Before I say what it is, I am NOT posting this as clickbait (which is how it's often used)! 😂 I'm posting this as a Maths teacher who knows this topic inside-out and wants to help people to understand it better. There are MANY mistakes that people make and get the wrong answer, and I'm going to cover them in bite-size chunks each week for a few weeks
I was going to start discussing this tonight, but was hoping for some more answers. I guess people don't want to risk a wrong answer in a #Maths teacher's post! 😂 Don't worry - I don't bite! 🙂 This is #Mastodon not the other place. This thread's here to help people learn, and there are MANY mistakes that can be made, and it helps to see some in action. You wanna learn where you might be going wrong with your #Mathematics ? Post here. 😉 I'll check in again tomorrow, and then start explaining
@DevWouter@mattgoldman both got the correct answer of 1 - well done! Let's look at the correct working out, then delve into where people go wrong with their #Maths#math
8/2(1+3)
=8/(21+23)
=8/(2+6)
=8/8
=1
Also acceptable is...
8/2(1+3)
=8/2(4)
=8/(2*4)
=8/8
=1
If you've seen someone say that it's ambiguous, it isn't - we shall see in coming weeks that #MathsIsNeverAmbiguous they have simply forgotten one or more #Mathematics rules
Reference in top left of screenshot. Note the use of the words "everything" and "must" - certainly no room for ambiguity there! Hence the Distributive Law, because must always be obeyed. AKA expand brackets, AKA expand and simplify.
Next, "if you want to remove the bracket" - in other words, you CAN'T remove the brackets UNTIL YOU HAVE DISTRIBUTED AND SIMPLIFIED. Mistake #1 removing the brackets before completing that.
Mistake #2 ignoring that the "multiplication" is INSIDE THE BRACKETS...
This leads people to treat it as multiplication "M", but it's expanding brackets, "B", so must be done first! See meme.
Lastly (for this week) note in our previous screenshot "number or a letter". By "letter" they mean pronumeral. "pro" means "substituted for" so it's literally a symbol (letter in this case) substituted for a numeral. What I'm getting at is THEY ARE THE SAME THING. Mistake #3 "that rule only applies to letters". No, it applies to letters AND numbers.
#MathsMonday week 2, Terms.
Simply put, in #Mathematics Terms are separated by operators and joined by brackets. The most common example is 2a=(2a). This makes it simpler to write fractions. e.g. 1/2a rather than 1/(2a). That also means 1/2a isn't mistaken as half a, which would actually be written as a/2. Notice in the latter that the a is to the left of the division, which means it's in the numerator, and vice-versa for the former (more about this aspect of #Maths#Math later, but first)...
I want to note that everything I say here applies equally to the more common forms of this problem you'll see...
8/2(2+2) and
6/2(1+2)
I just tweaked it to have unique numbers, 8/2(1+3) to be clearer about which number I was referring to.
In the Term 2(1+3) 2 is known as the coefficient. With no multiply sign, it is 1 Term. If I wrote 2*(1+3), then that is now 2 Terms! Look what happens...
8/2(1+3)=8/(2+6)=8/8=1
8/2*(1+3)=8/24=44=16
Different number of Terms - different answer! ...
So this is why you MUST NOT arbitrarily add multiply signs to Terms (which a lot of people do!). If you do, you can change the answer. Single Terms are inseparable. A VERY common mistake is to say that 2(4)=24. No, it isn't! It's equal to (24). You CANNOT remove the brackets until there is only a single Term left inside them. i.e. 8.
Also, because (2*4) is inside brackets, this is, as per earlier meme, part of "B" in BEDMAS, not "M" - there is no multiply sign outside brackets! More next week
2/4
We already know that (1+3) has to be multiplied by 2 - the coefficient - hence The Distributive Law (TDL). i.e. we MUST "distribute" the coefficient over whatever is inside the brackets, a(b+c)=(ab+ac). It's the reverse operation to factorising, so if we fail to obey TDL then we literally break factorised terms.
We can also see that 8 has to be divided by the factorised term, hence our rules for Terms - Terms are separated by operators (+, -, x, /÷), and joined by brackets...
3/4
So 1+3 has been joined by brackets, and the 2 has been joined to that due to not being separated by an operator, so our second term is 2(1+3) - the standard way of writing factorised expressions - and our 1st Term is 8,which is separated from the 2nd Term by a division. If we wrote out expanding the brackets in full, it would be 2(1+3)=(21+23). Note that the "multiplication" (distribution) is INSIDE the brackets, hence it is part of Brackets in order of operations, NOT "multiplication"...
4/4
Most common mistake I see is...
8/2(1+3)
=8/2(4)
=8/24
=44
=16
The mistake happens when 2 and 4 get separated, which moved 4 into the numerator, instead of staying in the denominator. i.e. ignored that 2(4) is a single Term.
The mistake is regarding distribution as "multiplication", and so lowering it from the high precedence of "brackets". "multiplication" ONLY refers literally to multiplication signs. More next week
1/7
This week for #MathsMonday we are going to debunk the "implicit multiplication" (IM) claims (and also look at the mnemonics). I say claims, because there is actually no such thing as IM in #Mathematics. I find invariably the people who say there is have forgotten The Distributive Law (TDL) and/or Terms, but most often both! As we have already seen, these are both rules of #Maths, taught in many #Math textbooks, so that right away debunks any claims that "there is no convention in Maths"...
2/7
and you may recall I talked about "there is no evidence" claims in my first #FactFriday post https://dotnet.social/@SmartmanApps/110869090048130445 and here we are with a "there is no convention" claims. ;-) So what they do is treat it like a more important form of multiplication, and so put it, not at Brackets, but just ahead of Multiplication... though this can STILL lead to wrong answers, because TDL and Terms - which they've lumped together - are quite different things...
3/7
I've even seen one say (of calculators) "sometimes it obeys (IM), sometimes it doesn't". That's because the calculator in question obeys Terms but NOT TDL, and it's behaviour is 100% consistent with that (but confusing to someone who has lumped them together as IM). They even went as far as claiming "PEMDAS is wrong", cos they were treating it like rules (whilst also #LoudlyNotUnderstandingThings like the actual rules TDL and Terms), and made up a new mnemonic, so let's look at mnemonics...
4/7
There are 4 mnemonics - BODMAS, BIDMAS, BEDMAS, and PEMDAS. The mnemonics aren't the rules, they are just a way to remember the rules, and, as per screenshot, it doesn't matter whether you do Division or Multiplication first, hence there is no conflict between the last and the first 3 (only in the mind of people who think they are the actual rules, and one or the other HAS TO be done first). This also debunks "it depends where you learnt" - the mnemonics aren't universal, but the rules are!
5/7
Note that "Brackets" means remove/solve/expand and simplify brackets. As we have seen this is taught in Year 7 (when we teach about coefficients of bracketed Terms). It does NOT mean "inside brackets", which is what's taught in Primary School (when bracketed Terms never have a coefficient), yet many people claim it does. I found it curious that so many people have a rock-solid memory of what they were taught in Primary School, but not High School, so I ran this poll https://dotnet.social/@SmartmanApps/110868343908844233
6/7
which was the closest similar scenario I could think of. We are told (disingenuously unfortunately, but that topic's for another week) that negative numbers don't have square roots... then in University "Hey! That wasn't right! We're now going to teach you about Complex numbers...". This poll is a bit different situation then, as not everybody went to University, but it does illustrate the lasting division over the issue (1/3rd of people still believe no square roots for negative numbers).
7/7
Note:University people teach things like Complex numbers, Matrix multiplication, etc, NOT order of operations - that is taught in High School (I've never seen a HS teacher get these questions wrong). If you wanna know about a topic, then ask a person who teaches that topic!
IM summary: Terms can be bracketed, so they are resolved at "B" with TDL, may include an exponent, resolved by "E", but either way fully resolved before "M", so people treating them both as IM already missed the boat! TBC
1/6 #MathsMonday
1917 (part 1) - Left Associativity
I have seen many people who argue about this #Mathematics issue that refer to changes in 1917, and I started to research it this week. The #Maths change I thought was made, I have not been able to find any reference for it, and that is about /2(1+3) vs. /2*(1+3). Apparently in pre-1917 #Math the latter was also considered to be a single term (in the denominator) but now is 2 terms. If anyone has a reference for that then please let me know...
2/6
So first I'll discuss a reference I did find, which relates to a letter from Lennes to The American Mathematical Monthly https://doi.org/10.2307/2972726. There's a lot to unpack, and the TL;DR is no, they did NOT change any rules, and this week I'll talk about left associativity, a mistake that I have seen people make in similar order of operations problems to the ones we have been discussing...
3/6
The problem in question included the expression +5-4+1. Left associativity means a term is associated with the sign to the left. i.e. 4 is associated with a minus, the other 2 terms with a plus. We have discussed previously that we can do addition and subtraction in any order, and this is true... provided you preserve left associativity! And one way to do that is always work left-to-right. But imagine we wanted to go right to left, and read it right to left! Then we would read it as...
4/6
1+4-5, which is clearly wrong, because we didn't preserve left associativity. To work right-to-left preserving left associativity, we would do +1-4+5, and we get the correct answer, because we correctly kept the minus sign with the 4
A mistake I have seen people make with 5-4+1 is to first calculate 4+1, then subtract that from 5! That's the same as doing 5-(4+1), which is NOT the same as 5-4+1 - you changed a sign and changed the answer!
5/6
If you did want to do 4 and 1 first, then you would have to do it as 1-4, and THEN you would still get the correct answer. It's important to keep the minus sign with the 4. i.e. left associativity.
So the easy way to do that is to work left-to-right, but within order of operations rules!
6/6
Cos the other mistake I see is people insisting left-to-right is a rule, and ignore order of operations! No, order of operations is the rule, and left to right is a rule WITHIN Division, and other operations - like add and subtract, and multiply - can be done in any order, provided you maintain left associativity for all signs. i.e. within add and subtract +5-4+1 can be done as +1-4+5, or even +5+1-4, but definitely never +1+4-5 or +5-(4+1).
On page 93 he says "it is agreed that each symbol applies only to the term immediately following it" which shows he understands #Maths Terms and Left Associativity...
2/7
He then, notably, never mentions Terms again! He goes on to say "each symbol applies only to the factor (or divisor) immediately following it" - also correct (we'll be coming back to this next week for reasons)
He then tries to interpret Terms as things that have to be multiplied (sigh), and admits "I have not been able to find a single instance where this is so interpreted" - should've taken that as a clue(!), but he persists with his "everyone is wrong except me" Youtu... I mean letter...
3/7
He also says "Note the sign X to indicate multiplication" - again, that was a clue for you, and again I will say, the "M" for Multiplication in the mnemonics means LITERALLY multiplication signs, as Lennes has just highlighted (and yet he still missed the point!).
From here he keeps referring to Terms as "products", which would be a valid thing to do... had he understood that a product in the RESULT of performing a multiplication! I'll illustrate with some algebra...
4/7
Using a=2 and b=3...
ab=23 (a multiplication, subject to multiplication precedence)
ab=6 (a Term/product - the result of a multiplication)
The reverse of products is factors - the product of 2 and 3 is 6, the factors of 6 are 2 and 3 (and 1 and 6). Eagle-eyed people will see a correlation here to Factorising and The Distributive Law being the reverse operations of each other (which we already discussed previously)
Note that ab (=6) is already fully-solved - there is nothing more to do...
5/7
Terms can be bracketed (B), and have exponents (E), and if either applies then we have to solve the Term(s), but ab has neither, it's fully solved. In other words, the first 2 steps in order of operations is solving Terms, which is then followed by solving operators/symbols.
So at a low-level, the order of operations mnemonics refer to symbols - Brackets, Exponents, Division, Multiplication, Addition, Subtraction.
6/7
Having covered that, I'll re-iterate that Terms are separated by operators and joined by brackets (ab is a single term, and so is 2(1+3))
Back to Lennes' letter - he claims the actual rule being used is "All multiplications are to be performed first and the divisions next", and I'm presuming this is the "change" that people refer to, but this was never made a rule (because Lennes was simply not understanding Terms to begin with, which as he noted was already the way all textbooks did it)...
7/7
And lastly he also said "The multiplications may be taken in any order, but the divisions are to be taken in the order in which they occur from left to right", and this is correct - division MUST be done left-to-right, but the other operations can be done in any order... provided you maintain left associativity!
Next week I'll wrap up 1917 with the thing I'm led to believe did change (but have yet to find any reference for it, but taking these claims in good faith...)
1/4 #MathsMonday
1917 (iii) - Terms included in denominator
My investigation into (alleged) changes in #Mathematics rules in 1917 started with claims that the number of terms included in the denominator of a #Math expression was changed in 1917 (though I've yet to find any actual evidence of this - let me know if you have a reference for it). Some mentioned Lennes' letter, yet his letter says nothing at all about this! For now, let's assume it's true and see what that would mean for #Maths...
2/4
Say we have a/b+c. Pre-1917 this was (apparently) interpreted as a/(b+c), and then in 1917 it was changed to mean (a/b)+c. The latter is certainly what we use now.
Some claim this means that 8/2(2+2) is now equal to 16. No, it doesn't mean that at all! 8/2*(2+2) is equal to 16, and pre-1917 was equal to 1, but 8/2(2+2) only has 1 term in the denominator anyway(!), so IS AND ALWAYS WAS equal to 1! i.e. these people are not understanding Terms - 2*(2+2) is 2 terms, but 2(2+2) is 1 term...
3/4
Again, I have found no reference to support that alleged rule change happened, but I DID find the opposite was proposed! On page 313 of Cajori's 1928 "A history of mathematical notations" https://ia600306.us.archive.org/6/items/historyofmathema031756mbp/historyofmathema031756mbp.pdf he says that Stokes PROPOSED that /b+c be interpreted as /(b+c). He says nothing further about it, however it's certainly not the way we interpret it now, so it came to nothing in the end, regardless of what may have initially happened at the time...
4/4
In fact, as far as I can tell, the rules for order of operations haven't changed ever in at least 400 years! So, if someone says "the rules changed 100 years ago", then ask them for a reference, cos I certainly can't find any! Also tell them that 2(2+2) is a single term anyway. ;-)
That's a wrap on the (alleged) events of 1917, but I still have some more topics to cover(!) on order of operations - see you next week. :-)
1/2
Postscript to Lennes' letter. My understanding of the alleged change to the #Mathematics rules was that /ab+cd was changed from meaning /(ab+cd) to /(ab)+cd. As I said, I could find no evidence of said change. Via another #Maths conversation today I came back to one of the sites which alleged this change to #Math as a result of Lennes' letter, and I suddenly realised what they ACTUALLY meant all along... though it STILL comes down to them not understanding Terms....
2/2
Lennes' letter was about the interpretation of /ab - does it mean /(ab) or /ab, and what I THOUGHT they'd been talking about was multiple Terms after the division - /ab+cd - but what they ACTUALLY meant was multiple pronumerals/numbers IN a Term! i.e. is it /ab or /ab. No wonder I couldn't find anything about the rule changing!
So,as has already been discussed, /ab does now and always has meant /(ab). It's in textbooks now, and Lennes' letter confirms it was in textbooks then. Case closed
P.S. I have no idea why anyone thinks the rule changed. Lennes PROPOSED a change, but there was no change because he was simply not understanding the existing rules around Terms to begin with (as demonstrated by what he said about the textbooks, who were all treating /ab as /(ab), as we still do now).
P.P.S. The new rule that Lennes proposed was to do all multiplication first, then all division (which is what he saw the textbooks as doing), which STILL means that /ab is interpreted as /(ab). But somehow the takeaway some people manage to get from his letter is that /ab means /a*b??? I'm at a loss as to how they can do that. ¯_(ツ)_/¯ The answer to 8/2(1+3) was 1, Lennes wanted a new rule to (in his eyes) formalise it to 1, and to this day it's still equal to 1 (but without any new rules).
1/6 #MathsMonday
I continue to see people who say that in #Mathematics ab=ab, and thought of a good #Maths example to illustrate why ab is a single #Math Term (i.e. ab=(ab))...
Let's say I was 2 metres tall (just for the sake of using whole numbers in the example). We write that as 2m - in this case m is short for metres, but it also looks like an algebraic term, right? 🙂 So let's say m is a pronumeral, and in this case m is equal to 1 metre. In other words in both cases, m is the units...
2/6
And the 2, which is known as the coefficient, says how many of those units tall I am - it is a scalar. I'll come back to this in a moment...
Now let's imagine I buy some mangoes... which we'll represent a mango with the pronumeral m. 🙂 Let's say I have 1 mango, and now I get a second mango. We can write that as 2m. i.e. I bought 2 separate mangoes, and 2m tells me I now have 2 total, but separate, mangoes.
3/6
I am not 2 halves of a human who stiches the halves together when I want to do something, I am one whole human, of height (in this example) 2m - there are no separate 1m's, just 1 human of a height measured in metres, and a coefficient/scalar of 2 of those units. And I say units, not metres, because we can change those units of the pronumeral - we could make m=100 centimetres, or even 1000 millimetres, and the coefficient would still be 2, and I would still be 1 whole person of that height 🙂
4/6
Now let's say I'm at an Olympic pool, and I want to work out how many of me I could lay end-to-end. That would be 50m/2m=50/2=25, so I could lay myself 25 times end-to-end.
BUT some people INSIST that 2m has to be written as 2m, so let's see what happens...
50m/2m=50/2m*m=25m^2, so laying myself end-to-end ends up with me being 50 metres squared... wait what?! Did I get splattered out into a big mess covering 50 square metres? 🙂 That can't be right. No, that's right, that's not right...
5/6
When you rewrote 2m as 2m, the m got flipped from being in the denominator to being in the numerator, and you ended up with the WRONG ANSWER. Welcome to why 2m=(2m) NOT 2m. It is a single Term - Terms are separated by operators (including multiply) and joined by brackets. Lennes' letter proves this was how it was done more than 100 years ago (and as far as I can tell it's been done this way for at least 400 years)
8/2(1+3)=8/24=44=16
8/2(1+3)=8/(2+6)=8/8=1
8/2(2+2)=1
6/2(1+2)=1
...
6/6
So remember, the "M" for multiplication in the mnemonics (BEDMAS, PEMDAS, whatever) refers LITERALLY to multiply signs. If there's no multiply sign, then it's NOT "multiplication", it's a Term. If there's a bracketed term which has a coefficient, then the coefficient has to be distributed, and that's part of solving Brackets/obeying The Distributive Law. Anyone who says anything different to that is just #LoudlyNotUnderstandingThings
1/9 #MathsMonday
Today I'm going to talk about adding/removing brackets, for 2 reasons...
people prematurely remove them
people adding them incorrectly
In #Mathematics we have many things which are the opposite of each other - add/subtract, multiply/divide, factorising/expanding - and so it is with brackets in #Math also. This is because if someone has put together an expression from a number, the rules of #Maths have to make sure we follow the exact opposite steps to get the same number...
2/9
I ran a poll this week to see where people would add brackets to 8÷2(1+3) - which I'll discuss shortly - and to know where to correctly add them, we first have to know what he rules are around removing them!
The first we have already discussed before, that being The Distributive Law. i.e. if a bracketed expression has a coefficient, then we MUST multiply what is INSIDE the bracket by what is outside the bracket. AKA Distribution. i.e. a(b+c)=(ab+ac)...
3/9
The next is we must solve innermost brackets first, then work our way out. So for (a*(b+c)) we would have to do (b+c) first, then multiply that by a. Of course, that gives a different answer than (ab+c), where ab would be done first, then add c to that.
The last is that we can't remove brackets until there's only 1 term left inside. So, for starters, you can't do 3(5)=35. You must do 3(5)=(35), and you can see we still have 2 terms inside the brackets, so we can't remove them yet...
4/9
But then (35)=(15)=15 - with 1 term left inside, we can NOW remove the brackets. We have discussed the impact of writing 3(5)=35 vs. 3(5)=15 when we discussed Terms. i.e. 2 terms vs. 1 term. In fact, the whole purpose for brackets to begin with is to show that everything inside, plus any coefficient, is all 1 term. Remember that Terms are separated by operators and joined by brackets, so ab is 2 separate terms, but (ab) and ab are single terms...
The first and last option were split evenly, which was expected - when-ever people claim that the expression is ambiguous, and that "you need to add brackets", they usually go for one of those 2 options (and sometimes 8÷(2*(1+3)) in the case of the last).
6/9
Remember that if any brackets are omitted, they can ONLY have been omitted if there was only 1 term left inside them, so right away we can rule out the first answer as wrong, as 8÷2 is 2 terms, so putting brackets around that firstly elevates it from being "division" in order of operations to "brackets", and secondly we just separated the coefficient from the actual bracketed term. When we add brackets, we must do so in a way that doesn't change the answer, and this (and Google) does...
7/9
The remaining options all give the correct answer, but one is more correct than the others in terms of making intent clear.
The last option, 8÷(2(1+3)) we would do innermost brackets first, 8÷(2(4))=8÷(8), but that's what The Distributive Law tells us to begin with anyway! 8÷2(1+3)=8÷(2+6)=8÷8, hence why we never put brackets there. In fact this is the standard way to write factorised expressions. e.g. 2a+6b=2(a+3b).
(8)÷2(1+3) does nothing to resolve the alleged ambiguity in 2(1+3)...
8/9
So that leaves us with 8÷(2)(1+3), so the answer which is most correct in my opinion came last in the poll, which in fact I predicted (hi @damovisa ;-) ).
So far we have discussed expressions with 1 set of brackets, and brackets within brackets, but there's also adjoining brackets. i.e. (a+b)(c+d). Everyone (hopefully) recognises that these need to be expanded. Many of you may even recall learning the acronym FOIL to keep track of which multiplications you've done...
9/9
This is typically taught in Year 8-9. Now imagine that b=0, that'd leave us with (a+0)(c+d)=(a)(c+d), and now a is a single term inside brackets, so those brackets can be removed, leaving us with a(c+d). AKA a(b+c). Oh look! We just arrived at The Distributive Law via expanding polynomials. 🙂
So I'd consider 8÷(2)(1+3) to be the most correct way to add brackets because it should remind people of expanding polynomials, and leave no doubt about the order. This will be discussed further later
1/10
This week for #MathsMonday I'm going to talk about #calculators, in particular the current topic of #Mathematics order of operations (which I am nearly finished with now), and e-calculators (I'm looking at you #Developers#Programmers).
It's important to know where brackets go in #Maths expressions, and after last week's topic I ran a follow-up poll https://dotnet.social/@SmartmanApps/111145907574869556 to see how many people could remember the #Math FOIL acronym from High School, because I sensed a deeper issue...
2/10
As you can see, less than half of people remembered FOIL, which relates to how to expand adjacent brackets. I had already noticed previously that when people said "it's ambiguous - add more brackets", in fact they often put brackets in the wrong place anyway (so still end up with the wrong answer), and my poll confirms that not only do people not remember Terms and The Distributive Law (which means the expression is unambiguous), they don't even remember how to expand brackets in general...
3/10
I've always used 8/2(1+3) to test calculators, and said in the poll that in my opinion the correct way to add brackets, if you were to do that, would be 8/(2)(1+3), because that might trigger one's memory of FOIL, but turns out most people don't remember that either(!), so #Education is really the only answer (welcome to why I'm writing this thread). Anyway, I tested the latter with my calculator, and (of course) it works, but also it pops up a (*) to show it just expanded brackets! Cool...
4/10
First, with physical calculators, I want to say I find it disingenuous when people say "even calculators don't agree". There was a brief period where Casio gave the wrong answer (no longer the case), but otherwise ONLY Texas Instruments calcs give the wrong answer, and it's right there in the manual why - they follow the Primary School order of operations, which doesn't work when bracketed expressions have a coefficient. Literally every other brand obeys Terms and The Distributive Law...
5/10
So all calcs bar one agree on the way to do this problem (and who on Earth knows why that one brand is doing something different - if you're going to allow people to enter a factorised expression then you have to make sure you obey The Distributive Law, as they are the reverse of each other).
Now, e-calculators (listen up #Developers#Programmers). I gave #Google a second chance, to see if it at least obeyed FOIL (i.e. expanding adjacent brackets), and... it doesn't (sigh)...
6/10
Even when I have specifically typed 8/(2)(1+3) it REMOVED MY brackets and added it's own brackets, again leading to the wrong answer. So not only do they not know The Distributive Law, they ALSO don't know FOIL (I'm so embarrassed when I see other programmers declare "you don't need to know Maths to be a programmer!" - well, look where THAT'S got us!).
At this point, I want to give a shout out to #MathGPT - literally the ONLY e-calculator I have ever found that gives the correct answer!
7/10 #Desmos USED TO give the correct answer (many teachers use it. I USED TO, but it's useless now - can't type in what's in the textbook, have to add extra brackets to make it work and do what's in the textbook correctly). I know one of the changes which broke it too. To make 8/2 a fraction (not division) you had to put brackets. i.e. (8/2). They changed it so that it automatically turns 8/2 into a fraction, which is cool, but unfortunately it ALSO turns 8÷2 into a fraction, which is wrong...
8/10
Of (some of) the other e-calcs, #Wolfram does the same as Desmos, #Android and #Excel both forcibly add a multiplication symbol (thus breaking the factorised term), and with the #Windows calculator, any coefficient you type in literally disappears! i.e. type in 8/2(1+3), and it gives you 8/(1+3). #Microsoft MathSolver is the buggiest. If you use ÷ then it essentially does the same as Google, but if you use / it turns it into a fraction and puts the whole 2nd term into the denominator...
9/10
So we get the right answer(!) UNLESS there's more than 1 term following the slash, in which case it puts ALL of them in the denominator, so we're back to wrong answers. So you CAN get the right answer, but ONLY if you use a slash, and ONLY if there's only 1 term following it.
So for those who claim "e-calculators do it correctly", how is it that they're all doing it DIFFERENT ways?? There is only 1 correct way to do it, and only #MathGPT does it that way (looking at you #Desmos)...
10/10
So #Developers#Programmers if your app/library isn't correctly parsing a(b+c) and/or (a+b)(c+d), preferably start by getting the latter working, by doing something like ((a+b)(c+d)) if needed, then changing the former to (a)(b+c) should also work. You could also turn a(b+c) into (a(b+c)). Never(!) add any other brackets or any other multiplication symbols anywhere else (that's the exact problem we have already - sigh).
1/12
I'm essentially at the end of my #MathsMonday series on #Mathematics order of operations (might tidy up some loose ends next week, maybe do a summary), but what I wanted to do this week was address some #Maths#textbook#authors and #Math#Teachers as I have seen issues with #Education also. i.e. as much as many people have misremembered what they were taught, I've seen cases of incorrect things being taught to begin with also...
2/12
In particular in regard to point 10 in these teaching rules - leave no room for mis-understanding. Of course we always try to follow 1 - teach the easy before the difficult - but sometimes we compromise to make it easy (more precisely to avoid hard questions I suspect!).
For example, I remember being told there was no such thing as a square root of a negative number, and yet we teach that there is if/when we go to University and learn about Complex numbers...
3/12
So I ran a poll - https://dotnet.social/@SmartmanApps/110868343908844233 - and as you can see, nearly 30% of respondents STILL believe that there is no such thing, so we have done those students a dis-service! I advocate that we need to be more honest with #students such as "we are only going to talk about square roots of positive numbers, but if you go to University then you will learn, if you are in a class about Complex numbers, that negative numbers can also have square roots,"....
4/12
In particular I wanted to address some little white lies that get told when teaching order of operations because, similarly, some former students still believe them! This is evident in the Texas Instruments calculator manual, where the person who programmed this has remembered being told in Primary School that "Brackets" means INSIDE the brackets, yet clearly forgotten being taught The Distributive Law in High School, where "Brackets" now means SOLVE the brackets...
5/12
And this is exacerbated by us always omitting coefficients of 1 (because it doesn't change the answer). Absolutely we start by teaching solving brackets with no (in fact 1) coefficient first, but I think we could do something like include the 1 the first few times, to get them ready for when they are going to learn about non-1 coefficients. But we should never say "inside the brackets", because we can see the result of that in the Texas Instruments manual.
6/12
The reason I am making a big deal about exactly what we tell students, is because sometimes, like in the case of Terms and omitted multiplication signs, students will say we're "just being lazy" when we leave out the multiplication sign. This gives rise to SEVERAL issues...
One little white lie here is that the multiplication sign is "optional" - no, if a and b are 2 separate Terms, then you MUST leave it in there, a*b, but if ab is a single Term, then you MUST leave it out! ...
7/12
I've seen people frequently just switch between them, which of course changes the expression, and frequently changes the answer, leading to the arguments we've all seen over the clickbait questions like 6/2(1+2). Many will say "that's just the same as 6/2*(1+2)". No, it isn't! When you added the multiplication sign you changed the expression and changed the answer. We need to be quite explicit that ab and ab are NOT the same thing. i.e. ab=(ab)... which leads to another consequence...
8/12
Some people also claim that we "are just being lazy" when we write a(b+c), rather than (a(b+c)), or worse other variations (like (d/a)(b+c)), and I have to point out no, that is absolutely the standard way to write a factorised expression, and the reason The Distributive Law exists is because it is the reverse operation to factorising (the definition of Terms also applicable here, with the deliberately omitted multiplication sign)...
9/12
So, again, we need to stress that a(b+c) is the standard way to write the factorised form of (ab+ac), which brings me to...
The reason I brought up "lazy" accusations (other than those particular ones being wrong) is because I HAVE seen some examples of laziness by authors (and potentially teachers too)
I know, and you know, all educators know that you cannot remove brackets unless there is only 1 term left inside them, and yet I've seen many textbook definitions which break that rule!
10/12
So rather than writing a(b+c)=(ab+ac), MANY textbooks write a(b+c)=ab+ac, which leads people to think they can removed the brackets as soon as they've distributed, which is wrong. This leads to people doing things like 8/2(1+2)=8/2+4=4+4=8. Writing ab=ab instead of ab=(ab) also goes in this category. This really IS a case of us being lazy, and we need to stop doing that. We KNOW the rule is don't remove brackets unless there's only 1 term inside, and we need to ALWAYS OBEY that rule...
11/12
Because otherwise people start prematurely removing brackets and end up with wrong answers. The classic example is 6/2(1+2)=6/23=33=9
Re screenshot. In this textbook, they've left out the brackets in the definition - "5(36) means 5x36" - but fortunately in this case they have a worked example where it matters and they DIDN'T remove the brackets prematurely. Not all textbooks have such a worked example, so we need to make sure the definition says the right thing i.e. "5(36) means (5x36)"
12/12
Failing to do what I have outlined has led to a situation where literally half of the population gets the wrong answer to 6/2(1+2), and even worse a lot of those people then try to #bully people who actually have the right answer (I know, because I've had some of them try to bully me, a Maths teacher, into "admitting" I was "wrong", and I've seen at least one time someone was #bullied into exactly that). We can help eliminate that by never being "lazy". e.g. always write a(b+c)=(ab+ac).
1/11
Well, I THOUGHT I was nearly done with this #MathsMonday topic. This week ANOTHER thing that the "#Mathematics is ambiguous" lot don't understand turned up - #Maths Expressions and #Math Unary/Binary operators. Today we'll prove the order of operations rules #MathsIsNeverAmbiguous
We've discussed before that Terms consist of pronumerals and/or numbers. Expressions consist of Operators (+,-,*,/) and Operands (Terms)...
2/11
I got sent down the garden path by someone, who made a (false) claim about the rules changing 100 years ago, because they were calling Terms "Expressions" - didn't realise for ages! Thanks for nothing (sigh, but at least that's been debunked now). See my posts about Lennes' letter, which start at https://dotnet.social/@SmartmanApps/110965810374299599
Next, addition and subtraction are unary operators. i.e. they are only associated with 1 number, the number that follows it...
3/11
...and note that it's the number that follows it. i.e. in the expression 6-5+2, the minus sign is associated with the 5, NOT the 6, hence left associativity - a term is associated with whatever sign is on it's left (and of course the 6 is associated with a +, but we don't normally write + at the start of an expression, only if it's a -, and even that we try to avoid. i.e. if we had an expression -a+b, we would usually write it as b-a)...
4/11
Now, some of my Tech followers would be aware that sometimes in programming we use postfix notation, which is right associativity - e.g 2+3 is stored as 23+ - but in Maths we only use left associativity.
Next, * and / are binary operators. i.e. they are associated with TWO terms - both the one that follows and the one that precedes it (whereas unary operators are only associated with the term that follows them).
So let's see what that all means for order of operations...
5/11
Consider 2+3*4. Some claim that the order of operations were "arbitrarily" made up in order to make this not "ambiguous". This isn't the case at all! The order of operations are the natural result of all that we have defined so far. They say "how do you know whether to do the addition first or the multiplication first without these arbitrary rules?". Easy...
We have 3 "terms" (if we accept "ambiguity") - 2 (which is associated with an unwritten +), +3, and 3x4. What to do with the 3?...
6/11
In 2+3*4, 3 is both associated with a + AND with *4. You know what it's NOT associated with - because + is a unary operator - the 2. So it IS associated with the 4, but not the 2, so we have to do the multiplication first! This is a natural consequence of * being a binary operator! It associates 2 numbers with each other, unlike a unary operator which is only associated with 1 number.
But let's put that aside for a moment and look at "addition first" from another angle...
7/11
Recall in The Distributive Law that we only multiply what is inside the brackets by the TERM which is outside the brackets. NOT the EXPRESSION which is outside the brackets, the term. i.e. in 8/2(1+3) the slash separates 2 terms - 8 and 2(1+3), so only 2 distributes over (1+3). To do any different would require writing it as (8/2)(1+3), where now we have an expression that we wish to distribute (made into a term by putting it in brackets).
8/11
If we did the addition first, then we're not multiplying the 4 by the term +3, but by the EXPRESSION 2+3. If we're going to multiply/divide by expressions rather than terms, then why did we use order of operations to break everything down into terms first? We would undo everything we just did! 2 and 3 are separate terms, so the only thing that 4 gets multiplied by is the term 3, THEN we can do the addition.
9/11
We have to solve binary operators (*,/) BEFORE we solve unary operators (+,-)! This isn't arbitrary - this is a natural consequence of associativity and the difference between terms and expressions! So to do addition first, it has to be in brackets, because that's how we say "this is 1 term, not 2", "this is an exception to the usual rules".
So BEDMAS from the top...
We have Brackets first, because they contain exceptions to the usual rules and/or is a factorised term to unfactorised...
10/11
Next is Exponents. Why? Recall that 5²=55. Recall ALSO that 5² is a single term, whereas 55 is 2 terms - you already know we can't go breaking up terms! So that means we have to solve exponents (single terms) BEFORE multiplication (multiple terms separated by *).
So with the first 2 steps that's all terms simplified, NOW we can move onto the operators...
We've seen now that binary operators have to be done before unary operators, so * / before + -...
11/11
And because 2/43=23/4, and 1+5-4=1-4+5, we can do those in any order (but within division itself we have to do that left-to-right, the other 3 don't matter).
And so, we have now proven that the order of operations, BEDMAS, is NOT arbitrary, but in fact a natural consequence of associativity and the difference between terms and expressions (solve terms, solve binary operators, solve unary operators). 🙂 See ya next week.
1/2
One more #MathsMonday post about order of operations - a short one this week(!) - and then I'll do a summary next week. This one is mainly just to address some objections I've seen to the correct answer, which amounts to "but that would mean a÷bc=a÷b÷c, and that can't be right!". That's exactly right actually(!), and is a #Math property that we use in #Maths things like factorising.
2/2
Let's say we have 1÷2÷3
1÷2÷3=½÷3=(1/6)=1/(2*3)
We could've also done
1÷2÷3=(1/2)(1/3)=(11)/(23)=1/(23)
Either way we have a÷b÷c=a/bc for any values of a, b, and c! Also a÷b÷c÷d=a/bcd, etc.
When we distribute a negative over a positive we get a negative (e.g. -2(+3)=-6). Similarly distributing a division over a multiply we get a division. i.e. ÷2(3)=÷6. So ÷(2(1+3))=÷2÷(1+3)
So yes, 8/2(1+3) is indeed equal to 8÷2÷(1+3)! (1+3) is in the denominator either way!
1/9 #MathsMonday
I have (unless someone comes up with yet ANOTHER way to get this wrong, which at this point wouldn't surprise me anymore!) finished covering #Math order of operations and will wrap this topic up with a summary of all relevant #Mathematics rules
@level98
Ah ok. Well I looked up VCE and it said Victoria. We use pronumeral in NSW, but the English mostly use variable, though I always taught my students pronumeral as variable is misleading - it may be a constant! Interestingly the pronumeral screenshot I used comes from a UK textbook!
Pro literally means "substituted", so a pronumeral is literally a symbol which has been substituted for a numeral. There may be something in Cajori if it was already in use then (1928).
@SmartmanApps I have a copy, and a digital copy on which I've searched... and it does not appear to be in there.
It would seem to have been introduced by an Australian middle-grade mathematics teacher / teachers.and a quick google seems to confirm that e.g. https://www.wordsense.eu/pronumeral/
It seems to be just a "mathematics education" term and I'm not convinced that introducing this additional terminology helps students.
@level98
"a digital copy on which I've searched" - ah ok. Well was worth a shot.
"additional terminology" - but it's not additional here. We don't use "variable" to start with here, since it may be a constant. U.K. uses both from what I've seen in textbooks. Not sure what other countries use. Hey @claudius you know a couple of German Maths teachers - do they use pronumeral or variable (or something else)?
@SmartmanApps@claudius It is "additional terminology" in the sense that: it was not used previously, and then it was introduced by mathematics educators.
It is also "additional" in that (as far as I am aware) it is not used outside of a relatively limited setting. 1/4
Mathematicians (and others, e.g. programmers) use "variable" to mean a symbol (e.g. the letter "x") representing a mathematical object. It does not mean "a number that varies in a particular context". It means the symbol may vary as to what it represents. So, for example, a letter, which at some point represents a constant value, is also a "variable". 2/4
@SmartmanApps On the other hand, maths is an eminently flexible in its use and people may specifically define terms and the use of terms etc. within a specific context in a particular way e.g. introducing the term "pronumeral" and defining a specific use of "variable" etc. But it should be understood that this is what is being done.
I mean, get a few mathematicians (physicists or whatever) in a room and they'll argue endlessly over the most useful use of terms, favourite notation etc. 3/4
@SmartmanApps But it should be clear that's what's being done. I'm not saying "pronumeral" is correct / incorrect but that it's good to understand its use is relatively limited and, I'd encourage teachers to check if it's use is helping, or not. Not only in their context, but also for students going on to study maths where the term is not used.
I'm not convinced its particularly useful - but, of course, that leaves open the possibility of someone convincing me otherwse in the future. 4/4
@level98
"not used previously" - previous to when? We've used it in NSW since before I went to high school
"programmers use variable" - ah yes, but we also have constants(!), so collectively they would be... pronumerals? 🙂 P.S. Maths was around before programming 😉 😂
"specifically define terms" - yes, Term itself has a specific definition in Maths which is important to understand. 😃 See Lennes 😂
"students going on to study maths where the term is not used" - but we always use it here
@SmartmanApps "We've used it in NSW since before I went to high school"
There was a time before you went to high school.
" 'students going on to study maths where the term is not used' - but we always use it here"
There is also maths beyond your "here".
Partly why I was originally curious about what grades you teach e.g. up to year 10?... as I say, I've not heard it used in Y11 & 12 or at a university level / among professional mathematicians.
@level98
"There was a time before you went to high school" - I know, that's why I asked what you meant by "previously" - previous to when?
"also maths beyond your here" - I know, but it gets used in the U.K. too (it's in my textbook screenshot on the term), but having not gone to high school there I don't know how long they've used it for 🙂
"what grades you teach", up to Year 12, then there's my University study as well - in all of that I've never once encountered "variable" being used here
@SmartmanApps I was born, raised and educated in the UK (school and Uni) and never came across "pronumeral". I've taught in Australian high schools and Uni's and only come across it in the middle years.
Of course, that's not definitive evidence of the breadth of it's use (hence why I'd been looking for references and asking others). But nothing I've found contradicts it... and even supports it. 2/3
@SmartmanApps It is not mentioned as a term in the VCE Study Design (and is not used in the exams... and not in any texts, as far as I'm aware, but I haven't exhaustively read all of them).
And I can't find it mentioned in the HSC syllabus etc. but I'm not so familiar with the documents etc. for that. 3/3
@level98
1/2
"whenever some math educator first used it" - ah ok. Well as far as I can tell in NSW that's somewhere back in the mists of time.
"but rarely used outside Australia", yes I saw that, but as someone who lived through Punk, I treated it with the same grain of salt as "the term Generation X was rarely heard of before the 90's" (which was clearly written by someone in the U.S.). i.e. people turn "I hadn't heard of it" into "it was rarely heard of"...
@SmartmanApps "The mists of time"... my guess is the 90's (not very misty).
Roughly when did you use it at high school?
I made clear I was not just relying on my own experience (though, adding it to the other evidence) but also multiple refs, curricula, exams, texts, asked many people in person, posted the question on social media etc.
Of course, if you can find refs to broad use elsewhere, I'd be interested. Otherwise, it'd seem sensible to go with available info rather than a "vibe".
@SmartmanApps Addendum: searching the "Australian National Curriculum" the term is only used twice... and not sure why they particularly chose to use it on those two occasions because, in the same "courses" the term variable is used extensively. I can only guess it was a quirk of the author. (Something I know of happening in other "curriculums".)
2/9
2. Terms are separated by operators and joined by grouping symbols (brackets and/or fraction bars). i.e. ab=(ab). ab is a single term, ab (without brackets) is 2 terms, but adding brackets, (ab) is again 1 term. Note: importantly this means that 1/ab=1/(ab)
2b. the implication of (2) is that whether you have a multiplication symbol or not is NOT "optional". If there is a term, you CAN'T just add in a multiplication, and if there are separate terms you can't just remove it...
You are saying that in the first case the 2 "binds to" the bracketed term and should be distributed first, before doing anything else. That means the first will evaluate to 8/(23+21) which is 1.
But in the second case the division and multiplication must be evaluated left to right, being of equal precedence. Therefore the second evaluates to:
(8 / 2) * (3+1)
which is 16.
So you are saying that the implicit multiplication in the first case has a higher precedence than the explicit multiplication in the second case.
@ColinTheMathmo@ingwar
1/3
"You are saying that in the first case the 2 "binds to" the bracketed term"
No, I'm saying that the 2 is PART OF the bracketed term - a term that includes brackets. Terms are separated by operators, and there is no operator separating them
2*(1+2)=2*3 (2 terms multiplied)
2(1+2)=6 (1 term, no multiplication)
See https://dotnet.social/@SmartmanApps/110846452267056791
@SmartmanApps Every practising research mathematician I've spoken with about this disagrees with you.
But I know that I won't convince you to reconsider. I know you have your approach, and your conventions, it's just that they do not accord with those of the mathematicians with whom I work.
Having a PhD in pure maths, and working with people who train maths olympiad candidates, and with people who have won the fields medal, we don't agree with you.
@ColinTheMathmo@ingwar
"disagrees with you" - textbooks. You mean disagrees with textbooks.
"I know you have your approach, and your conventions" - I have textbooks, with rules.
"Do you mean 1/(ab) or (1/a)b ?" - did you say "1 divided by ab" or "1 divided by a multiplied by b"?
"It's ambiguous" - read my thread and you'll find #MathsIsNeverAmbiguous In particular read the part about Lennes - he was saying the same thing as you in 1917, but Cajori was quite specific about it in 1928.
@SmartmanApps The frustration the university lecturers are expressing to me is that:
(a) students get it wrong, and
(b) what you are saying is wrong.
They are saying to me "No wonder the students are so confused when the educators are telling them things like this."
So we won't reach an agreement. I'm being told by professional mathematicians that what you are teaching is wrong. I think I'd rather believe them than you, especially given my own personal experience as a PhD in Pure Maths.
I will continue to tell people not to believe everything they see on the internet.
But for now I think there is nothing further to be gained from this exchange.
@ColinTheMathmo@ingwar
"students get it wrong", "students are so confused" - the teachers mark the students, and they (high school students) get it right - it's adults who get it wrong. My thread is for adults.
" I think there is nothing further to be gained from this exchange" - not while you keep pretending it's just me and not all textbooks. If you wanted a scholarly discussion on the topic then why all the ad hominems and appeals to authority?
@SmartmanApps I have no doubt that you are a great and effective teacher. I have no doubt that when you teach people things they are then able to recall them and use them.
But I provide this information for you to consider:
I am being told by professional working research mathematicians that part of what you are teaching is wrong.
I'm not an expert, I'm just a PhD in mathematics who works with these people, and I can see their frustration. All I wanted to do was to give you some feedback from actual, real, active research mathematicians.
But I can see that it's unwelcome. If you describe such feedback as ad hominems and appeals to authority, then they are clearly of negative value.
So I will shortly delete all my comments and leave you to it. I will also return to my colleagues and invite them to interact with you directly should they choose.
@ColinTheMathmo@ingwar
"I'm not an expert" - correct. High School Maths textbooks and teachers are. This is what we all teach - it's in the syllabus.
"invite them to interact with you directly should they choose" - well tell them to read my thread first and tell me where they think it's wrong, and provide textbooks that support their assertion that it's wrong. That's how debate works.
@SmartmanApps High School Maths Textbooks and Teachers are experts in teaching things, I have no problem with that.
But when current, active, research mathematicians say that the things being taught are wrong, and get frustrated with what students have been taught, then perhaps the textbooks and teachers really are teaching the wrong thing.
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