Cognitive Guided Instruction - Revision history

Khanh at 18:03, 9 January 2022

2022-01-09T18:03:42Z

Khanh at 18:02, 9 January 2022

2022-01-09T18:02:07Z

Khanh: /* Successorship and size */

2022-01-09T17:58:01Z

Successorship and size

Khanh: /* Counting */

2022-01-09T17:56:35Z

Counting

Khanh: /* Subtraction */

2022-01-09T17:55:43Z

Subtraction

Khanh: /* What does it mean to multiply two natural numbers? */

2022-01-09T17:52:42Z

What does it mean to multiply two natural numbers?

Khanh: Created page with " primary/elementary mathematics symbols '''Elementary arithmetic''' is the simplified portion of arithmetic that includes the operat..."

2022-01-09T17:50:00Z

Created page with " thumb|primary/elementary mathematics symbols '''Elementary arithmetic''' is the simplified portion of arithmetic that includes the operat..."

Show changes

@@ Line 1: / Line 1: @@
 [[File:Arithmetic symbols.svg|thumb|primary/elementary mathematics symbols ]]
-'''Elementary arithmetic''' is the simplified portion of [[arithmetic]] that includes the operations of addition, subtraction, multiplication, and division. It should not be confused with elementary function arithmetic.
+'''Elementary arithmetic''' is the simplified portion of arithmetic that includes the operations of addition, subtraction, multiplication, and division. It should not be confused with elementary function arithmetic.
 Elementary arithmetic starts with the natural numbers and the written symbols (digits) that represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.
@@ Line 447: / Line 447: @@
 ==Tools==
 The abacus is an early mechanical device for performing elementary arithmetic, which is still used in many parts of Asia. Modern calculating tools that perform elementary arithmetic operations include cash registers, electronic calculators, and computers.

@@ Line 1: / Line 1: @@
 [[File:Arithmetic symbols.svg|thumb|primary/elementary mathematics symbols ]]
-'''Elementary arithmetic''' is the simplified portion of [[arithmetic]] that includes the operations of [[addition]], [[subtraction]], [[multiplication]], and [[division (mathematics)|division]]. It should not be confused with [[elementary function arithmetic]].
+'''Elementary arithmetic''' is the simplified portion of [[arithmetic]] that includes the operations of addition, subtraction, multiplication, and division. It should not be confused with elementary function arithmetic.
-Elementary arithmetic starts with the [[natural numbers]] and the written symbols ([[Numerical digit|digits]]) that represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a [[multiplication table]] to assist with multiplication and division.
+Elementary arithmetic starts with the natural numbers and the written symbols (digits) that represent them. The process for combining a pair of these numbers with the four basic operations traditionally relies on memorized results for small values of numbers, including the contents of a multiplication table to assist with multiplication and division.
-Elementary arithmetic also includes [[Fraction (mathematics)|fractions]] and [[negative numbers]], which can be represented on a [[number line]].
+Elementary arithmetic also includes fractions and negative numbers, which can be represented on a number line.
 ==The digits==
-Digits are the entire set of symbols used to represent numbers. In a particular [[numeral system]], a single digit represents a different amount than any other digit, although the symbols in the same numeral system might vary between cultures.
+Digits are the entire set of symbols used to represent numbers. In a particular numeral system, a single digit represents a different amount than any other digit, although the symbols in the same numeral system might vary between cultures.
-In modern usage, the [[Arabic numerals]] are the most common set of symbols, and the most frequently used form of these digits is the Western style. Each single digit, if used as a standalone number, matches the following amounts:<br>
+In modern usage, the Arabic numerals are the most common set of symbols, and the most frequently used form of these digits is the Western style. Each single digit, if used as a standalone number, matches the following amounts:<br>
-<big>0</big>, [[0 (number)|zero]]. Used in the absence of objects to be counted. For example, a different way of saying "there are no sticks here", is to say "the number of sticks here is 0".<br>
+<big>0</big>, zero. Used in the absence of objects to be counted. For example, a different way of saying "there are no sticks here", is to say "the number of sticks here is 0".<br>
-<big>1</big>, [[1 (number)|one]]. Applied to a single item. For example, here is one stick: <big>I</big><br>
+<big>1</big>, one. Applied to a single item. For example, here is one stick: <big>I</big><br>
-<big>2</big>, [[2 (number)|two]]. Applied to a pair of items. Here are two sticks: <big>I I</big><br>
+<big>2</big>, two. Applied to a pair of items. Here are two sticks: <big>I I</big><br>
-<big>3</big>, [[3 (number)|three]]. Applied to three items. Here are three sticks: <big>I I I</big><br>
+<big>3</big>, three. Applied to three items. Here are three sticks: <big>I I I</big><br>
-<big>4</big>, [[4 (number)|four]]. Applied to four items. Here are four sticks: <big>I I I&nbsp; I </big><br>
+<big>4</big>, four. Applied to four items. Here are four sticks: <big>I I I&nbsp; I </big><br>
-<big>5</big>, [[5 (number)|five]]. Applied to five items. Here are five sticks: <big>I I I&nbsp; I I</big><br>
+<big>5</big>, five. Applied to five items. Here are five sticks: <big>I I I&nbsp; I I</big><br>
-<big>6</big>, [[6 (number)|six]]. Applied to six items. Here are six sticks: <big>I I I&nbsp; I I I</big><br>
+<big>6</big>, six. Applied to six items. Here are six sticks: <big>I I I&nbsp; I I I</big><br>
-<big>7</big>, [[7 (number)|seven]]. Applied to seven items. Here are seven sticks: <big>I I I&nbsp; I I I&nbsp; I</big><br>
+<big>7</big>, seven. Applied to seven items. Here are seven sticks: <big>I I I&nbsp; I I I&nbsp; I</big><br>
-<big>8</big>, [[8 (number)|eight]]. Applied to eight items. Here are eight sticks: <big>I I I&nbsp; I I I&nbsp; I I </big><br>
+<big>8</big>, eight. Applied to eight items. Here are eight sticks: <big>I I I&nbsp; I I I&nbsp; I I </big><br>
-<big>9</big>, [[9 (number)|nine]]. Applied to nine items. Here are nine sticks: <big>I I I&nbsp; I I I&nbsp; I I I</big>
+<big>9</big>, nine. Applied to nine items. Here are nine sticks: <big>I I I&nbsp; I I I&nbsp; I I I</big>
-Any numeral system defines the value of all numbers that contain more than one digit, most often by addition of the value for adjacent digits. The [[Hindu–Arabic numeral system]] includes [[positional notation]] to determine the value for any numeral. In this type of system, the increase in value for an additional digit includes one or more multiplications with the [[radix]] value and the result is added to the value of an adjacent digit. With Arabic numerals, the radix value of ten produces a value of twenty-one (equal to {{nowrap|2×10 + 1}}) for the numeral "21". An additional multiplication with the radix value occurs for each additional digit, so the numeral "201" represents a value of two-hundred-and-one (equal to {{nowrap|2×10×10 + 0×10 + 1}}).
+Any numeral system defines the value of all numbers that contain more than one digit, most often by addition of the value for adjacent digits. The Hindu–Arabic numeral system includes positional notation to determine the value for any numeral. In this type of system, the increase in value for an additional digit includes one or more multiplications with the radix value and the result is added to the value of an adjacent digit. With Arabic numerals, the radix value of ten produces a value of twenty-one (equal to 2×10 + 1) for the numeral "21". An additional multiplication with the radix value occurs for each additional digit, so the numeral "201" represents a value of two-hundred-and-one (equal to 2×10×10 + 0×10 + 1).
-The elementary level of study typically includes understanding the value of individual [[Integer|whole numbers]] using Arabic numerals with a maximum of seven digits, and performing the four basic operations using Arabic numerals with a maximum of four digits each.
+The elementary level of study typically includes understanding the value of individual whole numbers using Arabic numerals with a maximum of seven digits, and performing the four basic operations using Arabic numerals with a maximum of four digits each.
 ==Addition==
@@ Line 60: / Line 60: @@
 |9 ||10 ||11 ||12 ||13 ||14 ||15 ||16 ||17 ||style="background:#cff"|18
 |}
-When two numbers are added together, the result is called a ''[[Summation|sum]]''. The two numbers being added together are called ''addends''.
+When two numbers are added together, the result is called a ''sum''. The two numbers being added together are called ''addends''.
 ===What does it mean to add two natural numbers?===
@@ Line 71: / Line 71: @@
 *and even sometimes ''separating'' ("Tom had some apples. He gave 3 to Jane. Now he has 5. How many did he start with?").
-Symbolically, addition is represented by the "[[plus sign]]": +. So the statement "three plus five equals eight" can be written symbolically as {{nowrap|3 + 5 {{=}} 8}}. The order in which two numbers are added does not matter, so {{nowrap|3 + 5 {{=}} 5 + 3 {{=}} 8}}. This is the [[commutative]] property of addition.
+Symbolically, addition is represented by the "plus sign": +. So the statement "three plus five equals eight" can be written symbolically as 3 + 5 = 8. The order in which two numbers are added does not matter, so 3 + 5 = 5 + 3 = 8. This is the commutative property of addition.
-To add a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the sum of the two digits. Some pairs of digits add up to two-digit numbers, with the tens-digit always being a 1. In the addition algorithm the tens-digit of the sum of a pair of digits is called the "[[Carry (arithmetic)|carry]] digit".
+To add a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the sum of the two digits. Some pairs of digits add up to two-digit numbers, with the tens-digit always being a 1. In the addition algorithm the tens-digit of the sum of a pair of digits is called the "carry digit".
 ===Addition algorithm===

@@ Line 145: / Line 145: @@
 If a number is the successor of another number, then the first number is said to be ''greater than'' the other number. If a number is greater than another number, and if the other number is greater than a third number, then the first number is also greater than the third number. Example: five is greater than four, and four is greater than three, therefore five is greater than three. But six is greater than five, therefore six is also greater than three. But seven is greater than six, therefore seven is also greater than three&nbsp;... therefore eight is greater than three&nbsp; … therefore nine is greater than three, etc.
-If two non-zero natural numbers are added together, then their sum is greater than either one of them. Example: three plus five equals eight, therefore eight is greater than three ({{nowrap|8 &gt; 3}}) and eight is greater than five ({{nowrap|8 &gt; 5}}). The symbol for "greater than" is &gt;.
+If two non-zero natural numbers are added together, then their sum is greater than either one of them. Example: three plus five equals eight, therefore eight is greater than three (8 > 3) and eight is greater than five (8 > 5). The symbol for "greater than" is >.
-If a number is greater than another one, then the other is ''less than'' the first one. Examples: three is less than eight ({{nowrap|3 &lt; 8}}) and five is less than eight ({{nowrap|5 &lt; 8}}). The symbol for "less than" is &lt;. A number cannot be at the same time greater and less than another number. Neither can a number be at the same time greater than and equal to another number. Given a pair of natural numbers, one and only one of the following cases must be true:
+If a number is greater than another one, then the other is ''less than'' the first one. Examples: three is less than eight (3 < 8) and five is less than eight (5 < 8). The symbol for "less than" is <. A number cannot be at the same time greater and less than another number. Neither can a number be at the same time greater than and equal to another number. Given a pair of natural numbers, one and only one of the following cases must be true:
 *the first number is greater than the second one,
 *the first number is equal to the second one,

@@ Line 155: / Line 155: @@
 To count a group of objects means to assign a natural number to each one of the objects, as if it were a label for that object, such that a natural number is never assigned to an object unless its predecessor was already assigned to another object, with the exception that zero is not assigned to any object: the smallest natural number to be assigned is one, and the largest natural number assigned depends on the size of the group. It is called ''the count'' and it is equal to the number of objects in that group. Counting can also been seen as the process of tallying using tally marks.
-The process of [[counting]] a group is the following:
+The process of counting a group is the following:
 #Let "the count" be equal to zero. "The count" is a variable quantity, which though beginning with a value of zero, will soon have its value changed several times.
 #Find at least one object in the group which has not been labelled with a natural number. If no such object can be found (if they have all been labelled) then the counting is finished. Otherwise choose one of the unlabelled objects.
@@ Line 166: / Line 166: @@
 Often, when counting objects, one does not keep track of what numerical label corresponds to which object: one only keeps track of the subgroup of objects which have already been labelled, so as to be able to identify unlabelled objects necessary for Step 2. However, if one is counting persons, then one can ask the persons who are being counted to each keep track of the number which the person's self has been assigned. After the count has finished it is possible to ask the group of persons to file up in a line, in order of increasing numerical label. What the persons would do during the process of lining up would be something like this: each pair of persons who are unsure of their positions in the line ask each other what their numbers are: the person whose number is smaller should stand on the left side and the one with the larger number on the right side of the other person. Thus, pairs of persons compare their numbers and their positions, and commute their positions as necessary, and through repetition of such conditional commutations they become ordered.
-In higher mathematics, the process of counting can be also likened to the construction of a [[One to one correspondence|one-to-one correspondence]] (a.k.a. bijection) between the elements of a set and the set {1, ..., n} (where n is a natural number). Once such a correspondence is established, the first set is then said to be of size n.
+In higher mathematics, the process of counting can be also likened to the construction of a one-to-one correspondence (a.k.a. bijection) between the elements of a set and the set {1, ..., n} (where n is a natural number). Once such a correspondence is established, the first set is then said to be of size n.
 ==Subtraction==

@@ Line 177: / Line 177: @@
 As with addition, there are other possible interpretations, such as ''motion''.
-Symbolically, the [[Plus and minus signs#Minus sign|minus sign]] ("−") represents the subtraction operation. So the statement "five minus three equals two" is also written as 5 − 3 = 2. In elementary arithmetic, subtraction uses smaller positive numbers for all values to produce simpler solutions.
+Symbolically, the minus sign ("−") represents the subtraction operation. So the statement "five minus three equals two" is also written as 5 − 3 = 2. In elementary arithmetic, subtraction uses smaller positive numbers for all values to produce simpler solutions.
-Unlike addition, subtraction is not commutative, so the order of numbers in the operation might change the result. Therefore, each number is provided with a different distinguishing name. The first number (5 in the previous example) is formally defined as the ''minuend'' and the second number (3 in the previous example) as the ''subtrahend''. The value of the minuend is larger than the value of the subtrahend so that the result is a positive number, but a smaller value of the minuend will result in [[negative number]]s.
+Unlike addition, subtraction is not commutative, so the order of numbers in the operation might change the result. Therefore, each number is provided with a different distinguishing name. The first number (5 in the previous example) is formally defined as the ''minuend'' and the second number (3 in the previous example) as the ''subtrahend''. The value of the minuend is larger than the value of the subtrahend so that the result is a positive number, but a smaller value of the minuend will result in negative numbers.
-There are several methods to accomplish subtraction. The method which in the [[United States of America|United States]] is referred to as [[traditional mathematics]] teaches elementary school students to subtract using methods suitable for hand calculation. The particular method used varies from country to country, and within a country, different methods are in fashion at different times. [[Reform mathematics]] is distinguished generally by the lack of preference for any specific technique, replaced by guiding 2nd-grade students to invent their own methods of computation, such as using properties of negative numbers in the case of [[Investigations in Numbers, Data, and Space|TERC]].
+There are several methods to accomplish subtraction. The method which in the United States is referred to as traditional mathematics teaches elementary school students to subtract using methods suitable for hand calculation. The particular method used varies from country to country, and within a country, different methods are in fashion at different times. Reform mathematics is distinguished generally by the lack of preference for any specific technique, replaced by guiding 2nd-grade students to invent their own methods of computation, such as using properties of negative numbers in the case of TERC.
 American schools currently teach a method of subtraction using borrowing and a system of markings called crutches. Although a method of borrowing had been known and published in textbooks prior, apparently the crutches are the invention of William A. Browell, who used them in a study in November 1937 [https://web.archive.org/web/20170811133911/http://math.coe.uga.edu/tme/issues/v10n2/5ross.pdf]. This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.
@@ Line 187: / Line 187: @@
 Students in some European countries are taught, and some older Americans employ, a method of subtraction called the Austrian method, also known as the additions method. There is no borrowing in this method. There are also crutches (markings to aid the memory) which [probably] vary according to country.
-In the method of borrowing, a subtraction such as {{nowrap|86 − 39}} will accomplish the ones-place subtraction of 9 from 6 by borrowing a 10 from 80 and adding it to the 6. The problem is thus transformed into (70 + 16) − 39, effectively. This is indicated by striking through the 8, writing a small 7 above it, and writing a small 1 above the 6. These markings are called ''crutches''. The 9 is then subtracted from 16, leaving 7, and the 30 from the 70, leaving 40, or 47 as the result.
+In the method of borrowing, a subtraction such as 86 − 39 will accomplish the ones-place subtraction of 9 from 6 by borrowing a 10 from 80 and adding it to the 6. The problem is thus transformed into (70 + 16) − 39, effectively. This is indicated by striking through the 8, writing a small 7 above it, and writing a small 1 above the 6. These markings are called ''crutches''. The 9 is then subtracted from 16, leaving 7, and the 30 from the 70, leaving 40, or 47 as the result.
-In the additions method, a 10 is borrowed to make the 6 into 16, in preparation for the subtraction of 9, just as in the borrowing method. However, the 10 is not taken by reducing the minuend, rather one augments the subtrahend. Effectively, the problem is transformed into {{nowrap|(80 + 16) − (39 + 10)}}. Typically a crutch of a small one is marked just below the subtrahend digit as a reminder. Then the operations proceed: 9 from 16 is 7; and 40 (that is, {{nowrap|30 + 10}}) from 80 is 40, or 47 as the result.
+In the additions method, a 10 is borrowed to make the 6 into 16, in preparation for the subtraction of 9, just as in the borrowing method. However, the 10 is not taken by reducing the minuend, rather one augments the subtrahend. Effectively, the problem is transformed into (80 + 16) − (39 + 10). Typically a crutch of a small one is marked just below the subtrahend digit as a reminder. Then the operations proceed: 9 from 16 is 7; and 40 (that is, 30 + 10) from 80 is 40, or 47 as the result.
-The additions method seem to be taught in two variations, which differ only in psychology. Continuing the example of {{nowrap|86 − 39}}, the first variation attempts to subtract 9 from 6, and then 9 from 16, borrowing a 10 by marking near the digit of the subtrahend in the next column. The second variation attempts to find a digit which, when added to 9, gives 6, and recognizing that is not possible, gives 16, and carrying the 10 of the 16 as a one marking near the same digit as in the first method. The markings are the same; it is just a matter of preference as to how one explains its appearance.
+The additions method seem to be taught in two variations, which differ only in psychology. Continuing the example of 86 − 39, the first variation attempts to subtract 9 from 6, and then 9 from 16, borrowing a 10 by marking near the digit of the subtrahend in the next column. The second variation attempts to find a digit which, when added to 9, gives 6, and recognizing that is not possible, gives 16, and carrying the 10 of the 16 as a one marking near the same digit as in the first method. The markings are the same; it is just a matter of preference as to how one explains its appearance.
-As a final caution, the borrowing method gets a bit complicated in cases such as {{nowrap|100 − 87}}, where a borrow cannot be made immediately, and must be obtained by reaching across several columns. In this case, the minuend is effectively rewritten as {{nowrap|90 + 10}}, by taking a 100 from the hundreds, making ten 10s from it, and immediately borrowing that down to nine 10s in the tens column and finally placing a 10 in the ones column.
+As a final caution, the borrowing method gets a bit complicated in cases such as 100 − 87, where a borrow cannot be made immediately, and must be obtained by reaching across several columns. In this case, the minuend is effectively rewritten as 90 + 10, by taking a 100 from the hundreds, making ten 10s from it, and immediately borrowing that down to nine 10s in the tens column and finally placing a 10 in the ones column.
 ==Multiplication==

@@ Line 235: / Line 235: @@
 Suppose there are five red bags, each one containing three apples. Now grabbing an empty green bag, move all the apples from all five red bags into the green bag. Now the green bag will have fifteen apples.<br>
 Thus the product of five and three is fifteen.<br>
-This can also be stated as "five times three is fifteen" or "five times three equals fifteen" or "fifteen is the product of five and three". Multiplication can be seen to be a form of [[Multiplication and repeated addition|repeated addition]]: the first factor indicates how many times the second factor occurs in repeated addition; the final sum being the product.
+This can also be stated as "five times three is fifteen" or "five times three equals fifteen" or "fifteen is the product of five and three". Multiplication can be seen to be a form of repeated addition: the first factor indicates how many times the second factor occurs in repeated addition; the final sum being the product.
 Symbolically, multiplication is represented by the ''multiplication sign'': ×. So the statement "five times three equals fifteen" can be written symbolically as
 :<math>5 \times 3 = 15.</math>
-In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. {{nowrap|5 ⋅ 3}}. In some situations, especially in [[algebra]], where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g. ''xy'' means {{nowrap|''x'' × ''y''}}. The order in which two numbers are multiplied does not matter, so that, for example, three times four equals four times three.  This is the [[commutative property]] of multiplication.
+In some countries, and in more advanced arithmetic, other multiplication signs are used, e.g. 5 ⋅ 3. In some situations, especially in algebra, where numbers can be symbolized with letters, the multiplication symbol may be omitted; e.g. ''xy'' means ''x'' × ''y''. The order in which two numbers are multiplied does not matter, so that, for example, three times four equals four times three.  This is the commutative property of multiplication.
-To multiply a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the product of the two digits. Most pairs of digits produce two-digit numbers. In the multiplication algorithm the tens-digit of the product of a pair of digits is called the "[[Carry (arithmetic)|carry]] digit".
+To multiply a pair of digits using the table, find the intersection of the row of the first digit with the column of the second digit: the row and the column intersect at a square containing the product of the two digits. Most pairs of digits produce two-digit numbers. In the multiplication algorithm the tens-digit of the product of a pair of digits is called the "carry digit".
 ===Multiplication algorithm for a single-digit factor===