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.
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 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.
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.