1.4 subtraction w
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SubtractionTo subtract is to take away, or to undo an addition.
SubtractionTo subtract is to take away, or to undo an addition. We write “A – B” for taking the amount B away from A.
SubtractionTo subtract is to take away, or to undo an addition.
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).
We write “A – B” for taking the amount B away from A.
SubtractionTo subtract is to take away, or to undo an addition.
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,”
We write “A – B” for taking the amount B away from A.
SubtractionTo subtract is to take away, or to undo an addition.
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,”
We write “A – B” for taking the amount B away from A.
“B is subtracted, or is taken away, from A.”
SubtractionTo subtract is to take away, or to undo an addition.
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,”
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples”“five apples minus three apples,”
= 2 .
“B is subtracted, or is taken away, from A.”
SubtractionTo subtract is to take away, or to undo an addition.
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,”
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples”“five apples minus three apples,”
= 2 .
“B is subtracted, or is taken away, from A.”
SubtractionTo subtract is to take away, or to undo an addition.
Mayan numerals are visually instructive for subtraction of small numbers.
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,”
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples”“five apples minus three apples,”
= 2 .
“B is subtracted, or is taken away, from A.”
SubtractionTo subtract is to take away, or to undo an addition.
–
Mayan numerals are visually instructive for subtraction of small numbers.
= signifies 12 – 7 = 5,
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,”
For example,
or that
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples”“five apples minus three apples,”
= 2 .
“B is subtracted, or is taken away, from A.”
SubtractionTo subtract is to take away, or to undo an addition.
–
Mayan numerals are visually instructive for subtraction of small numbers.
= signifies 12 – 7 = 5,
– = signifies 13 – 6 = 7.
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,”
For example,
or that
We write “A – B” for taking the amount B away from A.
Hence the statements “five apples take away three apples,”
all mean 5 – 3
“three apples are taken away from five apples”“five apples minus three apples,”
= 2 .
“B is subtracted, or is taken away, from A.”
SubtractionWhen there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
Subtraction
For example, 11 – 4 is
– =
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
Subtraction
For example, 11 – 4 is
– = –
borrow
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
Subtraction
For example, 11 – 4 is
– = –
borrow
= = 7
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87:
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
1. lineup the numbers vertically to match the place values,
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is:
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
6 3 4 8 7–
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow14
2
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow14
2
7
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow14
2
7
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow14
2
7
12
5
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow14
2
7
12
5
4
Subtraction
For example, 11 – 4 is
– =
More subtraction examples using the Mayan pictorial method are given in the exercise to help some people to memorize them.
–
borrow
= = 7
For our base-10 numbers, each borrowed unit is exchanged to be 10 smaller units.
For example, 634 – 87 is: 6 3 4 8 7–
When there are not enough “ ‘s” to subtract, we have to convert a “ ” into “ .” This process is called “borrowing.”
To subtract,
2. then subtract the digits from right to left and “borrow” when it is necessary.
1. lineup the numbers vertically to match the place values,
need to borrow14
2
7
12
5
45
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
Step 1. Memorize the differences between two different digits.
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
Step 1. Memorize the differences between two different digits.
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.
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 78 – 30 = 94 – 20 =
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 35 – 7 = 63 – 8 =
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 63 – 8 =
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 =
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 = 93 – 57 =
For example,
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 = 93 – 57 = 53 – 20 – 8
For example,
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 =
= 33 – 8
93 – 57 = 53 – 20 – 8
For example,
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 =
= 33 – 8 = 25
93 – 57 = 53 – 20 – 8
For example,
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 =
= 33 – 8 = 25
93 – 57 = 53 – 20 – 8 93 – 50 – 7
For example,
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 =
= 33 – 8 = 25
93 – 57 = = 43 – 7
53 – 20 – 8 93 – 50 – 7
For example,
Step 1. Memorize the differences between two different digits.
For example, do the following calculation mentally,
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.
53 – 40 = 13 78 – 30 = 48 94 – 20 = 74
For example,
Step 3. Practice subtracting single digits from two-digit numbers,pay attention to the cases that require borrowing.
35 – 4 = 31 35 – 7 = 28 63 – 8 = 55
Stop 2. Practice subtracting multiples of 10 from two-digit numbers.
After Borrowing
After Borrowing
Subtraction
Step 4. Subtract two two-digit numbers in two steps: subtract the 10’s first, then subtract the unit-digits.
53 – 28 =
= 33 – 8 = 25
93 – 57 = = 43 – 7 = 38
53 – 20 – 8 93 – 50 – 7
For example,
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.
Subtraction
Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”
Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction.
Subtraction
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?
Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”
Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction.
Subtraction
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.
Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”
Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction.
Subtraction
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.
Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”
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.
Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction.
Subtraction
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.
Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”
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.
Therefore, unlike addition, because subtraction is not commutative, we have to establish to order of subtraction.
Subtraction
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.
Specifically, when subtracting two quantities A and B, we have to identify clearly that if the problem is “A – B” or “B – A.”
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,
SubtractionExample 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 floortop
SubtractionExample 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 floortop
1st hr 42th floor
SubtractionExample 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 floortop
1st hr 42th floor2nd hr 67th floor
SubtractionExample 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 andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
SubtractionExample 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.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
SubtractionExample 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.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
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.
SubtractionExample 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.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
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 wefrom the 108th floor? Write the answer as a subtraction.
SubtractionExample 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.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
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 wefrom the 108th floor? Write the answer as a subtraction.
Nth fl.
108th fl.
SubtractionExample 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.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
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 wefrom the 108th floor? Write the answer as a subtraction.
Nth fl.
108th fl.
?
SubtractionExample 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.
After the 1st hour, we still have 108 – 42 = 66 floors to the top.
a. How many floors were we away from the top after the 1st hour andhow many floors did we climb during the 2nd hour?
108th floortop
1st hr 42th floor2nd hr 67th floor
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 wefrom the 108th floor? Write the answer as a subtraction.We are on the Nth floor out of total 108 floors, so the number of remaining floors to the topis 108 – N as shown.
Nth fl.
108th fl.
?
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,
+ +
We say that “the addition is associative.”
=++
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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,
+ +
We say that “the addition is associative.”
=++
i.e. A + (B + C) = (A + B) + C where the “( )” means “do first.”
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
SubtractionSo 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
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.
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
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.
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).
= 6 – (3 + 2) = 6 – 5 = 1.In other words, 6 – 3 – 2
Subtraction
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.
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).
Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract.
= 6 – (3 + 2) = 6 – 5 = 1.In other words, 6 – 3 – 2
Subtraction
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.
A – B – C = A – (B + C)
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).
Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract. In symbols,
and in general, A – B – C – D – . . = A – (B + C + D + ..)
= 6 – (3 + 2) = 6 – 5 = 1.In other words, 6 – 3 – 2
Subtraction
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.
A – B – C = A – (B + C)
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,
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).
Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract. In symbols,
and in general, A – B – C – D – . . = A – (B + C + D + ..)
= 6 – (3 + 2) = 6 – 5 = 1.In other words, 6 – 3 – 2
Subtraction
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.
A – B – C = A – (B + C)
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 . . =
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).
Hence for a multi–subtraction, we may total the quantities that are to be taken away first then subtract. In symbols,
and in general, A – B – C – D – . . = A – (B + C + D + ..)
= 6 – (3 + 2) = 6 – 5 = 1.In other words, 6 – 3 – 2
– (a + b + c ..) (A + B + C ..)
SubtractionExample 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
SubtractionExample 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
82 – 16 – 44Do it in the given order.
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44= 82 – (16 + 44)
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44= 82 – (16 + 44)
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
= 82 – 60 = 22
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44= 82 – (16 + 44)
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
82 – 12 – 7 – 8 – 14 – 23Find the total reduction first.
= 82 – 60 = 22
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44= 82 – (16 + 44)
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
82 – 12 – 7 – 8 – 14 – 23Find the total reduction first.
= 82 – 60 = 22
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
= 82 – (12 + 7 + 8 + 14 + 23)
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44= 82 – (16 + 44)
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
82 – 12 – 7 – 8 – 14 – 23
20 30
Find the total reduction first.
= 82 – 60 = 22
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
= 82 – (12 + 7 + 8 + 14 + 23)
SubtractionExample 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
82 – 16 – 44Do it in the given order.
= 66 – 44= 22
Find the total reduction first.
82 – 16 – 44= 82 – (16 + 44)
82 – 12 – 7 – 8 – 14 – 23Do it in the given order.
82 – 12 – 7 – 8 – 14 – 23
20 30= 82 – 64= 18
Find the total reduction first.
= 82 – 60 = 22
= 70 – 7 – 8 – 14 – 23= 63 – 8 – 14 – 23= 55 – 14 – 23= 41 – 23 = 18
= 82 – (12 + 7 + 8 + 14 + 23)
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.
= 70 – 7 + 8 + 14 – 23
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.82 – 12 – 7 + 8 + 14 – 23
Group into two groups.
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.82 – 12 – 7 + 8 + 14 – 23
Group into two groups.
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
= 82 + 8 + 14 – (12 + 7 + 23)
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.82 – 12 – 7 + 8 + 14 – 23
90
Group into two groups.
30
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
= 82 + 8 + 14 – (12 + 7 + 23)
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.82 – 12 – 7 + 8 + 14 – 23
90
Group into two groups.
30
= 104 – 42= 62
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
= 82 + 8 + 14 – (12 + 7 + 23)
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.82 – 12 – 7 + 8 + 14 – 23
90
Group into two groups.
30
= 104 – 42= 62
Subtracting quantities in the wrong order is one of the most common mistakes in mathematics (addition requires no such fuss).
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
= 82 + 8 + 14 – (12 + 7 + 23)
Subtractionc. 82 – 12 – 7 + 8 + 14 – 23
82 – 12 – 7 + 8 + 14 – 23
Do it in the given order.82 – 12 – 7 + 8 + 14 – 23
90
Group into two groups.
30
= 104 – 42= 62
Subtracting quantities in the wrong order is one of the most common mistakes in mathematics (addition requires no such fuss).
= 70 – 7 + 8 + 14 – 23= 63 + 8 + 14 – 23= 71 + 14 – 23= 85 – 23 = 62
= 82 + 8 + 14 – (12 + 7 + 23)
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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