Tuesday, May 24, 2022

Math 7/8 Compacted: Rational vs. Irrational Numbers (11.3.21)

Four three two one oh listen up all right how many of us are feeling sick about these problems similarly to this very well-dressed woman here how many of us are feeling pretty much like this person is here .

What gives right so let's see what's happening yeah if you did this correctly then you might i'm hoping that you guys have noticed something i'm hoping that you guys have noticed something about these let's see here bad habits as well my goodness hollering out there no no no no they're .

Not they're not on equations just yet so let's talk about it if x squared equals 7 what am i going to do to find out what x is james yeah we're going to take the second root because that eliminates that x of the squared giving me x equals what about 2.646 how many of us got that x is .

Approximately equal to 2.646 are we forgetting something positive or negative yeah positive or negative 2.646 and then what is x equal to exactly james the second root of seven or the square root of seven right with a two or without i'm happy to accept either of .

Those okay how many people by raising hand were able to get those oh plus or minus thank you look at me forgetting that i have forgotten awesome very good now let's talk about number two how would you handle number two and let's see if i can get a little .

Help charlie what would you do for number two exactly yep yeah and that's just it guys a lot of the time people think what i mean by exact is the square root of 25 and what i mean by approximate is plus or minus five the truth is guys when you solve .

This we got an integer result which means that that's the exact and there's no need for an approximate like i guess i could approximate it with a 4.8 but why would i like i've got a perfectly good number right here because 5 times 5 is 25. points two point six four six times .

Itself is pretty close to seven how close to seven i wonder how close does that get us two point six four six squared is seven point zero zero one three one six pretty close to seven but not exactly as opposed to five times five which is twenty-five now let's check this last one here what .

Am i gonna do to solve out this last one shall i so we're going to add 15 to both sides and that's going to give me d to the third over 9 equals 24 i think because 15 i think it's uh i have a my ones look like a little bit like two yeah we're good keep going .

And that gets me d to the third and that one i don't have the top of my head 216 oh yeah of course it's 216. very good the third root of each side and that gets us what positive or negative 6 aha yes because a negative 6 times a negative 6 times a negative 6 would give us this .

A negative 2 16. well and that's just it i'm really glad we're talking guys here's the thing i totally get it that we're like okay so the exact as this thing and the approximate is this thing but actually that's not true six multiplied three times is 216. it's not close to 216 it is 216. .

Which means that the exact solution is 6 and there's no need to approximate it does that make sense when we get a nice number like this the we don't need an approximation when we get something horrible sometimes we need an approximation we also need to understand what it really is how many of us have been feeling kind of weird about .

These numbers that some that sometimes it comes out really nice and sometimes it comes out just like so horrible how many of us have noticed that and it kind of bothers them well let's see if we can pick that apart and figure out why that's happening today yeah go ahead yep that's why that's what we're gonna .

Do after our uh after our uh uh uh essential questions there we go we guys open up your planner page to 52 it's give someone a dollar a day someone was like no i wouldn't oh listen up perfect i thought it would be that would be an interesting social experiment i think to .

See if everybody brought a dollar and then just gave one dollar to somebody i know for a fact there would people be people who wound up leaving with zero dollars right yeah but there would be a lot of people leaving with one dollar and i better be people who are leaving with like six dollars like who do we i just .

Instinctively give that dollar to right like i think that would i think that would be way way interesting right i think it'd be way interesting anyway can i get you in five four three .

Two one don't let me forget this in fact maybe i need to talk about it right now well let's do this first and we'll do that okay let's see here uh what's our first essential question and i would love to get that from kai yeah let me write that down .

What is the difference between a rational number and an irrational number and how do i tell the difference let me get back in time for jeopardy all right art jeopardy today state parks of utah this state park isn't completed yet but .

Is named after a dinosaur named after our state but oddly enough it's not our state dinosaur uh-oh okay perfect by the way guys let's see here so does any so the correct answer is up there guys it .

Is utah raptor state park how many of you think it's weird the utah raptor is not our state dinosaur does anyone know what our state dinosaur actually is daniel what were you going to say it is another state fossils yes the state fossils the trilobite but the state dinosaur is different it does start with an a .

It is an allosaurus specifically allosaurus fragilis uh and then the pictionary that is indeed a stilts stilts before i take another step i forgot to do this in my first period because i am a fool guys your pm assembly schedule lunch is .

Not attached to your fourth period today it is attached to your fifth period so i want you to think it's different i want you to think you get a decision to make buddy do you want to eat lunch i guess that means you have first lunch or no none lunch you just eat lunch and first lunch anyway the point is this guys .

Everybody look at your uh or think about your fifth period teacher and be ready to write this down in your planner because i'm gonna tell you what lunch your fifth period teacher has you have first lunch if your fifth period teacher is .

Alan badger burroughs creveling crookston evans gustafson hamilton hendry jardine johnson miner niederhauser qua rongstead taylor or walker if you have one of those teachers guys you have first lunch you have second lunch meaning you go to fifth period and then go to lunch afterwards so if i just read your fifth period .

Teacher after fourth go to lunch and then go to that fifth period teacher if i if you have one of these people after fourth you're gonna go to fifth period and then after fifth period you're gonna go to lunch and then you're gonna go to sixth period your uh uh your lunch second lunches so you go to fifth period and then lunch is .

Your fifth period is alvarado dalamore dougherty featherstone gleason hanson mcgowan miller mott mudrow perkins me snyder thompson whiting right or yund is there anybody who's not sure what lunch they have kai i think you just have first lunch man unless you don't want to and then yeah sounds did you make a call .

Absolutely that sounds like a really good idea i just wanted to say i think that's from it makes it easy right listen up perfect lucy it was easier before please go ahead kai you can take that that's okay that's okay you gotta do do what you gotta do i .

It used to i just need to get the right flow because i remember it used to be that way the way that you they were the teachers that we used to have but i think it was who's i think it's because i think because carling left and .

Then the replacement has one syllable in their name and so i wasn't able i'm not so i have to like get that back anyway i digress we'll be here all day if i start talking about that underneath your warm up got a pencil all right let's see what we got out there .

What the huh wow we have 10 minutes that is crazy that is way shorter than i was expecting we better hurry then got a pencil so today's topic is rational or irrational .

Go ahead and put that tech deck away please i want to make sure we're focusing here especially with as little time as we've got lest i send it back to the 1990 1990s from once it came now here we go guys .

It turns out that there are that the kinds of numbers that have like when we take those when we take the root of something and get a nice number that is a rational number and rational numbers have really been all the numbers that we've been talking about mr mcgowan does anyone remember which what i said for stuart doesn't have stuart fifth .

Oh oh yeah is he on there stuart nothing about to talk to him at the office but anyway so here we go guys numbers all pretty much every number you've ever seen before is rational with .

A few exceptions of the ones that we've been seeing like every number that you had to do a reduced root for those are irrational numbers yes ma'am let's define them let's make let's make a real definition for this here we go guys let's write down the definition of a rational number a rational number .

Is a number duh but a number that can and the key word here is can be written as a fraction of integers can be written as a fraction of integers that's why for example the square root of 9 .

Is rational because it's 3. how could i write 3 as a fraction of integers yeah go ahead 3 over 1 right 3 over 1 is rational because i can write it as a fraction of integers why is 0.38 rational because you can write it as 38 hundredths .

Why is negative 4 17 rational because it's literally a fraction of integers right integers meaning the whole numbers and the opposite of whole numbers like negative 8 42. this is rational because it can be written as a fraction of integers weirdly enough the rational like if i have negative 4 .

17 0.38 that's also rational can anyone explain why why would that be rational it's weird but but it totally is jane what are you thinking yeah because what would happen is i could turn that into 38 hundredths and then you flip and then multiply and then .

Suddenly it's a fraction of integers again so rational numbers are pretty much all these things there's a little side effect to that so since rational numbers can be written as a fraction of integers rational numbers must have decimals .

That terminate or repeat so repeating decimals are also rational that's another way to check you could say well look 0.38 it's a terminating decimal so it must be rational 0.5 repeating is a repeating decimal it must be rational and that's exactly where we're headed .

Yeah lucy yeah let's give that a name yeah so if it's rational it can be written as a fraction of integers and one another way to check that is if it has a decimal that terminates or repeats and what lucy's trying to get it i think is that irrational numbers exactly irrational numbers .

Cannot be written go ahead and write this down cannot be written as a fraction of integers we're running out of time here and have decimals that do not terminate .

Uh or repeat and owen came up with a number that does exactly that owen can you think of a number that does exactly that pi is a perfect example of an irrational number let's change this up guys this he's got first okay guys this uh this .

That i got kind of sidelined i wasn't expecting that to take quite as long and i think is it because we actually don't get extra time for the announcements i think that's what happened here because i didn't happen last time it is it's because we didn't get time okay that makes me feel less bad okay folks .

We're not going to do that self-assessment watch i'm going to un-publish it right now i know unless somebody actually already started it in which case i can't unpublish it bam haha sweet okay there's no self-assessment uh guys we'll talk about .

This tomorrow and we'll probably have time to work on that stuff so it's like the it's so fast we're gonna finish this idea tomorrow so for now just sleep on it and we'll get on there okay sorry about that everybody uh put your calculator's way and i'll just

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