For ten to twelve year olds
skills to check or develop
Place Value and Counts from 1 to 1 000 000 000
Site folder
Decimal Place Value cover the following and more.
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Dictation Exercise: Write numbers between 1 and 1000000 as
decimals when described in words.
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Writing Exercise: Given numbers between 1 as 1000000 in
written or oral form, produce their decimal representation.
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Skip Counting Continued: Add Counting by 10000, 50000 and
100000 to and from 1 000,000 to the skip counting skill list of
previous levels.
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Place Value Comprehension- USA and Modern British, Short Scale
Option: To understand the place value in long multidigit decimals
from 1 sextillionths to 999 sextillions, we rewrite the numbers in a
mixed 3-digit and word format in accordance with North American values
for billions, trillions, quadrillions, quintillions and sextillions.
Here
1 thousand = 1000 ones = 103 ,
1 million = 1000 thousand = 100, 000 = 106 ,
1 billion = 1000 million = 100, 000, 000 = 109
Give a lesson on powers of 10 before there presentation as above..
Digits Right of the Decimal Point: In reading aloud, decimal
places to the right of the decimal should be read in groups of three,
with extra padding on the right as needed to make a full group of
three: 422.345 678 890 would be read aloud or written in mixed
word & decimal format as 422 ones, 345 thousandths, 678
millionths, 890 billions Notice how 89 has become 890. Here the number
of ones, thousandths, millionths and on is kept in decimal form when
written, while being read aloud in expanded. So 422 is written as
shown but read aloud as four hundred and twenty-two.
Digits to Left of actual or implied Decimal Point: In the
decimal 185,501,456,423 the place value of the leading three digits
is not immediately obvious. The digits after it have to group into
threes and counted (there are 12) or the place value has to be found
in a backwards manner. We do that next not necessarily for the sake
of efficiency, but for the sake of student amusement in explaining
place value.
A Second Place Value Comprehension Example: Write
185,501,456,423 backward first - least important groups of 3
digits first - in the following mixed word & decimal format:
423 ones, 456 thousands, 501 millions and 185 billions
Following place value determination, we write the latter
forwards and so obtain the following place value interpretation
185 billions, 501 millions, 456 thousands and 423 ones.
for 185,501,456,423.
|
Digits on both sides of decimal point: When a decimal has a
digit on both sides of a decimal point, determine the place value of
those on the left first using the backward and forward method above:
Example: For the the decimal, 43,487, 044, 009 .
435 432 435 find the place value of the digits to the left of the
decimal in groups of three and left-overs,
9 ones, 44 thousands, 487 millions and 43 billions
-- most important last --
With the with place value of the leading part known,
43 billions, 487 millions, 44 thousands, 9 ones, 435
thousandths, 432 millionths and 400 billionths.
|
-
Place Value Comprehension- Traditional British, Long Scale
Option:
In the traditional British nomenclature,
1 thousand = 1000 ones = 103 ,
1 million = 1000 thousand = 100, 000 = 106 ,
1 Billion = 1000, 000 million = 100, 000, 000 = 1012 ,
with capitals used to distinguish the long scale values from the short
scale ones.
Digits Right of the Decimal Point: In reading aloud, decimal
places to the right of the decimal should be read in groups of three,
with extra padding on the right as needed to make a full group of
three: 422.345 670 would be read aloud or written in mixed word
& decimal format as 422 ones, 345 thousandths, 670 millionths
or as 422 ones, 345 670 millionths Here the number of ones,
thousandths (?) and millionths is kept in decimal form when written,
Digits to Left of actual or implied Decimal Point: In the
decimal 3, 345,085,501,456,423 the place value of the leading digits
is not immediately obvious. The digits after it have to group into
threes and counted - there are 12 - or the place value has to be
found in a backwards manner. We do that next not necessarily for the
sake of efficiency, but for the sake of student amusement in
explaining place value.
A second Place Value Comprehension Example: Write
3,345,085,501,456,423 backward first - least important groups
of 6 digits first - in the following mixed word & decimal
format:
456, 423 ones; plus 850,501 millions, 3,345 Billions,
Following place value determination, we rewrite the latter
forwards and so obtain the following place value interpretation
3,345 billions, 850,501 millions, and 456, 423 ones.
for 3,345,085,501,456,423.
|
Digits on both sides: When a decimal has a digit on both
sides of a decimal point, determine the place value of those on the
left first using the backward and forward method above:
Example: For the the decimal, 44,789,043,487, 044,
009.435 43 find the place value of
the digits to the left of the decimal in groups of three and
left-overs,
44,009 ones plus 43, 487 millions and 44,789 Billions
- most important last
With the with place value of the leading part known,
44,789 Billions, plus 43 487 millions plus 44,009 ones
and 435,430 millionths
|
Remark - a cosmetic preference: Six digits at time may be too
many for most students to grasp quickly. Three digits at a time
appears easier. The Standard International system with its 3 digit at
a time grouping may be better for UK students.
For example 437, 345,567, 670 may be read aloud backward
as 670 units, 567 kilounits 345 megaunits and 437 nanounits.
This SI or metric way could coexist or supplant the Canadian-American
way described above.
More on Addition and Subtraction of Decimals
Arithmetic with multidigit Decimals
You child should be add columns of multidigit decimals without and with
decimal points.
Add two to five multidigit decimal numbers with and without carries,
with places before and after the decimal point.
To do these or like exercises, students will have to be shown how to
align and place decimal points.
Subtraction Problems without and with multiple conversions
Learners ages 11 to 13 should be able to subtract decimals with digits
before and after decimal points without and with multiple conversions.
See the site folder
Decimal Comparison and Subtraction Methods
Reference: See the site folder
Decimal Comparison and Subtraction.
Subtract or find the difference between pairs of numbers 1 to 1000000
by decimals column methods? Test with examples that do not and then do
involve single and multiple conversions, especially in the case of
subtraction. See explanations in previous level and in the site folder
with
Decimal Comparing and Subtracting Methods of J-concept and notation
for conversions needed in comparison and subtraction.
Check the result of subtraction via an addition.
Emphasize again that when a check fails, the mistakes or mistakes are
between the the start of the solution and the end of the check, and may
occur in one or both.
Subtraction Shortcut: Given the sum of a first and second number
equals a third, observe that the first gives the value of the third minus
the second, and that the second gives the value of the third minus the
first.
The Comparison and Subtraction Connection
-
Show how to say which the most or least in a pair of numbers? Review
or explain again the use of the more and less than signs > and
< to indicate when one number is more than or less than another.
Show how to use conversions to compare numbers and to calculate by
how much one is more than or less than the other.
-
Extension: Explain again the use of the composite signs
> and < to indicate when one number is more than
or equal, or less than or equal another.
-
When one number is more than another, find out how much more by
subtraction.
-
When one number is less than another, find out how much less by a
subtraction.
Calculate Products under 10000 of multidigit numbers
Reference: Site folder
Decimal Multiplication Methods
-
The addition of 845 like terms 1323 may be computed by
summing the first five, then and finally 800 terms. That is done next
in the next example of decimal column method for multiplication
323
×845
1615
12920
258400 +
272935
|
Drill and Practice required to perfect this skill
-
Multiplication by four Digit Whole Numbers - Move Level VI or
VII: The addition of 845 like terms 1323 may be computed by
summing the first five, then and finally 800 terms. That is done next
in the next example of decimal column method for multiplication
Drill and Practice required to perfect this skill
Division Results Revisited
Here we will avoid ambiguous notation while buildinging fraction skills and sense. Observe the following.
-
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 may be 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 and improper fractions 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
Long Division
Aim Divide numbers in the range 1 to 100000 by single digit divisors
via short or long division methods, and thus find quotients and remainder.
-
Show learners know that the remainder plus (quotient times divisor)
yields the "original" number.
Teach the Short or Long Division methods for showing that for each
whole number N and each whole number divisor d, there is a unique
whole q - the quotient and a unique natural number r - the remainder
with 0 < r < d, such that N = q × d +r. For example for
the whole number N = 17 and divisor d = 3,
17 = 5 × 3 +2
So the formula N = q × d +r holds with q =5 and r = 2.
At this level, the short and/or long division methods with be taught
by example and rote. But the results can be checked by verifying the
calculated numbers q and r thus obtained satisfy
N = q × d +r
Reference: The format and procedure for long division is
shown in the site folder 1.
Decimal Long Division methods in (12 lessons) b>Note: Mastery
of long division with single digit divisors with result checking
requires and shows mastery of addition, subtraction and single
Note 1 : In the high school study of polynomials, a senior high
school mathematics required for calculus, long division for
polynomials (not decimals) appears. Long division for decimals is
preparation for that which should not be missed
Site folder
long division video-based lessons provides a thought based
development of long division methods for multidigit divisors with some
explanation of how or why long division works. The explanations or
additional insights are optional. The format in the videos. That format
(or similar one) will help develop skills and confidence.
Long Division methods employ multiplication and subtraction to arrive at
quotients and remainders.
As preparation for the example below we list the first multiples 1 to 10
times the divisor.
1 2 3 4 5 6 7 8 9 10
23 46 69 92 115 137 161 184 207 230
For 1 and 2 digit divisors, such preparation exercises multiplication
skills. The list may written horizontally as here, or vertically on one side or
another of the long division calculation as in site webvideo examples.
20789 | 20789
--------- | --------
23 | 478155 | 23 |478155
- 460000 as 23 x 2 = 46 --> 20000 | 46 (23 x 2 -> 2)
------ | - --
18155 |* 18
-16100 23 x 7 = 161 --> 700 | 00 (23 x 0 -> 0)
----- | ---
2055 |* 181
-1840 23 x 8 = 184 --> 80 | 161 (23 x 7 -> 7)
----- | ---
215 23 x 9 = 207 --> 9 | 205
-207 | 184 (23 x 8 -> 8)
---- | ---
8 Last Leftover or | 215
remainder is 8. | 207 (23 x 9 -> 9)
| ---
| 8 (less than 23)
| Stop.
|
The example here illustrates long division methods with and without extra
zeroes to serve as place holders. Long division process requires multiplication
and subtraction skills, and in doing so tests them. Difficulty here points to a
check orreview of such skills. The remainder has to be less than the divisor.
All arithmetic methods can be checked. The check for long division
consist of the verifying the following:
Dividend = quotient × divisor + remainder
Here we need to verify
478155 = 20789 × 23 + 8
While students on paper figuring skill is not certain we require the
following the calculation of 20789 × 23 via a column method
20789 ×23 62367
+415780
4748147
Addition of the remainder 8 gives the original dividend 478155
For if or when the check fails, tell students the error in their
figuring will lie between their reading of the question and the end of
the check. Tell students that on homework and in class activities or
tests, that checks should done before written work is submitted, so
they can identify or correct their mistakes before handing their work.
Further tell students not to erase their written work is a check fails.
Tell them instead to cross it out lightly, and to submit it and a new
solution with their homework or test answers. Explain that the failure
of check may be due to a fault in the check. Explain too that your need
to see their written work not just for marking, but to better identify
their skill level. Explain too that lightly crossed out work, if neatly
done, may get credit. Finally, the last reason for handing in work for
a which a check fails is the possibility that the work is correct
because of a fault in the check.
Divisibility and Remainder Calculation Rules
Teach some by rote for now. The results can be verified directly. That
lessens the need for explanation how and why these rules or methods work.
Examples or rule usage and Explanations or hints as to why these rules
work appear in the site folder
Remainder Arithmetic and Divisibility Calculators or
quotient-remainder long or short division methods can be used to check
results.
Last Digit Rule: For factors 2 and 5 of 10, and for 10 itself,
the remainder of a whole number on division by each equals the remainder
on division of the last digit.
Examples:
2349 for division by 5 has the same remainder as its last digit 9 on
division by 5. So the remainder is 4.
2349 for division by 2 has the same remainder as its last digit 9 on
division by 2. So the remainder is 1.
493 for division by 5 has the same remainder as its last digit 3 on
division by 5. So the remainder is 3.
23 for division by 2 has the same remainder as its last digit 3 on
division by 2. So the remainder is 1.
2349 for division by 10 has the same remainder as its last digit 9 on
division by 5. So the remainder is 9.
In consequence:
(a) the remainder on division by 5 will be zero if the last digit is a 0
or 5.
(b) the remainder on division by 2 will be 0 and the number will said to
be even if the last digit is 0, 2, 4, 6 or 8 - in other words if the last
digit is even.
(c) the remainder on division by 2 will be 1 and the number will be said
to be odd if the last digit is 1, 3, 5, 7 or 9 - in other words if the
last digit is even.
(d) the remainder on division by 10 will be zero if the last digit is a
0.
Question: Which two last digits will give a remainder of 2 for
division by 5.
The Sum of Digits Rule for 3 and 9: A whole number has the same
remainder on division by 3 and by 9 as the sum of the digits in its
decimal representation.
The sum of the digits is much smaller than the original number. So it is
easier to determine the remainder of the sum.
Going Further: A whole number has the same remainder on division
by 9 (or 3) as the sum of the remainders for each of its digits on
division by 9 (respectively 3). Further, the remainder of a whole number
after division by 3 equals the remainder after division by 3 of the
remainder after division by 9.
Primes and Composite Numbers
The lesson below illustrates the following points:
-
With the aid of area calculation examples imply that the product of
two whole numbers where both factors are more than one is also more
than both of the factors. Thus a product of two whole numbers > 1
is more than both.
-
Say a whole number is composite if it is the product of two smaller
whole numbers. Use 10, 12, 15 and even 20 times table to identify
composite numbers.
-
Say a whole number is prime if it is not the product of two smaller
whole numbers. Use 10, 12, 15 and even 20 times table to identify the
all primes less than 10, 12, 15 and 20 respectively.
-
Show how to obtain the prime factorization or decomposition of whole
numbers with the the aid of the divisibility rules.
-
Show how to the use the squares of leading primes 2, 3, 5, 7 and 11
to identify all composite and prime whole numbers less than 169, and
to obtain the prime factorization of the composite whole numbers
-
In the development of fraction skills, show how to use primes and
prime factorization of numerators and denominators to simplify
fractions and products quickly by cancellation of common prime
divisors and any larger common factor of opportunity that appears.
Lesson: Introduction to Primes and Composites
Prime and Composite Whole Numbers less than 16
A whole number is composite if is given by the producd of
smaller whole numbers, with each factor greater than one. A whole
number is prime if it is greater than one and it is not
composite.
Examples of Composite Numbers
The blue part of the times table consists of composite numbers.
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1
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2
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3
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4
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5
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7
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7
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21
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49
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56
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48
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72
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80
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9
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27
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45
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63
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81
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All the numbers in the blue and grey cells of the above table are
composite. They are product of smaller whole numbers.
Identifying Primes with the 10 times table
The products of all pairs of whole numbers <5 and > 1 appear
in the grey cells. None of those products equals 5. So the number 5
is not composite. It is prime. By inspection, we like wise observe
the whole numbers 7, 3 and 2 are also prime.
The 10 times table gives products of all pairs of whole numbers
< 11 and > 1. Since 11 does not appear in the 10 table table,
eleven is prime.
Conclusion: From the 10 times table, the numbers 2, 3, 5,
7 and 11 are prime.
Identifying Primes with the 15 times table
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From the larger 15 times table, we observe that the 13 are also
prime because it does not appear among the products of the numbers
2 to 12. The 12 times table would have sufficient here to give that
result.
Conclusion: From the 12 times table, the numbers 2, 3, 5,
7, 11 and 13 are prime.
|
Two webpages
-
Prime Factorization method [Square Based]
-
examples
finish the primary and high school development of primes. Most likely,
the lesson ideas in these pages are best reserved for older students. But
you may see some merit in the earlier use.
Motivation for Prime Number Skills
Prime number factorization - also called decomposition - can be of
service in the exact arithmetic find in algebra in the expression of
fractions and of roots in - what we call by convention - simpler forms.
Exact arithmetic with fractions with or without the future service of
prime numbers develops fractions skills and sense - that may have take
home value. The mastery of primes and prime factorization, and the
associated exact arithmetic represents the earliest skill in pre-college
mathematics whose value is intellectual and/or preparation for college
studies in business, science, technology, engineering and mathematical
fields. In general mastery of figuring skills with decimals and fractions
exposes students to the domino effect of mistakes in multi-steps written
methods, and exposes students to the strong role notation in writing and
doing the steps in a way that the writer and others may see and check as
done or later. With that arithmetic skill may be seen and verified or
corrected.
Fraction and Measurement Skills
Students may cover the following between ages 9 and 10.
Arithmetic With Fractions and Mixed Numbers
Arithmetic and Comparison Questions with Like Denominators
The following examples describe with words operations and
indicate their geometric illustration.
Show how improper fractions are possible
proper fractions result from the division
of a single object into pieces of equal value: halves, thirds,
quarters and so on. With such divisions it is possible to have
upto two halves, three thirds, four quarters and so on. But in
measuring lengths and areas with units of measure, we have 3 ×
&\frac12; units is 1 & ½ units. So improper
fractions may appear in counting how many half units, third
units, quarters units there are in a length or measure which is
more than one unit.
|
Addition:
3 times a quarter + 2 times a quarter = [3+2] times a
quarter = 5 quarters
In fraction the foregoing quarters counting argument becomes
\[ \frac34 + \frac24 = \frac{3+2}4 = \frac54 \]
Geometric Model: Declare the length
of a line segment (measured or not) to be a unit length.
Illustrate the foregoing by drawing three quarters of this
length and two quarters side by side and show by counting
quarter lengths, the result is 5 quarters long.
In Fraction Notation:
Subtraction:
7 times a third - 5 times a third = [7-5] times a
third = 2 thirds
In fraction the foregoing thirds subtraction argument becomes
\[ \frac73 - \frac53 = \frac{7-3}3 = \frac23 \]
Geometric Model: Declare the length
of a line segment (measured or not) to be a unit length.
Illustrate the foregoing by drawing 7 thirds and
subtracting 5 thirds. show by counting quarter lengths,
the result is 2 thirds long.
Multiplication by a whole number:
3 times [4 fifths] = [3 times 4] fifths = 12 fifths.
In fraction the foregoing counting argument becomes \[ 3
\times \frac45 = \frac{3 \times 4}5 = \frac{12}5 \]
The right hand side is just $\frac45+\frac45+\frac45$
Geometric Model: Declare the length
of a line segment (measured or not) to be a unit length.
Illustrate the foregoing by drawing 4 fiths, three times
side-by-side. Counting, repeated addition or
multiplication gives 12 fifths.
Comparison:
10 sevenths is more than 4 sevenths by 6 sevenths.
since 10 its = 4 its + 6 its for any kind of its.
Reading the symbol $\gt$ as more than we may write
\[ \frac{10}7 \gt \frac 47 \quad \mbox{ by } \quad 67 \]
since \[ \frac{10}7 = \frac 47 + \quad 67 \]
Geometric Model: Declare the length
of a line segment (measured or not) to be a unit length.
Illustrate the foregoing by drawing 4 fiths, three times
side-by-side. Counting, repeated addition or
multiplication gives 12 fifths.
Multiplication by a Unitary Fraction: What is a
quarter of 36 tenths?
The question is like asking what is a quarter of 36 ones or
36 its. Their answers are 9 ones or 9 its because 36 = 4 × 9.
Answer: 36 tenths = 4 times 9 tenths. So a
quarter of 36 tenths is 9 tenths.
Note again The calculation is possible and easy as
the numerator 36 is a multiple of 4.
In fraction notation \[ \frac14 \times \frac{36}{10}
=\frac9{10} \]
Geometric Model: Declare the length
of a line segment (measured or not) to be a unit length.
Illustrate the foregoing by drawing 36 tenths of it.
Then show that grouping the tenths into 9 at
a time divides the 36 tenths long line segment into four
equal parts.
Multiplication by a Simple Fraction: What is 5
eighths of 24 sevenths?
Here an eigth of 24 ones or its is 3 ones or its. In
consequence, five eigths of 24 ones or its will be five times
as many ones or its. That gives 15 ones or its.
Answer: 24 sevenths = 8 times 3 sixteenths. So
an eighth of 24 sevenths is 3 sevenths, and hence 5
eighths of 24 sevenths would be 5 × 3 sevenths = 15
sevenths.
Note again:" The calculation is possible as the
numerator 24 is a multiple of 8.
In fraction notation, we may record the foregoing reasoning
as follows.
\begin{eqnarray*} \quad \\ \frac58 \times \frac {24}7
&=& 5 \times \left[\frac18 \times \frac {24}7
\right] \\ &=& 5 \times \frac37 \\ &=&
\frac{15}7\\ &=& 2 +\frac17 \\ \quad
\end{eqnarray*} Geometric Model:
Declare the length of a line segment [measured or not] to
be a unit length. Illustrate the foregoing by drawing 24
seventhss of it. Then show that
grouping the sevenths into threes at a time divides the 24
sevenths long line segment into 8 equal parts - eighths of
it. Now five of those eighths would 5 x 3 sevenths = 15
sevenths.
|
Division
Leading Questions and Examples:
-
What fraction of 2 is 1? Answer one-half or $\frac12$
-
What fraction of 3 is 1? Answer one-third or $\frac13$
-
What fraction of 4 is 1? Answer one-quarter or $\frac15$
-
What fraction of 5 is 1? Answer one-fifth or $\frac15$
-
What fraction of 3 is 2? Answer two-third or $\frac23$
-
What fraction of 3 is 3? Answer three-third or $\frac33$
or 1.
-
What fraction of 4 is 2? Answer two-quarter or $\frac24$
or $\frac12$ - a half
-
What fraction of 8 is 5? Answer five eighths or $\frac58$
More Leading Questions and Examples:
-
How many whole times does 4 go into 12 and what is the
remainder? Here the number 4 is the divisor, and the
number 12 is the dividend.
Answer: 3 whole numbers exactly - exactly because the
remainder is zero.
-
How many whole times does 5 go into 14 and what is the
remainder?
Answer: 2 whole numbers with a remainder of 3 as $14 = 2
\times 5 + 4$
Observe the remainder 4 is 4-fifths of 5.
-
How many whole times does 3 go into 25 and what is the
remainder?
Answer: 8 whole numbers with a remainder of 1 as $24 = 8
\times 3 + 1$
Observe the remainder 1 is one-third of 3.
-
How many whole times does 5 go into 33 and what is the
remainder?
Answer: 6 whole numbers with a remainder of 3 as $33 = 6
\times 5 + 3$
Observe the remainder 3 is 3-fifths of 5.
Still More Leading Questions and Examples:
-
How many times does 5 go into 20? Here the number 5 is
the divisor, and the number 20 is the dividend.
Answer: Exactly 4 times since $4 \times 5 = 20$
-
How many times does 4 go into 26?
Solution: $26 = 6\times 4 + 2$ where 2 is 2-quarters of
4. Thus the answer is $4 +\frac24 = 4+\frac12$ times
exactly.
-
How many times does 10 go into 46?
Solution: $46= 4\times 10 + 6$ where 6 is 6-tenths of 10
or in 3-fifths of 10 since one-fifth of 10 is 2. Thus the
answer is $4 +\frac6{10} =4 +\frac35$ times exactly.
-
How many times does 7 go into 18?
Solution: $18 = 2\times 7 + 4$ where 4 is 4 sevenths of
4. Thus the answer is $2 +\frac47$ times exactly.
All answers but the first are mixed numbers. We may
rewrite all as improper fractions.
Rewriting the answers as improper fractions
-
How many times does 5 go into 20?
Answer: Exactly 4 times since $4 \times 5 = 20$
Raising terms gives $4 =\frac41 =\frac{4 \times 5}5 =
\frac {20}5$
Thus the answer is equivalent to an improper fraction -
the dividend 20 over the divisor 5. Lowering terms in the
latter fraction gives the mixed number answer.
-
How many times does 4 go into 26?
Solution: $26 = 6\times 4 + 2$ where 2 is 2-quarters of
4. Thus the answer is $6 +\frac24 = 4+\frac12$ times
exactly.
Observe the number
\begin{eqnarray*} 6 +\frac24 & = &\frac {6\times
4}4+ \frac24 \\ &=& \frac{6\times 4 + 2}4 \\
&=& \frac{26}4 \end{eqnarray*} Thus the answer is
equivalent to an improper fraction given by the dividend
26 over divisor 4. Lowering terms in the latter fraction
would give and would have given the mixed number answer.
-
How many times does 10 go into 46?
Solution: $46= 4\times 10 + 6$ where 6 is 6-tenths of 10
or in 3-fifths of 10 since one-fifth of 10 is 2. Thus the
answer is $4 +\frac6{10} =4 +\frac35$ times exactly.
Observe the number
\begin{eqnarray*} 4 +\frac4{10} & = &\frac
{4\times 10}{10}+ \frac6{10} \\ &=& \frac{4\times
10 + 6}{10} \\ &=& \frac{46}{10} \end{eqnarray*}
Thus the answer is equivalent to an improper fraction
given by the dividend 46 over divisor 10. Lowering terms
in the latter fraction would give and would have given
the mixed number answer.
-
How many times does 7 go into 18?
Solution: $18 = 2\times 7 + 4$ where 4 is 4 sevenths of
4. Thus the answer is $2 +\frac47$ times exactly.
Observe the number
\begin{eqnarray*} 2 +\frac4{7} & = &\frac
{2\times 7}{7}+ \frac4{7} \\ &=& \frac{2\times 7
+ 4}{7} \\ &=& \frac{18}{7} \end{eqnarray*} Thus
the answer is equivalent to an improper fraction given by
the dividend 18 over divisor 7. Lowering terms in the
latter fraction would give and would have given the mixed
number answer.
|
|
The next step in fraction skill development follows.
Addition, Comparison and Subtraction with unlike denominators
The addition, comparision and subtraction of the fractions \[
\frac{7}{11} \qquad \frac{5}{11} \qquad \mbox{ and }\qquad \frac{23}{11} \]
is easy due to the presence of like denominators. Now imagine in counting
or measuring, we obtain the following mixed numbers and fractions \[
\def\u{\mbox{ its}} \frac{2}{3} \u \qquad \frac{5}{4}\u \qquad \mbox{ and
}\qquad 2+ \frac{5}{6}\u \]
How do we add, compare and subtract these counts or measures? The first
fraction is 2 times one-third of a unit, the second fraction is 5 times a
quarter of a unit, and the last fraction is 2 units plus 5 times a sixth
of a unit, or [12 + 5] times a sixth of unit. But 12 is a common multiple
of all three denominators 3, 4 and 6 since \[ 12 = 3 \times 4 = 4 \times
3 = 2 \times 6 \]
All the previous measures can be converted from multiples of $\frac13$,
$\frac14$ and $\frac16$ to multiplies of $\frac1{12}$ by raising terms.
This raising of terms may come from first principles: The observations
[explain why to students]
\begin{eqnarray*} \frac14 \mbox{ of } \frac 13 &= & \frac1{12} \\
\frac13 \mbox{ of } \frac 14 &= & \frac1{12} \\ \frac12 \mbox{ of
} \frac 16 &=& \frac1{12} \end{eqnarray*}
gives
\begin{eqnarray*} \frac 13 &= 4 \times \frac1{12}& =\frac4{12} \\
\frac 14 &= 3 \times \frac1{12} &=\frac3{12} \\ \frac 16 &= 2
\times \frac1{12} & =\frac3{12} \end{eqnarray*}
Therefore
\begin{eqnarray*} \frac{2}{3} \u & = 2 \times \frac 13 \u = 2 \times
\frac 4{12} \u & = \frac {8}{12} \u \\ \frac{5}{4}\u & = 5 \times
\frac 14 \u = 5 \times \frac 3{12} \u & = \frac {15}{12} \u \\
\frac{17}6 &= 17 \times \frac 16 \u = 17 \times \frac 2{12} \u &
= \frac {34}{12}\u \end{eqnarray*}
So these counts or measures are now expressed as multiples of $\frac
1{12}\u$. In that form the counts or measures are easily added, compared
and subtracted. For addition, comparision and subtraction, conversion
into other forms as well will work. But this conversion works for us.
Above we used \[ \left(2+ \frac{5}{6}\right)\u = (12+ 5)\times \frac 16
\u 17 \times \frac 16 \u \] Often, the reasoning above would be followed
without writing a unit $\u$ for counting or measurement. But our notions
of equivalent fractions, equivalent mixed numbers and later decimals
equivalent to them reflects their use as multipliers of a unit.
Conversion to add, compare or subtract is not new. For example the count
or number 121 is more than the count 63 despite the first number having
fewer ones and tens. That is because the 100 in 121 can be expressed as 9
tens plus 9 ones plus 1 one. Addition, comparison and subtraction
operations on decimals and on fraction employ conversions as needed.
In the foregoing, the number 12 is the least common multiple of 3, 4 and
6. That is given. In showing students how to find a common denominator
for the addition, comparision or subtraction of a pair of fractions, the
list method may be employed. The benefit of using the least common
multiple leads to smaller in the calculation of sum or difference where
by convention the addition or subtraction with the aid of a like or
common denonominator is followed by simplification. One needs to say to
student that addition followed by simplification together usually
requires less work if one employs a least common denonominator in place
of any other common denominator. With the decimal representation of whole
numbers in the numerators and denominators, exceptions can be found. With
practice, students may learn to recognize and exploit the exceptions. The
usual case is the one to teach first.
Raising terms could also have been done mechanically as follows
\begin{eqnarray*} \frac{2}{3} \u & = \frac {2 \times 4}{3 \times 4}
\u & = \frac {8}{12} \u \\ \frac{5}{4}\u & = \frac{5 \times 3}{3
\times 4} \u & = \frac {15}{12} \u \\ \frac{17}6 \u & = \frac{17
\times 2}{2 \times 6} \u & = \frac {34}{12}\u \end{eqnarray*}
Most students will like find this mechanical raising of terms easier to
follow than the fuller and likely more overwhelming explanatiosn above.
But some students will not understand nor accept the mechanical approach
without a fuller explanation. Most likely in place of the exposition
above, explaining how to raise terms mechanically should be done first by
rote or with explanation in accordance with the wants, needs and
abilities of your students, individually or in groups.
Mutliplication
Product \[ \frac34 \times \frac{20}{7} = 3 \times \mbox{ one quarter of }
20 \times \frac17 \] \] is easily found because a quarter of 20 is 5.
Thus, writing $\times$ instead of the word "of" gives \[ \frac34 \times
\frac{20}{7} = 3 \times 5 \times \frac17 = \frac{15}7 \] Or \[ \frac34
\times \frac{20}{7} = \frac34 \times \frac{5 \times 4}7 = \frac{3 \times
5}7 = \frac{15}7 \] The calculation can be done mechanically by
cancelling the factor 4 common in the denominator of the first factor
with the factor 4 in the numerator of the second factor. By raising
terms, all products of fractions can be calculated mechanically. An
example follows. \begin{eqnarray*} \frac58 \times \frac73 & =&
\frac58 \times \frac{8 \times 7}{8 \times 3}\\ & =& 5\times
\frac{ 7}{8 \times 3}\\ & =& \frac{5\times 7}{8 \times 3} =
\frac{35}{24} = 1 + \frac9{24} \end{eqnarray*}
Several examples like this will suggest the mechanical rule
Multiply the numerators, multiply the denominators
or
Multiply the tops, multiply the bottoms
for the calculation of products of two or more fractions. For two
fractions described using the shorthand role of letters, \[ \frac AB \times
\frac CD = \frac {A \times C}{B \times D} \] In the second example above 5
would give the value of A, 8 would give the value of B, 7 would give the
value of C, and 3 would give the value of D. The foregoing represents an
algebraic description of the more general, multiply the tops, multiply
the bottoms rule. Understanding the formula would be a small, optional
step for student of this age level, in the introduction of algebra.
Division with Like Denominators
The question of how many whole times 5 its goes into 18 its has the
answer 3 whole times. The remainder is 18 its - 3 × 5 its = [18 -15] its
= 3 its. The remainder is 3 fifths of 5 its.
The related question question of how many times 5 its goes into 18 its
has the mixed number answer: 2 and 3 fifth whole times. Here \[ 3 + \frac
35 = \frac{5 \times 3}5 + \frac 35 = \frac{18}5 \] Because of that we may
say and write 18 its divided by 5 its is \[ \def\its{ \mbox{ its }} [18
\its] \div [5 \its] = \frac {18}5\] The latter equals $3 + \frac 35.$ Now
the use of the word times is agrees with \[ \frac {20}5 \times 5 \its =
10 \times \frac 15 \mbox{ of } 5 \its = 20 \its \] Thus 5 its goes into
20 its, exactly $\frac{18}5 = 3 + \frac35$ times.
Replacing the it
The it in the foregoing can be any count, amount or measure. In the
equation \[ [18 \its] \div [5 \its] = \frac {18}5\] we may take the its
to be halves, thirds, quarters, fifths, sixths, 99ths, and so on. One
example would be \[ [18 \frac 17] \div [5 \frac17] = \frac {18}5\] The
latter can be written as \[ \frac {18}7 \div \frac 57 = \frac {18}5 \] In
general, \[ \frac {18}m \div \frac 5m = \frac {18}5 \] for each whole
number we may choose to substitute for m. One it here would be one m-th.
The general rule is as follows.
When a first fraction has the same denominator as a second, nonzero,
fraction, the first fraction divided by the second is a fraction whose
numerator is that of the first, and whose denonominator is given by that
of the second.
In shorthand notation, this slogan or rule says \[ \frac Am \div
\frac Cm = \frac AC \quad \mbox{ provided } C \ne 0 \] That provides a
mechanical pattern to follow in the like-denominator case
For addition, comparison and subtraction with unlike denominators, we
raise terms to obtain like denominators. Raising terms to obtain like
denominators can also be done to show and say how to divide fractions
with unlike denominators. That process is left for next year. It leads to
the mechanical rule, to divide by a nonzero fraction, multiply by its
reciprocal.
Evaluation of Area Formulas
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.
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.
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.
For exercise asking for the value of A where A = an arithmetic
expressions instruct students to use a similar vertical alignment of
equal signs:
A = an arithmetic expression
= the formula with lengths replaced by their values
= a simplified arithmetic expression
= another simplification
= ...
= simplified results
Here evaluation consist of a sequence of calcuation in which
subexpressions may be replaced by simpler ones, or values, one at a time,
one after another. See the study tip:
Written formats for formats for developing and showing skill.
End Notes A and B.
A. Mixed Measures:
In describing lengths of time, we may talk about days, hours,
minutes and even seconds, and by convention all mixed units in this
description. Thus we speak of 2 hours and 15 minutes without
convention requiring a conversion into a large number of minutes,
or into a small mixed number 2¼ of hours. The length seven
quarters of a meter describes the same length as 1.75 meters and 1
meter 75 cm. In describing long thin rectangular, we say the length
is 5 m and the width is 80 cm. For some that description may be
more pleasing than saying 500 cm by 80 cm, or 5 m by 0.8 m. How
we describe measures does not affect them but the numbers in the
description depend on the choice of units. The area of the
foregoing rectangular is given by the product of it dimensions in
square meters, in square meters or even as a number of meter-cm.
The latter would be the area of a rectangle with length one meter
and width one cm. The area of the rectangular is given by three
different expressions
A = 5 m × 80cm
A = 5 m × 0.80 m
A = 500 cm × 80 cm
The foregoing leads to three answers: 400 m × cm, 4
m2 and 40000 cm2 for the area. Each has the
same value since 1 m2 = 10000 cm2 and 1 m ×
cm = 100 cm2
There is no harm in using mixed units of measures in evaluating
formulas as long as the unit carried through the steps. Conversion
of the the different units for a measure may be done in any step or
in the original data. What is important is that a measure be
describe as a number of units and not by by a number alone. In
general, arithmetic with measures may be done with mixed unit of
measures alone or multiplied and divided by others. While the form
of a result may vary, its values will not. The following are
example of addition using and keeping mixed units of time
measure:
5 hours, 30 minutes + 4 hours, 20 minutes = 9 hours, 50
minutes
2 hours, 50 minutes + 3 hours, 50 minutes = 6 hours, 40
minutes
In the latter, a conversion of 100 minutes into 1 hour, 40 was
don.
An operational mastery of fractions with units will help
Note: While pure mathematics may avoid the carrying of units
in and through calculations by selecting a a consistent system of
units for calculations, the intellectual overhead in selecting that
consisting system and converting all units of measure to it may be
avoided by using and carrying units of measure through
calculations. That is the practice in senior high school and
college courses in chemistry and physics. Moreover, fraction skill
with units present, and manipulations with products and quotients
of units of measure in general, useful for the description of
speed, rates and proportionality constants. There is more
immediate motivation and context to this manipulations with units
of measure as is or multiplied by numbers, than there is to the
combination of monomials in letters w, x, y and z in products and
quotients.
B. Mixed Numbers
In the early development of decimal notation, the digits 1 to 9
represent simple numbers while two or three digit numbers like 42
and 368 represented mixed or compound numbers. The latter
represent the sum of 4 tens and 2 ones, and the sum of 3 hundreds,
6 tens and 8 ones respectively. So we count in mixed groups:
hundreds, tens and ones. Now the fraction 5 quarter- meters
represent a whole number of the unit one quarter meter. We may
write 5 quarter meters as one meter and one quarter meters. That
mixes the unit of length measure one meter with the unit one
quarter meter. Now the mixed number 4½ stands for 4 wholes and one
half a whole. The mixed units of counting here are ones and one
half. Now in counting or measuring we may find ourselves with 4
wholes, ¾ of a whole, and ½ a whole. The total count or measure
will be 5¼ wholes. The underlying notion here is that we may count
and measure with whole and with multiples of unit-numerator
fractions, more easily written here in word form as one half, one
third, one fourth, one fifth and so on. A mixed number or measure
is equal to a whole number of ones plus a proper fraction:
multiples of fractional units. In general, we add, subtract,
multiply and even divide mixed numbers and measures of ones and
units where the units of counting and measure may be different.
The practice of raising terms for the sake of addition, subtraction
or comparison is resembles the conversion of mixed units of measure
into multiples of a common unit. For example - given in a long
format chosen to illustrate ideas)
2
3
|
+
|
3
4
|
=
|
2
|
×
|
1
3
|
+
|
3
|
×
|
1
4
|
|
|
|
|
|
=
|
2
|
×
|
4
12
|
+
|
3
|
×
|
3
12
|
|
Convert one third units and one quarter wholes
into one twelfth units
|
|
|
|
=
|
|
|
17
12
|
|
|
|
|
|
Count number of twelfths : 8 = 9 = 17
|
|
|
|
=
|
|
1
|
5
12
|
|
|
|
|
|
Convert 12ths into wholes
(that resembles the conversion of more than 10 tenths
in wholes in addition with decimals.
|
Remark: Notions of mixed numbers and measures underlying
many arithmetic operations with counts, decimals, fractions and
measures. Clarification of those notions may help us to decide
whether or not, or to what extent we should discuss them in
developing mathematics skills.
|
More
-
Units of Measure for Quantities, physical or monetary. Metric.
Imperial. Origins of Various Units [Consult Dictionary or
Encyclopedia]. Conversion of physical units. Carrying units in
calculation. A child familiar with more than one system of units has a
cultural advantage over a child indoctrinated with only one system.
From the cultural perspective, describing the history of units of
measurements implies there is more to arithmetic and computation than
following rules for this or that. Discussion of units and their
origins, and their different types, is part of the history of
mathematics, science, technology and society.
-
Metric Regulations. Regulations requiring the use of metric [ok]
but also regulations forbidding use of imperial [bad]. The olde
addition of English pounds, shilling and pence provided a non-metric
& non-decimal example of how to carry or convert units from one
column into another. Miles, Yards, Feet, inches provide another
non-metric and non-decimal example of how to carry. Familiarity with
these non-decimal units may give a better understanding of the carrying
operations in decimal operations and in metric operations. Operations
with nonnumeric units can be employed to explain the advantages of
metric units [simplification in unit carries/conversion] and to
illustrate the carry/unit conversion process. See the previous item.
|
|
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
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
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
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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