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B Decimal Comparing and Subtracting Methods

Notes, Lessons 1 to 8

are described next. Conversions here are also called borrows.

• Lessons 1 & 2: Direct comparison and subtraction of whole numbers where no conversions (borrows) are needed. The more than symbol > and the less than symbol < are introduced to compare numbers. Comparisons and subtractions can be done directly one the subtracted number has fewer ones, tens and hundreds etc than the decimal from which it is subtracted. In this case no knowledge of lexicographic ordering of decimals is required. Exercises are included.
  
  Teachers and Tutors: In this section, the symbol > is called the more than symbol here, as that is consistent with calling the symbol <. the less than symbol. More the name more than for the symbol > avoids or minimizes a later confusion in the comparison of signed numbers, a confusion that results from a divergence between the common meaning of greater than as a comparison of size and the technical definition of more than (or greater than).
  
• Lessons 3 to 5: These lesson explains when and how conversions (borrows) can be done to extend decimal notation through the introduction of a J symbol (value 10), so that comparison and subtraction can be done directly after a conversion. Exercises are included.
  
  Teachers and Tutors: While students may still learn or be taught how to compare decimals lexicographically, the conversion approach given here justifies the latter and provides a common path for comparison and subtraction, one that makes "borrows" easy to understand and explain.
  
  The conversion results in an add 2, subtract one, arithmetic problem. Exercises are included. Both examples and exercises involve decimal counts (whole numbers) and decimal fractions (mixed numbers with fractional parts provided by tenths and/or hundredths). Borrows or conversion appear on both sides and across decimal points in examples, if not in the exercises. Exercises are included. (Students may want to see Lesson 5 first - It covers a method to make subtractions easier.)
  
  Teachers and Tutors: Here is the first instance of a conversion in arithmetic to allow an operation to be done. Further conversions will be met in the addition, subtraction, comparison and division of fractions. There-in lies a unifying thread or theme for showing and saying how arithmetic will be done with decimals and fractions. Operations are based on conversions. Instead of writing Convert to Compare/Subtract, lesson labels could have used the phrase Compare/Subtract Indirectly. .
  
• Lesson 6: The how-to Subtract Efficiently, Option. Calculating 9 + 3 - 5 as 12 -5 is less efficient in the eye of this beholder than calculating 9- 5 +3 = 4 + 3 - 3. The latter avoids subtraction of a single digit number from a 2 digit number. Exercises are included.
  
  Teachers and Tutors: When numbers are converted to make subtraction more direct, expressions of the form 9 + A - B and J + A - B appear where A < B are digits 0 to 9, and J has the value 10. Evaluation of the two expressions 9 + A - B and J + A - B may be obtained more quickly from the left to right calculation of 9 - B + A and J - B + A, respectively.
  
• Lesson 7: Subtraction of Decimal Fractions involving tenths and hundreths with Conversion before and after decimal point.
  
• Lesson 8: Working with Mixed Units: Lesson 7 gives several examples of column methods for subtraction of mixed unit of measures for money, length, time, working hours, and angles. Exercises are included.
  
• Lesson 9: Here is a review or repeat of ideas in lessons 1 to 4 and 6. View this video to set expectations for arithmetic subtraction skills.

Teachers Tutors: The older site area Number Theory Area explores three methods for subtraction. In the latter page, conversion or borrows when needed, the first method, is illustrated with examples. I suspect that the convert to compare and subtract method here might help in the explanation of (1) how electronic chips store numbers and perform remainder arithmetic while catching under- and over-flows; and (2) the use of the Abacus.

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