Arithmetic Ends and Values by Age 13-14

By ages 13 to 14 years of age, students should be able to master the following skills and practices, and underlying concepts.

1. Counts - Directly and Reliably Given a set of objects on paper or in a space, count them in sequence one by one, or in groups of two, three, four and five, etc. The resulting should be independent of the counter. All students in a class should get the same result.

2. Count by adding or subtotaling subcounts : Given a set of marks on paper, group them into disjoint - that is non-overlapping subsets - find the total count by adding subcounts. Again the result should be independent of the division or partition into disjoint sets. All students should get the same result. Further, this method or practice can be applied iteratively. That is the subcounts may be obtained by adding subsubcounts.

3. Totals of whole numbers and finite decimals with subtotals: In counting or keeping track of money with whole units (dollars, pounds, etc) and pennies, amounts may be given by decimals with none or two decimal places after the decimal points. The two decimal after the decimal point represent pennies. Given a collection of amounts scattered over a page, the amounts may be grouped into disjoint sets for the formation of subtotals - subamounts. Then the totals added to find the grand total. All students should obtain the same result. The subtotals too can be found by adding subsubtotals. This method or practice can also be applied or illustrated in column methods for addition by grouping vertically adjacent decimals to form subtotals. For whole numbers and decimals scattered across a page, the numbers may be grouped in opportunistic ways to make the resulting subtotals easier to add. For example, the observation that certain groupings of amounts lead to integral multiples of 1, 10, 100, 1000 and so on may be exploited to make the further addition of subtotals easier - some would call this practice part of the development of mental arithmetic skills. If the amounts or decimals are arranged in tables, addition by subtotal rows should give the same result as addition by subtotal columns. Finally, averages can be calculated by adding subtotals and dividing by the count of amounts totals. That count may also be given by adding subcounts.

4. Totals of mixed numbers and fractional amounts by adding subtotals: In measurement to the nearest half, third, quarter, tenth or hundredth may result in integral and fractional amounts, alone or together as in mixed numerals. Here \[ \frac74 = 1+\frac34 = 1\frac34 \] gives an example. Given a collection of mixed numerals, they may be totalled by adding subtotals. The resulting should be independent of the division of these numbers or measures into disjoint sets. The division into disjoint sets may be done opportunisitically to form subtotal - more integral or fractional counts - easier to add. Results should be the same for all computers or students regardless or independent of the division into subsets for the calculation of subtotals. The subtotals themselves may be given by totaling sub-subtotals. In the case of tables of numbers or measures, addition of row subtotals and column subtotals should be given by the same amount.

   Remark. Given a set of mixed numerals, we may transform them into proper and improper fractions with like denominators $n.$ Each fraction then represents a multiple of the same unit numerator fraction $\frac1n$. The multiple "counts" how many time the unit numerator fraction is present in one of the proper or improper fractions. Adding the multiples directly or with grouping than counts how many times the unit fraction is present in the sum. The result should be independent of how grouped in accordance with our counting methods for whole numbers. That being said, each simplification of fractions and each formation of mixed numerals during the summing process only changes the form of the count, not its substance, and represents a form of opportunistic grouping. This remark as is or better put and further illustrated might help some understand how counting principles and practices for whole numbers implies like principles and practices for mixed numerals as is, or expressed as proper and improper fractions. Decimals with two decimals places after the decimal point, zeroes allowed in one or both of those places, may be viewed as integral multiples of one hundredth $\frac1{100}.$ Whether one adds money in terms of pennies - one hundredths - or in terms of whole units - dollars and pennies - should not affect the result.

5. Calculating Average Height or Position: Given a list of unsigned hieghts or coordinates, students should be able to compute averages by dividing the sum by the count of heights or coordinates in question. The sum may be added by subtotaling. Students should be shown that adding the same number to each height adds that number to the average. The added number in question should be chosen in the first instance, so that each modified height is non-negative. Similar patterns and experience may be then be provided in finding the average height when some or all are negative. Here adding the same number, be it positive or negative, adds that number to the average.

   The sign of coordinates depend on the location and choice of origin along a coordinate line, be it vertical or horizontal. The foregoing implies the average of a set of coordinates - the average position - does not depend on the choice of origin for a coordinate line. The position is invariant. The foregoing observation is consistent with the practice that the sum of coordinates employed in the calculation of their average value may be computed by adding subtotals. Emprirically, the subtotalling method for finding totals works for firnite sets of negative and positive numbers, and for finite sets of numbers or coordinates not all of the same sign. See the next item.

6. Total by Subtotalling positive, zero and negative amounts or numbers: In tracking the money value of assets and debts, positive amounts represent assets and negative amounts represent debts. As an exercise, imagine or give a small to large set of positive and negative amounts in dollars and cents scattered over a page. The net or total value may be obtained by grouping into disjoint subsets for the formation of subtotals. The latter may positive or negative. The total should be independent of how the amounts are divided for subtotalling. The subtotals themselves may be obtained by adding subtotalling. The division of amounts for subtotalling may be done opportunistically for the sake of easier subtotals to add or interpret. For example the total may be given by the sum of assets - a positive amount; plus the sum of debts - a negative amount. Each sum or subtotal can also be obtained by subtotals. When the assets and debts are located in a table, the total or net value may be obtained by adding row sums or column sums. The result should be independent of each summation or totalling methods. So adding twice in two different ways or at two different times may provide a check.

   Calculating Net Worth: Assets and debts may be represented by signed decimal numbers with digits in the tenth and one-hundreth places. Net worth may be find by adding these numbers in some sequence or via some convenient subtotals. Computations when done exactly - with no rounding and mistakes - should all lead to the same result. The foregoing implies individual net worth does not depend on how debts and assets are summed. Counting in different orders will not improve nor diminish one net worth, be it positive or negative. That pattern has take value in handling money matters at home, work and school.

7. Measuring and Figuring Practices: Electronic devices for finding length, mass-weight, time and volume make measurement too simple. Skill development depends on hands-on experience with analogue devices - rules, tape measures, balances, clocks and graduated cylinders. In this, measurements may be done directly or computed with the aid of arithmetic - addition, subtraction and multiplication if not division. Measurements are given by denominate numbers - that is an integral and/or fractional count times a unit of measure. Different units of measure may be present, even mixed in the direct measurement of numbers, or their calculutions. Measures may be obtained by adding submeasures. Measures may also be obtained by subtraction. The foregoing provides room for discussion of measurement and figuring errors, and approximate practices for minimizing and estimating errors in direct and computed measures. Students should be shown metric and non-metric units of measure to indicate the advantage of standardization with both kinds, and to make students aware of the mix of both kinds of units in practice. For example, even in countries which are officially metric, the day has 24 hours, the hour has 60 minutes, and each minute has 60 seconds. Degrees are still measured with minutes and seconds. So metric is not universal. Conversions would be required.

   Measuring and figuring practices have value for home, work and school. Construction with wood, metal and fabric all require measurment. Students may be shown by measurements how the volume or capacity of cone is one third that of a cylinder or container with the same base and height. Students may also be shown by measurement how the volume or displacement of a sphere plus twice that of circular cone of the same base and hieght equals that the volume or capacity of a cylinder whose hieght and diameter equals the sphere diamater. \[\frac 43 \pi r^3 + \frac13 (2r)\pi r^2 = (2r)\pi r^2 \] Questions about painting and covering walls, floors and cielings may lead to measurement by adding and subtracting submeasures. The submeasures themselves may also be obtained directly or by adding and subtracting sub-submeasures.

8. Buying and Selling Goods and Services: In shopping, children and young teenagers will see themselves and parents buying goods and services. Mastering cost calculation and comparison practices with decimals, fractions and percentages in school would have take-home value. Measurement and figuring practices would combine here with the purchasing power of money. Reciprocal rates in the form of how many units of measure per unit or dollar of currency, and how many units of currency per unit would appear here. The rates may be given verbally using phrases such as per or for each. But they may also be given using denominate numbers. Arithmetic with donominate numbers alone and in fractions would have practical use here, and also be of service in later representation of proportionality constants, and in scientific calculation practices with denominate numbers. While pure mathematics deals with pure numbers, denominate numbers are needed in practice at home and in quantitative disciplines. So their use should be shown in late primary and early secondary mathematics lessons. Students should be shown how to calculations with decimals, fractions and percentages in routine problems.

   More Money Matters. Not every one has work or has a salaried job. But we can show students how to calculate their salary given an hourly rate, talk about saving accounts without and with interest, simple or compound, and talk about the cost of living. In latter, students may be shown how to calculate the cost of food and shelter. The dependence of food prices on seasons, how price vary in stores, and how for-profit and non-profit sellers must mark-up prices to cover costs and earn income. Advice such as living in one means, and a penny saved, is a penny earnt may be given. Dicken's wrote Annual income twenty pounds, annual expenditure nineteen six, result happiness. Annual income twenty pounds, annual expenditure twenty pound ought and six, result misery. Showing young teens how to budget for living expenses and how to run a small business - cover costs etc - would provide a practical framework for counting and figuring with money in a be-prepared-for-adult-responsibilities, responsibilities that will strike sooner or later.

9. Elementary Arithmetic : Counting and measuring may be done directly or via arithmetic. For most people, decimal place and decimal methods for arithmetic can be met and mastered step by step with partial justification. Detailed explanations are available. But they are likely to overwhelm students and their adult tutors or teachers. In decimal arithmetic, student ages 9 to 12 may be shown how to do and record work in steps that can be seen and hence confirmed or corrected. Practice in this is valuable because it will reveal the domino effects of care and mistakes figuring. Care to avoid mistakes in multistep methods in arithmetic is a sign of practical intelligence. People who figure well are likely to watch and avoid the domino effects of mistakes in further multistep methods met at home, school and work. Avoiding the domino effect of mistakes provides an end, a tool and value for skill mastery in general.

   Skill and confidence in arithmetic may come from learning to do by rote or with some comprehension in a way that leads to repeatable and reproducible results. This approach would be simpler for children, their teachers and their parents. Here the take-home value of learning to do exceeds the value of comprehension in full or part. Comprehension of why methods work can be left for later or skipped completely. But explanations of why should always be available in references for those students uncomfortable without.

10. Long Division Revisited

    • In the context 23 ÷ 4 = 5 R 3, the expression has 5 R 3 has one meaning - here, 5 times 4 is three less than 23.

    • In the context 33 ÷ 6 = 5 R 3, the expression has 5 R 3 has another meaning - here, 5 times 6 is 3 more than 30.

    At the primary school level, the two meanings are easily understood from the context. But in further mathematics, we avoid expressions with ambiguous meaning. To remove the ambiguity or dependence on context for expression like 5 R 3, where is a simple remedy: avoid the remainder notation, and use mixed numbers to describe the result exactly

    As part of the development of fractions, students may learn that 3 = ¾ of 4 = ¾ × 4. To avoid and end the use of the mathematical ambiguous notation 5 R 3 in primary school mathematics, I would rewrite 23 = 5 × 4 + 3 as

    23 = 5 × 4 + 3 = 5 × 4 + ¾ × 4 = 5¾ × 4

11. Fractions : Arithmetic with fractions can be learnt after or besides arithmetic with decimals. The latter is a pre-requisite. Here again methods can be learnt by rote. However, methods for adding, comparing and subtracting fractions can be introduced and justified through raising terms. That is standard. Some students and teachers may find the explanation comforting. Other will find the explanation a source of discomfort. There is no pleasing all. That being said, raising terms can also be used to develop and justify fraction multiplication and division methods. So mastery with comprehension becomes an easy or easier option. Details appear in the fraction section of my website: www.whyslopes.com

12. Fractions and Mixed Numeral: In counting with decimal notation, we group and convert each occurrence of ten ones into one ten, each occurrence of ten tens into a single hundred, and each occurrence of ten hundreds into a single thousand. Thus we count whole units in terms of ones, tens, hundreds, thousands and even larger groups. The decimal 345 involve mixed units of counting: ones, tens and hundreds.

    Likewise, we may group ten tenths into a single unit, ten one-hundreths in to a single tenth, and ten one-thousandths into a single one-hundredth. The decimal 45.67 may be regarded as [i] 45 whole units plus 67 hundreths, or it may be regarded as [ii] the improper fraction 4567 hundreths. Option [i] represents a mixed numeral. Option [ii] represent a count of hundreths. More generally the mixed numeral 2¾ may be regarded as the improper fraction 11 quarters and thought of a count of quarters. Examples may and should lead students to thing of mixed numerals and improper fractions as equivalent ways of expressing a count. As a further example, a given mass may be described as 5.455 kilograms - a multiple of a single unit of measure. The same mass may be described as 5 killograms + 455 grams. The latter involves mixed units of quantity. Or, it may be described a 5455 grams. A long duration given as multiple of seconds may be expressed in terms of improper fractional multiplies minutes or hours alone, or some of the seconds may be grouped into minutes, and the minutes grouped into hours. The foregoing indicates the presence of mixed units of counting and measure in describing counts, describing sums of integral and fractional amounts, and in describing quantities - here mass and time. Students need to be shown how to do arithmetic and conversion with mixed numerals and mixed measures in casual but efficient ways. Examples should show that results do not depend on the selection of units for counting and measure. For instance the number 545 units may be described as 5.45 hundreds

13. Reading Decimals Aloud: Canadian-American versus UK-German system. Metric System recommended. Understanding place value.

>Decimal Operatons: Moving decimal points - the fraction connection
14. Evaluating Arithmetic Expressions and Formulas:

15. Statistics: In talking about buying and selling goods, averages may be introduced as a way for sellers to anticipate future sales and expenses. Apart from selling, the division of circles to form pie charts offers an opportunity to geometrically illustrate mastery of fractions and percentages less than one or one hundred percent.

For more details, see site pages Ages 10 to 12 Arithmetic and      Ages 12 to 14 Arithmetic.

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