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D Decimal Long Division Methods

Notes

Long Division (LD) Examples and Theory

• Lesson 1. Where Does Division Appear? The division question of asking how many times does one quantity, number or unit go into another appears in counting, grouping and measurement.

  1. For Distances and Length Measurement: How many times does a shorter length (a unit length perhaps) go into (go into) a longer length. The longer length is view a whole number multiple of the shorter length with a remainder.

  2. For Dot Counting and Grouping: How many same size groups of say three dots can be formed from sixteen dots provide a first example.

  3. For Volume Measurement: How many times does a small unit of volume go into a larger volume.

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• Lessons 2 to 4 - Worked Examples of Long Division with 1 and 2 digit divisors.
  
  (2) 6847 = 2282 × 3 + 1 (Dividend: 6847, Divisor: 3, Quotient: 2282, Remainder: 1)
  (3) 7834 = 1119 × 7 + 1 (Dividend: 7834, Divisor: 7, Quotient: 1119, Remainder: 1)
  (4) 89463 = 6881 × 13 + 10 (Dividend: 7834, Divisor: 7, Quotient: 1119, Remainder: 1)
  
  These examples illustrate the long division method and format met in the site author's school days. These examples introduce the helpful practice of listing the first 10 multiples of the divisor. Single digit multiples are needed in the method. These examples also show how to check (test) that a computed or given quotient and remainder are correct.
  
  Quotable quotes: Even if we don't know why long division works, we can check its results by testing whether or not
  
  Dividend = Quotient × Divisor + Remainder.
  
  Remember: If a check fails, there is an error between the start of the long division itself and the end of the check. And if the check fails, you need double check the check and the long division.
  
  For single digit divisors, there is a short division format that is more compact (uses less space) than the long division format. Learn it if you wish.
  

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• Lesson 5: Examples in which LD (long division) is done with & without extra zeroes to further indicate how, if not why, LD works.
  

• Lesson 6: Why the long division methods works. Here is a closer inspection illustrated with the long division (with extra zeroes) of 7275 by 2 to 7275 = 3637 x 2 + 1. This first explanation and lesson 8 are for people who like to understand long division fully as well as learning to do it and check results.
  

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• Lesson 7. Long Division With a Mistake - How to handle mistakes in mathematics solutions
  
  Long Division Method here suggests
  
  5626 = 308 × 15 + 6
  
  Observe how the error is spotted and ignore the extra zeroes that appear in the solution. Their presence will be explained below.
  
  Tip 1: In class, when a check spots an error, the error itself will between the start of your solution and the end of your check. The error could be in your check.
  
  Tip 2: In a test, do not erase your solution if you a spot an error or think you have made one. Do not erase your solution until you have written and checked a replacement. Better yet, cross out the solution instead of erasing it when you have the replacement. And if on a test, you do not have time to provide a replacement, give the solution and check, and write Oops! do not have time to correct in large print or boldly besides the check. The student who submits wrong a clearly written solution with a clearly written check, right or wrong, gets more respect and show a greater command of subject matter than the students who does not present work clearly. The advice do not erase your solution may help as well with instructors who are looking for reasons to reward your work. Good luck.
  
  Tip 3: In homework, if you have time to correct your work, do so. If not follow the advice in tip 2, and change your error declaration to : Oops! Did not make time to correct.
  

• Lesson 8. Correcting the Error. Doing long division properly (with extra zeroes) to get
  
  5626 = 375 × 15 + 1
  
  and then checking the results. It works.
  

• Lesson 9. Yet another insight into why long division method works. This lesson inspects the numbers and subtractions that appear in long division of the dividend 5626 by a divisor 15 to show why 5626 = 375 × 15 + 1
  

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• Lessons 10 and 11: More examples of long division.

• Lesson 12. The long division method in expanded form gives yet another explanation of why the LD method works.

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