Primary School -  Counting and  Arithmetic Skills

Parallel to learning about the alphabet,  children may learn about the digits 1 to 9, and the digit 0, and their significance or meanings.   In the process,  students may be shown sets of 1 to 3, 1 to 6 and 1 to 9 units (pennies, animals, etc) to the connect single digit whole numbers with counting units that are identical or of equal value.  Counting with two and three digit decimals may follow as students learn how to count to 10 and beyond in hundreds, tens and single digits.  The question of addition for single digit numbers stems from hands one experience with the questions of the form, how units do we get if a pile of 3 is added to a pile of 4.  The result pattern and regularity implies 3 + 4 = 7. The same or similar pattern land regularity. leads to the addition table for single digit and small double digit numbers.  The times table for pairs of single digit numbers follow from use of counting to say for example how units are there in 3 row of 4 units.  The units may be dots or unit squares. The latter provides a first connection between multiplication and area calculation. 

Students ability to understand is strengthened and developed by

1. mastery of multi-step methods for arithmetic addition, subtraction, multiplication or long division with whole numbers and fractions with exact (finite) decimal representations.
2. the further use of rules, methods and patterns in sequence to arrive at solution of puzzles or define further pathways and methods for figuring, geometry or drawing. 
3. meeting short chains of reason where rules of the form A if B and B if C are combined to conclude A if C.
4. learning how to put jigsaw puzzles together by trail and error, helped or guided by strageties, the foremost example of which is starting with the edge pieces.

The ability to combine rules and patterns needs to be sown and cultivated before being required in explanations. The aim is sow the seeds, not to destroy the garden. For example, in presenting decimal methods for addition, comparison and subtraction, some students may be aided and others bothered or confused by talk of why conversion methods (carries and borrows) work. So explanations may help some and confuse others.   

Counting in groups of hundreds, groups of tens and units where the number of hundreds, tens and units is given by a single digit permits the decimal place representation of numbers or counts.  In counting single, double and even triple digit number of units,  students see and expect that order of counting does not affect the result, that is the number of hundreds, tens and units in the result.  When counts disagree, there is a recount. That practice of counting again when counts disagree, points to or implies a principle: counts should be independent of how a set of elements is counted. 

When sets of units are pre-arranged into groups of one hundreds, tens and ones (no more than 9 of each),  counting need not be one by one.  The decimal count may take advantage of the arrangement.  Addition of counts or whole numbers takes advantage of the latter to imply decimal methods for addition without and then with carries or conversions. The foregoing provides a thought-based development of decimal place methods for addition which some students may need or appreciate, which some students will not see a needed. 

The practice of counting again when counts disagree, points to or implies a principle: counts should be independent of how a set of elements is counted. Now the number of elements of a set can be counted one by one in many sequences. The number of elements can also be counted by division into two or more groups - disjoint subsets, and then summing the numbers in each subset. There are many ways in which those numbers can be added. The principle or assumption that all ways should lead to the same count can and should be emphasized. The latter property or assumption implies the associative and commutative laws for  addition of counts or whole numbers  in secondary school mathematics.  Emphasizing the latter sets the stage for this. 

Decimal methods for addition may be learnt by rote or with explanation. Like wise, with more explanation, decimal methods for subtraction and comparison (two sides of the same coin) learning by rote or with explanation of why the methods work is possible. The latter imparts a variable mix of rote and thought-based mastery in mathematics training.  More rote learning may follow and be necessary in the mastery of decimal methods for multiplication and long division.  Explanations why the latter work are available in site arithmetic videos. They would be above the heads of most students. So should not be imposed on students.

Rote learning appears to be a necessary part of primary and secondary mathematics - unavoidable. Explanations of why methods work should be offered only when they might aid the operational command.   Young minds are impressionable. Most students will be satisfied with the following justification of arithmetic methods: They work, they are prescribed by authorities.  Students may take the view that a school authorities would not ask teachers to mislead them. So proofs or explanations are not necessary.  From an operational viewpoint, the students are right.  In learning to do, in learning to follow arithmetic step carefully to get repeatable, reproducible results, understanding why the steps work is optional. Skill and confidence in doing may follow when students obtain repeatable and reproducible results. But logic or the thought- and pattern-based development or introduction of ideao can appear and should where it aids or enables the operational command. 

Counting, Geometry and Algebra

Primary school mathematics may introduce student to the counting of squares in rectangles with sides that are integral multiples of a unit length.  When the lengths are  W and L,  the count may be viewed a L rows of W squares or W columns of L squares. Counting principles imply the total number of squares is L times W and W times L. Tha should be emphasized. In secondary school, that observations can be recast as the commutative law for multiplication in secondary school. 

Algebra begins in primary school with the statement of formulas for areas of rectangles, right triangles and scalene triangles, and showing students how to evaluate them.  To get student in the habit of writing more than just an answers, whenever one of this formulas is employed, the geometric region in question should be drawn, the formula should be written with an equal sign in it, and evaluation should be proceed as follows.

The evaluation of an area A should proceeds

A = the formula
    = the formula with lengths replaced by their values
    = a simplified arithmetic expression 
    = another simplification 
    = ...     
    = 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.

The introduction of geometric formulas and their evaluation represents a first step in algebra.  To say (i) that the area A of a rectangle is given by  the formula A = W x L where W and L are the widths and length of a rectangle which has been drawn is far easier to grasp than to say (ii) let  m and n be whole numbers. 

Fractions

Reciprocals of whole numbers - Fraction with unit denominators

Primary school students may grasp the concept of division into like parts or into parts of equal value.  So apples, pears, pies, circles and rectangles may be divided into two like (congruent, isometric) parts, called halves; or divided in three like parts (thirds), or four (fourths) or five (fifths) and so on. And 60 pennies may divided into two, three, five, six, twelve, twenty or thirty parts. So division of one unit or several yields parts that be described as a  half, third, quarter, fifth, sixth,  and so on. The unit fraction is associated or obtained from division into like parts or parts of equal value.

Fractions in general 

Then two-thirds is two times a third,  three fifths is three times a fifth.   Whence students can learn to multiply unit fractions to obtain multiples of unit fractions.  Then student can learn to add, subtract and compare multiples of a unit fractions - the like denominator case. The introduction of equivalent fractions allows students (a) to add, subtract and compare fraction by raising numerators and denominators to obtain like denominators; (b) to simplify fractions by lowering numerators and denominators; and finally (c) to make a habit of following addition and subtraction of fractions with simplification of the resulting sum or differences. The latter may be called arithmetic or addition and subtraction with simplification. 

Multiplication of Fractions - Special Case

The question what is a half and what is a third of six fifths may introduce product of fractions. 

• a half of six units is three units.  So half of six fifths is three fifths.
• a half of six units is two units.  So half of six fifths is two fifths.

The foregoing suggest a pattern for products where the numerator of the multiplicand (second factor) is a multiple of denominator of the first factor.  

(I) one N-th of  P×N  Q- ths with be P Q-ths

(II)  M one N-th of  P× N  Q- ths will  be M × P Q-ths or  (M × P) Q-ths 

Multiplication of Fractions - Product Formula

That leaves the question of how to calculate

The solution follow by raising terms

Thus  the product is obtained by raising terms and then using the special case calculation. After that the calculation can be combined with simplification of the result.  That leads to multiplication with simplification.  Student may be introduced for efficient methods for that via the introduction of cancellation of common factors to lower  the product numerator and M × P and  denominator  N × B.

Appendix:  Division of Fractions

Division of Fraction by Whole Numbers

One N-th of a fraction is equivalent to dividing the fraction into N equal like parts or N parts of equal value. 

Thus N× P  Q- ths divided by N  would be  P Q-ths

Now 

We observe  

Division of Fractions with Like Denominators

We may ask how many times does 3 ITs go into 15 ITs.  The answer is 5 whole times.

We may ask how many times does 3 ITs to into 17 ITs.  One whole number based  answer is 5 whole times with 2 left over.

Now ask how times does 3 go into 2 ITs  The answer could be 2 third times since

So a second answer to the question how many times does 3 ITs to into 17 ITs is answer is 5 whole and 2 thirds whole times  Observe

When we allow fractional replies,  the answer to the question 3 ITs to into 17 ITs is the improper fraction 

Moving On

Above IT stands for a measure of something.  But the measure does not have to be a whole unit. 

Re-read the above with IT given by a half, a third, a quarter, a tenth  or an M-th.  That suggests

In general, a formula for division of fractions with like denominators would be given by

We apply that below. 

Division of Fractions with Unlike Denominators

Saying or showing how to do operation defines.  The operation follows again by raising terms to obtain like denominators.  

Check:  Quotient times divisor = Divisee 

The quotient 

gives the multiplier of 

that yields 

Aside: In the language of proportionality, 

is proportional to the divisor with constant of proportionality

Question: How many times does a fraction go into one?

In particular 

and 

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