# 1.4 subtraction w

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Published on February 6, 2014

Author: Tzenma

Source: slideshare.net

Subtraction http://www.lahc.edu/math/frankma.htm

Subtraction To subtract is to take away, or to undo an addition.

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A.

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference).

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,”

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away, from A.”

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away, from A.” Hence the statements “five apples take away three apples,” “three apples are taken away from five apples” “five apples minus three apples,” all mean 5 – 3 =2 .

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away, from A.” Hence the statements “five apples take away three apples,” “three apples are taken away from five apples” “five apples minus three apples,” all mean 5 – 3 =2 .

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away, from A.” Hence the statements “five apples take away three apples,” “three apples are taken away from five apples” “five apples minus three apples,” all mean 5 – 3 =2 . Mayan numerals are visually instructive for subtraction of small numbers.

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away, from A.” Hence the statements “five apples take away three apples,” “three apples are taken away from five apples” “five apples minus three apples,” all mean 5 – 3 =2 . Mayan numerals are visually instructive for subtraction of small numbers. For example, – = signifies 12 – 7 = 5, or that

Subtraction To subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A. We call the outcome “the difference of A and B” and it is often denoted as D and we write that A – B = D (the Difference). The following phrases are also translated as “A – B”: “A subtracts B,” “A minus B,” “A less B,” “A is decreased or reduced by B,” “B is subtracted, or is taken away, from A.” Hence the statements “five apples take away three apples,” “three apples are taken away from five apples” “five apples minus three apples,” all mean 5 – 3 =2 . Mayan numerals are visually instructive for subtraction of small numbers. For example, – = signifies 12 – 7 = 5, or that – = signifies 13 – 6 = 7.

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.”

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – =

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow –

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, For example, 634 – 87:

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, For example, 634 – 87: 634 87 –

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” when it is necessary. For example, 634 – 87 is: 634 87 –

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” when it is necessary. need to borrow For example, 634 – 87 is: 634 87 –

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” 14 when it is necessary. 2 need to borrow For example, 634 – 87 is: 634 87 –

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” 14 when it is necessary. 2 need to borrow For example, 634 – 87 is: 634 87 – 7

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” 14 when it is necessary. 2 need to borrow For example, 634 – 87 is: 634 87 – 7

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” 12 14 when it is necessary. need to borrow 5 2 For example, 634 – 87 is: 634 87 – 7

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” 12 14 when it is necessary. need to borrow 5 2 For example, 634 – 87 is: 634 87 – 47

Subtraction When there are not enough “ „s” to subtract, we have to convert a“ ” into “ .” This process is called “borrowing.” For example, 11 – 4 is – = borrow – = =7 More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them. For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.To subtract, 1. lineup the numbers vertically to match the place values, 2. then subtract the digits from right to left and “borrow” 12 14 when it is necessary. need to borrow 5 2 For example, 634 – 87 is: 634 87 – 5 47

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45.

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that.

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits.

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers.

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 78 – 30 = 94 – 20 =

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing.

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. For example, 35 – 4 = 63 – 8 = 35 – 7 =

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. For example, 35 – 4 = 31 63 – 8 = 35 – 7 =

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After For example, Borrowing 35 – 4 = 31 63 – 8 = 35 – 7 = 28

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits.

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 93 – 57 =

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 53 – 20 – 8 93 – 57 =

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 53 – 20 – 8 93 – 57 = = 33 – 8

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 53 – 20 – 8 93 – 57 = = 33 – 8 = 25

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 53 – 20 – 8 93 – 57 = 93 – 50 – 7 = 33 – 8 = 25

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 53 – 20 – 8 93 – 57 = 93 – 50 – 7 = 43 – 7 = 33 – 8 = 25

Subtraction One should be comfortable with subtracting two two-digit numbers such as finding the difference between \$28 and \$45. Here is one approach that will help one to do that. Step 1. Memorize the differences between two different digits. Stop 2. Practice subtracting multiples of 10 from two-digit numbers. For example, do the following calculation mentally, 53 – 40 = 13 78 – 30 = 48 94 – 20 = 74 Step 3. Practice subtracting single digits from two-digit numbers, pay attention to the cases that require borrowing. After After For example, Borrowing Borrowing 35 – 4 = 31 63 – 8 = 55 35 – 7 = 28 Step 4. Subtract two two-digit numbers in two steps: subtract the 10‟s first, then subtract the unit-digits. For example, 53 – 28 = 53 – 20 – 8 93 – 57 = 93 – 50 – 7 = 43 – 7 = 38 = 33 – 8 = 25

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 =

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction.

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. + +

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. + = +

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. As noted before that adding two apples to three apples is the same as adding three apples to two apples – we get five apples. + = +

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. As noted before that adding two apples to three apples is the same as adding three apples to two apples – we get five apples. + = + We say that addition is commutative, i.e. A + B = B + A.

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. As noted before that adding two apples to three apples is the same as adding three apples to two apples – we get five apples. + = + We say that addition is commutative, i.e. A + B = B + A. It makes physical sense to remove two apples from a pile of five apples – we are left with three apples. –

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. As noted before that adding two apples to three apples is the same as adding three apples to two apples – we get five apples. + = + We say that addition is commutative, i.e. A + B = B + A. It makes physical sense to remove two apples from a pile of five apples – we are left with three apples. But we can’t do the reverse, i.e. remove five apples from a pile of two apples. –

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. As noted before that adding two apples to three apples is the same as adding three apples to two apples – we get five apples. + + = We say that addition is commutative, i.e. A + B = B + A. It makes physical sense to remove two apples from a pile of five apples – we are left with three apples. But we can’t do the reverse, i.e. remove five apples from a pile of two apples. – – ?

Subtraction Your Turn: Do the following subtraction mentally in two steps. 72 – 48 = 72 – 40 – 8 = 84 – 36 = 84 – 30 – 6 = 41 – 28 = 92 – 64 = This brings up the issue of the order of subtraction. As noted before that adding two apples to three apples is the same as adding three apples to two apples – we get five apples. + + = We say that addition is commutative, i.e. A + B = B + A. It makes physical sense to remove two apples from a pile of five apples – we are left with three apples. But we can’t do the reverse, i.e. remove five apples from a pile of two apples. – – Hence subtraction is not commutative, i.e. A – B ≠ B – A. ?

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction.

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction. Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction. Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.” Example A. Translate each problem into a subtraction expression using the given numbers or symbols. a. The listed price of a Thingamajig is \$500. How much money do we save if we buy one for \$400 at Discount Joe?

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction. Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.” Example A. Translate each problem into a subtraction expression using the given numbers or symbols. a. The listed price of a Thingamajig is \$500. How much money do we save if we buy one for \$400 at Discount Joe? \$500 is more than \$400, hence we save 500 – 400 = \$100.

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction. Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.” Example A. Translate each problem into a subtraction expression using the given numbers or symbols. a. The listed price of a Thingamajig is \$500. How much money do we save if we buy one for \$400 at Discount Joe? \$500 is more than \$400, hence we save 500 – 400 = \$100. b. If L is the Listed price and D is the Discount Joe’s price, what are values of L and D in part a. In terms of L and D, how much do we save if we buy the item at Discount Joe.

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction. Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.” Example A. Translate each problem into a subtraction expression using the given numbers or symbols. a. The listed price of a Thingamajig is \$500. How much money do we save if we buy one for \$400 at Discount Joe? \$500 is more than \$400, hence we save 500 – 400 = \$100. b. If L is the Listed price and D is the Discount Joe’s price, what are values of L and D in part a. In terms of L and D, how much do we save if we buy the item at Discount Joe. With the information from part a. L is the \$500 and D is \$400.

Subtraction Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction. Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.” Example A. Translate each problem into a subtraction expression using the given numbers or symbols. a. The listed price of a Thingamajig is \$500. How much money do we save if we buy one for \$400 at Discount Joe? \$500 is more than \$400, hence we save 500 – 400 = \$100. b. If L is the Listed price and D is the Discount Joe’s price, what are values of L and D in part a. In terms of L and D, how much do we save if we buy the item at Discount Joe. With the information from part a. L is the \$500 and D is \$400. The amount saved is 500 – 400 = 100,

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor. 108th floor top

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor. 108th floor top 42th floor 1st hr

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor. 108th floor top 67th floor nd 2 hr 42th floor 1st hr

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. After 1 hour we were at the 42nd floor. After two hours, we were at the 67th floor. a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? 108th floor top 67th floor nd 2 hr 42th floor 1st hr

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. 108th floor top After 1 hour we were at the 42nd 67th floor 2nd hr floor. After two hours, we were at 42th floor the 67th floor. 1st hr a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? After the 1st hour, we still have 108 – 42 = 66 floors to the top.

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. 108th floor top After 1 hour we were at the 42nd 67th floor 2nd hr floor. After two hours, we were at 42th floor the 67th floor. 1st hr a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? After the 1st hour, we still have 108 – 42 = 66 floors to the top. During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour.

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. 108th floor top After 1 hour we were at the 42nd 67th floor 2nd hr floor. After two hours, we were at 42th floor the 67th floor. 1st hr a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? After the 1st hour, we still have 108 – 42 = 66 floors to the top. During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour. b. We are on the Nth floor, how many floors are we from the 108th floor? Write the answer as a subtraction.

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. 108th floor top After 1 hour we were at the 42nd 67th floor 2nd hr floor. After two hours, we were at 42th floor the 67th floor. 1st hr a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? After the 1st hour, we still have 108 – 42 = 66 floors to the top. During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour. 108th fl. b. We are on the Nth floor, how many floors are we from the 108th floor? Write the answer as a subtraction. Nth fl.

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. 108th floor top After 1 hour we were at the 42nd 67th floor 2nd hr floor. After two hours, we were at 42th floor the 67th floor. 1st hr a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? After the 1st hour, we still have 108 – 42 = 66 floors to the top. During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour. 108th fl. b. We are on the Nth floor, how many floors are we from the 108th floor? Write the answer as a subtraction. ? Nth fl.

Subtraction Example B. We climbed the 108-floor Sears Tower in Chicago. 108th floor top After 1 hour we were at the 42nd 67th floor 2nd hr floor. After two hours, we were at 42th floor the 67th floor. 1st hr a. How many floors were we away from the top after the 1st hour and how many floors did we climb during the 2nd hour? After the 1st hour, we still have 108 – 42 = 66 floors to the top. During the 2nd hour we climbed from the 42nd floor to the 67th floor hence we climbed 67 – 42 = 25 floors during the 2nd hour. 108th fl. b. We are on the Nth floor, how many floors are we from the 108th floor? Write the answer as a subtraction. Nth We are on the floor out of total 108 floors, so the number of remaining floors to the top is 108 – N as shown. ? Nth fl.

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first,

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, + +

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, + +

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, + + + +

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, + + + = +

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + = +

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + = We say that “the addition is associative.” +

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals.

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – –

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – – –

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – – –

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – – – – –

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – – – – – –

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – – – – ≠ – –

Subtraction If we are gathering three piles of apples, it does not matter which two piles we group together first, i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.” + + + + = We say that “the addition is associative.” But the results of “subtracting” three piles of apples depends on the order of the removals. – – – – ≠ – – So subtraction is not associative, i.e. (A – B) – C ≠ A – (B – C).

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2).

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2). We treat 6 – 3 – 2 as (6 – 3) – 2, i.e. take away 3, then take away another 2, therefore we take away 5 in total.

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2). We treat 6 – 3 – 2 as (6 – 3) – 2, i.e. take away 3, then take away another 2, therefore we take away 5 in total. In other words, 6 – 3 – 2 = 6 – (3 + 2) = 6 – 5 = 1.

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2). We treat 6 – 3 – 2 as (6 – 3) – 2, i.e. take away 3, then take away another 2, therefore we take away 5 in total. In other words, 6 – 3 – 2 = 6 – (3 + 2) = 6 – 5 = 1. Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract.

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2). We treat 6 – 3 – 2 as (6 – 3) – 2, i.e. take away 3, then take away another 2, therefore we take away 5 in total. In other words, 6 – 3 – 2 = 6 – (3 + 2) = 6 – 5 = 1. Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract. In symbols, A – B – C = A – (B + C) and in general, A – B – C – D – . . = A – (B + C + D + ..)

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2). We treat 6 – 3 – 2 as (6 – 3) – 2, i.e. take away 3, then take away another 2, therefore we take away 5 in total. In other words, 6 – 3 – 2 = 6 – (3 + 2) = 6 – 5 = 1. Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract. In symbols, A – B – C = A – (B + C) and in general, A – B – C – D – . . = A – (B + C + D + ..) Furthermore, for a mixed problem, we may separate the addition and the subtraction into two groups, find the total of each group, then find the difference of two totals, or that,

Subtraction So when subtracting two or more numbers in a row, we can’t arbitrarily subtract the ones in the back as shown that (6 – 3) – 2 ≠ 6 – (3 – 2). We treat 6 – 3 – 2 as (6 – 3) – 2, i.e. take away 3, then take away another 2, therefore we take away 5 in total. In other words, 6 – 3 – 2 = 6 – (3 + 2) = 6 – 5 = 1. Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract. In symbols, A – B – C = A – (B + C) and in general, A – B – C – D – . . = A – (B + C + D + ..) Furthermore, for a mixed problem, we may separate the addition and the subtraction into two groups, find the total of each group, then find the difference of two totals, or that, A – a + B – b + C – c . . = (A + B + C ..) – (a + b + c ..)

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 b. 82 – 12 – 7 – 8 – 14 – 23

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 b. 82 – 12 – 7 – 8 – 14 – 23

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44 = 82 – (16 + 44)

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44 = 82 – (16 + 44) = 82 – 60 = 22

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44 = 82 – (16 + 44) = 82 – 60 = 22 Find the total reduction first. 82 – 12 – 7 – 8 – 14 – 23

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44 = 82 – (16 + 44) = 82 – 60 = 22 Find the total reduction first. 82 – 12 – 7 – 8 – 14 – 23 = 82 – (12 + 7 + 8 + 14 + 23)

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44 = 82 – (16 + 44) = 82 – 60 = 22 Find the total reduction first. 82 – 12 – 7 – 8 – 14 – 23 = 82 – (12 + 7 + 8 + 14 + 23) 20 30

Subtraction Example C. Calculate each of the following problems using two different ways. Do it from left to right in the order given and do it by calculating the total reduction first. a. 82 – 16 – 44 Do it in the given order. 82 – 16 – 44 = 66 – 44 = 22 b. 82 – 12 – 7 – 8 – 14 – 23 Do it in the given order. 82 – 12 – 7 – 8 – 14 – 23 = 70 – 7 – 8 – 14 – 23 = 63 – 8 – 14 – 23 = 55 – 14 – 23 = 41 – 23 = 18 Find the total reduction first. 82 – 16 – 44 = 82 – (16 + 44) = 82 – 60 = 22 Find the total reduction first. 82 – 12 – 7 – 8 – 14 – 23 = 82 – (12 + 7 + 8 + 14 + 23) = 82 – 64 = 18 20 30

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. 82 – 12 – 7 + 8 + 14 – 23

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 63 + 8 + 14 – 23 = 71 + 14 – 23 = 85 – 23 = 62

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. Group into two groups. 82 – 12 – 7 + 8 + 14 – 23 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 63 + 8 + 14 – 23 = 71 + 14 – 23 = 85 – 23 = 62

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. Group into two groups. 82 – 12 – 7 + 8 + 14 – 23 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 82 + 8 + 14 – (12 + 7 + 23) = 63 + 8 + 14 – 23 = 71 + 14 – 23 = 85 – 23 = 62

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. Group into two groups. 82 – 12 – 7 + 8 + 14 – 23 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 82 + 8 + 14 – (12 + 7 + 23) = 63 + 8 + 14 – 23 30 90 = 71 + 14 – 23 = 85 – 23 = 62

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. Group into two groups. 82 – 12 – 7 + 8 + 14 – 23 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 82 + 8 + 14 – (12 + 7 + 23) = 63 + 8 + 14 – 23 30 90 = 71 + 14 – 23 = 104 – 42 = 85 – 23 = 62 = 62

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. Group into two groups. 82 – 12 – 7 + 8 + 14 – 23 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 82 + 8 + 14 – (12 + 7 + 23) = 63 + 8 + 14 – 23 30 90 = 71 + 14 – 23 = 104 – 42 = 85 – 23 = 62 = 62 Subtracting quantities in the wrong order is one of the most common mistakes in mathematics (addition requires no such fuss).

Subtraction c. 82 – 12 – 7 + 8 + 14 – 23 Do it in the given order. Group into two groups. 82 – 12 – 7 + 8 + 14 – 23 82 – 12 – 7 + 8 + 14 – 23 = 70 – 7 + 8 + 14 – 23 = 82 + 8 + 14 – (12 + 7 + 23) = 63 + 8 + 14 – 23 30 90 = 71 + 14 – 23 = 104 – 42 = 85 – 23 = 62 = 62 Subtracting quantities in the wrong order is one of the most common mistakes in mathematics (addition requires no such fuss). When reading mathematical expressions or translating real life problems involving subtraction into mathematics, always ask the question “who subtracts whom?” Answer it clearly, then proceed.

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January 18, 2019

January 18, 2019

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January 18, 2019

January 18, 2019

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