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#MathsMonday

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💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br><a href="https://dotnet.social/tags/Chinese" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Chinese</span></a> <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a><br>I've seen this video a couple of times, but I never saw any explanation.... so I decided to work it out for myself how they did that! 🙂 Feel free to watch it and see if you can work it out yourself, but if you can't (or don't want to), then here's how this works (scroll down for reveal)...</p><p>First some pronumerals, then the steps - a=97, b=94, c=3, d=6<br>- c=100-a<br>- d=100-b<br>- the first 2 digits is a-d<br>- the last 2 digits is cxd</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/2<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br><a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <br>This week mainly just a new resource to share with you all that I came across this week, as a result of another of those ridiculous claims make in arguments about <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a>. In this case there was a familiar "division is just the inverse of multiplication" claim, which I knew wasn't right. I mean we know that multiplication is repeated addition - 2x3=2+2+2 - so does that mean division is repeated subtraction? 😂 Or...</p>
PJ "chinga la migra" Coffey<p>Ooh that's quite nice. </p><p>A trigonometry proof of Pythagoras. That's unexpected.</p><p><a href="https://www.cbsnews.com/news/teens-come-up-with-trigonometry-proof-for-pythagorean-theorem-60-minutes-transcript/" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://www.</span><span class="ellipsis">cbsnews.com/news/teens-come-up</span><span class="invisible">-with-trigonometry-proof-for-pythagorean-theorem-60-minutes-transcript/</span></a></p><p><a href="https://mastodon.ie/tags/maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>maths</span></a> <a href="https://mastodon.ie/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a></p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/6<br>I have a <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> <a href="https://dotnet.social/tags/FactFriday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>FactFriday</span></a> cross-over event post for you this week 🙂 </p><p>The <a href="https://dotnet.social/tags/fact" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>fact</span></a> is that <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> is universal. We may use different notation in different countries, but the underlying <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> is the same everywhere. i.e. there is nothing "arbitrary" about <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> </p><p>I saw someone question if there might be "different" Maths somewhere else in the universe. No. If I have 1 thing, and I get another thing, I now have 2 things. i.e. 1+1=2 (or your country's notation)...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/9<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br>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 <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> order of operations and will wrap this topic up with a summary of all relevant <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> rules</p><p>First a reminder that the index for this thread is at <a href="https://dotnet.social/@SmartmanApps/110897908266416158" rel="nofollow noopener" target="_blank"><span class="invisible">https://</span><span class="ellipsis">dotnet.social/@SmartmanApps/11</span><span class="invisible">0897908266416158</span></a>, where these <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> issues are discussed in depth</p><p>1. a pronumeral is literally a substitute for a numeral, and as such all arithmetic rules apply to them...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/2<br>One more <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> 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 <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> property that we use in <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> things like factorising.</p><p>Let's illustrate this with an example...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/11<br>Well, I THOUGHT I was nearly done with this <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> topic. This week ANOTHER thing that the "<a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> is ambiguous" lot don't understand turned up - <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> Expressions and <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> Unary/Binary operators. Today we'll prove the order of operations rules <a href="https://dotnet.social/tags/MathsIsNeverAmbiguous" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsIsNeverAmbiguous</span></a> </p><p>We've discussed before that Terms consist of pronumerals and/or numbers. Expressions consist of Operators (+,-,*,/) and Operands (Terms)...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/12<br>I'm essentially at the end of my <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> series on <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> 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 <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://dotnet.social/tags/textbook" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>textbook</span></a> <a href="https://dotnet.social/tags/authors" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>authors</span></a> and <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://dotnet.social/tags/Teachers" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Teachers</span></a> as I have seen issues with <a href="https://dotnet.social/tags/Education" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Education</span></a> 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...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/10<br>This week for <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> I'm going to talk about <a href="https://dotnet.social/tags/calculators" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>calculators</span></a>, in particular the current topic of <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> order of operations (which I am nearly finished with now), and e-calculators (I'm looking at you <a href="https://dotnet.social/tags/Developers" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Developers</span></a> <a href="https://dotnet.social/tags/Programmers" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Programmers</span></a>).</p><p>It's important to know where brackets go in <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> expressions, and after last week's topic I ran a follow-up poll <a href="https://dotnet.social/@SmartmanApps/111145907574869556" rel="nofollow noopener" target="_blank"><span class="invisible">https://</span><span class="ellipsis">dotnet.social/@SmartmanApps/11</span><span class="invisible">1145907574869556</span></a> to see how many people could remember the <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> FOIL acronym from High School, because I sensed a deeper issue...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/9<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br>Today I'm going to talk about adding/removing brackets, for 2 reasons...<br>- people prematurely remove them<br>- people adding them incorrectly<br>In <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> we have many things which are the opposite of each other - add/subtract, multiply/divide, factorising/expanding - and so it is with brackets in <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> also. This is because if someone has put together an expression from a number, the rules of <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> have to make sure we follow the exact opposite steps to get the same number...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/6<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br>I continue to see people who say that in <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> ab=a*b, and thought of a good <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> example to illustrate why ab is a single <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> Term (i.e. ab=(a*b))...</p><p>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...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/4<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br>1917 (iii) - Terms included in denominator<br>My investigation into (alleged) changes in <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> rules in 1917 started with claims that the number of terms included in the denominator of a <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> 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 <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a>...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/7<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br>1917 (ii) - Lennes' letter (Terms and operators)<br>I've seen many refer to <a href="https://www.jstor.org/stable/2972726" rel="nofollow noopener" target="_blank"><span class="invisible">https://www.</span><span class="">jstor.org/stable/2972726</span><span class="invisible"></span></a> as a change in <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> precedence rules, and yet nothing at all actually changed! It does show that 100 years ago though, even then there were people <a href="https://dotnet.social/tags/LoudlyNotUnderstandingThings" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>LoudlyNotUnderstandingThings</span></a> <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> terms and rules! 😂</p><p>On page 93 he says "it is agreed that each symbol applies only to the term immediately following it" which shows he understands <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> Terms and Left Associativity...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/6<br><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a><br>1917 (part 1) - Left Associativity<br>I have seen many people who argue about this <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> issue that refer to changes in 1917, and I started to research it this week. The <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> 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 <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> 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...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/7<br>This week for <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> 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 <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a>. 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 <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a>, taught in many <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> textbooks, so that right away debunks any claims that "there is no convention in Maths"...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p><span class="h-card"><a href="https://mastodontech.de/@denki" class="u-url mention" rel="nofollow noopener" target="_blank">@<span>denki</span></a></span> <span class="h-card"><a href="https://defcon.social/@fortyseven" class="u-url mention" rel="nofollow noopener" target="_blank">@<span>fortyseven</span></a></span> <span class="h-card"><a href="https://mamot.fr/@clement_la_baleine" class="u-url mention" rel="nofollow noopener" target="_blank">@<span>clement_la_baleine</span></a></span> <span class="h-card"><a href="https://wasnever.cool/@schmutzie" class="u-url mention" rel="nofollow noopener" target="_blank">@<span>schmutzie</span></a></span> <br>2(2+2) is a single, factorised term, which must be expanded according to The Distributive Law. For full coverage of everything read my (still in progress) <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> thread on order of operations <a href="https://dotnet.social/@SmartmanApps/110807192608472798" rel="nofollow noopener" target="_blank"><span class="invisible">https://</span><span class="ellipsis">dotnet.social/@SmartmanApps/11</span><span class="invisible">0807192608472798</span></a><br>TL;DR Google is wrong</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>1/4 <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> <br>Now we'll tie the previous 2 <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> rules together and see why <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> order of operations means <a href="https://dotnet.social/tags/MathsIsNeverAmbiguous" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsIsNeverAmbiguous</span></a> provided you <a href="https://dotnet.social/tags/DontForgetDistribution" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>DontForgetDistribution</span></a> </p><p>Start with 1, also the result of any number divided by itself, so let's write 8/8</p><p>We can factorise, and 8 is even, so let's take out a 2, 8/2(4)</p><p>We can rewrite the stuff in brackets without changing the answer, so let's write 8/2(1+3)</p><p>I started with 1,so we have to make sure the rules of <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> get us back there...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p><a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a> week 2, Terms.<br>Simply put, in <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> Terms are separated by operators and joined by brackets. The most common example is 2a=(2*a). This makes it simpler to write fractions. e.g. 1/2a rather than 1/(2*a). 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 <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://dotnet.social/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> later, but first)...</p>
💡𝚂𝗆𝖺𝗋𝗍𝗆𝖺𝗇 𝙰𝗉𝗉𝗌📱<p>Thread index <a href="https://dotnet.social/@SmartmanApps/110897908266416158" rel="nofollow noopener" target="_blank"><span class="invisible">https://</span><span class="ellipsis">dotnet.social/@SmartmanApps/11</span><span class="invisible">0897908266416158</span></a></p><p>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</p><p>So 8÷2(1+3)=? What is the answer (bonus: and WHY is it the answer)? <a href="https://dotnet.social/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://dotnet.social/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://dotnet.social/tags/MathsMonday" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MathsMonday</span></a></p>
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