, the sqrt(2) - the concept - may be introduced and characterized as the number on real number line whose square is two.  Then sqrt(2) may be found to 4 or more decimal places - the practice - with the aid of a table or calculator.   That provides a decimal approximation to sqrt(2),  one that may be improved.  We speak of sqrt(2) and its properties, decimal approximation included, describing what its properties should be, and providing students with an operational handle on the concept - the notion that there a sequence of decimals whose square can be made arbitrarily close to 2. The same sequence of decimals multiples of a unit length, the side of right isoceles triangle, further gives a sequence of physical approximations to the hypotenuse.  All  the foregoing are manifestations (shadows from a fire in a cave discussed by a greek philosopher __) of the concept of sqrt(2) - the manifestations limit and define the concept. 

B.  Working with Signed Decimals

1. Show (Suggest) how signed decimals, may multiply displacement vectors - act as multipliers. 
2. Show how to add and subtract multiples of a single vector and how that leads to another multiple  - Extrinsic motivation for definition of sum of signed decimals - saying how add them the multipliers defines the sum.
3. Show how to subtract multiples of a single vector and how that leads to another multiple - Extrinsic motivation for definition of difference of signed decimals -  saying how calculate the difference (subtract) defines the operation. 
4. Show how to divide a longer vector by a shorter collinear vector. 
5. ???? Show how to divide a multiple of a vector by a shorter multiple of the vector.  Latter gives extrinsic viewpoint of division of decimals ????

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science