Help for Saxon Algebra I

[ifsogirl73]

My 8th grader is currently completing Saxon Algebra I. The spiral approach has worked well for her this year and she is able to get through all problems well throughout the lessons. However, we are stuck with distributive property with exponents and it's various problems. We're a bit more than halfway through the book and we've taken some days to use youtube, Khan academy and other resources to help her figure this out but she's stuck. She uses Nicole the Math Lady and Saxon Dive with her daily lessons but it's not helping with this particular section. We have gone over it more than a few times (it's pretty clear to me to at this point) but she still has days where she's looking at the problem like it's the first time she's seeing it. Alternatively, there are days where she will only miss one problem (argghh!!!!).

Does anyone have any advice on what else we can use that will help get through this? I'm hoping someone has had this issue and was able to resolve it with a resource I can look up. 

Edited May 6, 2021 by ifsogirl73

[EKS]

Can you give an example of a problem she is having trouble with and what she is doing wrong?

[ifsogirl73]

Problems 22-24.

[Lecka]

All those problems have a variable raised to a negative power.  Does she know how to do that with simpler problems?  I remember it being confusing.

Could she write things out as “x • x • x” instead of x^3 and then see that the manipulations are the same (as with adding the exponents to multiply).  I remember that helping me.

Good luck!

[Lecka]

I think negative exponents take a while to get the hang of?  
 

I think they can look intimidating, too.  
 

Have you tried doing similar problems with substituting numbers (easy numbers like 2 or 10) to see how it goes with numbers instead of variables?  I think it can seem more connected to familiar numbers that way.

 

These are things I remember from when I learned Algebra.  Good luck!

[ifsogirl73]

1 hour ago, Lecka said:

All those problems have a variable raised to a negative power.  Does she know how to do that with simpler problems?  I remember it being confusing.

Could she write things out as “x • x • x” instead of x^3 and then see that the manipulations are the same (as with adding the exponents to multiply).  I remember that helping me.

Good luck!

She does know how to do it with simpler problems. We've tried it several times and it make sense to her when we substitute similar problems but when she sees the problem in the book, it's another matter. 

You may be right, it just make take lots of review. I'm being patient and not pushing so we'll spend more time reviewing and practicing problems.

Thanks! 

 

[Not_a_Number]

2 hours ago, ifsogirl73 said:

She does know how to do it with simpler problems. We've tried it several times and it make sense to her when we substitute similar problems but when she sees the problem in the book, it's another matter. 

You may be right, it just make take lots of review. I'm being patient and not pushing so we'll spend more time reviewing and practicing problems.

Thanks! 

I'm curious how she'd explain x^3*x^4? Mind asking her and recording it? 

[EKS]

What is she doing wrong?  Do you have an example?  My suggestions are going to depend on where she is getting hung up.

[Lecka]

For talking through it....

The first thing I look for is to see if there is addition or subtraction in part of the problem.  
 

There is not addition or subtraction, so it’s just reducing the fraction.

 

Then I would go one letter at a time, and say “first do m, then do n,” and just do one letter at a time.

If there is addition or subtraction, then you would not reduce that way.  Then it is more likely to be looking for a way to factor or use the distributive property.  

[Lecka]

https://www.mathsisfun.com/definitions/term.html
 

It matters to know what a “term” is.  The ones that do not have addition or subtraction are terms.  
 

You will look at them differently than problems that have a term as part of them.

 

It can be good to first look for all the terms and identify them.

 

For the distributive property — at some point you are looking for “like terms.”

If you are looking at one term, you are probably looking to reduce that one term.

If there is more than one term, you are probably looking to reduce terms or else look for like terms.  
 

 

[Lecka]

If you read the directions for the problems — number 19 and 20 say to factor.  They are literally saying — for this problem, factor.

Then it goes to some problems where you aren’t factoring, you are just reducing.

Then number 23 says to expand using the distributive property.

Then 24 literally says to add using *like terms.*  First, this means each term is going to have a like term in common with the other terms.  So look for that.  Then when you see that — what if you just said “m^2k^5” could just be x — if it is just a like term, let it just be x or the number 5 or something that will look familiar and easy to work with.  Then it can look like “5x - 3x + 4x + 3x.”  (Aka substitute x for m2k5, since they are going to both just be representing one thing... if you substituted numbers for m and k, you would get a number and it would be the same number — so you can look at m2k5 that way to make it easier to work with.)

I think it is important to know the vocabulary words and identify what type of problem it is.

For now they are mixing it up but still saying “here is the kind of problem it is” in the directions.  
 

It is definitely harder to go back and forth between problem types — because you have to identify the problem type and not just do the same strategy in every problem.  It is hard to know what to look for.  This is just part of doing mixed review with Saxon.  

For number 24, a lot of times when you find like terms in a problem that looks that way, it is going to be to find a common denominator to use to add or subtract — and the answer will look like: terms added or subtracted on top of the fraction line, over a common denominator — and the common denominator is the like term that was factored out.  So that tricked me a little bit.  AND it can right after a problem about expanding using the distributive property so it seemed like it might be the reverse of the previous problem.  But — not really.  That is also a Saxon kind of thing.

I would check myself in the answer key before I was sure I was doing these right, this is what I think, though.

Something to know about terms and think about with terms — you can often think of them as just x or just a number, when manipulating them.  It’s not always helpful to look at them that way, but it can be if there is a step of adding like terms and it looks confusing.  Because — if you are manipulating a term that is a big fraction with lots of variables, but it is a term (no addition or subtraction), then it is going to behave like a term the same as if it were x, so you can substitute x.  
 

They are just things to get used to looking for because they will keep coming up, there is kind-of a checklist to go through to look forward things saying “do I factor? Do I reduce? Do I look for a common denominator?  Do I look for a like term?” And eventually you just notice things like that.  In the problems that say the problem type it is important to, to some extent, use the vocabulary term and look for how the vocabulary term matches that problem and what process goes with that vocabulary term.  
 

My daughter is doing Saxon 7/6 and it is definitely more work to go through this in the mixed reviews where there are similar looking problems but then you actually are doing different things.  She often misses problems that she does know how to do, because she kept doing the same thing she did on the problem before.  
 

I think Saxon does this on purpose to point out how you need to notice the difference and know which thing to do.

I think it is part of the learning process and there is an expectation to learn by having made a mistake and then going back to see why it was a mistake — and sometimes catch those tricky things the first time through, but other times — they are made to be tricky and when you see the trick you can look for it next time.  

[Lecka]

When looking at mistakes — if the wrong strategy was used — it is good to say “I thought it was this kind of problem, but it was really that kind of problem.” And then say why, what makes them different kinds of problems.  That is how to know what to notice foe next time.  

[Lecka]

https://www.aft.org/ae/spring2020/agarwal_agostinelli
 

http://uweb.cas.usf.edu/~drohrer/pdfs/Interleaved_Mathematics_Practice_Guide.pdf

These are articles about interleaving in math practice problems.

 

In theory it is harder to do mixed practice.  It is expected to make more mistakes from not identifying the problem type properly.  It is expected to be a part of the learning process.
 

But it is also supposed to be an effective way to learn to tell apart problem types!!!!!!!!

 

I tell my daughter they put in tricks to try to trick her but also to help her learn.  
 

I don’t know if you are new to Saxon and that part of Saxon.  If you have been in it a while — how nice your daughter hasn’t dbeen getting tricked since Saxon 7/6!  But she might not know they are trying to trick her if she is not used to getting tricked 😉

[Not_a_Number]

2 hours ago, Lecka said:

When looking at mistakes — if the wrong strategy was used — it is good to say “I thought it was this kind of problem, but it was really that kind of problem.” And then say why, what makes them different kinds of problems.  That is how to know what to notice foe next time.  

I have to say... working this procedurally has never stuck for people I’ve worked with. I know that Saxon is quite procedural already, so maybe you HAVE to do it, but I far prefer there to be more sense-making in math.

[EKS]

I'll give this a shot without knowing what the specific problem is.

One way to make problems with negative exponents easier is to turn them into positive exponents.  All this means is that if a negative exponent is in the numerator, you put it (and its base) in the denominator, and if it is in the denominator, you put it (and its base) in the numerator.  The reason this works is because x^(-1) is, by definition, 1/x.  And if that is true, then 1/x^(-1) is 1/(1/x), which is simply 1 divided by 1/x, which is 1 times x/1, which is x.  It's important that before she uses this technique that she understand why it works.  For the problem where it asks that the result contain all negative exponents, once you get to the end, you just flip the fraction and then make all of the exponents negative.  But, again, it's important to understand why this works.

But there are multiple ways that she could be having trouble with these problems, not just with the negative exponents.  The best thing would be is if someone could sit with her and try to untangle where her confusion is coming from.  

Edited May 7, 2021 by EKS

[Lecka]

On 5/7/2021 at 6:28 AM, Not_a_Number said:

have to say... working this procedurally has never stuck for people I’ve worked with. I know that Saxon is quite procedural already, so maybe you HAVE to do it, but I far prefer there to be more sense-making in math.

You know, it happened yesterday.... my daughter missed a problem in math, and then I watched her work it, and then I helped her, and then I came to a conclusion ------ she has no idea how this kind of problem works or why, and I had better go back and spend some time going over the entire concept with her.  

But it is so odd to me to think that anytime there is some confusion in math it must be because the program used is too procedural.  

 

 

[Not_a_Number]

3 minutes ago, Lecka said:

You know, it happened yesterday.... my daughter missed a problem in math, and then I watched her work it, and then I helped her, and then I came to a conclusion ------ she has no idea how this kind of problem works or why, and I had better go back and spend some time going over the entire concept with her.  

But it is so odd to me to think that anytime there is some confusion in math it must be because the program used is too procedural.  

What can I say? That was emphatically my experience when teaching college calculus 😞. It's also my experience working with AoPS kids.  

Don't get me wrong: I teach my kiddos procedures when their understanding is sound. I am not someone who thinks the standard algorithms are useless or anything like that. I have them all memorized myself and I'm pretty quick at arithmetic both mentally and on paper 😉. 

But what I've found is that when you build procedures without first getting a robust understanding you lose a lot of a child's natural sense-making. I've been working on a series of blog posts about this, actually, because it comes up so much in conversations: 

http://mentalmodelmath.com/2021/04/28/what-is-a-mental-model-anyway/

[LinRTX]

@Lecka And the funny thing is that my daughter who is a junior math major and is praised for her understanding of math was raised on Saxon. She did Saxon 2 through Calculus. It really is not just procedural. It does teach more than that but you have to actually use the entire program and not eliminate some of the problems. 

[Not_a_Number]

2 minutes ago, LinRTX said:

@Lecka And the funny thing is that my daughter who is a junior math major and is praised for her understanding of math was raised on Saxon. She did Saxon 2 through Calculus. It really is not just procedural. It does teach more than that but you have to actually use the entire program and not eliminate some of the problems. 

A program doesn't make the student. Some kids can learn math that way and some can't. You really can't generalize from one student. 

A procedural program is just fine when a kid is motivated to understand, retains concepts easily, and has a teacher who makes sure to reinforce the conceptual bits themselves. But not all students are like that and not all teachers are like that. 

What you don't want in math is purely procedural LEARNING. Will you get that from Saxon? Not necessarily. But if that's what one gets out of Saxon, then in my opinion, Saxon is doing the student a disservice. 

I've seen purely procedural learning come out of using AoPS, too, lol. Just because a program is labelled as "conceptual" doesn't mean all the learning will be conceptual... but the point is that this part of learning is essential. 

[LinRTX]

2 hours ago, Not_a_Number said:

A procedural program is just fine when a kid is motivated to understand, retains concepts easily, and has a teacher who makes sure to reinforce the conceptual bits themselves. But not all students are like that and not all teachers are like that. 

I can tell you this student was not a motivated math student. She hated it until her junior year of high school, in Calculus. Every math program requires a teacher. Very few children will learn by themselves. I have seen Saxon disasters, mainly when the child was handed both the book and the answer key.

[Not_a_Number]

Just now, LinRTX said:

I can tell you this student was not a motivated math student. She hated it until her junior year of high school, in Calculus. Every math program requires a teacher. Very few children will learn by themselves. I have seen Saxon disasters, mainly when the child was handed both the book and the answer key.

I agree that programs generally require teachers, yes 🙂 . But having watched people teach math, some people do NOT add the conceptual component to procedural programs successfully. And I think it's hard for people to pinpoint what it is they need to do to add that component. Plus, it simply depends on the kid: some kids naturally pick up the concepts in early math and some are absolutely able to plug and chug without doing any thinking. 

I have to say that I would also probably personally not be comfortable using a math program that resulted in a kid hating math until calculus. I understand that for some kids that turns out well but it would make me very nervous. (As a mathematician, I can tell you that the most common thing people say to you when you say you do math is "I hated math at school.") 

[EKS]

It isn't that Saxon doesn't teach concepts--because it absolutely does, many times in exactly the same way that Singapore (for example) does. 

It's that all that review tends to reinforce the procedures over the concepts.  One way to reinforce concepts is to need to reconstruct how to do something using the concept.  All that review in Saxon doesn't allow this to happen.

Having said that, the way the problem sets are designed, if you follow one problem type from one bit of instruction about it to the next, you will find that there is a progression, very gradual and very subtle, that leads the student into the next infusion of instruction.  This is why it is critical, if you use Saxon, to do all of the problems, because (I believe) that they have designed it to include some discovery.  When they skip problems, students don't get the benefit of discovering at least some things for themselves.

I wish I still had my Saxon books because I'd look for an example of what I'm talking about and post it.

[Not_a_Number]

7 minutes ago, EKS said:

It's that all that review tends to reinforce the procedures over the concepts.  One way to reinforce concepts is to need to reconstruct how to do something using the concept.  All that review in Saxon doesn't allow this to happen.

Thank you. That's an excellent way to put it. 

During my years teaching math, I've become convinced that the best way to learn concepts is to actually USE the concepts. And if you've drilled a procedure really well, you actually don't need the concept for any of the questions (especially if the program mostly involves rote questions that can be done with standard procedures.) 

I had kids in college calculus who could graph any line in the planet but really didn't understand that a line is a collection of points satisfying a certain equation. And they /really/ didn't understand how that interacted with graphs of much more complicated equations. This was because they were intensely overtrained on lines. When they graphed lines, they never had to access the "conceptual" part of their brain at all. This didn't work for them at all. 

[Lecka]

I don’t think you can take those students and say “they were taught by Saxon, that is the problem.”

I agree that is a problem, but I don’t think the solution is to dislike Saxon.

I think students can try to memorize answers and not grapple with actually thinking regardless of how they are taught.  
 

I think addressing that issue is separate from a curriculum choice. 

[Not_a_Number]

2 minutes ago, Lecka said:

I don’t think you can take those students and say “they were taught by Saxon, that is the problem.”

I agree that is a problem, but I don’t think the solution is to dislike Saxon.

I think students can try to memorize answers and not grapple with actually thinking regardless of how they are taught.  
 

I think addressing that issue is separate from a curriculum choice. 

I think some curricula encourage it and some don’t. Just like some teachers encourage it and some don’t.

[Lecka]

I think it is a complex issue and I think there are a lot of contributing factors over a period of years.  
 

I think some of it is just giving up on things making sense.  Well — I think that can happen with any program and any teacher.  It *can* happen.  
 

What makes it more likely?

 

Well, I have chosen Saxon thinking it would make sense to my daughter and be something she could be successful with.  
 

For my situation — the program where she will do well and feel like she understands is the one where I think she will feel successful and engender a virtuous cycle of positive reinforcement for her with math.

 

I think this is what will make her want to engage more with math.

 

So far — while not perfect, I am happy with how it is working out.  
 

She is much more willing to engage in thinking and trying than she was before.

 

I started helping her after school after she scored below grade level on an ILearn assessment last year during 5th grade.  She is at the end of 6th grade and she has made a lot of improvement and her confidence and attitude are way up.

 

I did not think I could get her to engage with some of the more conceptual programs I looked at, and some of them frankly have a higher difficulty level.  Well — I don’t think I could get her to get to a place of saying “I want to make sense” coming from that place if I chose some other programs I looked at.

 

I don’t think it is uncommon for students to make sense and want to make sense, when they are experiencing success and understanding.  I think when it is going well — Saxon delivers this for some/many students.  
 

(Edited.)

 

 

Edited May 9, 2021 by Lecka

[Not_a_Number]

1 hour ago, Lecka said:

I did not think I could get her to engage with some of the more conceptual programs I looked at, and some of them frankly have a higher difficulty level.  Well — I don’t think I could get her to get to a place of saying “I want to make sense” coming from that place if I chose some other programs I looked at.

Oh, I agree with you -- you don't want a harder program for a kid who's struggling. And success breeds success. So I absolutely understanding why you chose Saxon.

But personally, I don't sequence things procedurally 🙂 . I do make sure to sequence them so that a child can be successful, though. 

Edited May 10, 2021 by Not_a_Number

[daijobu]

20 hours ago, Lecka said:

I think some of it is just giving up on things making sense.  Well — I think that can happen with any program and any teacher.  It *can* happen.  

 

Exactly this.  Somehow students internalize the idea that they can't believe their own brains and can't logic their way out of a hairy problem.