Oy! I'll be semi-happier after the April 25 week when our math TAKS tests are over. For the last forever amount of weeks, I've been tutoring 3 sets of kids, once a week, during advisory, strictly for this test. And last Friday some words came out of my mouth to my group of tutees that now have me thinking.
I didn't want them to think they were just in there because we thought they'd have a high chance of failing. That's definitely not the case. And, I guess I'm a product of the "feel good about yourself" school of growing up .... though from what-all I'm reading, I have to think about how I offer THOSE words of praise. So after we went over some material, I said something to the effect of, "okay! we're going for commended!" and I went on with going over problems.
Well, a 10th grader in that group came up to me afterward. She's a diligent, hard-working, come-in-for-tutoring if she thinks she needs it kind of math student. It does not come easily to her at ALL, and I had her last year for algebra and this year for geometry, and I've seen a great improvement in her algebra skills this year after her having time to absorb it.
Anyway, she came up to me and said, I'm really discouraged about this math TAKS. I haven't been commended on the math portion since the 5th grade. She's passed every year, but here we are, a higher performing school, and the message we're encouraged to give is, "don't just PASS, go for commended!". So not getting commended is apparently the new failing for these students. Crap! It didn't occur to me until that moment how students may feel every year when, yes, they may pass, but, oh no, I've let down my teachers by not being commended.
I don't know how successfully, but I told her that one of the main reasons we've been saying that (in my opinion) is to get the kids to try and do their best, and I already see she puts out her effort on every question she goes through, and that's the type of diligence that's going to get her far in the world, not the "oh yay! I passed and got commended on a state test." ... It's even telling that she's stressing about the 10th grade TAKS, and it's not really as high stakes for the kids as the exit level in 11th grade.
Regardless, here's another bad side effect of these tests and what we may seem like we expect from the kids all in the name of some outside random measure of "are we doing our jobs as teachers" test.
Sunday, March 27, 2011
Tuesday, March 22, 2011
Trig and Teaching and Knocking
TEACHING:
Lately I've been noticing in various forms of teaching or discussions about teaching, sometimes the "speaker/blogger/writer", either intentionally or not, conveys the message that, "if you don't do things THIS way, you're doing it wrong!" ... or ... "why in the WORLD would you even CONSIDER teaching so-and-so THAT way, you hack!". Or maybe it's just how I'm reading into things.
I think for the most part we're all trying to do the best job we can, and obviously there's no magic bullet that's going to work for all topics and all days and all teachers all of the time. We (and by "we" I obviously mean "me" because that's obviously who it's all about) should just stop beating ourselves up when someone speaks/writes/communicates how they do something, and it sounds so perfect and rosy, and my (our?) first thought is, "sheesh, THAT person has it all figured out. What's wrong with me?". Okay, my self-pep talk is over.
KNOCKING:
Anti-Joke of the moment I love:
Knock, Knock.
Who's there?
To.
To Who?
To "Whom".
TRIGONOMETRY:
As with all things we do the 1st time through (or maybe it's just me), there are kinks to be worked out with the activity I wrote about last time. The last 2 days I took my students to the computer lab, and they self-taught basic trigonometry ratios with my GeoGebra file and the 4 page packet. I realize I'm fortunate to (this year) have frequent access to a computer lab.
I know that that was one of the BIG reasons before that I didn't like Geometer's Sketchpad or even bother to learn GeoGebra. Once the kids grasped it on the one time a semester I actually got to go to the lab, then they forgot it all the next semester when we could go back, so basically what was the point. I guess that's why I'm liking this GeoGebra, because if a student has computer access at home (and I know that's not always the case), at least this is an extra resource for absorbing things. Anyway, choir .... preaching .... bla bla bla
I would do the activity again next year, but here are some things that happened and how I'd revamp. I teach a 1.5 hour class, and after we went over homework and did another small activity, we had about an hour for this activity ... WAY too little time for them to be set on their own and finish up.
The first thing I changed was I had them cross out 2 of the cases for each of the 6 tables. So they were only checking sin/cos/tan twice for each acute angle. Next year, I think I'd change it even more: check just once total and then check what a neighbor or 2 got and notice.
I also would take out the ratio of sides on my GeoGebra activity. Invariably, once a kid had to fill in column 2, they didn't read very carefully, and just used my #s above, even though they may not be the correct ratio of sides ... but HEY, all the letters started to blend together.
Other than that, with periodic "giddyup-ing" by me in the form of "you have 30 more seconds to finish up to ______". "Okay, the main point of that was _______. Now do the next section." ..... I think it went well ... Especially, since I seem to have lost my voice YET again this year. Chronic sick-fest.
Lately I've been noticing in various forms of teaching or discussions about teaching, sometimes the "speaker/blogger/writer", either intentionally or not, conveys the message that, "if you don't do things THIS way, you're doing it wrong!" ... or ... "why in the WORLD would you even CONSIDER teaching so-and-so THAT way, you hack!". Or maybe it's just how I'm reading into things.
I think for the most part we're all trying to do the best job we can, and obviously there's no magic bullet that's going to work for all topics and all days and all teachers all of the time. We (and by "we" I obviously mean "me" because that's obviously who it's all about) should just stop beating ourselves up when someone speaks/writes/communicates how they do something, and it sounds so perfect and rosy, and my (our?) first thought is, "sheesh, THAT person has it all figured out. What's wrong with me?". Okay, my self-pep talk is over.
KNOCKING:
Anti-Joke of the moment I love:
Knock, Knock.
Who's there?
To.
To Who?
To "Whom".
TRIGONOMETRY:
As with all things we do the 1st time through (or maybe it's just me), there are kinks to be worked out with the activity I wrote about last time. The last 2 days I took my students to the computer lab, and they self-taught basic trigonometry ratios with my GeoGebra file and the 4 page packet. I realize I'm fortunate to (this year) have frequent access to a computer lab.
I know that that was one of the BIG reasons before that I didn't like Geometer's Sketchpad or even bother to learn GeoGebra. Once the kids grasped it on the one time a semester I actually got to go to the lab, then they forgot it all the next semester when we could go back, so basically what was the point. I guess that's why I'm liking this GeoGebra, because if a student has computer access at home (and I know that's not always the case), at least this is an extra resource for absorbing things. Anyway, choir .... preaching .... bla bla bla
I would do the activity again next year, but here are some things that happened and how I'd revamp. I teach a 1.5 hour class, and after we went over homework and did another small activity, we had about an hour for this activity ... WAY too little time for them to be set on their own and finish up.
The first thing I changed was I had them cross out 2 of the cases for each of the 6 tables. So they were only checking sin/cos/tan twice for each acute angle. Next year, I think I'd change it even more: check just once total and then check what a neighbor or 2 got and notice.
I also would take out the ratio of sides on my GeoGebra activity. Invariably, once a kid had to fill in column 2, they didn't read very carefully, and just used my #s above, even though they may not be the correct ratio of sides ... but HEY, all the letters started to blend together.
Other than that, with periodic "giddyup-ing" by me in the form of "you have 30 more seconds to finish up to ______". "Okay, the main point of that was _______. Now do the next section." ..... I think it went well ... Especially, since I seem to have lost my voice YET again this year. Chronic sick-fest.
Sunday, March 20, 2011
Learning Trigonometry
Now I'm almost ready to start my trigonometry unit. Here is the 4-page self-teaching packet I'll hand my students tomorrow, along with a graphing calculator and an e-mail message that has all the links they need. My goals for them are to be able to: identify opposite and adjacent sides of an angle in a right triangle, find the sine, cosine, and tangent ratios of a triangle given the 3 side lengths, and link these skills to the special triangles they just "learned". Once I make up a homework sheet (or use the textbook), I'm good to go.
Also, on a cheery, vacationy note, here are some Marfa and Fort Davis, Texas pictures from our short Spring Break trip.
Also, on a cheery, vacationy note, here are some Marfa and Fort Davis, Texas pictures from our short Spring Break trip.
Saturday, March 19, 2011
Quiz Anyone?
I'm writing up my worksheet to go along with my geogebra "baby trigonometry" applet, and I found I needed/wanted a quiz for them to take online. I couldn't find any with the right number of questions or types of questions, so I made one up. We'll see Monday if it's a success or a stinker.
Wednesday, March 16, 2011
SO excited about Geogebra
A nice person commented from my last post and showed me some applets she'd made for Geogebra (thank you Lsquared). This must have planted a seed in my mind, because today I was tinkering around with how to start up my trigonometry unit. I went to the Geogebra site. I clicked "download". Then I clicked "Applet Start". I figured my kids don't have downloading capabilities on the school computers, so this would be fine.
I tinkered around with things and came up with this activity. I'm going to make a worksheet that goes with it to guide them through a self teaching, first day of trig, what is sine/cosine/tangent activity. I think some of the questions may even be of the effect of, "set your triangles, set up your calculator just so, press cos ____ (degree shown), compare it to ratio."
I would love feedback on this "non applet". Effective? Hard to manage? Fine?
Note: I tested out opening the file on another computer. It "failed" with Firefox, but worked fine with Explorer. I DID have to change the window size to 200% or something like that. But there is a way to show WHOLE picture.
Anyway! Off to write the worksheet. I'll share. Okay, really off to dinner then Lindy Hop, then Marfa, THEN to write the worksheet.
I tinkered around with things and came up with this activity. I'm going to make a worksheet that goes with it to guide them through a self teaching, first day of trig, what is sine/cosine/tangent activity. I think some of the questions may even be of the effect of, "set your triangles, set up your calculator just so, press cos ____ (degree shown), compare it to ratio."
I would love feedback on this "non applet". Effective? Hard to manage? Fine?
Note: I tested out opening the file on another computer. It "failed" with Firefox, but worked fine with Explorer. I DID have to change the window size to 200% or something like that. But there is a way to show WHOLE picture.
Anyway! Off to write the worksheet. I'll share. Okay, really off to dinner then Lindy Hop, then Marfa, THEN to write the worksheet.
Tuesday, March 15, 2011
Spring Break
Phew! I can't tell you what a consistent 9 hours of sleep each night can do for a person ... or maybe I can. I feel great! Here are some pressing activities on my plate:
1. Watch cheesy bad movies on Netflix streaming (Blackbelt Jones, Cherry 2000, ....)
2. Read book(s) (Saving Fish from Drowning)
3. Catch up on grading (test corrections)
4. Sign up for the Digital Electronics STI workshop for PLTW (soooooo excited)
5. Prep myself for the Cardboard Chair Project for my engineering class (design, test, create a cheap durable cardboard chair for a dorm room ... cool "hook" video)
6. Prep myself for my PBL unit (project based learning) of trigonometry
7. Take a blanket out to local parks and laze around and soak up sun and outdoors
8. Go to Marfa, TX (Thursday)
9. Do my "train" project on Inventor for IED (I'm doing the homework with my students this year ... keeps me honest)
10. Puzzles, puzzles, puzzles (hiyawake puzzles my new favorite)
11. Get ready for the PLTW certification visit on 4/1/11 (eek! hope that's not portentous)
12. Sleep more than 7 hours a night (or did I mention that already?)
13. Get my IED students to write thank you e-mails for a recent speaker/friend that came to visit and talk about his engineering experiences. (It was useful to send my students THIS e-mail as a primer):
Hey! Let's be green (and timely since it's Spring Break). Here is your EXTRA homework to turn in to me NEXT Monday 3/14/11 .... and by that I mean 3 days from now .... OVER spring break.
Please send me an e-mail, either with an attachment of a thank you card for Mr. H., or your thank you words inside the e-mail message.
I will then gather them all up together in one e-mail and send them along to him.
Thank YOU for saying more than just "thank you for coming to speak to us". Maybe indicate what you liked or found interesting or learned or whatever.
Example, here's my "thank you" to YOU, my IED students:
Dear IED students of 2010-2011,
I love teaching you because you're so creative and make me think of new things and ways to teach you. I also love that you're excited about the various things we do and learn. I look forward to your questions because they make me think through the why's and how's of various concepts.
Signed,
Ms. ______
I look forward to your thank yous for Mr. H.
Ms. _____
(I've gotten 6 out of 12 back so far)
Anyway .... must go exercise with my other latest craze (as of 4 weeks).
1. Watch cheesy bad movies on Netflix streaming (Blackbelt Jones, Cherry 2000, ....)
2. Read book(s) (Saving Fish from Drowning)
3. Catch up on grading (test corrections)
4. Sign up for the Digital Electronics STI workshop for PLTW (soooooo excited)
5. Prep myself for the Cardboard Chair Project for my engineering class (design, test, create a cheap durable cardboard chair for a dorm room ... cool "hook" video)
6. Prep myself for my PBL unit (project based learning) of trigonometry
7. Take a blanket out to local parks and laze around and soak up sun and outdoors
8. Go to Marfa, TX (Thursday)
9. Do my "train" project on Inventor for IED (I'm doing the homework with my students this year ... keeps me honest)
10. Puzzles, puzzles, puzzles (hiyawake puzzles my new favorite)
11. Get ready for the PLTW certification visit on 4/1/11 (eek! hope that's not portentous)
12. Sleep more than 7 hours a night (or did I mention that already?)
13. Get my IED students to write thank you e-mails for a recent speaker/friend that came to visit and talk about his engineering experiences. (It was useful to send my students THIS e-mail as a primer):
Hey! Let's be green (and timely since it's Spring Break). Here is your EXTRA homework to turn in to me NEXT Monday 3/14/11 .... and by that I mean 3 days from now .... OVER spring break.
Please send me an e-mail, either with an attachment of a thank you card for Mr. H., or your thank you words inside the e-mail message.
I will then gather them all up together in one e-mail and send them along to him.
Thank YOU for saying more than just "thank you for coming to speak to us". Maybe indicate what you liked or found interesting or learned or whatever.
Example, here's my "thank you" to YOU, my IED students:
Dear IED students of 2010-2011,
I love teaching you because you're so creative and make me think of new things and ways to teach you. I also love that you're excited about the various things we do and learn. I look forward to your questions because they make me think through the why's and how's of various concepts.
Signed,
Ms. ______
I look forward to your thank yous for Mr. H.
Ms. _____
(I've gotten 6 out of 12 back so far)
Anyway .... must go exercise with my other latest craze (as of 4 weeks).
Saturday, March 12, 2011
Special Quadrilaterals
We finished studying special quadrilaterals a while ago, took a test, and now the kids are trickling in for their retests. Ew. Something didn't stick or even get through to some students, and I'm trying to process how to improve it for next year.
Here is how I taught it:
parallelograms: I had 4 already printed on a paper, and they were to use their rulers and protractors to measure various things to discover the facts about the angles and diagonals.
squares/rhombi/rectangles: I had them neatly draw one each in their grid notebooks (I gave the instructions on where to put the vertices). Then I had them again use their protractors and rulers to measure various things and fill out a chart as to which had which properties (diagonals congruent, diagonals perpendicular, diagonals bisecting opposite angles).
trapezoid/kite: I had a sheet with some drawn we worked through the logic of things to get to their properties.
All this went in their notes. They had review problems. At no time did I have them (and apparently they didn't think they needed to) gather all the information into one place (foldable? 1/2 page) to have a quick summary. (I will change this for next year).
Okay, then test time. The usual suspects did well, but too many kids had no idea about things they should have. For example, when a student was coming in for a retest, and I asked her about what a rhombus was, she was silent. Oy! Now I know this particular student coasts by and can look like she's playing school, but .... Regardless, I want them to learn despite themselves.
I think something happened at another tutoring session that will make me either add a day to my lessons, or replace the previous lessons with this. I'm leaning towards add to the lesson.
Another student was studying rhombuses, so that she could take the retest, and she still wasn't understanding or processing things she should know. She couldn't even draw a vaguely accurate rhombus. I took a piece of colored construction paper, and pretty quickly used a ruler to cut it into a rhombus (it turned out to be approximately 8.5" per side). I liked that it was big enough to see things on and to write on and play with. Then I had her folding it to make the diagonals. Then we stared at it. It was clear the diagonals were perpendicular. It was clear that they were bisected. It was clear which angles were congruent (you could match them up and check). It was clear that the diagonals bisected the angles (again you could fold and check).
Then as she was trying to redo a test problem, I had her write the given information on the rhombus, and then figure out the rest. ... I don't know if this will stick. I guess we'll see when she actually does a retest, but maybe the extra visual and tactile properties of the shape will be clearer in her mind.
Here is how I taught it:
parallelograms: I had 4 already printed on a paper, and they were to use their rulers and protractors to measure various things to discover the facts about the angles and diagonals.
squares/rhombi/rectangles: I had them neatly draw one each in their grid notebooks (I gave the instructions on where to put the vertices). Then I had them again use their protractors and rulers to measure various things and fill out a chart as to which had which properties (diagonals congruent, diagonals perpendicular, diagonals bisecting opposite angles).
trapezoid/kite: I had a sheet with some drawn we worked through the logic of things to get to their properties.
All this went in their notes. They had review problems. At no time did I have them (and apparently they didn't think they needed to) gather all the information into one place (foldable? 1/2 page) to have a quick summary. (I will change this for next year).
Okay, then test time. The usual suspects did well, but too many kids had no idea about things they should have. For example, when a student was coming in for a retest, and I asked her about what a rhombus was, she was silent. Oy! Now I know this particular student coasts by and can look like she's playing school, but .... Regardless, I want them to learn despite themselves.
I think something happened at another tutoring session that will make me either add a day to my lessons, or replace the previous lessons with this. I'm leaning towards add to the lesson.
Another student was studying rhombuses, so that she could take the retest, and she still wasn't understanding or processing things she should know. She couldn't even draw a vaguely accurate rhombus. I took a piece of colored construction paper, and pretty quickly used a ruler to cut it into a rhombus (it turned out to be approximately 8.5" per side). I liked that it was big enough to see things on and to write on and play with. Then I had her folding it to make the diagonals. Then we stared at it. It was clear the diagonals were perpendicular. It was clear that they were bisected. It was clear which angles were congruent (you could match them up and check). It was clear that the diagonals bisected the angles (again you could fold and check).
Then as she was trying to redo a test problem, I had her write the given information on the rhombus, and then figure out the rest. ... I don't know if this will stick. I guess we'll see when she actually does a retest, but maybe the extra visual and tactile properties of the shape will be clearer in her mind.
Tuesday, March 08, 2011
Pythagorean Theorem Video
Today we started reviewing radicals, and talking about Pythagorean Theorem. I know they've seen it before, but they still need help on the harder problems. I found this cool, short video that I showed a few times in a row until they understood it. It was a fun discussion on what he was trying to do.
Then we built up their skills with individual problems of varying difficulties:
(the terms listed are "a", "b", and "c" for a right triangle with hypotenuse "c"):
FIND x
1. x, 4, 10
2. 3, 9, x
3. 2*sqrt(3), x, 4
4. x, 3, sqrt(17)
5. x, 3x, 20
6. some sort of trapezoid with 90 degree angles on one side.
Anyway, I just wanted to post about the water video. Done. Dinner.
Then we built up their skills with individual problems of varying difficulties:
(the terms listed are "a", "b", and "c" for a right triangle with hypotenuse "c"):
FIND x
1. x, 4, 10
2. 3, 9, x
3. 2*sqrt(3), x, 4
4. x, 3, sqrt(17)
5. x, 3x, 20
6. some sort of trapezoid with 90 degree angles on one side.
Anyway, I just wanted to post about the water video. Done. Dinner.
Friday, March 04, 2011
Room Fustiness
Phew! Stinky Room Alert! I don't know what it is .... well, maybe I do, but the last few weeks while I'm teaching in my room for hours on end, I don't notice anything. But step out of the room for a few minutes and walk back in after a gaggle of teenagers have been in there, and Stink City.
It's not that way in the morning when I first get to work. It's not that way right before 1st period. But after I teach a class or two and leave the room and come back. Ew. Not HORRIBLE horrible, but noticeably smelly/musty.
So my question is, is it just my particular group of teens? They don't really smell strongly, but I guess get enough of them together and have them in there for an hour and a half and watch out. I don't like the fake smell of room deodorizers. I can't open my windows. I have one plant in there (which sometimes causes its own problems with teeny bugs that flutter around and like to hover around the vacuum of your nose only to be accidentally sucked in and EWWWW). Do we think more plants would help? Certain plants? Has anyone successfully overcome this TRAGIC situation?
Signed,
Singed Nosed Sufferer
It's not that way in the morning when I first get to work. It's not that way right before 1st period. But after I teach a class or two and leave the room and come back. Ew. Not HORRIBLE horrible, but noticeably smelly/musty.
So my question is, is it just my particular group of teens? They don't really smell strongly, but I guess get enough of them together and have them in there for an hour and a half and watch out. I don't like the fake smell of room deodorizers. I can't open my windows. I have one plant in there (which sometimes causes its own problems with teeny bugs that flutter around and like to hover around the vacuum of your nose only to be accidentally sucked in and EWWWW). Do we think more plants would help? Certain plants? Has anyone successfully overcome this TRAGIC situation?
Signed,
Singed Nosed Sufferer
Wednesday, March 02, 2011
Algebra/Basics Review in Geometry
I'm basically teaching two levels of geometry this year even though we call them both "preAP". I have the 8th and 9th graders in one class, and for the most part they whiz through everything and delight in the challenging problems. I'm also teaching 10th graders that are in other sections, and they approach math in a different way. I have to keep reminding myself not to rush rush rush through topics and to keep remembering that they need more practice on everything to make it stick and to make it make sense.
I am doing one thing, though, to help them keep their algebra skills fresh: ALGEBRA :). For example, for the last few topics, I made sure to make up geometry problems that ended up having a factorable quadratic equation in it at some point. This led us to recall FOILing and ("claw"ing .... Thanks, Mrs. H) and factoring. Now I'm making sure to include quadratics that CAN'T be factored (hello quadratic formula review).
BUT. Here are some things that I didn't think would come up, but did/do, and REALLY I should make a mental (or better) note to myself of where things can go askew, so that I can make more problems throughout the year to have such examples, so that we can have a discussion about them and keep them fresh in our minds.
Example 1:
A student is solving: 2x + 4 = (1/2)x + 8, for example.
Hmmmm, I don't like that (1/2)x,
so I'll just remember that I can do the opposite to x to undo it,
and I multiply ONLY the 2x and the (1/2)x by 2 to end up with:
4x + 4 = x + 8. Eeeeek.
Example 2:
A student gets to a point in an equation where they have (13/2)x = 14. Well, heaven forbid we keep things in fractions and go the easier route of multiplying both sides by 2/13 to get x = 28/13. Boom. Done.
No.
We convert to 6.5x = 14.
We stress that 14 is not easily divided by 6.5.
We chug through and do long division and create pain and suffering for ourselves.
We curse the teacher for such a hard problem and no calculator.
Example 3:
Teacher takes the expedient route frequently and makes up problems where the answers are integers. This saves time, she thinks, so that they're not struggling with messy things and they're concentrating on new material.
Students freak out the first instance an answer is not integer.
Gasp! I must have done something wrong. Things ALWAYS work out nice and pretty in "math world". Fractions are not REAL numbers, no matter WHAT my teacher says. Oh, and by the way, "your answer key says 3/2. Is it okay if I write it as 1.5? As 1 1/2?"
Example 4:
A problem comes up where you're asked to find the height of a person. The decimal answer is 4.666666666 feet. You think this either means 4' 6" or 4' 7" (if you round). You DON'T think that 2/3 of a foot is not either of these answers.
I am doing one thing, though, to help them keep their algebra skills fresh: ALGEBRA :). For example, for the last few topics, I made sure to make up geometry problems that ended up having a factorable quadratic equation in it at some point. This led us to recall FOILing and ("claw"ing .... Thanks, Mrs. H) and factoring. Now I'm making sure to include quadratics that CAN'T be factored (hello quadratic formula review).
BUT. Here are some things that I didn't think would come up, but did/do, and REALLY I should make a mental (or better) note to myself of where things can go askew, so that I can make more problems throughout the year to have such examples, so that we can have a discussion about them and keep them fresh in our minds.
Example 1:
A student is solving: 2x + 4 = (1/2)x + 8, for example.
Hmmmm, I don't like that (1/2)x,
so I'll just remember that I can do the opposite to x to undo it,
and I multiply ONLY the 2x and the (1/2)x by 2 to end up with:
4x + 4 = x + 8. Eeeeek.
Example 2:
A student gets to a point in an equation where they have (13/2)x = 14. Well, heaven forbid we keep things in fractions and go the easier route of multiplying both sides by 2/13 to get x = 28/13. Boom. Done.
No.
We convert to 6.5x = 14.
We stress that 14 is not easily divided by 6.5.
We chug through and do long division and create pain and suffering for ourselves.
We curse the teacher for such a hard problem and no calculator.
Example 3:
Teacher takes the expedient route frequently and makes up problems where the answers are integers. This saves time, she thinks, so that they're not struggling with messy things and they're concentrating on new material.
Students freak out the first instance an answer is not integer.
Gasp! I must have done something wrong. Things ALWAYS work out nice and pretty in "math world". Fractions are not REAL numbers, no matter WHAT my teacher says. Oh, and by the way, "your answer key says 3/2. Is it okay if I write it as 1.5? As 1 1/2?"
Example 4:
A problem comes up where you're asked to find the height of a person. The decimal answer is 4.666666666 feet. You think this either means 4' 6" or 4' 7" (if you round). You DON'T think that 2/3 of a foot is not either of these answers.