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AoPS PreAlg Ch2 (Exponents)


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DS11 finished SM through 5B, including IP & CWP.  His arithmetic skills are quite good (he's fast and accurate) and his mental math is also solid.  So - we moved next to AoPS pre-algebra.

 

Ch 1 took him about a week and a half and he did fairly well.  I let him work through each section of the book on his own, then reviewed his answers on the problems together and we talked through what he missed.  After some discussion to explain a few points I found he picked it up quite well.  There were even a few 'ah ha!' moments where he experienced the joy of leading himself to real comprehension (he came running into the living room exclaiming that he understood why you couldn't divide by zero because it has no recriprocal because there's nothing you can multiply it by and get 1'.  This was very exciting to him, which is good.)

 

 

I thought - 'wow, this is going just swimmingly!!!'

 

 

Then came chapter 2.

 

 

For each section, individually, he seemed to get the property being discussed…but he missed a decent number of the problems.  I walked him through the derivations (why a^n * a^m = a^(n+m), etc.) and he's following along..but..come to the review problems at the end and he crashed and burned.  He's not got a tight enough grasp of what's going on to really use this.  I'm not concerned about pace, just comprehension…so clearly we need to fix this before moving on.

 

 

I'm tempted to have him work through some sheets like

 

http://www.mcckc.edu/common/services/BR_Tutoring/files/math/expon_logar/Exponent_Rules_&_Practice.pdf

 

which require that you KNOW the relevant rules for exponents (even if you don't understand them) and then after he's learned them and can use them go back and say 'now - explain to me why 2^3^4 is 2^12 to ensure he understands the concept.

 

Anyone know of more practice pages like this one?

 

 

Or, we could try another run through AoPS chapter 2.

 

 

Or…anyone have an alternate suggestion?

 

thanks.

 

-andy

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Then came chapter 2.

 

 

Oh, yes.   :glare:  I'm convinced that chapter 2 is the hardest chapter in the book.  We actually skipped chapter 2 and went to 4 after 1 (my daughter's been skipping all over the book).  Would going through the Khan Academy videos help?  

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I'd see absolutely nothing wrong with doing a whole bunch of exponents worksheets and then going back to redo chapter 2.

 

One thing that may work is to just pick up a random algebra or pre-algebra textbook from your library (they should have one), do the chapter on exponents, and then go back and redo that chapter.

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Chapter 2 was tough for DS as well. I think it was so because it required him to think about numbers in new ways. We continually reviewed what exponents are but he would still confuse when to add/subtract and when to multiply/divide the exponents. So I made him stop and think about what the problem was "saying". Was it three 2s multiplied by five 2s (2^3*2^5)? Or was it five groups of three 2s multiplied ((2^3)^5)?

 

He also was resistant to thinking about numbers as multiplicative parts. I finally paused for a day and i made him do nothing but prime factorizations. As we moved into ch 3 and 4, I'm glad I did because it made those chapters easier.

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which require that you KNOW the relevant rules for exponents (even if you don't understand them) and then after he's learned them and can use them go back and say 'now - explain to me why 2^3^4 is 2^12 to ensure he understands the concept.

 

Anyone know of more practice pages like this one?

 

 

Or, we could try another run through AoPS chapter 2.

 

 

Or…anyone have an alternate suggestion?

 

thanks.

 

-andy

I re-read your post and wanted to address this part. I expected DS to understand the rules. On the white board, I had him continually write out exponents.

 

5^9/5^4

= (5*5*5*5*5*5*5*5*5)/(5*5*5*5) [definition of exponents]

= ((5*5*5*5)/(5*5*5*5))*(5*5*5*5*5) [associative property]

= 1*(5*5*5*5*5) [identity property]

= 5^5 [identity property]

 

Then I'd ask, what could be done to make this easier? DS would say, subtract the powers. We did this, before opening the book, for several days while working on Chapter 2. He worked through when to add, subtract, and multiply powers by writing out all the numbers and "discovering" the shortcut at the end. I think for DS it was a more thorough approach than working through several worksheets and memorizing the rules.

 

This made the math much easier when he worked with variables. He knew that x^5 was five X's multiplied together so there wasn't any confusion on the rules at that point.

 

ETA: correct terminology

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We had some issues with that chapter also, so I took a break from it, did the exponent sections in Dolciani Prealgebra (drill & kill), and then went back to AoPS. The chapter went SOOOOO much better! He just needed more time practicing exponent basics to feel comfortable using them.

 

Dolciani is cheap. I picked it up off Amazon for $6 shipped, but it may be $8 or $10 now. Still cheap. I assigned only the odd problems in the sections that I thought we could use more practice in. My son had no problem with the review section or the challenge section (well, he needed help on some problems, but he was able to do several without help, which is quite good for the challenge section!) after doing the exponent sections of Dolciani.

 

The good news is... Chapter 3 is a whole lot easier! Divisibility rules, factoring, primes, stuff like that.

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Chapter 2 was hard here, too-but chapter 3 was downright fun, and 4-6 have been smooth sailing ever since (I think I'm going to wait on chapter 7 until after Christmas-December is a MESS here!). I think it's a combination of fairly hard content that usually hasn't been covered too in-depth previously, plus just plain getting used to AOPS. Don't give up!

 

I did pull out some easier, more straightforward exponent activities to do when DD was getting frustrated. I think that helped a lot.

 

 

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It'll be ok.  Have him watch the videos and do Alcumus.  Sometimes it also takes some time to sink in and become a part of their problem-solving toolbox, even when they understood the concept initially.  I think it would be fine to move on to ch 3 and 4, which are easier, and come back to ch 2.

 

When my oldest child got to the end of the book, I went back and assigned some review problems - even the challenge problems seemed so. much. easier by then :)

 

Warning, my three kids who have been through the Prealgebra book found ch 5 to be difficult also, perhaps more difficult than ch 2.  We used additional resources for practice (Dolciani and/or Jacobs).  Especially my third one - he understood but it took time for it to sink in.  That kiddo has a tendency to want to turn his brain off and resort to a procedure, and I am planning on having him go back through parts of ch 5 again later.  (He started the book last spring but has been on hiatus, long story; he recently started back up again and is finally getting to the fun stuff, ch 10)

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  • 3 weeks later...

 

Warning, my three kids who have been through the Prealgebra book found ch 5 to be difficult also, perhaps more difficult than ch 2.  We used additional resources for practice (Dolciani and/or Jacobs).  Especially my third one - he understood but it took time for it to sink in.  That kiddo has a tendency to want to turn his brain off and resort to a procedure, and I am planning on having him go back through parts of ch 5 again later.  (He started the book last spring but has been on hiatus, long story; he recently started back up again and is finally getting to the fun stuff, ch 10)

Thank you for the bolded.  I feel so "not alone".  lol  I am working through Chapter 5 with my boys right now, and although I understand it and can do it easily, it has moved too quickly for them.  We have watched the videos, and I have explained and worked through many problems.  It is just not quite clicking.  We are most of the way through, slogging it out, but I am clear that they have not mastered it nearly enough to move on.  I have decided to take a detour to dive into linear equations with Key to Algebra (book 3, I think...I forget, as I have all 10).  My take on Chapter 5 is that it moves too quickly from the simple to the complex.  My guys cannot seem to keep all of the various math rules in mind when working the problems.  And negative numbers are often their undoing.  (one of my biggest compaints about Singapore US Edition - no negative numbers).

 

I am not a math gal so I often second guess myself, and I find it reassuring to read this.  :)

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Thank you for the bolded.  I feel so "not alone".  lol  I am working through Chapter 5 with my boys right now, and although I understand it and can do it easily, it has moved too quickly for them.  We have watched the videos, and I have explained and worked through many problems.  It is just not quite clicking.  

 

 

 

Wow, weirdly, we are also working on chapter 5 of prealgebra.  I've found that if algebraic manipulations of linear equations is second nature to you, it's important to remember to articulate each step ad nauseum until your student is ready to skip it.

 

For example, when you add 4 to both sides, rewrite the equation with the +4 to each side.  

 

5x - 4 = 7

5x - 4 + 4 = 7 + 4

 

 

Then observe that -4 +4 =0.  We do this all the time because she really needs to see it.  

 

Likewise with negatives, my dd needs to rewrite the equation:

 

5x - 4 = 7

5x + (-4) = 7

 

I figure she can do that as much as she needs to until she's ready to skip that step.  But until then, we'll keep rewriting our equations.  Good luck!  

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If you happen to have Zaccaro's Real World Algebra lying around . . . the first 3 chapters do a really wonderful job at a step-by-step explanation of solving equations! I've been working through it myself, and ran into some tough problems in Chapter 3 Einstein section, but I think it is really well-taught and well-explained.  I think it might be a nice complement.  I think it explains how to do it in a step-by-step way better than AoPS does.

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Wow, weirdly, we are also working on chapter 5 of prealgebra.  I've found that if algebraic manipulations of linear equations is second nature to you, it's important to remember to articulate each step ad nauseum until your student is ready to skip it.

 

For example, when you add 4 to both sides, rewrite the equation with the +4 to each side.  

 

5x - 4 = 7

5x - 4 + 4 = 7 + 4

 

 

Then observe that -4 +4 =0.  We do this all the time because she really needs to see it.  

 

Likewise with negatives, my dd needs to rewrite the equation:

 

5x - 4 = 7

5x + (-4) = 7

 

I figure she can do that as much as she needs to until she's ready to skip that step.  But until then, we'll keep rewriting our equations.  Good luck!  

 

One very useful technique, which is taught in schools in Germany and which we have insisted our children use, is to draw a short vertical line behind the equation and indicate behind this line the operation you are planning to perform to both sides of the equation (for example, the +4 in your example).

In the next step, the operation is done to both sides and terms are simplified.

We found it very valuable to have our beginning algebra students show the operation like this.

When they became proficient, they were no longer required to do this. I still find myself doing the same in longer calculations, and it definitely helps to keep track.

 

 

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ALWAYS has him write it out(a^3=axaxa) if he cannot intuitively solve it. It is not a hard concept, just.. Well ... Different. When DS went through it, I have to keep remind him to write it out. It is very important to have those solid, it gonna show up again and again

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One very useful technique, which is taught in schools in Germany and which we have insisted our children use, is to draw a short vertical line behind the equation and indicate behind this line the operation you are planning to perform to both sides of the equation (for example, the +4 in your example).

In the next step, the operation is done to both sides and terms are simplified.

We found it very valuable to have our beginning algebra students show the operation like this.

When they became proficient, they were no longer required to do this. I still find myself doing the same in longer calculations, and it definitely helps to keep track.

 

 

I want to make sure I understand this correctly.

 

Original equation:

 

8X + 2 = 4X + 10

 

Then:

 

8X + 2 = 4X + 10 / -2

 

Is that correct?

 

I do have my boys write out every operation and not just do them mentally, even if they feel it is a simple one.  I told them that they can do them mentally when they have much more practice and are consistently getting correct answers when writing them.

 

Since negative numbers really trip them up, I have them draw a box around each integer, constant or variable, so they can associate the positive or negative sign with the integer (or the "understood positive sign at the beginning of an equation).  Then, they can reorder that side of the equation to put the like parts of it together in order to simplify. 

 

One of the benefits of not being a mathy person is that I still approach these equations in a very basic way so I can teach the boys how to do the same.  As I explain to them, my way may be a bit slower, but I consistently get the correct answer.

 

I find teaching writing and literature analysis to come naturally.  I do it intuitively and easily.  Because I lack this skill with math, I often feel like I am bumbling around in the dark.  I appreciate any help from you mathy board folks.  Nothing is too basic for me.  :D  I often fear that I am using the wrong math terms here on the board.  Really.  However, I have a very good understanding of fractions and linear equations so I hope these will take me far.  (Just being honest here...I have always felt like a math dunce even though I was clearly not a dunce in other areas.  Teaching my own kids has made me face this monster - not the math but my perception of my skills.)  I have more upper math curricula on my shelf than any other subject, simply due to my own anxiety! 

 

We will start with the Key to Algebra book today and put AoPS pre-A aside for the moment.  I anticipate working through the entire Chapter 5 again.  (Shhh, don't tell my boys.  lol)  The Zaccaro book is on its way from Amazon and will be here during our break, and I am looking forward to using it, as well.  I looked at my Dolciani book last week and did not find an equivalent chapter.

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I find teaching writing and literature analysis to come naturally.  I do it intuitively and easily.  Because I lack this skill with math, I often feel like I am bumbling around in the dark.  I appreciate any help from you mathy board folks.  

 

 

If it makes you feel better, I am just the opposite.  Literature leaves me feeling frustrated and annoyed.  I outsource most of language arts.  

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It will be very helpful to us new to AoPS Prealgebra to have some people whose kids have done the whole book make a list of which chapters are difficult and may need extra practice or sections in Dolciani Prealgebra (or other math resources) to cement the concepts. We just started today after doing three days of Dolciani. Ds11 prefers the big AoPS Prealgebra book, but I will definitely use Dolciani to fill in for the harder chapters in AoPS. Thanks to those who already gave valuable suggestions.

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I want to make sure I understand this correctly.

 

Original equation:

 

8X + 2 = 4X + 10

 

Then:

 

8X + 2 = 4X + 10 / -2

 

Is that correct?

 

 

Yes. In the next step, the equation would become

 

8x =4x+8     | -4x

4x=8         | :4 

x=2

 

I also make sure they align the equal signs.

Possibly the most important thing to instill is to have only one equation per line. one of the most common mistakes is writing of run-on expressions that are connected with equal signs, but are not really equations, because students do operations in between.

For example, they might write

8x=4x+8=4x=8=x=2 which is complete nonsense, but very, very common. Nip this in the bud.

 

 

 

 

Since negative numbers really trip them up, I have them draw a box around each integer, constant or variable, so they can associate the positive or negative sign with the integer (or the "understood positive sign at the beginning of an equation).  Then, they can reorder that side of the equation to put the like parts of it together in order to simplify. 

 

We had, with both kids, a period of frequent careless mistakes related to signs. What helped very much was to use colored pencils and have them color plus and minus signs.

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We had, with both kids, a period of frequent careless mistakes related to signs. What helped very much was to use colored pencils and have them color plus and minus signs.

I appreciate the explanation posted above and wanted to say that, interestingly, older ds asked for a red pencil to mark the negative signs since he tends to overlook them.  Also, he dislikes "t" as a variable because it is too similar to a plus sign so he asked for permission to change it to "c", which of course I gave.

 

I need to be more diligent about seeing that he has a red pencil in hand prior to starting the lessons.

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I appreciate the explanation posted above and wanted to say that, interestingly, older ds asked for a red pencil to mark the negative signs since he tends to overlook them.  Also, he dislikes "t" as a variable because it is too similar to a plus sign so he asked for permission to change it to "c", which of course I gave.

 

I need to be more diligent about seeing that he has a red pencil in hand prior to starting the lessons.

 

I recently learned that you can buy erasable colored pencils.  I've been thinking about using them for geometry problems.  

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One very useful technique, which is taught in schools in Germany and which we have insisted our children use, is to draw a short vertical line behind the equation and indicate behind this line the operation you are planning to perform to both sides of the equation (for example, the +4 in your example).

 

 

 

I occasionally do something similar.  For example, if I want to divide both sides of an equation by 4, I write, "D4" on the next line.

 

      4x = 28

D4:  x =   7

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For the OP: I had to bring in extra practice problems for the exponents chapter. I happen to have Forgotten Algebra so I used problems from it. We are in Chapter 5 now, and I still give him a couple of exponent drill problems a couple times a week from there.

 

And speaking of Chapter 5...DS was suddenly struggling. I too had to put it away. Expanding the product was confusing him to no end. Math Drill has some good worksheets in its Algebra section. Search for distributive property. And we are going through Kahn Academy (manipulating expressions, maybe-not sure). I sure hope we can start redoing Ch5 after the holiday.

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Thanks for the excellent suggestions (and commiseration).  We've been on vacation (a great thing about homeschooling is never having to travel amid the crowds!) and have recently gotten back into things.

 

To my surprise, the drill pages didn't help.  We talked through & re-did the derivation of the 'rules', he tried the drill page I linked to…and got 50% of them wrong.  We worked through the ones he got wrong and the next day he re-did the drill page….and got 50% wrong (partially, but not completely, the same 50%   :huh: )

 

I think we'll take a look at the videos and I'll have him write out the detailed expansion & steps, like:

 

On the white board, I had him continually write out exponents.

5^9/5^4 
= (5*5*5*5*5*5*5*5*5)/(5*5*5*5) [definition of exponents]
= ((5*5*5*5)/(5*5*5*5))*(5*5*5*5*5) [associative property]
= 1*(5*5*5*5*5) [identity property]
= 5^5 [identity property]

 

 

 

He will hate that (whine, whine, complain...it's so much writing, this takes forever… :crying: ) but I think it'll help.

 

 

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To my surprise, the drill pages didn't help.  We talked through & re-did the derivation of the 'rules', he tried the drill page I linked to…and got 50% of them wrong.  We worked through the ones he got wrong and the next day he re-did the drill page….and got 50% wrong (partially, but not completely, the same 50%   :huh: )

 

I just looked at that drill page. I tried something like that with my son, and it was too much too soon. He needed SLOWER drill. That's where Dolciani was fabulous. It wasn't as deep as AoPS - not even close. It was very basic, and had a lot of problems at the basic level. Once he got comfortable at that very basic level, then he was able to move on to more complex.

 

If you're going to have him write everything out, can you use a white board? That makes the physical aspect of writing easier. My son does all his AoPS on the white board right now.

 

In my son's case, he understood the rules, but applying them just wasn't automatic at all, and he needed lots of basic problems applying the rules (not mixing them and doing tricky things) to become automatic. So backup, drill one basic rule at a time, keep the problems easy, and then move to the next rule.

 

If you don't want to purchase another prealgebra book, look at one that is free online. I know someone has posted links to CA math textbooks on the Logic board. You could have him do some pages from the exponents sections of one of those books. He may think the problems are really easy (after working AoPS ones), but that's ok! Have him do them anyway, and that will help with the automaticity. You want the basics of exponents to be easy. ;)

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I appreciate the explanation posted above and wanted to say that, interestingly, older ds asked for a red pencil to mark the negative signs since he tends to overlook them.  Also, he dislikes "t" as a variable because it is too similar to a plus sign so he asked for permission to change it to "c", which of course I gave.

 

I need to be more diligent about seeing that he has a red pencil in hand prior to starting the lessons.

 

I write script t's as variables for this reason.

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Possibly the most important thing to instill is to have only one equation per line. one of the most common mistakes is writing of run-on expressions that are connected with equal signs, but are not really equations, because students do operations in between.

For example, they might write

8x=4x+8=4x=8=x=2 which is complete nonsense, but very, very common. Nip this in the bud.

 

+100. This makes it very, very difficult when a student needs to use the transitive property of equality, because he cannot distinguish between a chain of valid equalities and his own confused reasoning.

 

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Thanks for the excellent suggestions (and commiseration). We've been on vacation (a great thing about homeschooling is never having to travel amid the crowds!) and have recently gotten back into things.

 

To my surprise, the drill pages didn't help. We talked through & re-did the derivation of the 'rules', he tried the drill page I linked to…and got 50% of them wrong. We worked through the ones he got wrong and the next day he re-did the drill page….and got 50% wrong (partially, but not completely, the same 50% :huh: )

 

I think we'll take a look at the videos and I'll have him write out the detailed expansion & steps, like:

 

 

 

 

He will hate that (whine, whine, complain...it's so much writing, this takes forever… :crying: ) but I think it'll help.

DS hated it, too. In my mind, it didn't matter. The practice really helped him understand what exponents are. After all, he practiced the addition fact families for years. Taking a few days to really think about exponents was time well spent. It made equations with variables so much easier.

 

ETA: we used the white board. I'd work one problem as an example then I'd have him work only 4-5 problems, explaining the steps to me. The first day, adding and subtracting exponential powers. The second day, multiplying powers. The remaining days of the chapter, giving him a few problems for review then releasing him for his work. This took five to ten minutes each day. I neither drilled him nor did I give him a page of problems. If he can't do 4 problems, he wouldn't be able to do 20.

Edited by ErinE
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  • 3 weeks later...

DS hated it, too. In my mind, it didn't matter. The practice really helped him understand what exponents are.

 

ETA: we used the white board. I'd work one problem as an example then I'd have him work only 4-5 problems, explaining the steps to me. The first day, adding and subtracting exponential powers. The second day, multiplying powers. The remaining days of the chapter, giving him a few problems for review then releasing him for his work. This took five to ten minutes each day. I neither drilled him nor did I give him a page of problems. If he can't do 4 problems, he wouldn't be able to do 20.

 

following up....

 

This worked really really well.  He didn't totally hate it, either. :)

I pitched this as a game...the rules are:

1) given a list of transforms (initially not including anything about exp expect 'definition of exponent'..just things like 'arithmetic', 'assoc of mult', etc) and a complex expression

2) simplify the expression using a sequence of transforms.  Each transform goes on a new line, you can use only one at a time, and you have to identify which one you used.

 

as you did, we did this on the whiteboard.  After we'd done enough of them w/o exponent transforms, those got added to the list but the rules stay the same.

 

After a few days of this game, he tried the drill sheet I posted awhile ago - with the same rules - and got nearly all of them.  We're now going back to the Ch2 review problems in AoPS and it's MUCH better.

 

Thanks for the suggestion!

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just wanted to say thank you all for the helpful discussion thread.  i logged on tonight to see if i could figure out what in the world to do about DS10 struggling so much with chapter 2 in Aops Pre-Alg.  sooo thankful that i have some new ideas for tomorrow's lessons :)  glad to not be alone.  most of all, i'm glad he'll feel encouraged that *he* is not the only one struggling 

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