I recall being taught the flip and multiply as the result of three easy tricks
You can multiply by one and not change the result,
You can multiply in any order and not change the result. This includes doing a division with the number on its left before other left multiplies.
Any non zero number divided by itself is one.
3/4 / 2/3 is the same as
3/4 x 1 / 2/3 is the same as
3/4 x (3/2 x 2/3) / 2/3 is the same as
3:4 x 2/3 x (2/3 / 2/3) is the same as
3:4 x 3/2 x 1 is the same as
3/4 / 3/2
These three are easy to grasp and if you have to write out all the steps 20 times or so you will remember them for a long time.
I don’t recall naming them at the time.
But I’d love to have a conversation with your colleague about why he understands any of the. Are true. Do all the people asking for teaching understanding know why multiplication is associative and division is left associative?
How far do they go before they are at an axiom and it’s all just memorize these axioms or pretty deep philosophy on why one set of axioms is better.
When we were studying vector spaces at university and became stuck on some proof or other, my friend and I would invoke an axiom of our own invention which was "Axiom 0: Because I say so". :-)
Another observation is how many people treat math sort of like physics in that understanding seems to be based on experimental results or a good diagram.
That’s a great way to understand some things but it is not mathematics.
I am not referring to actual proof by diagram here but the diagram of an example which is the flimsy proof by anecdote.
There are two separate issues here. One is the obsession over "understanding" which in its current state can hold students up when they're ready to move on. E.g., the student may say after spending a week "understanding" what slope is "Can we move on, we get what slope is." And the response is "But do you REALLY get it?" and the teacher spends another three weeks. Even in freshman calc, students are given an intuitive definition of limits/continuity and move quickly to the powerful problem solving applications of derivatives and integrals.
The second issue which you bring up is the Common Core induced focus on inductive reasoning. (Specifically, the CC Standard of Mathematical Practice to "look for and make use of structure.) Inductive reasoning is fine for initial reasonable guesses, but ultimately one has to use deductive reasoning in order to prove the conjecture. In elementary math, showing some examples of how fractional division leads to the invert and multiply rule via diagrams is appropriate for that level. So on the one hand the reformers wring their hands and say "Students are doing but not understanding", and on the other praise inductive reasoning to the skies, because then "they're thinking like mathematicians." (See https://barrygarelick.substack.com/p/growing-tile-problems-misplaced-emphasis and https://barrygarelick.substack.com/p/growing-tile-problems-vs-equity-and )
There is a simpler, though admittedly not as formally complete, way to get this across to slightly younger students. Basically an expansion on saying "it's the inverse." Division by a number is a way to "undo" multiplication by that same number, to go back to what you started with before the multiplication.
Multiplying by a/b (use any actual numbers here) is the same as either multiplying by a and then dividing by b, or first dividing by b and then multiplying by a. They both yield the same answer. So, once that has been done to a starting number, how do you undo it? You multiply by b and divide by a, which is exactly what it means to multiply by b/a.
Some kids will get that, but most will glom onto the procedure in a "just tell me what to do" type attitude. More important for me is that they understand the context of what fractional division is. With more tools under their belt, then a more formal explanation can be made in algebra classes. John Mighton starts off with showing whole number divided by fraction. First with 5 oranges cut into halfs, means 5 ÷ 1/2, which also means you have 5 x 2 = 10 halves. He then extends that to something like "how many 3/4 servings of an orange are there in 12 oranges?" Start with 12 ÷ 1/4, to get 12 x 4 = 48 quarter pieces. Then assemble those 48 quarter pieces into groups of 3-quarters, or 48/3. What you've done is multiply by 4 and divide by 3, which is the same as multiplying by 4/3 (i.e., 3/4 inverted).
Singapore's math books do the same. The process of inversion is then extended to problems like 5/6 ÷ 2/3 without further explanation. When they're more familiar with algebraic operations you can use 2/3(x) = 5/6 to show that the solution is to multiply the coefficient of x by its reciprocal, and then the other side as well: 3/2 ∙ 2/3(x) = 5/6 ∙ 3/2, etc. (As explained in "Traditional Math")
Yes, these are all good. I do believe that the idea, that multiplication by a fraction involves both a multiplication and a division, is an essential part of number sense that should be introduced as early as possible in the contexts you've mentioned.
I'm still reeling over latest changes to CA Math standards which have eliminated Algebra as a precursor to many higher level college and high school courses. We truly are dumbing down our kids and this mamby pamby commentary about what kids need is horrific. Just teach them math. Then we can have the debate once our kids are back on track with their global competitors.
I'm puzzled what it means to understand "why" a mathematical concept is true. When professional mathematicians use this term, it usually means that they want a formal proof of the result. Is that what people are asking for -- a rigorous proof of why the "invert and multiply" rule is correct? Of course such a proof is possible, but it's necessarily a little technical, and if a student is having problems with fractional division then I doubt that seeing a proof will make things better. So my suspicion is that whatever it is that would constitute a explanation of "why" the rule works, it's not a formal, rigorously correct proof.
On the other hand, if a proof is not what constitutes an answer for explaining "why" the rule works, then what does? Worked examples? I'm genuinely mystified.
I think what they mean is being able to visualize what is happening. John Mighton uses an explanation that is also used in Singapore's "Primary Math" series. Typically, it's easier to show what's going on with a whole number divided by a fraction. Once that is done, one can see that the divisor is inverted and multiplied. Then this is extended to fraction divided by fraction without explanation of "why". (See https://www.youtube.com/watch?v=e1gcBP2TmPk&t=217s)
I was able to show why it works in algebra class, once students had the tools necessary to understand.
I agree with you that at the lower grades (5th and 6th, say), seeing such a demonstration will not make things better. What they really are saying is students should understand the context; i.e., what it means when we divide by a fraction.
A more formal explanation in algebra classes is that: Since subtraction of a number means adding the additive inverse of that number, then division by a number means multiplying by the multiplicative inverse of that number. So a/b = a x 1/b. Students at this point know what a multiplicative inverse is, and that m/n x n/m = 1. So m/n = 1/(n/m).
I recall being taught the flip and multiply as the result of three easy tricks
You can multiply by one and not change the result,
You can multiply in any order and not change the result. This includes doing a division with the number on its left before other left multiplies.
Any non zero number divided by itself is one.
3/4 / 2/3 is the same as
3/4 x 1 / 2/3 is the same as
3/4 x (3/2 x 2/3) / 2/3 is the same as
3:4 x 2/3 x (2/3 / 2/3) is the same as
3:4 x 3/2 x 1 is the same as
3/4 / 3/2
These three are easy to grasp and if you have to write out all the steps 20 times or so you will remember them for a long time.
I don’t recall naming them at the time.
But I’d love to have a conversation with your colleague about why he understands any of the. Are true. Do all the people asking for teaching understanding know why multiplication is associative and division is left associative?
How far do they go before they are at an axiom and it’s all just memorize these axioms or pretty deep philosophy on why one set of axioms is better.
When we were studying vector spaces at university and became stuck on some proof or other, my friend and I would invoke an axiom of our own invention which was "Axiom 0: Because I say so". :-)
Another observation is how many people treat math sort of like physics in that understanding seems to be based on experimental results or a good diagram.
That’s a great way to understand some things but it is not mathematics.
I am not referring to actual proof by diagram here but the diagram of an example which is the flimsy proof by anecdote.
There are two separate issues here. One is the obsession over "understanding" which in its current state can hold students up when they're ready to move on. E.g., the student may say after spending a week "understanding" what slope is "Can we move on, we get what slope is." And the response is "But do you REALLY get it?" and the teacher spends another three weeks. Even in freshman calc, students are given an intuitive definition of limits/continuity and move quickly to the powerful problem solving applications of derivatives and integrals.
The second issue which you bring up is the Common Core induced focus on inductive reasoning. (Specifically, the CC Standard of Mathematical Practice to "look for and make use of structure.) Inductive reasoning is fine for initial reasonable guesses, but ultimately one has to use deductive reasoning in order to prove the conjecture. In elementary math, showing some examples of how fractional division leads to the invert and multiply rule via diagrams is appropriate for that level. So on the one hand the reformers wring their hands and say "Students are doing but not understanding", and on the other praise inductive reasoning to the skies, because then "they're thinking like mathematicians." (See https://barrygarelick.substack.com/p/growing-tile-problems-misplaced-emphasis and https://barrygarelick.substack.com/p/growing-tile-problems-vs-equity-and )
Part of my point was to address the first issue by suggesting those raising it may not actually understand the subject themselves.
They may have some cargo cult style ways to imitate a Polya but don’t see to travel far on their on path to understanding.
There is a simpler, though admittedly not as formally complete, way to get this across to slightly younger students. Basically an expansion on saying "it's the inverse." Division by a number is a way to "undo" multiplication by that same number, to go back to what you started with before the multiplication.
Multiplying by a/b (use any actual numbers here) is the same as either multiplying by a and then dividing by b, or first dividing by b and then multiplying by a. They both yield the same answer. So, once that has been done to a starting number, how do you undo it? You multiply by b and divide by a, which is exactly what it means to multiply by b/a.
Some kids will get that, but most will glom onto the procedure in a "just tell me what to do" type attitude. More important for me is that they understand the context of what fractional division is. With more tools under their belt, then a more formal explanation can be made in algebra classes. John Mighton starts off with showing whole number divided by fraction. First with 5 oranges cut into halfs, means 5 ÷ 1/2, which also means you have 5 x 2 = 10 halves. He then extends that to something like "how many 3/4 servings of an orange are there in 12 oranges?" Start with 12 ÷ 1/4, to get 12 x 4 = 48 quarter pieces. Then assemble those 48 quarter pieces into groups of 3-quarters, or 48/3. What you've done is multiply by 4 and divide by 3, which is the same as multiplying by 4/3 (i.e., 3/4 inverted).
Singapore's math books do the same. The process of inversion is then extended to problems like 5/6 ÷ 2/3 without further explanation. When they're more familiar with algebraic operations you can use 2/3(x) = 5/6 to show that the solution is to multiply the coefficient of x by its reciprocal, and then the other side as well: 3/2 ∙ 2/3(x) = 5/6 ∙ 3/2, etc. (As explained in "Traditional Math")
Yes, these are all good. I do believe that the idea, that multiplication by a fraction involves both a multiplication and a division, is an essential part of number sense that should be introduced as early as possible in the contexts you've mentioned.
I'm still reeling over latest changes to CA Math standards which have eliminated Algebra as a precursor to many higher level college and high school courses. We truly are dumbing down our kids and this mamby pamby commentary about what kids need is horrific. Just teach them math. Then we can have the debate once our kids are back on track with their global competitors.
I'm puzzled what it means to understand "why" a mathematical concept is true. When professional mathematicians use this term, it usually means that they want a formal proof of the result. Is that what people are asking for -- a rigorous proof of why the "invert and multiply" rule is correct? Of course such a proof is possible, but it's necessarily a little technical, and if a student is having problems with fractional division then I doubt that seeing a proof will make things better. So my suspicion is that whatever it is that would constitute a explanation of "why" the rule works, it's not a formal, rigorously correct proof.
On the other hand, if a proof is not what constitutes an answer for explaining "why" the rule works, then what does? Worked examples? I'm genuinely mystified.
I think what they mean is being able to visualize what is happening. John Mighton uses an explanation that is also used in Singapore's "Primary Math" series. Typically, it's easier to show what's going on with a whole number divided by a fraction. Once that is done, one can see that the divisor is inverted and multiplied. Then this is extended to fraction divided by fraction without explanation of "why". (See https://www.youtube.com/watch?v=e1gcBP2TmPk&t=217s)
I was able to show why it works in algebra class, once students had the tools necessary to understand.
I agree with you that at the lower grades (5th and 6th, say), seeing such a demonstration will not make things better. What they really are saying is students should understand the context; i.e., what it means when we divide by a fraction.
A more formal explanation in algebra classes is that: Since subtraction of a number means adding the additive inverse of that number, then division by a number means multiplying by the multiplicative inverse of that number. So a/b = a x 1/b. Students at this point know what a multiplicative inverse is, and that m/n x n/m = 1. So m/n = 1/(n/m).