A PEMDAS issue request for explanationA simple explanation or derivation of Cramer's rule, suitable for secondary Algebra 2?Method for teaching factorizationWhat is the pedagogical justification and history for using mnemonics to teach order of operations?Mindless use of “antisimplifications” such as $1/x+1/y=(x+y)/xy$ and $1/sqrt2=sqrt2/2$How to use these actions words for subtraction?How to teach multiplication between integers for the first timeHow to teach a student algebra who misses too much previous knowledge?How to present the order of factors and summands for the usual multiplication procedureCould this visual explanation of horizontal shift be helpful ? …( if not beautiful…)Real-life exceptions to PEMDAS?
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A PEMDAS issue request for explanation
A simple explanation or derivation of Cramer's rule, suitable for secondary Algebra 2?Method for teaching factorizationWhat is the pedagogical justification and history for using mnemonics to teach order of operations?Mindless use of “antisimplifications” such as $1/x+1/y=(x+y)/xy$ and $1/sqrt2=sqrt2/2$How to use these actions words for subtraction?How to teach multiplication between integers for the first timeHow to teach a student algebra who misses too much previous knowledge?How to present the order of factors and summands for the usual multiplication procedureCould this visual explanation of horizontal shift be helpful ? …( if not beautiful…)Real-life exceptions to PEMDAS?
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
This question made the rounds recently -
$8÷2(2+2)=?$
Now, I glanced at this, answered "1" and then saw the full article pinted in the New York Times, The Math Equation That Tried to Stump the Internet.
The article concludes that 16 is the correct answer, citing that
$8÷2 x 4$
when approached from left to right, results in 16.
My disagreement lies in the dismissal of the parentheses, as my explanation would be that
$8÷2(x+x)$ would, as a first step, simplify to
$8÷2(2x)$ and then
$8÷4x$
in which case, if I were to offer this last bit and asked for a value when x=2, few would argue that division comes first.
Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority?
(If you agree with the articles' '16', I am happy to hear the reasoning.)
The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.
Note: PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. I don’t know how common it is outside the US. Or even in different locations within the US, for that matter.
algebra arithmetic-operations
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
$endgroup$
add a comment |
$begingroup$
This question made the rounds recently -
$8÷2(2+2)=?$
Now, I glanced at this, answered "1" and then saw the full article pinted in the New York Times, The Math Equation That Tried to Stump the Internet.
The article concludes that 16 is the correct answer, citing that
$8÷2 x 4$
when approached from left to right, results in 16.
My disagreement lies in the dismissal of the parentheses, as my explanation would be that
$8÷2(x+x)$ would, as a first step, simplify to
$8÷2(2x)$ and then
$8÷4x$
in which case, if I were to offer this last bit and asked for a value when x=2, few would argue that division comes first.
Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority?
(If you agree with the articles' '16', I am happy to hear the reasoning.)
The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.
Note: PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. I don’t know how common it is outside the US. Or even in different locations within the US, for that matter.
algebra arithmetic-operations
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
$endgroup$
1
$begingroup$
This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority?
$endgroup$
– Joseph O'Rourke
9 hours ago
$begingroup$
For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division.
$endgroup$
– JoeTaxpayer
8 hours ago
2
$begingroup$
When you wrote "$8div 2x4$", did you mean to write "$8div 2times 4$"?
$endgroup$
– Joel Reyes Noche
6 hours ago
add a comment |
$begingroup$
This question made the rounds recently -
$8÷2(2+2)=?$
Now, I glanced at this, answered "1" and then saw the full article pinted in the New York Times, The Math Equation That Tried to Stump the Internet.
The article concludes that 16 is the correct answer, citing that
$8÷2 x 4$
when approached from left to right, results in 16.
My disagreement lies in the dismissal of the parentheses, as my explanation would be that
$8÷2(x+x)$ would, as a first step, simplify to
$8÷2(2x)$ and then
$8÷4x$
in which case, if I were to offer this last bit and asked for a value when x=2, few would argue that division comes first.
Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority?
(If you agree with the articles' '16', I am happy to hear the reasoning.)
The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.
Note: PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. I don’t know how common it is outside the US. Or even in different locations within the US, for that matter.
algebra arithmetic-operations
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
$endgroup$
This question made the rounds recently -
$8÷2(2+2)=?$
Now, I glanced at this, answered "1" and then saw the full article pinted in the New York Times, The Math Equation That Tried to Stump the Internet.
The article concludes that 16 is the correct answer, citing that
$8÷2 x 4$
when approached from left to right, results in 16.
My disagreement lies in the dismissal of the parentheses, as my explanation would be that
$8÷2(x+x)$ would, as a first step, simplify to
$8÷2(2x)$ and then
$8÷4x$
in which case, if I were to offer this last bit and asked for a value when x=2, few would argue that division comes first.
Now, my question - for those who agree with me, what is it about PEMDAS that misses this issue, that the number right before the parentheses multiplies the contents with a higher priority than the division to its left? How do we address that priority?
(If you agree with the articles' '16', I am happy to hear the reasoning.)
The third option is also fine, that such a set of numbers and symbols is ambiguous, and a second set of parentheses is required for clarification.
Note: PEMDAS is an acronym for the words parenthesis, exponents, multiplication, division, addition, subtraction. I don’t know how common it is outside the US. Or even in different locations within the US, for that matter.
algebra arithmetic-operations
algebra arithmetic-operations
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
edited 8 hours ago
JoeTaxpayer
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
asked 9 hours ago
JoeTaxpayerJoeTaxpayer
5,68620 silver badges47 bronze badges
5,68620 silver badges47 bronze badges
1
$begingroup$
This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority?
$endgroup$
– Joseph O'Rourke
9 hours ago
$begingroup$
For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division.
$endgroup$
– JoeTaxpayer
8 hours ago
2
$begingroup$
When you wrote "$8div 2x4$", did you mean to write "$8div 2times 4$"?
$endgroup$
– Joel Reyes Noche
6 hours ago
add a comment |
1
$begingroup$
This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority?
$endgroup$
– Joseph O'Rourke
9 hours ago
$begingroup$
For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division.
$endgroup$
– JoeTaxpayer
8 hours ago
2
$begingroup$
When you wrote "$8div 2x4$", did you mean to write "$8div 2times 4$"?
$endgroup$
– Joel Reyes Noche
6 hours ago
1
1
$begingroup$
This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority?
$endgroup$
– Joseph O'Rourke
9 hours ago
$begingroup$
This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority?
$endgroup$
– Joseph O'Rourke
9 hours ago
$begingroup$
For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division.
$endgroup$
– JoeTaxpayer
8 hours ago
$begingroup$
For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division.
$endgroup$
– JoeTaxpayer
8 hours ago
2
2
$begingroup$
When you wrote "$8div 2x4$", did you mean to write "$8div 2times 4$"?
$endgroup$
– Joel Reyes Noche
6 hours ago
$begingroup$
When you wrote "$8div 2x4$", did you mean to write "$8div 2times 4$"?
$endgroup$
– Joel Reyes Noche
6 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.
If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4𝑥.
I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)
Back to your 8÷4𝑥. If we write it as 8/4𝑥, can you see the ambiguity? Is that x on the top or bottom of the fraction?
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
If the expression were, say, $48div 4times 12$, there would not be much disagreement (multiplication and division are performed from left to right).
But an expression such as $48div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying.
If one interprets the parentheses as a set of grouping symbols, then the "P" in "PEMDAS" is used and the "correct" answer is $1$. If one interprets the parentheses as a multiplication, then the "M" in "PEMDAS" is used and the "correct" answer is $144$.
As far as I know, there is no "authority" that declares which of these two interpretations is correct. As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
$endgroup$
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.
If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4𝑥.
I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)
Back to your 8÷4𝑥. If we write it as 8/4𝑥, can you see the ambiguity? Is that x on the top or bottom of the fraction?
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.
If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4𝑥.
I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)
Back to your 8÷4𝑥. If we write it as 8/4𝑥, can you see the ambiguity? Is that x on the top or bottom of the fraction?
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
$endgroup$
add a comment |
$begingroup$
The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.
If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4𝑥.
I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)
Back to your 8÷4𝑥. If we write it as 8/4𝑥, can you see the ambiguity? Is that x on the top or bottom of the fraction?
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
$endgroup$
The issue is the implied multiplication in 8÷2(2𝑥). Different calculators actually resolve this differently, so in that sense we would want to say this is ambiguous.
If implied multiplication works the same way explicit multiplication does, then we do the division first (left to right) and get 4(2𝑥). If implicit multiplication has a higher priority than explicit operations, we would get what you want: 8÷4𝑥.
I hate these examples, because they lead people to think math is very silly. The conventions (of PEMDAS) are usually reflected in the typesetting conventions we use. You would never see someone using the ÷ symbol in a complicated expression like this. It would be written 8/2(2+2), and then it would feel right to do the division first. (I believe our subconscious has internalized PEMDAS as matching the typesetting, loose for + and -, and tighter for multiplication and division, and even tighter for exponentiation.)
Back to your 8÷4𝑥. If we write it as 8/4𝑥, can you see the ambiguity? Is that x on the top or bottom of the fraction?
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
answered 8 hours ago
Sue VanHattum♦Sue VanHattum
10.2k1 gold badge22 silver badges64 bronze badges
10.2k1 gold badge22 silver badges64 bronze badges
add a comment |
add a comment |
$begingroup$
If the expression were, say, $48div 4times 12$, there would not be much disagreement (multiplication and division are performed from left to right).
But an expression such as $48div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying.
If one interprets the parentheses as a set of grouping symbols, then the "P" in "PEMDAS" is used and the "correct" answer is $1$. If one interprets the parentheses as a multiplication, then the "M" in "PEMDAS" is used and the "correct" answer is $144$.
As far as I know, there is no "authority" that declares which of these two interpretations is correct. As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
$endgroup$
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
add a comment |
$begingroup$
If the expression were, say, $48div 4times 12$, there would not be much disagreement (multiplication and division are performed from left to right).
But an expression such as $48div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying.
If one interprets the parentheses as a set of grouping symbols, then the "P" in "PEMDAS" is used and the "correct" answer is $1$. If one interprets the parentheses as a multiplication, then the "M" in "PEMDAS" is used and the "correct" answer is $144$.
As far as I know, there is no "authority" that declares which of these two interpretations is correct. As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
$endgroup$
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
add a comment |
$begingroup$
If the expression were, say, $48div 4times 12$, there would not be much disagreement (multiplication and division are performed from left to right).
But an expression such as $48div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying.
If one interprets the parentheses as a set of grouping symbols, then the "P" in "PEMDAS" is used and the "correct" answer is $1$. If one interprets the parentheses as a multiplication, then the "M" in "PEMDAS" is used and the "correct" answer is $144$.
As far as I know, there is no "authority" that declares which of these two interpretations is correct. As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
$endgroup$
If the expression were, say, $48div 4times 12$, there would not be much disagreement (multiplication and division are performed from left to right).
But an expression such as $48div 4(12)$ results in disagreement because the parentheses could mean one of two different things: a way of grouping or a way of multiplying.
If one interprets the parentheses as a set of grouping symbols, then the "P" in "PEMDAS" is used and the "correct" answer is $1$. If one interprets the parentheses as a multiplication, then the "M" in "PEMDAS" is used and the "correct" answer is $144$.
As far as I know, there is no "authority" that declares which of these two interpretations is correct. As you can see from the images above, even engineers working in the same calculator company do not seem to have a common interpretation.
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
answered 6 hours ago
Joel Reyes NocheJoel Reyes Noche
5,9392 gold badges18 silver badges56 bronze badges
5,9392 gold badges18 silver badges56 bronze badges
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
add a comment |
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
$begingroup$
"that declares which of these two interpretations is correct." This is clear in a programming context, because the operator precedence must be unambiguous: the parentheses are never ambiguous. Maybe the programming language precedences are seeping into clarify the mathematics issues?
$endgroup$
– Joseph O'Rourke
5 hours ago
add a comment |
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1
$begingroup$
This is where I lose you: "$8÷2(2x)$ and then $8÷4x$." Why do you prioritize evaluating $2(2x)$ before $8÷2$, which has L-to-R priority?
$endgroup$
– Joseph O'Rourke
9 hours ago
$begingroup$
For a reason I struggle to articulate, it seems to me the multiplying of what’s in parentheses should take priority to the division.
$endgroup$
– JoeTaxpayer
8 hours ago
2
$begingroup$
When you wrote "$8div 2x4$", did you mean to write "$8div 2times 4$"?
$endgroup$
– Joel Reyes Noche
6 hours ago