3. Harder Cases - Convert to Compare and Subtract
The three videos below with some overlap show how to
convert decimals form of numbers so that comparison and direct
subtraction may done directly. The conversion leads to a modified
decimal notation with a signal letter J standing for the number
10. The letter J, the tenth letter in the Roman alphabet,
resembles an Aramaic symbol for 10. The capital letter X was considered
as alternative to J, but there are too many small letters x in
mathematics. The overlap between the videos stems from three
efforts or three starts to the video making process. Tip:
Read the questions below before watching the first video twice. View
the others if you need more help to do the exercises below.
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Video Display Area
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View First Flash Video
(Has a few seconds of silence at
the start. Wait for the silence to end)
Summary: This first, 5:31
minute video introduces a single letter J to denote the number
10, and employs J as another digit besides 0 to 9 in a
"decimal" representation of whole numbers. The
J-conversion of decimals (another way to borrow) leads to
place value comparison of whole numbers and leads to a
borrow-free (no more conversion) view of subtraction. More than
one J-conversion may be done in a single
subtraction.
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Summary: This second, 7:32
minute video provides a quick word about comparison of whole
numbers. (1) Expands decimal representation of whole numbers to
compare number of units, tens and hundreds in the first
instance where a place-by-place comparison of whole numbers
given in decimal form. (2) Compares 7864 with 10000 and
introduces J (value 10) as the tenth eleventh
digit and to provide an alternate way to give a "decimal"
representation of whole numbers.
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Summary: This duplicate lesson
employs a single letter J to denote the number 10, and employs
J as another digit besides 0 to 9 in a "decimal" representation
of whole numbers. The J-conversion of decimals (another
way to borrow) leads to place value comparison of whole
numbers and leads to a borrow-free (no more conversion) view of
subtraction. More than one J-conversion may be done in a single
subtraction.
5:35 minutes Comparison: Introduction of J
conversion for decimals with and without decimal point
Optional Lesson
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Try the first video first. and the remaining two, if need-be.
(As the author, I should return to this page, and pick one, or
replace all three by another. Will do that when I have time.)
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Notes
The above lessons employ direct digit-by-digit
comparison and subtractions for decimal counts and fractions where
possible, and failing that employ conversions (borrows) in a way that
makes direct or nearly direct digit-by-digit comparison and subtraction
possible. All the above explore and express these ideas in a longer
way, more accessible to students - we hope.
Comparison without and with Conversions:
- 956 is greater than 436 has the first number has the same number of
units, but more tens and more hundreds - no conversion required.
- 10000 is greater than 8269 as 1000 = 9999 + 1 = 9990 + 10 has more
thousands, more hundreds, more tens and more ones.
- 3234635 is greater than 18722 as 32465 = 3233635 + 1000 = 323365 +
999990 +10 has more units, more tens, more hundreds, more thousands and
so on than 18742 (maximal conversion method for comparison)
- 32 is greater than 18722 as 34635 = 24635 + 1000 = 24635 + 9900 + 100
has more units, more tens, more hundreds, more thousands and so on than
18722 (minimal conversion method for comparison)
Teaching Tip: Comparison by conversion (when necessary)
provides an alternative to the lexicographic
comparison of decimal counts and fractions. This alternative sets
the stage for subtraction.
Associated Subtraction by Conversion Method:
No conversions (borrows) are required to do the following
subtraction
5 6 4 9 9
3 8
- 2 8 7 1 6
as the first number has more units, tens, hundreds, thousands, and tens
of thousands, digit by digit, place value by place value.
The minimal conversion method for comparison implies the common (minimal)
conversion method for subtraction.
3 2 3 4 6 3 5
- 1 8 7 2 2
by conversion, that is by writing 30000 as 29000 + 1000 gives
2 9 10
3 2 3 6 6 3 5
- 1 8 7 2 2
Note: Comparison-Conversion lesson in this site area use a letter J
to stand for the number 10.
Here the 3 in the ten thousand position is crossed-out and replaced
by 2, 9 and 10 in the top row to indicate the
replacement or conversion of 30000 by/to 29000 + 1000. Here
position indicates place value. Following the conversion, in each column,
the rows to be added combined have more ones, tens, hundreds and so
the minuend than the subtrahend. So the difference is easily
computed
2 9 10
3 2 3 6 6 3 5
- 1 8 7 2 2
3 2 1 7 9 1 3
Remark: If a conversion method other than
minimal is employed, the subtraction will result in carries.
Exercise Set.
Teaching Tip: For some or many students, which
one is more than the other will be obvious in each pair from their
earlier mastery of counting and decimal notation.
Numbers 10 to 15 are labeled optional for students who have a previous
knowledge of how to compare.
Express each of the following numbers using J-Notation as in the above
lesson.
- A = 300
- B = 500
- C = 6000
- D = 5000
- E = 10000
- F = 80000
- G = 50
- I = 20
- H = 10
Use conversion and J notation when needed to compare the following
pairs of numbers
- 235 and 456
- 345 and 789
- 1000 and 673
- 1238 and 549
- 356 and 260
More than one solution may be correct.
15. For each of the following, compare the two numbers with the help of
digit-by-digit place value, or with the help of the J-conversion
presented in the above and further video lessons.
- Do you see the connection or difference between (E) and (F)?
- Do you see the connection or difference between (G) and (H)?
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
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if one or more explanations is not to liking, try another. It may
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
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Appetite.
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