For twelve to fourteen year olds

skills to check or develop

1. Dictation Exercise: Write  numbers between decimals when described in words. 
   
   The next two items provide two options for describing and expressing the decimal form of numbers in terms of  words.
   

2. Place Value Comprehension- USA and Modern British, Short Scale Option: To understand the place value in long multidigit decimals from 1 sextillionths to 999 sextillions,  we rewrite the numbers in a  mixed 3-digit and word format  in accordance with North American values for billions, trillions, quadrillions, quintillions and sextillions. Here
   
   1 thousand = 1000 ones = 10³ ,
   1 million = 1000 thousand = 100, 000 = 10⁶ ,
   1 billion = 1000 million = 100, 000, 000 = 10⁹ ,
   1 trillion = 1000 billion =  100,000,000 = 10¹² ,
   1 quadrillion = 1000 trillion =100,000, 000,000 = 10¹⁵ ,
   1 quintillion = 1000 quadrillion  =1000, 000,000, 000,000 = 10¹⁸ , 
   1 sextillion = 1000 quintillion  =1000, 000, 000,000, 000,000 = 10²¹ , 
   
   Give a lesson on powers of 10 before there presentation as above.. 
   
   In the physical sciences, students will meet Avogadro's number. The latter is approximately 6.02 × 10²³    = 602 × 10²¹ = 602 sextillion.  That provides one reason for helping students understand place value in the range  from 1 sextillionths to 999 sextillions, that is from the smallest value 21 places to right of the decimal point to the largest value 24 places to the left of the decimal point. Tutors and teachers with a sense of humor can show students how to read multidigit decimals in this range, not all at once, but gradually. 
   
   Digits Right of the Decimal Point:  In reading aloud, decimal places to the right of the decimal should be read in groups of three, with extra padding on the right as needed to make a full group of three:  422.345 678 893 23  would be read aloud or written in mixed word & decimal format as 422 ones, 345 thousandths, 678 millionths, 893 billions and 230 trillionths. Notice how 23 has become 230.  Here the number of ones, thousandths, millionths and on  is kept in decimal form when written, while being read aloud in expanded.  So 422 is written as shown but read aloud as four hundred and twenty-two.   

   Digits to Left of actual or implied Decimal Point: In the decimal 345,085,501,456,423 the place value of the leading three digits is not immediately obvious. The digits after it have to group into threes and counted (there are 12) or the place value has to be found in a backwards manner.  We do that next not necessarily for the sake of efficiency, but for the sake of student amusement in explaining place value. 

   A second Place Value Comprehension Example:  Write  345,085,501,456,423  backward first - least important  groups of 3 digits first - in the following mixed word & decimal format: 423 ones, 456 thousands, 501 millions, 85 billions and 345 trillions, Following place value determination, we write the latter  forwards and so obtain the following place value interpretation 345 trillions, 85 billions, 501 millions, 456 thousands and 423 ones. for 345,085,501,456,423.

   Digits on both sides:   When a decimal has a digit on both sides of a decimal point,  determine the place value of those on the left first using the backward and forward method above: 

   Example:  For the the decimal,  43,487, 044, 009.435 432 4 find the place value of the digits to the left of the decimal in groups of three and left-overs, 9 ones, 44 thousands, 487 millions and 43 billions - most important last. With the with place value of the leading part known,   43 billions, 487 millions, 44 thousands, 9 ones, 435 thousands, 432 millions and 400 trillionths.

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3. Place Value Comprehension- Traditional British, Long Scale Option:
   In the traditional British nomenclature, 
   
   1 thousand = 1000 ones = 10³ ,
   1 million = 1000 thousand = 100, 000 = 10⁶ ,
   1 Billion = 1000, 000  million = 100, 000, 000 = 10¹² ,
   1 Trillion = 1000, 000  Billion =  100,000,000 = 10¹⁸ ,
     
   with capitals used to distinguish the long scale values from the short scale ones. These terms could help with place value comprehension in the range 1 millionth to 999,999 Trillions. In the physical sciences, students will meet Avogadro's number. The latter is approximately 6.02 × 10²³    = 602,000 × 10¹⁸ = 602,000 Trillions.  
   
   Digits Right of the Decimal Point:  In reading aloud, decimal places to the right of the decimal should be read in groups of three, with extra padding on the right as needed to make a full group of three:  422.345 670   would be read aloud or written in mixed word & decimal format as 422 ones, 345 thousandths, 670 millionths or as 422 ones, 345670 millionths   Here the number of ones, thousandths (?)  and millionths  is kept in decimal form when written, 

   Digits to Left of actual or implied Decimal Point: In the decimal 3, 345,085,501,456,423 the place value of the leading  digits is not immediately obvious. The digits after it have to group into threes and counted (there are 12) or the place value has to be found in a backwards manner.  We do that next not necessarily for the sake of efficiency, but for the sake of student amusement in explaining place value. 

   A second Place Value Comprehension Example:  Write  3,345,085,501,456,423  backward first - least important  groups of 6 digits first - in the following mixed word & decimal format: 456, 423 ones; plus 850,501 millions, 3,345 Billions, Following place value determination, we  rewrite the latter  forwards and so obtain the following place value interpretation 3,345 billions,  850,501 millions, and 456, 423 ones. for3,345,085,501,456,423.

   Digits on both sides:   When a decimal has a digit on both sides of a decimal point,  determine the place value of those on the left first using the backward and forward method above: 

   Example:  For the the decimal, 44,789,043,487, 044, 009.435 43 find the place value of the digits to the left of the decimal in groups of three and left-overs,  44,009 ones plus 43, 487 millions and 44,789 Billions - most important last. With the with place value of the leading part known,   44,789 Billions, plus 43 487 millions plus 44,009 ones and 435,430 millionths

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   Remark - a cosmetic preference: Six digits at time may be too many for most students to grasp quickly. Three digits at a time appears easier. 
   

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   Mutliplication - Review

   Product \[ \frac34 \times \frac{20}{7} = 3 \times \mbox{ one quarter of } 20 \times \frac17 \] is easily found because a quarter of 20 is 5. Thus, writing $\times$ instead of the word "of" gives \[ \frac34 \times \frac{20}{7} = 3 \times 5 \times \frac17 = \frac{15}7 \] Or \[ \frac34 \times \frac{20}{7} = \frac34 \times \frac{5 \times 4}7 = \frac{3 \times 5}7 = \frac{15}7 \] The calculation can be done mechanically by cancelling the factor 4 common in the denominator of the first factor with the factor 4 in the numerator of the second factor. By raising terms, all products of fractions can be calculated mechanically. An example follows. \begin{eqnarray*} \frac58 \times \frac73 & =& \frac58 \times \frac{8 \times 7}{8 \times 3}\\ & =& 5\times \frac{ 7}{8 \times 3}\\ & =& \frac{5\times 7}{8 \times 3} = \frac{35}{24} = 1 + \frac9{24} \end{eqnarray*}

   Several examples like this will suggest the mechanical rule

   Multiply the numerators, multiply the denominators
   or
   Multiply the tops, multiply the bottoms

   for the calculation of products of two or more fractions. For two fractions, multiplication may be described using the shorthand role of letters, \[ \frac AB \times \frac CD = \frac {A \times C}{B \times D} \] In the second example above 5 would give the value of A, 8 would give the value of B, 7 would give the value of C, and 3 would give the value of D. The foregoing represents an algebraic description of the more general, multiply the tops, multiply the bottoms rule. Understanding the formula would be a small, optional step for student of this age level, in the introduction of algebra.

   The algebra shorthand description of provides a formula for multiplication. Besides the mastery of given formulas for areas and perimeters, the description can be cast as just another formula.

   Division with Like Denominators - Review

   The question of how many whole times 5 its goes into 18 its has the answer 3 whole times. The remainder is 18 its - 3 × 5 its = [18 -15] its = 3 its. The remainder is 3 fifths of 5 its.

   The related question question of how many times 5 its goes into 18 its has the mixed number answer: 2 and 3 fifth whole times. Here \[ 3 + \frac 35 = \frac{5 \times 3}5 + \frac 35 = \frac{18}5 \] Because of that we may say and write 18 its divided by 5 its is \[ \def\its{ \mbox{ its }} [18 \its] \div [5 \its] = \frac {18}5\] The latter equals $3 + \frac 35.$ Now the use of the word times is agrees with \[ \frac {20}5 \times 5 \its = 10 \times \frac 15 \mbox{ of } 5 \its = 20 \its \] Thus 5 its goes into 20 its, exactly $\frac{18}5 = 3 + \frac35$ times.

   Replacing the it

   The it in the foregoing can be any count, amount or measure. In the equation \[ [18 \its] \div [5 \its] = \frac {18}5\] we may take the its to be halves, thirds, quarters, fifths, sixths, 99ths, and so on. One example would be \[ [18 \times \frac 17] \div [5 \times \frac17] = \frac {18}5\] The latter can be rewritten as \[ \frac {18}7 \div \frac 57 = \frac {18}5 \] In general, \[ \frac {18}m \div \frac 5m = \frac {18}5 \] for each whole number we may choose to substitute for m. One it here would be one m-th. The general rule is as follows.

   When a first fraction has the same denominator as a second, nonzero, fraction, the first fraction divided by the second is a fraction whose numerator is that of the first, and whose denonominator is given by that of the second.

   In shorthand notation, this slogan or rule says \[ \frac Am \div \frac Cm = \frac AC \quad \mbox{ provided } C \ne 0 \] That provides a mechanical pattern to follow in the like-denominator case

   For addition, comparison and subtraction with unlike denominators, we raise terms to obtain like denominators. Raising terms to obtain like denominators can also be done to show and say how to divide fractions with unlike denominators. That process is left for next year. It leads to the mechanical rule, to divide by a nonzero fraction, multiply by its reciprocal.

   Inversion of Fractions and Mixed Numbers - Reciprocals

   Interchanging [switching, swapping] the numerator and denominator of a nonzero fraction gives another nonzero fraction. Examples of this inversion operation follow.

   Fraction or Mixed Number	its Reciprocal
   \[\frac34\]	\[\frac43 = 1 +\frac13\]
   \[\frac{11}{25}\]	\[\frac{25}{11}= 2+\frac4{11}\]
   \[1+\frac57 =\frac{12}{7}\]	\[\frac7{12}\]
   \[\frac AB \]	\[\frac BA \]

   Here inverting a mixed number or the equivalent improper fraction gives a proper fraction. While inverting a proper fraction gives an improper one.

   Observe \begin{eqnarray*} \frac{7}{4} \times \frac 4 7 &=&\frac {7 \times 4}{4 \times 7} \\ &=& \frac{28}{28} \\ &=& 1 \end{eqnarray*}

   In general, the product of a fraction with its reciprocal has the value 1. In the foregoing, we could have simplified the fraction \[\frac {7 \times 4}{4 \times 7} \] to obtain 1 via cancellation of common factors or divisors. That would have avoided the calculation of the product $7 times 4.$

   Division with unlike denominators

   Recall the following:

   When a first fraction has the same denominator as a second, nonzero, fraction, the first fraction divided by the second is a fraction whose numerator is that of the first, and whose denonominator is given by that of the second.

   In shorthand notation, this slogan or rule says \[ \frac Am \div \frac Cm = \frac AC \quad \mbox{ provided } C \ne 0 \] That provides a mechanical pattern to follow in the like-denominator case

   First Example

   This example shows how raising terms permits one fraction to be divided by another.

   \begin{eqnarray*} \quad \\ \frac45 \div \frac 37 &=& \frac{4\times 7}{ 5 \times 7} \div \frac{ 5 \times 3} { 5 \times 7} \\ &=& \frac{4\times 7}{ 35} \div \frac{ 5 \times 3} { 35} \\ &=& \frac{4\times 7} { 5 \times 3} \\ &=& \frac{28} { 15} \end{eqnarray*} because raising term leads to a like-denominator case.

   Observe the same result can be achieved mechanically by calculating the first fraction by reciprocal of the second: \begin{eqnarray*} \frac45 \times \frac 73 &=& \frac{4\times 7} { 5 \times 3} \\ &=& \frac{28} { 15} \end{eqnarray*}

   Check the Result: \begin{eqnarray*} \frac37 \times \frac{28} { 15} &=& \frac{3\times 28}{ 7 \times 15} \\ &=& \frac{3\times 4 \times 7 }{ 7 \times 5 \times 3} \\ &=& \frac{4 } { 5 } \end{eqnarray*}

   To multiply, we put the product of the numerators over the product of the denominators, but instead of calculating the products, we factor their factors in the hope of finding common divisors to cancel. The hoped for cancellation occurs.

   Second Example

   This example shows how raising terms permits one fraction to be divided by another. \begin{eqnarray*} \quad \\ \frac{8}{11} \div \frac {12}{13} &=& \frac{8\times 13}{ 11 \times 13} \div \frac{ 11 \times 12} { 11 \times 13} \\ &=& \frac{4\times 7}{m} \div \frac{ 5 \times 3} { m} \mbox { where } m = 11 \times 13 \\ &=& \frac{8\times 13} { 11 \times 12} \\ &=& \frac{104} { 132} \end{eqnarray*} because raising term leads to a like-denominator case.

   Observe the same result can be achieved mechanically by calculating the first fraction by reciprocal of the second: \begin{eqnarray*} \frac{8}{11} \times \frac {13}{12} &=& \frac{8\times 13} { 11 \times 12} \\ &=& \frac{104} { 132} \end{eqnarray*}

   Check the Result: \begin{eqnarray*} \frac {12}{13} \times \frac{104} { 132} &=& \frac{12\times 104}{ 13 \times 132} \\ &=& \frac{12 \times 8 \times 13}{ 13\times 11 \times 12} \\ &=& \frac{8} { 11 } \end{eqnarray*}

   Replacing Division by a Multiplication

   The two examples above suggest the division of a first fraction by a second equals the fraction given by the multiplication of the first fraction by the inversion or reciprocal of the second. Thus division is replaced by an multiplication.

   \begin{eqnarray*} \quad \\ \frac56 \times \frac 34 &=& \frac56 \times \frac 43 \\ &=& \frac{5\times 4 }{ 6 \times 3} \\ &=& \frac{5 \times 2 \times 2 } {2 \times 3 \times 3 } \\ &=& \frac{5 \times 2 } { 3 \times 3 } \\ &=& \frac{10} { 9 } \\ \quad \end{eqnarray*} Raising terms to get like denominators would give the same result.

   A second example: \begin{eqnarray*} \left[2+\frac12\right] \times \left[1+\frac 23\right] &=& \frac{5}2 \div \frac53 \\ &=& \frac52 \times \frac 3 5 \\ &=& \frac{5\times 3 }{ 2 \times 5} \\ &=& \frac{3} 2 \\ &=& 1+ \frac12 \end{eqnarray*}

   Perfecting Fraction Skills

   Consolidation: The development of fractions skills may be extended and refined with the efficient use of least common denominators and the recognition and cancellation of common factors in addition, subtraction, multiplication and division operations, operations whose completion by convention requires the simplification of proper fractions and the expression of impropert fractions as mixed numbers. In the first instance, least common denominators in raising term steps for addition and subtraction may be found by a list method, while recognition and cancellation of common divisors or factors in the simplification step of completed arithmetic operations may be done with the aid of decimal-based rules for recognizing multiples of 2, 3, 5 and so on.

   Using Primes: Knowledge of fractions and mixed numbers has take home value for counting, accounting and measurement. More generally, prime number factorization may be employed to do exact arithmetic with fractions and mixed numbers efficiently. In particular, there is a square method with gives students a means to quickly obtain by hand or with the aid calculators, the prime factorization of whole numbers less than a thousand by knowledge of the 11 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 with squares less than 1000. The take-home value of learning about primes and how prime factorization help with fractions may be marginal, but the latter is of true or ritual importance in the further study of exact or pure mathematics. Site material shows how to introduce primes and prime factorization in ways easily followed and repeated. Skill with prime factorization and it use provides the first taste of skills without immediated take-home value required by college programs in business, science, technology, engineering and mathematics

   Shifting of Decimal points in Long Division

   To do long division with divisor that has digits after a decimal point the decimal places or points can be shifted in both divisor and dividend. Such shifting of decimal points can be taught by rote or be justified in special cases through operations with the aid of fractions. \begin{eqnarray*} 2.34 \div 2.4 &=& \frac{234}{100} \div \frac {24}{10} \\ &=& \frac{234}{100} \times \frac {10}{24} \\ &=& \frac{234 \times 10 }{ 100 \times 24} \\ &=& \frac{234 \times 1 }{ 10 \times 24} \\ &=& \frac{234} { 24 } \end{eqnarray*} Thus division by 2.4 can be replaced by division by 24.

   Another example: \begin{eqnarray*} 23.4 \div 8.456 &=& \frac{234}{10 } \div \frac {8456}{1000} \\ &=& \frac{234}{10} \times \frac {1000} {8456} \\ &=& \frac{234 \times 1000 }{ 10\times 8456} \\ &=& \frac{234 \times 100 }{ 1 \times 8456} \\ &=& \frac{23400} { 8456 } \end{eqnarray*} Thus division by 8.456 can be replaced by division by 8456.

   A third, somewhat awkward, example: \begin{eqnarray*} 3.56 \div .4 &=& \frac{356}{100 } \div \frac {4}{10} \\ &=& \frac{356}{100 } \times \frac {10}{4} \\ &=& \frac{356 \times 10 }{ 100\times 4} \\ &=& \frac{356 \times 1}{ 10 \times 4} \\ &=& \frac{356} { 40 } \end{eqnarray*} Likewise, \begin{eqnarray*} 35.6 \div 4 &=& \frac{356}{10 } \div \frac {4}{1} \\ &=& \frac{356}{10 } \times \frac {1}{4} \\ &=& \frac{356 \times 1 }{ 10\times 4} \\ &=& \frac{356} { 40 } \end{eqnarray*} Therefore $ 3.56 \div .4 = 35.6 \div 4$ and division by 0.4 can be replaced by division by 4. Question: How can one show that more directly?

   With or without those asides, the quotient × divisor plus remainders calculation can employed to check the results of long division.

Scientific Notation

So far we have covered most methods of how to add,compare, subtract and multiply with decimal with a few digits before and after a decimal point. Scientific notation will set the stage for multiplication and long division with decimals several orders of magnitude greater or smaller than 1, numbers that do not have several significant digits. In the case of long division, rules for shifting decimal points of divisors are based based on operations with fractions, operation in which mixed decimal numbers are expressed as whole numbers over powers of ten.

The foregoing role for scientific notation extends or departs from the role of providing a platform for the discussion of significant digits - a view of error analysis - in calculations.

How to Show Work in Evaluation of Arithmetic Expressions

Skill and written work has to be seen to be credible. The show work formats below require you to do and record given data, formulas and figuring steps, etc, one at a time, one after another, so that you and others have a story or path to see and check, as done or later.

The calculation of arithmetic expressions now and algebraic expressions later may employ a vertical alignment of equal signs as follows. The aim here is show how to do and record evaluation steps, one at a time, one after another, in which the the steps can be seen and corrected as done or later

The equal sign here means has the same value as. The repeated use of the equal sign above and below gives the mathematical version of a sentence that goes on and on, with clauses and subclauses.

In the following sentence a word, a verb, is missing.

The dog ____ the cat up the tree.

The missing verb here is chased>

writing an arithmetic or algebraic expression without using equal signs to show the steps is like writing a sentence with a verb.

Integers

Show students how to interpret integers and signed numbers, and how to do arithmetic with integers and signed numbers. .      1 Integers as Coordinates
     2 Integers Multiplies of a Unit Moverment
     3 Adding Movements with same direction
     4 Adding Movements wiht opposite directions
     5 Zero Movement and Additive Inverses
     6 Multiplication by Natural Numbers
     7 Multiplication by Signs
     8 Multiplication by Signed Numbers - Integers
     9 Multiplying Integers
     10 Integer Multiplication Formulas
     11 Adding Integers - Formulas and Examples
     12 Adding Integers - More Examples
     13 Subtraction with Additive Inverse

Geometric Motivation for Arithmetic with integers:
Describing Addition of Movements along a line

Addition of Movement in Same Direction: 

Suppose the movement A to B and B to C are collinear and in the same direction:

Diagram 1.

Then the distance of C to A will be  a + b  if the distance of A to B is little a and the distance of B to C is little b.   The same conclusion holds if the the arrows are horizontal and pointing in the opposite direction:

Diagram 2.

In the foregoing A, B and C are collinear with  B between A and C 

Addition of Movement in With Opposite Directions

In the following A, B and C are collinear with  B not between A and C.  The direction of the resulting movement is the direction of the longest.

diagram 3.

The distance or the length of the resulting movement is the length a of the longest minus the length b of the shortest.  The direction of the longest gives the direction of the result.  Here is another illustration:

diagram 4.

Remark:  The foregoing could have been covered with two diagrams if a slanted line had been used.

Scalar Multiplication of Movements (Keeping the same initial point)

• The product of a movement of length r with a positive number +a has the same direction and length ar.
• The product of a movement of length r with a positive number -a has the opposite direction and length ar again.
  

H. Addition of Signed Numbers:

The previous topic sets the stage for the following

Length and sign

The magnitude or length of a signed number is given by removing its sign prefix. The result is an unsigned number. Thus the length (or absolute value) of the signed number -10 is 10; the magnitude of the signed number +8.5 is 8.5; and the magnitude of -5 is 5; and the magnitude of 0 (zero) is 0 (zero). Here +0 = -0 = 0 all have the same value. 

The sign of a real number is given by the value of the prefix use to indicate the sign. So the sign of -10 is -; the sign of +8.5 and 8.5 is +. The sign of 0 need not be defined, but it can be taken to be +. 

We could say length instead of magnitude. The actual length of a multiple -10k of a unit vector k would be 10 units, while the length relative to the unit vector k would be 10, and the sign relative to k would be the minus or negative sign  - .

Adding with Like Signs

The sum of two real numbers  P = +a  and Q =  +b is found as follows

The sum of two real numbers P=  -a  and  Q = -b is found as follows

Like Signs Addition Rule for Real Numbers

If P and Q are real numbers the same sign then 

P + Q = (common sign)( length(P) + length(Q)) 
= (common sign)(sum of the addend's lengths)

Here the magnitudes are unsigned real numbers given by decimal or fractions etc.

Adding with Unlike Signs

The case where the sign of the longest is + follows.

Unlike Signs Addition Rule for Real Numbers

If P and Q have opposite signs and are unequal in length then 

P + Q = (sign of longest) (Longest - Shortest)

The case where the sign of the longest is + is shown below. The case where the sign of the longest is - is similar. Tutors should give that case or examples of it. 

Opposite Signs and Equal Length 

If P and Q have opposite signs and are equal in magnitude (length) then P and Q are additive inverses, and 

P + Q = 0

Exercise:  Explain how the rules for multiplying signed numbers follows from the  rule given above for the Scalar Multiplication of Movements (Keeping the same initial point) The product of a movement of length r with a positive number +a has the same direction and length ar. The product of a movement of length r with a positive number -a has the opposite direction and length ar again.

First Arithmetic Check List

1. Decimals: Decimal Representations of Whole Numbers 0 to 1 million

2. Decimals:  Decimal representation of tenths, hundredths and thousandths and ten thousandths alone and with whole numbers - mixed decimals to four decimal places.  Include here the approximation 3.1416 of $\pi$to 4 decimal places and emphasize that it is an approximation to 4 decimal places - the nearest ten-thousandth.

3. Decimals: Column or place value methods for Addition, Subtraction, Comparison, Multiplication and Long Division for whole numbers and mixed decimals with a knowledge of how check them via subtraction, addition, division and the rule dividend = quotient times; divisor + remainder, and knowledge of how to place the decimal point in the calculation of sums, differences, products and quotients.

4. Number Theory: Know how to calculate least common denominators of a pair of  small numbers m and n from listing the first m multiples of n and the first n multiples of m.

5. Number Theory: : Know how to calculate great common divisor using Euclid's Algorithm.

6. Number Theory: : Define proper and improper whole number factors for a whole numbers. The number 1 and the whole number are improper factors. All other factors are proper. 

7. Declare (define) a whole number is prime if it is not the product of two proper factors less than  than it. Declare (define) a whole number to be composite if it is  the product of two proper factors less than  than it. The words less than allow the times tables 10 by 10 and 12 by 12 to allow students recognize primes by inspection. 

8. Number Theory:  apply the quick prime identification rule: A number less than169 is prime if it is not a multiple of 2, 3, 5, 7 or 11. This is especially useful with decimal notation based rules for identifying multiples of 2, 3 and 5.   The multiples of 11 less than 100 are easily recognized. The further multiples of 11 less than 169 and the multiples of 7 less than 169 which are not multiples of 2,3, 5 nor 11 may be listed and memorized. 

9. Number Theory: Calculate least common multiples and greatest common divisors from prime decompositions. 

10. Fractions: Form a unit numerator  fraction of easily divided geometric object or of a group of objects whose count is a multiple of the unit fraction denominators.  Connect the foregoing with division by denominator.

11. Fractions: Add, compare and subtract fractions with like denominators. 

12. Fractions: See how the notion of equivalent description of fractions - when used as measure of length etc - leads to lowering and raising terms. 

13. Fractions: See how to raise terms to add, compare and subtract fractions with unlike denominators, and introduce the convention that sums and differences should be expressed in lowest terms. 

    Teachers may give cross-multiplication rules for fraction comparison,  but the explanations here are easy and should not overwhelm students - if or when they do, focus on the ability to compare and make explanation available for students willing to follow it.

14. Fractions: Know how to use least common multiples, greatest common divisors, along with raising and lowering term methods to add, subtract, compare and multiply fractions, so the convention of expressing sums, differences and products in lowest terms may be done efficiently - that is with the avoidance of number larger than need-be in the intermediate steps. 

15. Vertical  Aligned = sign Format:  Do arithmetic with fractions and evaluate geometric formulas in a well-formatted step by step manner with equal signs present and aligned vertically with one step per line (except at the bottom of a page), and in each line,  equal signs, addition and subtraction signs, and division bar aligned horizontally.  The aim here to have a format that provide a standard and an aid for doing and recording work step by step. 

16. Give rules for recognition of  odd and even numbers, and explain why?

17. Give rules for recognition of multiples of 2, 3, 5 or 10?

Second Arithmetic Check list

1. Compare and order proper and improper fractions using the least common denominator or comparison of integer parts, as appropriate.

2. Read and write improper fractions.

3. Can you charge produce the 10 times table on demand?

4. Multiply 2 to 4 digit numbers by 2 digit numbers?\

5. Divide 1 to 5 digit numbers by 1 or 2 digit numbers and find the remainder using the long division algorithm?

6. Use the rules for recognizing multiples of 2, 3, 5, 9, 10 and 11?

7. Rewrite fractions as percentages or decimals, finite or repeating.

8. Express a decimal as a percentage, and vice-versa.

9. Recognize through its prime factorization, when a fraction will a finite decimal expansion..

10. Round decimals to the nearest tenth, hundredth or thousandth.

11. Explain the use of decimals to one, two, three or four decimal places.

12. Compare and order decimals.

13. Multiply and divide decimals by whole numbers or decimals.

14. Convert decimals into percents or fractions.

15. Solve for an unknown given equations with whole number coefficients.

16. Express an infinite, repeating decimal expansion as a fraction?

17. Express a fraction as a percentage, and vice-versa.

18. Express a fraction as finite or repeating decimal ?

19. Does your charge know that a fraction has a finite decimal expansion when and only when the denominator is equal to a product of 2s and 5s with no other primes in the prime decomposition/factorization of the denominator?

20. Understand powers, that is exponents,  in arithmetic?

21. Give the prime decomposition of a whole number?

22. Recognize multiples of 2, 3, 5, 10 and 11 with the aid of rules for this recognition?

23. Find the greatest common multiple and least common divisors using the prime decompositions for whole numbers in question? 

24. Simplify square roots using factorization into squares or primes?

25. Can you charge use the greatest common divisor for a pair of whole numbers to compute their least common multiple? 

26. show in simple examples why fractions resulting from simplification or introduction of higher terms are  equivalent?

27. Powers of Ten: Write 10, 100, 1000, 10000, etc as powers of ten?

28. Scientific Notation: Write a decimal as the product of a power of ten with a number between 1 and 10?  Use scientific notation to estimate the size of products and ratios of numbers written in scientific notation.

29. Write a number given in Scientific Notation as a decimal?

30. Signed Numbers: Identify where signed numbers appear - position along a line, thermometers, negative assets or debts. Say how to add and subtract signed numbers.  Say how to multiply and divide whole numbers. State the law of signs.

31. Given the first term in a arithmetic sum and an additive constant, compute the further terms, one at a time, and one after another.

32. Find the sum of a finite arithmetic sum. Justify the formula by writing the finite sequence forward and backwards (Gauss's method).

33. Given the first term in a geometric sum and an multiplier, compute the further terms, one at a time, and one after another.

34. Find the sum of a finite geometric sum by means of a formula. Justification reserved to a future lesson on mathematical induction.

Extra/Enriched Arithmetic

1. Use Euclid Algorithm to find the greatest common divisor for a pair of numbers? Euclid Algorithm for this provides the quickest way to simplify  fractions - reduce to lower terms. This method - not commonly taught - provides the quickest way to simplify fractions and their products, and to find the least common multiple multiple of a pair of numbers or the least common denominator.

2. Explain why column methods for addition, subtraction and multiplication of decimals work? 

3. Show how the numbers appearing in decimal long division, or the work needed by it, imply a number is equal to a remainder plus quotient times remainder? A similar reasoning applies to the polynomial long division method to be met later.

4. Visualize the addition, multiplication, division and subtraction of lengths where the lengths are whole number or  fractional multiples, proper or not,  of a unit length?

5. Explain how the former gives a geometric viewpoint and motivation for arithmetic?

Divisibility and Remainder Calculation Rules

1. Last Two Digit Rule:  For factors of 100, and 100 itself,  the remainder of a whole number on division by each equals the remainder on division of the last digit. Here are some factors of 100
   
   2, 4, 5, 10, 20, 25, 50, 100
   
   Example:  The remainder of 360  on division by 25  has the same value as the remainder on division of the last two digits 60 on division by 25. The latter remainder is 10.
   

2. Last Three Digit Rule:  For factors of 1000, and 1000 itself,  the remainder of a whole number on division by each equals the remainder on division of the last digit. Here are some factors of 1000
   
   500, 250, 125, 200, 50, 40, 20, 10.
   

3. Backward Pair Sum Rule:  The remainder after division by 11 of a whole number has the same value as the remainder after division by 11 of the backward sum of pairs of digits in the decimal representation.
   
   Even number of digits:   145671 has the same remainder on division by 11 as the backward sum
   71 + 56 + 14
   
    Odd number of digits:  51655 has the same remainder on division by 11 as the backward sum
   
   55 + 66 + 5
   
   Alternate Rule:  If the decimal representation has an even number of digits, the remainder after division by 11 equals the remainder after division by 11 of the sum of pairs of digits.  If the decimal representation has an odd number of digits, the remainder after division by 11 equals the remainder after division by 11 of the first (highest value) digit plus the remainder on division by 11 on the rest of the the decimal representation of the whole number. 
   

Arithmetic with multidigit Decimals Exercises

Add columns of multidigit decimals without and with decimal points.

Subtraction Problems without and with multiple conversions

YSubtract decimals with digits before and after decimal points without and with multiple conversions. 

Multiplication Problems

Multiply decimals and locate the decimal point properly in their products. 

Students should be able to locate the decimal point properly.

Long Division Problems with whole number quotients and remainders

Students should be able to do long division with whole numbers and obtain an integral remainder less than the divisor.

Student should know how to check results by verifying

dividend = quotients × divisor + the remainder

holds for the calculated quotient and remainder. 

Remark: Familiarity with long division with whole numbers makes long division with polynomials easier. 

Long Division Problems to finitely many places

Continue the long division process to obtain 2, 3 or several decimal places. 

The expression of 4 elevenths as a decimal by long division leads to an infinite decimal with a repeating tail. 

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