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# those chapters at the end of the math book no one gets to in school

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BalletBoy did really well with Dolciani Pre-Algebra - the percents and problem solving stuff was all hard for him, but he flew through most of the book and hasn't complained much at all even though he's my slow worker, which is good. But now he's on his final chapter and it's one of the last two in the book and, oh my gosh, it's a slog. I'm a bit determined to get through it. But I must say, it's the Pythagorean Theorem and all kinds of rules about special triangles and *none* of this was in Mushroom's pre-algebra and everything about it is brand new to BalletBoy.

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You guys, I had to sit with him and derive the information to solve the problem (the hypotenuse of a isosceles right triangle - he refused to use it otherwise because he didn't trust it) and then solve the same problem FOUR TIMES IN A ROW. :banghead:

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But then I was thinking today that, good grief, I'll bet when this text was still being used regularly in classrooms that *no one* ever made it to these final two chapters (the other one is statistics and probability, but we did something else for that - Dolciani managed to make it dry, which is astounding for such a fun middle school topic).

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Tell me I'm right. That they didn't really intend that kids master this stuff in pre-algebra. They just stuck it in there because. And that no one ever got to it. I don't remember ever having finished a math textbook at any point in my school career.

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Also, wish me luck. The next section in this chapter introduces trig. :blink:Â  :glare:

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But then I was thinking today that, good grief, I'll bet when this text was still being used regularly in classrooms that *no one* ever made it to these final two chapters (the other one is statistics and probability, but we did something else for that).

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Tell me I'm right. That they didn't really intend that kids master this stuff in pre-algebra. They just stuck it in there because.

They finished probability and statistics here because it was in the pre-common core annual state testing.

Pythagorean theorem was in middle school math before prealgebra and probably in algebra too for the pre-common core California edition for Holt and Pearson. It wasn't the last few chapters, more like in the middle.

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ETA:

MEP has decent trigonometry chapters.

Pythagorean theorem unit 3 in book 8A http://www.cimt.org.uk/projects/mepres/book8/book8.htm

Edited by Arcadia
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The Pythagorean Theorem itself was okay. It was more the special triangles stuff. Sigh. It's a lot of memorization. I definitely remembered the Pythagorean Theorem, but I never could have pulled out the 30-60-90 triangles with the longer leg being aĂ¢Ë†Å¡3 and all that. I'm almost positive that there's no trig required of middle school math in Common Core.

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I'm happy to introduce it. I think it's probably good so it's less scary when he meets it again.

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MEP seems to have trig as a 9th grade topic, which seems more right to me than 7th grade pre-A for sure. It's literally two pages in Dolciani, so I can't believe kids were really meant to master much of anything about it.

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You will see it all again in geometry, and it's not foundational to Algebra, so I say if it's driving him crazy, skip it for now. Just a warning - I know very little about math. Ă°Å¸Ëœâ€°

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The goal is to intro the material just enough so that when they see it again, they aren't starting from scratch. Â Don't bother with memorization, go for awareness and general "hey, this thing is out there, you'll get the details in coming years, don't be scared of it".

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I definitely remembered the Pythagorean Theorem, but I never could have pulled out the 30-60-90 triangles with the longer leg being aĂ¢Ë†Å¡3 and all that.

For the 30-60-90 triangle, draw an equilateral triangle of sides 2 units. Cut in half to get two 30-60-90 triangles.

Hypotenuse becomes 2 units

Sides become 1 unit and Ă¢Ë†Å¡3 unit

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For the 45-45-90 triangle, draw an issoceles triangle with equal sides of 1 unit. Hypotenuse becomes Ă¢Ë†Å¡2 unit.

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If he forgets the sine,cosine and tangent of 30, 45, 60 and 90 degrees during a test or an exam, he can draw on the spot and work it out.

Edited by Arcadia
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Pre-A? Just throwing it in there for a great-to-have-seen-it-before! I wouldn't memorize any if those, but it will be good later (geo, alg, or even alg 2) to really understand why those numbers are what they are.

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But, for now, move on. :)

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The Pythagorean Theorem itself was okay. It was more the special triangles stuff. Sigh. It's a lot of memorization. I definitely remembered the Pythagorean Theorem, but I never could have pulled out the 30-60-90 triangles with the longer leg being aĂ¢Ë†Å¡3 and all that. I'm almost positive that there's no trig required of middle school math in Common Core.

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You don't need trigonometry to do that - just simple geometry.

Take two 30-60-90 triangles and stick them together to form an equilateral triangle. You see that the short side of the triangle is half the hypotenuse, and the length of the longer side follows from Pythagoras.

There should be no memorization about this.

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I did not go to school in the US. This was covered in 6th grade math class. Trig was not introduced until 9th or 10th grade.

Edited by regentrude
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Making note to self about this for next year when Dd is doing Dolciani PreA.

She will take the class with Wilson Hill. Interesting to see if they get all the way through the book or skim/skip the end stuff.

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Those special right triangles used to be frequent fliers on the SAT (less so, now).

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You don't need trigonometry to do that - just simple geometry.

Take two 30-60-90 triangles and stick them together to form an isoceles triangle. You see that the short side of the triangle is half the hypotenuse, and the length of the longer side follows from Pythagoras.

There should be no memorization about this.

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I did not go to school in the US. This was covered in 6th grade math class. Trig was not introduced until 9th or 10th grade.

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Yes, that's how it's taught. But there's a bunch of problems - different types of triangles, different sides given, find x or y or whatever. It's a lot easier if they either memorize the relationships or refer back to the formula/chart for them. Honestly, I *do* think it's a lot of memorization. This is just what geometry is like - you have to remember that because of this, then that, and because of that, then this other rule, and then you can figure out the length of that side or whatever the problem is asking for. Having that stuff memorized is like knowing your times tables, IMHO. You can go back through and use repeated addition to figure out the multiplication facts and you can go back to the most basic rules and extrapolate everything again from that, but it makes it a lot faster if you don't have to remember that, oh, I can turn this into an equilateral triangle and then I can plug the numbers in and then plug them in again, and so forth but instead just say, oh, the longer side is this times the square root of that or whatever.

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I know trig isn't required. The *next* section teaches some basic trig. We definitely didn't do that before algebra I when I was a kid.

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I know he's going to have to learn this stuff eventually. I just can't think of any of it as needing to be mastered now. But it sounds like everyone is saying I'm wrong. That it does. That going into algebra and geometry, kids will need to have this stuff mastered?

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He can master that in geometry instead of now. Pythagorean triples such as 3,4,5 and 5,12,13 are useful to remember but I won't purposely memorize those.

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The thing with purposely memorizing is that it is easy to panic and forget in an exam situation. It is like the trigonometry identities. As long as I can remember sine squared theta + cosine squared theta = 1, I can derive the other two identities.

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Knowing how the graphs of sine, cosine and tangent look like is also useful.

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Yes, that's how it's taught. But there's a bunch of problems - different types of triangles, different sides given, find x or y or whatever. It's a lot easier if they either memorize the relationships or refer back to the formula/chart for them. Honestly, I *do* think it's a lot of memorization. This is just what geometry is like - you have to remember that because of this, then that, and because of that, then this other rule, and then you can figure out the length of that side or whatever the problem is asking for. Having that stuff memorized is like knowing your times tables, IMHO. You can go back through and use repeated addition to figure out the multiplication facts and you can go back to the most basic rules and extrapolate everything again from that, but it makes it a lot faster if you don't have to remember that, oh, I can turn this into an equilateral triangle and then I can plug the numbers in and then plug them in again, and so forth but instead just say, oh, the longer side is this times the square root of that or whatever.

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I know trig isn't required. The *next* section teaches some basic trig. We definitely didn't do that before algebra I when I was a kid.

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I know he's going to have to learn this stuff eventually. I just can't think of any of it as needing to be mastered now. But it sounds like everyone is saying I'm wrong. That it does. That going into algebra and geometry, kids will need to have this stuff mastered?

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No, it does not need to be mastered before going into algebra. He will get it again in geometry.

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I see this as first exposure.

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But I completely disagree on the memorizing. Of course, once the student has worked a sufficient number of problems, he will remember certain strategies that worked for certain classes of problems, but I see absolutely no value in making a child memorize a lot in geometry. This is NOT what "geometry is like" - there are a few basic principles, and from these, everything can be derived. The value is in understanding the relationships and being able to see the connections and derive what you need, not in memorizing a bunch of things to crank out numbers fast.

The beauty of geometry is that there is almost no memorization required - just thinking. Memorized stuff can be forgotten if not used; understood concepts can always be re-derived and will never be lost.Â

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ETA: sin and cos of 30 and 60 are actually a great example. Often, students who have memorized the values by rote will mix up which of them is 1/2 and which is (sqrt 3)/2 - but the students who have understood sine and cosine conceptually will be able to recall by means of a quick sketch of the triangle or the unit circle.

Edited by regentrude
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First exposure. Okay, that's exactly what I said and hoped.

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There's so much vocabulary in geometry. And so many formulas. Obviously getting to the underlying meaning is best, but I don't know how one does proofs and solves problems without having the vocabulary and some of the formulas memorized. Obviously, there's a great deal of logic involved as well and solid math skills in general, but... if you don't know what an isosceles triangle is vs. a scalene triangle, then how do you prove it or utilize the information in the problem to apply it? And while it would be great if kids could rederive the volume of a sphere every time... I think most of us are just going to memorize it. What am I missing here? Honestly, geometry was always my least favorite part of math so it's entirely possible I'm missing something here.

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First exposure. Okay, that's exactly what I said and hoped.

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There's so much vocabulary in geometry. And so many formulas. Obviously getting to the underlying meaning is best, but I don't know how one does proofs and solves problems without having the vocabulary and some of the formulas memorized. Obviously, there's a great deal of logic involved as well and solid math skills in general, but... if you don't know what an isosceles triangle is vs. a scalene triangle, then how do you prove it or utilize the information in the problem to apply it? And while it would be great if kids could rederive the volume of a sphere every time... I think most of us are just going to memorize it. What am I missing here? Honestly, geometry was always my least favorite part of math so it's entirely possible I'm missing something here.

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The volume of the sphere and the circumference and area of a circle are actually among the few formulas that have to be memorized, because they cannot be derived using basic geometry.

But what else is there? Most things are rather logical and can often be seen from a brief sketch. And again, it will become automatic through use, not through memorizing formulas.Â

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There is vocabulary, but one does not need to sit down and "memorize" vocabulary with flash cards - one reads, hears, and uses the words often enough so the terms become part of one's vocabulary. We have never used a vocab program, just Â learned words from reading them in context multiple times. So, the student will encounter the terms every time they appear in a Â problem, and will come to associate them with the object. Don't we all learn new words like this all the time in various areas of life, without drilling vocab - just by using them? Learning a word for a specific kind of triangle is no more difficult than remembering a word for a new ethnic dish or a bird species or a tool.

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And a lot of words also just make sense if you know some basic word roots. You don't need to wonder what an equilateral triangle could possibly be since it literally means equal-sided. So since "iso" is same, isosceles has to be the one with "same sides".

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I would not consider a geometry program that heavily emphasized memorization.

Edited by regentrude
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First exposure. Okay, that's exactly what I said and hoped.

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There's so much vocabulary in geometry. And so many formulas. Obviously getting to the underlying meaning is best, but I don't know how one does proofs and solves problems without having the vocabulary and some of the formulas memorized. Obviously, there's a great deal of logic involved as well and solid math skills in general, but... if you don't know what an isosceles triangle is vs. a scalene triangle, then how do you prove it or utilize the information in the problem to apply it? And while it would be great if kids could rederive the volume of a sphere every time... I think most of us are just going to memorize it. What am I missing here? Honestly, geometry was always my least favorite part of math so it's entirely possible I'm missing something here.

I agree with you that there is a ton of vocabulary in geometry. And a lot of the terms have other meanings that seem to have to relation to their geometry definitions. Vertical angles? Corresponding angles? I have to make up stories about letters (correspondence) getting mis-delivered to the same house on the next street.

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We tried a geometry program that went straight into proofs and assumed you already had lists of theorems and postulates memorized. There were "cheat sheets" in the back of the book, but it was still overwhelming. We switched to informal geometry with MUS.

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I think some people who have done it forever or who use it a ton don't have the same perspective, don't realize how abstract it can be to someone who is learning or teaching geometry without a lot of prior knowledge. I mean, I did fine in high school math but that was 20 years ago and I haven't used it since, with the exception of teaching my own kids. ;). I get it.

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I absolutely know what you mean by there being a lot of formulas & vocabulary in geometry. Those of us who do not learn things by doing them just a few times struggle with this part of geometry. It takes me and my kids hundreds or thousands of times to get something ingrained to the point of knowing it as well as some kids can with only five or fifteen or even fifty repetitions.

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My goal with some of these geometry concepts is mastery, However, other things, like some of the theorems, will be learned for exposure. When DD was trying to prove stuff in geometry last year, I encouraged her to think about "what do I need" to get it done. Often, that would be enough to jog her memory that there was a theorem that would help. She'd look up exactly what the wording was and be on her way. I'm not a stickler on having everything memorized.

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Quark, I think, talked about making a reference notebook of theorems & postulates that the kid could refer to when doing their homework & tests. She said that eventually her son didn't need it anymore. I think all of my kids would benefit from making something like that - and most would have to reference it the whole year. The important thing is that you know they exist - or you can re-derive them from what you DO remember. But, there is always a base of information that you will need to know and understand first.

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Farrar, Â

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if it helps you feel any better, we are almost to the end of Dolciani Pre-A and I haveÂ ruthlessly cut and slashed wisely chosen to stress some parts of the last few chapters more than others. ;) Â I am allowing some time on Khan Academy to touch on some of those topics instead of allowing an otherwise good book to completely pulverize my boy's love of math. Â He has done very well with most of the book, even a majority of the C problems, but some of the lessons just seem like nit-picky overindulgence *at this level, for most kids*. Â I know that this is an accelerated book by well regarded math teachers, but there can be too much of a good thing, sometimes. Â  :leaving:

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I think he'll enjoy Dolciani Algebra next year, even if I didn't get to every thing in every lesson in the Pre-A book. Â

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This kid is an average math kid. I honestly haven't had him do that much of the C problems most of the way through. My approach has been to test him on the material first. If he mostly had it, to assign fewer problems and C problems. But on the topics that he needed a lot of work on - mostly percents and problem solving and the geometry topics, which were new to him - I don't know that he's really done any of the C problems at all. Just A and B ones. Sometimes the extra A and B ones in the teacher manual.

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I'm good with this being just introductory. I find it a bit surprising that anyone would think you'd need to master the introductory trig stuff during a PRE algebra course. I told him specifically today that I didn't expect mastery. I'm sure that to some people the fact that he kept getting turned around in that section about how the longer leg of the 30-60-right triangle has the formula with a square root and the hypotenuse of the isosceles right triangle has the square root and which square root it is - 2 or 3 - means that he's an utter dolt. But it was just a huge information dump from what I could see. There's a lot in there. And a lot of steps of reasoning it out if you didn't follow the little cheat sheet to re-figure it all out.

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When you've got books giving you a cheat sheet and people keeping notebooks of lists of all the formulas and so forth... I don't know how to conceive of that as anything other than memorization based. I know that the end goal is to be able to derive all this stuff and apply it and that's the heart of the subject... but surely most people spend much of their geometry education memorizing formulae and theorems and vocabulary so forth. I mean, to me, one of the lovely things about arithmetic and early algebra is that while it's important to know the vocabulary and be able to talk about it clearly in correct terms, you don't actually have to know the vocabulary in order to gain a deeper understanding of the math and the relationships between the numbers and so forth. But when it comes to measurement and geometry, without the vocabulary, you simply can't do anything or go anywhere. So I see that this is one of my kid's weaknesses with... well, everything, honestly - he is just not a kid with a giant vocabulary and memorizing stuff is not his strength - and it makes the geometry harder than it makes other topics in math. We had the same issues with measurement. Without the words, there's nothing.

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So I see that this is one of my kid's weaknesses with... well, everything, honestly - he is just not a kid with a giant vocabulary and memorizing stuff is not his strength - and it makes the geometry harder than it makes other topics in math. We had the same issues with measurement. Without the words, there's nothing.

Geometry is very pictorial and logical. Maybe google for geometry in ballet to help him visualize e.g. http://www.savannahballettheatre.org/about/education-outreach/the-geometry-of-dance-powerpoint

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Lots of rough paper and lots of colored pens/pencils to draw. We actually carried BIC pens in my backpacks and in the car which we use for geometry and German (masculine, feminine, neuter gender for nouns). https://www.amazon.com/BIC-4-Color-Medium-Assorted-3-Count/dp/B002JFR8Q8

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For example my DS11 may not remember the definition of supplementary angles, complementary angles, corresponding angles, alternate angles but he can label all the angles correctly. For example he can mark out all corresponding angles correctly on a diagram but forgot that those angles are called corresponding angles. He can mark out all the alternate interior and exterior angles with colored pencils/pens but forgot the name for it. He knows supplementary angles add to 180 degrees but forgets if they are called supplementary or complementary. He remembers his multiplication table after he was almost done with aops prealgebra.

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Geometry and trigonometry does come in again in Physics' mechanics sections. Trigonometry is in algebra 2, precalculus and calculus. So lots of spiraling practice.

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How is his visual spatial skills? My husband who has a hard time with geometry is the one that needs the GPS even for regular routes.

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Geometry is very pictorial and logical.Â

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Lots of rough paper and lots of colored pens/pencils to draw. We actually carried BIC pens in my backpacks and in the car which we use for geometry and German (masculine, feminine, neuter gender for nouns)

For example my DS11 may not remember the definition of supplementary angles, complementary angles, corresponding angles, alternate angles but he can label all the angles correctly. For example he can mark out all corresponding angles correctly on a diagram but forgot that those angles are called corresponding angles. He can mark out all the alternate interior and exterior angles with colored pencils/pens but forgot the name for it. He knows supplementary angles add to 180 degrees but forgets if they are called supplementary or complementary.

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The bolded.

The important thing is to recognize which angles are "of the same kind", which are identical and which add to 180. Memorizing what they are called is not as important as being able to work with them and prove stuff, using those properties.

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Colored pencils and lots of paper- that's how we got through geometry.Â

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When you've got books giving you a cheat sheet and people keeping notebooks of lists of all the formulas and so forth... I don't know how to conceive of that as anything other than memorization based. I know that the end goal is to be able to derive all this stuff and apply it and that's the heart of the subject... but surely most people spend much of their geometry education memorizing formulae and theorems and vocabulary so forth.

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That is not how we were taught in our schools, nor how we were teaching our own kids.

I find it very unfortunate that the curricula that are prevalent here seem to take this approach and create this impression that geometry is basically about memorization - when geometry is about discovering relationships between lines and shapes.

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I mean, to me, one of the lovely things about arithmetic and early algebra is that while it's important to know the vocabulary and be able to talk about it clearly in correct terms, you don't actually have to know the vocabulary in order to gain a deeper understanding of the math and the relationships between the numbers and so forth. But when it comes to measurement and geometry, without the vocabulary, you simply can't do anything or go anywhere. So I see that this is one of my kid's weaknesses with... well, everything, honestly - he is just not a kid with a giant vocabulary and memorizing stuff is not his strength - and it makes the geometry harder than it makes other topics in math. We had the same issues with measurement. Without the words, there's nothing.

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I disagree. He can draw a picture of parallels intersected by a line and see the different types of angles, see which ones are identical, which ones are in corresponding locations, which ones are sort of across from each other and add to 180. He can observe these relationships without knowing the specific words for the types of angles. He can prove that angles of "a type" are identical by calling them alpha and beta. The name is secondary, and I find it important to keep that in mind.

A large assortment of colored pencils is your best friend.

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Knowing that the word for a triangle with two identical sides is "isoceles" does not mean knowing any geometry. Working with the triangle, finding out things about it, using its properties is geometry - and for that purpose, calling it "a triangle with two identical sides" suffices completely. You can even use the property in formal proofs: "Because a=b, it follows that...." .

Eventually, the term will become part of his vocabulary - but the important thing is that knowing a term is not geometry.

Just like the vast memorization of "science vocabulary" does not mean a student has learned anything about science.

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Correct terminology is important, but if an overemphasis on terminology and formulas creates the impression that THIS is what geometry is about, that is very unfortunate and joy killing.

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Edited by regentrude
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Geometry is very pictorial and logical. Maybe google for geometry in ballet to help him visualize e.g. http://www.savannahballettheatre.org/about/education-outreach/the-geometry-of-dance-powerpoint

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Lots of rough paper and lots of colored pens/pencils to draw. We actually carried BIC pens in my backpacks and in the car which we use for geometry and German (masculine, feminine, neuter gender for nouns). https://www.amazon.com/BIC-4-Color-Medium-Assorted-3-Count/dp/B002JFR8Q8

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For example my DS11 may not remember the definition of supplementary angles, complementary angles, corresponding angles, alternate angles but he can label all the angles correctly. For example he can mark out all corresponding angles correctly on a diagram but forgot that those angles are called corresponding angles. He can mark out all the alternate interior and exterior angles with colored pencils/pens but forgot the name for it. He knows supplementary angles add to 180 degrees but forgets if they are called supplementary or complementary. He remembers his multiplication table after he was almost done with aops prealgebra.

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Geometry and trigonometry does come in again in Physics' mechanics sections. Trigonometry is in algebra 2, precalculus and calculus. So lots of spiraling practice.

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How is his visual spatial skills? My husband who has a hard time with geometry is the one that needs the GPS even for regular routes.

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His spatial and reasoning skills are fine. Like, he breezed through Dragonbox Elements and can do visual logic puzzles or the sort of thing that you mention.

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It's just that if you don't remember supplementary or complementary, then every single question is going to trip you up as you go back and check which is which. Who cares if you can tell the difference if you can't remember the words very well? And there's typically a dozen different concepts that are dependent on vocabulary like that in a single set of problems.

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I don't think this is about the program used. I mean, I think of Dolciani as being a really solid, conceptual program and it has lots of little boxes with the formulas and the vocabulary. Are people really telling me that Dolciani is not a conceptual program? I'm not sure what this kid will use for geometry (that's over a year in the future, after all). Ds who is doing Jacob's Algebra now will almost certainly use Jacobs, which I already have. But it has these sorts of little lists and so forth. I would need to do some vocabulary drilling for him with that program I'm sure (he's, if anything, worse than his brother at vocabulary type stuff, though he's stronger at logic and problem solving across the board). I don't think that's because the program is requiring it. I think it's because the subject requires it for a kid who isn't a natural at remembering specific terms. I mean, my kids still sometimes get turned around about things like pentagon and hexagon. They can obviously count to five and six, and we've obviously drilled this. But if we haven't done it in awhile, they still forget sometimes. If memorization of vocabulary comes naturally to you, maybe you just don't see how dependent geometry is on this stuff.

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I had Dolciani from 7th grade on, in the late 70s. It is easy to skip learning terminology if you are visual and studying independently, but if you are in the habit of reading and studying the text rather than jumping to exercises, it becomes natural. I peer taught that in high school,because people want to rush and pick out just enough to do the exercises,which meant they couldn't get the grade they wanted or the understanding they needed.

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Back in the day for me, in an independent study class, preA was every chapter with a 90 avg or higher on the publishers test for the A. No A was given to those who opted to do less, even with a 100 avg. Our assignment was every odd problem, but only enough in oral and A to get the point. Missing anything in B or C meant also doing the evens. In later years, C was cut to min of 2...back then only college prep students took Alg 1 and up. I was well prepared for college.

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I believe preA was intended to be mastered. The boys did use some of the info in Scouting at that age.

Edited by Heigh Ho
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I am NOT a math person, but I do think that geometry and algebra are very different in the way we think in order to do them. In my experience, most people prefer one to the other because it fits more naturally with how they think. I am a geometry person. ;) Â Maybe this doesn't hold for people who are all-around great at math, like Regentrude, but it does seem to be true of most people I know.

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Interestingly, I found the same thing to be true in grad school of Greek and Hebrew. Almost all students found one to be much easier and more natural than the other while a few linguistics-types loved it all.

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I am NOT a math person, but I do think that geometry and algebra are very different in the way we think in order to do them. In my experience, most people prefer one to the other because it fits more naturally with how they think. I am a geometry person. ;) Â Maybe this doesn't hold for people who are all-around great at math, like Regentrude, but it does seem to be true of most people I know.

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Absolutely.

I struggle with 3d geometry because my spatial visualization skills in three dimensions are lousy. (I recall one geometry problem I did with my kids, where I made playdough models of cubes with corners cut off because I could not "see" the result in my mind.)

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For a linear thinker, algebra is often preferred because it is formulaic; if you follow a prescribed procedure, you will arrive at the answer.

Geometry is more like an Art and requires much more creativity to see a way to approach a problem or proof, because there is no algorithm for that.

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It's just that if you don't remember supplementary or complementary, then every single question is going to trip you up as you go back and check which is which. Who cares if you can tell the difference if you can't remember the words very well? And there's typically a dozen different concepts that are dependent on vocabulary like that in a single set of problems.

Supplementary and complementary angles were in California's state testing. If my kids have forgotten on test day when they were in public school, so be it since we weren't aiming for perfect scores for elementary school state testing. The math section in SAT and ACT doesn't test definitions so I won't focus on memorizing.

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For test prep, the public school math teacher told the kids who couldn't remember to memorize that supplementary angles is like Superman flying through the sky so those angles add up to 180 degrees. Complementary angles by elimination becomes the angles that add to 90 degrees.

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It is like my annoyance at kids memorizing PEMDAS (teacher taught Please Excuse My Dear Aunt Sally) and sohcahtoa (sine, cosine, tangent).

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For things like pentagon and hexagon, root words study help.

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My husband and I are from the do every single problem in the book followed by Schaum series followed by ten years series (past Cambridge exams) generation. It is a lot of drill. My husband and my DS11 have to work hard at remembering which is why DS11 does all three science every year. DS11 can't even remember all the months of the year in English (he knows in German). DS11 needs much more drill than DS12, that is just how it is.

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ETA:

For 3D geometry, if you have ZomeTools or any chemistry modeling kit, make use of them if your child like math manipulative.

Edited by Arcadia
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His spatial and reasoning skills are fine. Like, he breezed through Dragonbox Elements and can do visual logic puzzles or the sort of thing that you mention.

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It's just that if you don't remember supplementary or complementary, then every single question is going to trip you up as you go back and check which is which. Who cares if you can tell the difference if you can't remember the words very well? And there's typically a dozen different concepts that are dependent on vocabulary like that in a single set of problems.

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I don't think this is about the program used. I mean, I think of Dolciani as being a really solid, conceptual program and it has lots of little boxes with the formulas and the vocabulary. Are people really telling me that Dolciani is not a conceptual program? I'm not sure what this kid will use for geometry (that's over a year in the future, after all). Ds who is doing Jacob's Algebra now will almost certainly use Jacobs, which I already have. But it has these sorts of little lists and so forth. I would need to do some vocabulary drilling for him with that program I'm sure (he's, if anything, worse than his brother at vocabulary type stuff, though he's stronger at logic and problem solving across the board). I don't think that's because the program is requiring it. I think it's because the subject requires it for a kid who isn't a natural at remembering specific terms. I mean, my kids still sometimes get turned around about things like pentagon and hexagon. They can obviously count to five and six, and we've obviously drilled this. But if we haven't done it in awhile, they still forget sometimes. If memorization of vocabulary comes naturally to you, maybe you just don't see how dependent geometry is on this stuff.

I totally agree with you. The vocabulary is tricky. And honestly, every geometry program On the shelf is going to expect you to memorize definitions of terms, theorems, axioms, and postulates. If someone doesn't think this involves deliberate memorization, then I suspect they either naturally learn vocabulary effortlessly, or math is such a part of their lexicon that they don't remember not knowing these terms.

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For my son who has trouble with vocabulary (and who is doing geometry, biology and psychology this term - flash cards are his friends!) I made up stories.

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Joe walks around a corner and bumps into his friend Steve. Joe gives him wave and a compliment (nice shirt!). Compliment to your friend that you bump into as you turn a corner --> complementary angles add up to 90 degrees.

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Mike had bad posture so his mom made him take vitamin supplements to help him grow straight and tall. Supplements make your back straight --> supplementary angles add up to 180 degrees.

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Vertical angles form two v's.

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And I already wrote about my corresponding angles story.

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My son has had three years of pre-geometry (integrated into his middle school math) and this never made it into long-term understanding. He is fine with math concepts, but geometry adds such a language element to that it takes the fun out of it for him. And he still hasn't seen a drop of trig in any of it. It wasn't even in algebra 1. I mean, yes, he worked with the Pythagorean theorem and distance formula, not that as far as what you are describing. And we DID finish every book!

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It is like my annoyance at kids memorizing PEMDAS (teacher taught Please Excuse My Dear Aunt Sally) and sohcahtoa (sine, cosine, tangent).

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Boy, soh-cah-toa was my crutch when learning trig for the first time in high school. Â I remember being skeptical when I was first taught, but boy did it come in handy. Â I was pleasantly surprised to see that it's still taught in the AoPS books. Â

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But I've never heard of PEMDAS. Â What does that possibly mean? Â

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Boy, soh-cah-toa was my crutch when learning trig for the first time in high school. Â I remember being skeptical when I was first taught, but boy did it come in handy. Â I was pleasantly surprised to see that it's still taught in the AoPS books. Â

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But I've never heard of PEMDAS. Â What does that possibly mean? Â

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Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

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My big issue with it is that a lot of students who are "applying the rule" and don't understand the rule will look at something like 3-4+5 and say "Well, I do addition before subtraction, so 4+5 is 9 and then 3-9 is -6".Â

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Boy, soh-cah-toa was my crutch when learning trig for the first time in high school. Â I remember being skeptical when I was first taught, but boy did it come in handy. Â I was pleasantly surprised to see that it's still taught in the AoPS books.Â

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I was in my 30s and teaching physics here at the university when I first encountered SOHCAHTOA - several students had it scribbled at the margin of their exam. I had no idea what that meant ;)

We don't have a mnemonic like this in German. (My impression is that we seem to have fewer mnemonics overall.)

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Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

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My big issue with it is that a lot of students who are "applying the rule" and don't understand the rule will look at something like 3-4+5 and say "Well, I do addition before subtraction, so 4+5 is 9 and then 3-9 is -6".Â

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Now, THIS is one where we do have a great mnemonic in German that does not have this issue.

We use a colon as the sign for division and a dot as the sign for multiplication, and the mnemonic is: "Dot operations before line operations"

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We don't have a mnemonic like this in German. (My impression is that we seem to have fewer mnemonics overall.)

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Huh. We do in Dutch - SOS-CAS-TOA. Turns out that the Mnemonics in trigonometry Wikipedia page only has a translation in Dutch:Â

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https://en.wikipedia.org/wiki/Mnemonics_in_trigonometry

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I never learned a mnemonic for PEMDAS though, though apparently there do exist at least a couple - this one is kind of funny:

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HoeÂ moetenÂ wijÂ vanÂ deÂ onvoldoendesÂ afkomen Â (How can we get rid of failing grades)

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I never learned a mnemonic for PEMDAS though, though apparently there do exist at least a couple - this one is kind of funny:

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HoeÂ moetenÂ wijÂ vanÂ deÂ onvoldoendesÂ afkomen Â (How can we get rid of failing grades)

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Popcorn Every Monday Donuts Always Sunday

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My kids like this because we have donuts at church on Sundays!Â

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The volume of the sphere and the circumference and area of a circle are actually among the few formulas that have to be memorized, because they cannot be derived using basic geometry.

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I'm not sure I even buy that assertation. Pi is defined as a ratio between circumference and diameter so that formula is a given. If you decompose the area of a circle into an arbitrary number of concentric circles or rings and then slice those rings into rectangles or lines, it is geometrically obvious that the resulting triangle has a base length of 2(pi)r and the height is the radius r. By making the width arbitrarily small you can make that triangle arbitrarily smooth giving a total area of 1/2 [2(pi)r]r = (pi)r^2.

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Since the volume of a sphere is analogous to the volume of a pyramid, I assume it can likewise be derived from first principles based solely on a definition of pi.

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Take a look at page 36 and 44 of the SAT practice test 1. Some area and volume formulas, Pythagoras Theorem and the 30-60-90 triangle, 45-45-90 triangle are provided. https://collegereadiness.collegeboard.org/pdf/sat-practice-test-1.pdf

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ETA:

ACT does not have those formulas provided.

Everyone is still awake *sigh*

Edited by Arcadia
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Popcorn Every Monday Donuts Always Sunday

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My kids like this because we have donuts at church on Sundays!Â

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Thanks, but I meant I didn't learn a mnemonic for it in Dutch. PEMDAS is actually easy enough to remember without coming up with a sentence for it, since you can pronounce it, unlike HMWVDOA (the W is for roots, in case you're wondering about the extra letter). My point was I didn't learn any mnemonic for it, but it doesn't take much to learn the order of operations (well, until you get to computer science, and the table of order of operations for a programming language is a page long).Â

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I'm not sure I even buy that assertation. Pi is defined as a ratio between circumference and diameter so that formula is a given. If you decompose the area of a circle into an arbitrary number of concentric circles or rings and then slice those rings into rectangles or lines, it is geometrically obvious that the resulting triangle has a base length of 2(pi)r and the height is the radius r. By making the width arbitrarily small you can make that triangle arbitrarily smooth giving a total area of 1/2 [2(pi)r]r = (pi)r^2.

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Since the volume of a sphere is analogous to the volume of a pyramid, I assume it can likewise be derived from first principles based solely on a definition of pi.

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But the concept of making the width arbitrarily small is really a calculus concept. You are basically integrating.

I would not consider this "basic geometry" in the sense of something a geometry learner would be familiar with.

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You can, of course derive the volume of the sphere by integration as well.

Edited by regentrude
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I'm not sure I even buy that assertation. Pi is defined as a ratio between circumference and diameter so that formula is a given. If you decompose the area of a circle into an arbitrary number of concentric circles or rings and then slice those rings into rectangles or lines, it is geometrically obvious that the resulting triangle has a base length of 2(pi)r and the height is the radius r. By making the width arbitrarily small you can make that triangle arbitrarily smooth giving a total area of 1/2 [2(pi)r]r = (pi)r^2.

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Since the volume of a sphere is analogous to the volume of a pyramid, I assume it can likewise be derived from first principles based solely on a definition of pi.

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And this kid isn't even up to Geometry yet. He's still in *PRE* algebra. Thus my feeling that they shouldn't have to *master* this stuff yet. Honestly, when put this way, it almost seems silly that they cover this concept at all.

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And this kid isn't even up to Geometry yet. He's still in *PRE* algebra. Thus my feeling that they shouldn't have to *master* this stuff yet. Honestly, when put this way, it almost seems silly that they cover this concept at all.

I have no experience with Dolciani but my kids public school math textbooks are definitely spiral. There is lots of exposure to the same concepts every year before the algebra and geometry stage.

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Jacobs Mathematics a Human Endeavor is a nice book though if you want something for review over summer. I read it at one of my local library and I forgot which edition I read.

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My younger learn by "pegs" in math. More exposure for him is a good thing.

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I have no experience with Dolciani but my kids public school math textbooks are definitely spiral. There is lots of exposure to the same concepts every year before the algebra and geometry stage.

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Jacobs Mathematics a Human Endeavor is a nice book though if you want something for review over summer. I read it at one of my local library and I forgot which edition I read.

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My younger learn by "pegs" in math. More exposure for him is a good thing.

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I've been using parts of Human Endeavor with both of them off and on for awhile. I am a little obsessed with that book.

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Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

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My big issue with it is that a lot of students who are "applying the rule" and don't understand the rule will look at something like 3-4+5 and say "Well, I do addition before subtraction, so 4+5 is 9 and then 3-9 is -6".Â

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PEMDAS makes my daughter very angry. The first time she saw it was in a test prep book a few weeks ago. The problem is not her understanding of the order of operations, which she has down cold, but is exactly what you mention above. She insists that PEMDAS is NOT accurate and wants to know why anyone would teach that to unsuspecting math students.

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I refer to PEMDAS and PEDMAS interchangeably, just because I can never remember it. :p But when I taught my kids the order of operations, I would write it out and put a small arrow pointing right under the MD and another under the AS, and I would always read it as "in the order they occur." My kids know when to go out of order for PEMDAS and when to go in order for multiplication/division and addition/subtraction.

Edited by Alte Veste Academy
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PEMDAS makes my daughter very angry. The first time she saw it was in a test prep book a few weeks ago. The problem is not her understanding of the order of operations, which she has down cold, but is exactly what you mention above. She insists that PEMDAS is NOT accurate and wants to know why anyone would teach that to unsuspecting math students.

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To be fair, you can do addition in any order as long as you carry the sign. Â So in the example mentioned, 3-4+5, I would teach combining positive 3 with negative 4 first, or positive five with negative four and then the positive 3. Â Once a student grasps negative numbers I like to get them out of the subtraction mindset altogether. Â It's much easier to think of 4-(-9) as "positive four combined with the opposite of negative 9" than "four minus a negative nine"...which is essentially meaningless in concrete terms.

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To be fair, you can do addition in any order as long as you carry the sign. Â So in the example mentioned, 3-4+5, I would teach combining positive 3 with negative 4 first, or positive five with negative four and then the positive 3. Â Once a student grasps negative numbers I like to get them out of the subtraction mindset altogether. Â It's much easier to think of 4-(-9) as "positive four combined with the opposite of negative 9" than "four minus a negative nine"...which is essentially meaningless in concrete terms.

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Yes.

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"I just want one rule."Â

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LOL

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