Pre-A to Algebra question

[mirabillis]

Figuring out what to do for my 7th grader. (I'll cross-post this in Middle School too)

She'll finish Dolciani Pre-A by Christmas break. She won't start Algebra I until Fall (outsourced with Myhomeschoolmathclass - Jan in TX). So I have a semester to do what... ? She definitely needs reinforcement as math does not come naturally and we tirelessly work through Pre-A together - it doesn't come super easily. We are looking forward to this 'bonus' time to reinforce.

Should I find another Pre-A to go through for the remainder of the year? If yes, ideas?

Or should I start an Algebra program (could be Dolciani's Algebra) so she'll have a solid 1-1/2 years of Algebra under her belt?

Ideas? All my other older kids finished Pre-A at end of year, transitioning to Alg following year - so this 1/2 year finish is new territory for us.

Thanks!

Edited December 4, 2020 by mirabillis

[BusyMom5]

I've got boys the same age and level.  Here are a few ideas I've got:

1. MUS Algebra 1

2. CLE - pick a few light units 

3.  Keys To Algebra 

4.  Arbor School books- Crocodiles and Coconuts, Chuckles the Rocket Dog

5.  Life of Fred PreAl 

 

Right now they say that just wantvto start Saxon Algebra 1.  We shall see!  Ive used Arbor school and Fred with my oldest, MUS with second DD.  

[alisha]

I really liked Balance Math teaches Algebra (Critical Thinking Company).

[cintinative]

I would be tempted to start into Alg I with Dolciani if she likes that teaching method. There are lots of practice problems in the back in addition to the ones in the text. But if she would hate seeing the same things again in Jann's class, maybe that would be bad?

Have you considered asking Jann what she would recommend? Maybe there is a good precursor text she could recommend.

[forty-two]

Hands On Equations, maybe?  I've just paused Alg 1 to do a quick run-through of HoE, to help oldest dd better wrap her brain around how equations work.  I wish I'd done it over the summer,  or even before Pre-Alg (which is what I'm doing with my middle).  Dd has great intuition, but her ability to break things into explicit steps, and explicitly name what's going on in each step, is weak.  Two days of HoE has already done more to bridge the gap between her intuitive approach (which makes sense to her) and explicitly going through the problem step-by-step than 3mo of the Alg text.  She was excited about "having math that made sense".  I have her tell me explicitly what she is doing each step and tell me which property justifies each step (the latter is not in HoE but was something covered in our Pre-Alg (also Dolciani) and in our Alg), to help her bridge her intuitive understanding of using the pawns to the thinking called for in alg that hasn't yet made sense to her.

We're moving very fast at the moment - did level 1 (of 3) in two days; but I think the verbal problems book - where you apply the basic algebra taught to word problems - will slow us down some, especially as it adds the challenge of figuring out how to set up the problem, translating the word problem into an equation.  Dd can solve the problems in her head, but figuring out how to model them in an alg eq is much harder for her.

[EmilyGF]

My daughter has really enjoyed the first Arbor Math book (Jousting Armadillos) and learned a lot. It has really fit her learning style. I'd say it is more pre-pre-Algebra, though, than actual Prealgebra. Maybe the later books are more what you are looking for.

If your child does well with writing their own definitions by noticing patterns and investigating interesting questions, Arbor Math might be a great fit. My daughter was super-frustrated with math before, but now brings problems to the dinner table to talk about. She is the type who draws comics for fun; now she can write comics for math definitions. She's so happy! 

[Not_a_Number]

On 12/4/2020 at 5:05 PM, EmilyGF said:

My daughter has really enjoyed the first Arbor Math book (Jousting Armadillos) and learned a lot. It has really fit her learning style. I'd say it is more pre-pre-Algebra, though, than actual Prealgebra. Maybe the later books are more what you are looking for.

If your child does well with writing their own definitions by noticing patterns and investigating interesting questions, Arbor Math might be a great fit. My daughter was super-frustrated with math before, but now brings problems to the dinner table to talk about. She is the type who draws comics for fun; now she can write comics for math definitions. She's so happy! 

What does Jousting Armadillos cover, out of curiosity? 

[Shoeless]

10 hours ago, Not_a_Number said:

What does Jousting Armadillos cover, out of curiosity? 

Here's a link to see the entire book via google books. Jousting Armadillos.  The table of contents is the first few pages (obviously. 😉 )

[Not_a_Number]

12 hours ago, MissLemon said:

Here's a link to see the entire book via google books. Jousting Armadillos.  The table of contents is the first few pages (obviously. 😉 )

Thanks! 

Skimming through, I see that they talk about fractions being division... it's interesting to me that this is news for some kids. That's the only way I ever did fractions with DD8 -- I told her they were division 😛 . I figured that the fewer different concepts she had to remember, the better. 

Edited December 6, 2020 by Not_a_Number

[Shoeless]

7 hours ago, Not_a_Number said:

Thanks! 

Skimming through, I see that they talk about fractions being division... it's interesting to me that this is news for some kids. That's the only way I ever did fraction with DD8 -- I told her they were division 😛 . I figured that the fewer different concepts she had to remember, the better. 

I was expressly taught that I should not think of fractions as division, that while yes, 3/4 will equal 0.75 if you divide 3 by 4, a fraction was something different from division.  Thinking about fractions as division was called incorrect and lazy. 

That's about when I gave up on really understanding math. It seemed like the definitions were incredibly nuanced, and unless I was born with a talent for math (which I was not), math would be beyond my grasp. 

[Not_a_Number]

8 minutes ago, MissLemon said:

I was expressly taught that I should not think of fractions as division, that while yes, 3/4 will equal 0.75 if you divide 3 by 4, a fraction was something different from division.  Thinking about fractions as division was called incorrect and lazy. 

That's about when I gave up on really understanding math. It seemed like the definitions were incredibly nuanced, and unless I was born with a talent for math (which I was not), math would be beyond my grasp. 

What?????!!!!!! 

OK, I'm sorry your teachers were goobers. That's ridiculous. 

[Wheres Toto]

4 minutes ago, Not_a_Number said:

What?????!!!!!! 

OK, I'm sorry your teachers were goobers. That's ridiculous. 

 I agree.  Basically all advanced math represents division as a fraction.  You pretty much don't ever see the division sign (the line with dots above and below one, that is) once you get to Algebra.  

Edited December 6, 2020 by Wheres Toto

[Not_a_Number]

Just now, Wheres Toto said:

 I agree.  Basically all advanced math represents division as a fraction.  You pretty much don't ever see the division sign once you get to Algebra.  

You say "Where's Toto?", I say "Where's the apostrophe?".... 😉 

[Shoeless]

54 minutes ago, Wheres Toto said:

 I agree.  Basically all advanced math represents division as a fraction.  You pretty much don't ever see the division sign (the line with dots above and below one, that is) once you get to Algebra.  

It will likely be no surprise that I did not perform well in advanced math. 

[Not_a_Number]

2 minutes ago, MissLemon said:

It will likely be no surprise that I did not perform well in advanced math. 

Well, with teaching like that, I'm surprised you managed to learn that 3/4 was 0.75... 

[Not_a_Number]

As someone on FB pointed out, the division symbol is even SHOWING a fraction. The dot above is where the number you're dividing goes, the dot below is where the number you're dividing by goes... it's letting you write a fraction without having to use vertical space, that's all. 

[Shoeless]

1 minute ago, Not_a_Number said:

Well, with teaching like that, I'm surprised you managed to learn that 3/4 was 0.75... 

Well, that's an easy example, because I can relate it to money. Three quarters of a dollar is 75 cents, or 0.75. 

[Not_a_Number]

Just now, MissLemon said:

Well, that's an easy example, because I can relate it to money. Three quarters of a dollar is 75 cents, or 0.75. 

You have no idea how many people I've met who can't add 1/2 to 1/4 😛 . 

[Lecka]

I think some of these things that get brought up as “why don’t kids know it,” I learned in middle school but not in elementary school.  I always think — well, what age of kids?  Because often I wouldn’t have known in 4th grade, but would have known in 8th grade.  

[Not_a_Number]

Just now, Lecka said:

I think some of these things that get brought up as “why don’t kids know it,” I learned in middle school but not in elementary school.  I always think — well, what age of kids?  Because often I wouldn’t have known in 4th grade, but would have known in 8th grade.  

You mean like the fact that fractions are division? 

[Shoeless]

2 minutes ago, Not_a_Number said:

As someone on FB pointed out, the division symbol is even SHOWING a fraction. The dot above is where the number you're dividing goes, the dot below is where the number you're dividing by goes... it's letting you write a fraction without having to use vertical space, that's all. 

This is literally the first time in my life I have heard that. It makes sense, though! 

I think whenever I get hung up on something mathematical, I'm going to now try everything I was told not to do, lol. 

[Not_a_Number]

Just now, MissLemon said:

This is literally the first time in my life I have heard that. It makes sense, though! 

I think whenever I get hung up on something mathematical, I'm going to now try everything I was told not to do, lol. 

You can ask me first, lol. I'll either give you permission or act horrified 😉 . 

[Lecka]

Yes — I am sure I didn’t learn about fractions representing division when I first learned fractions.... but I’m sure I had it introduced in middle school and knew it already in 8th grade.  I think I learned it in 6th or 7th grade, when I learned to convert fractions to decimals.  
 

Because I think I learned that when learning to convert fractions to decimals.  
 

 

[Not_a_Number]

Just now, Lecka said:

Yes — I am sure I didn’t learn about fractions representing division when I first learned fractions.... but I’m sure I had it introduced in middle school and knew it already in 8th grade.  I think I learned it in 6th or 7th grade, when I learned to convert fractions to decimals.  

Because I think I learned that when learning to convert fractions to decimals.  

See, this is weird to me. One of my goals with math teaching was to cut down on the number of concepts that a kid had to internalize... so, for instance, I really didn't teach decimals except as fractions, and I never taught fractions except as division, and I taught fraction division to be completely THE SAME as division by integers... 

So it's funny to me when people are introduced to fractions in some other way. I don't think they even make sense in any other way but as division!! 

[Noreen Claire]

How about Mathematics A Human Endeavor?

[Lecka]

Fractions primarily make sense to me as a number, as a point on a number line.  Not as division.

Especially familiar fractions.  I associate them with a quantity.  
 

Edit:  I think that’s how I learned fractions.  Then later I learned about the division part.

Though — I do think I always knew, that a division problem could be written as a fraction and that meant divide.  But never went anywhere with that meaning much of anything until later.  
 

 

Edited December 7, 2020 by Lecka

[Not_a_Number]

Just now, Lecka said:

Fractions primarily make sense to me as a number, as a point on a number line.  Not as division.

Especially familiar fractions.  I associate them with a quantity.  

But division does result in a quantity, right? Like, 4 divided by 2 is 2. That's a quantity, a point on the number line.... 

[Lecka]

Yes.  Of course it does.  
 

But if you say — you don’t understand how someone can understand a fraction without thinking of it as division — well, I think I understood fractions as representing numbers, before I understood them as representing division.  
 

I have a 6th grader who made a mistake in the past week with saying “4/16 = 4” — so I can see there she has got some confusion.  But I think she does have a lot of understanding of the fraction 4/16 — she just doesn’t have it all tied together yet.

 

And I think she’s an average math student.  

[Not_a_Number]

2 minutes ago, Lecka said:

Yes.  Of course it does.  

But if you say — you don’t understand how someone can understand a fraction without thinking of it as division — well, I think I understood fractions as representing numbers, before I understood them as representing division.  

Oh, I absolutely understand how someone can understand a fraction without thinking of it as division. I just don't understand TEACHING fractions in a way other than division. But of course, if a kid isn't taught that way, they won't know, unless they are super mathy. 

 

Quote

I have a 6th grader who made a mistake in the past week with saying “4/16 = 4” — so I can see there she has got some confusion.  But I think she does have a lot of understanding of the fraction 4/16 — she just doesn’t have it all tied together yet.

And I think she’s an average math student.  

Yeah, I understand 🙂 . I do think it's easier if you think of it as division, because then it's clear that 4/16 is a lot less than 4... 

Edited December 7, 2020 by Not_a_Number

[Shoeless]

10 minutes ago, Not_a_Number said:

See, this is weird to me. One of my goals with math teaching was to cut down on the number of concepts that a kid had to internalize... so, for instance, I really didn't teach decimals except as fractions, and I never taught fractions except as division, and I taught fraction division to be completely THE SAME as division by integers... 

So it's funny to me when people are introduced to fractions in some other way. I don't think they even make sense in any other way but as division!! 

Cutting down on the number of concepts to internalize would have been considered "cheating" or "taking shortcuts" by my elementary and middle school math teachers. 

How Lecka thinks of fractions is how I was expected to think of them. They were specific points on a number line and *not* division. We were also discouraged from drawing the actual number line or any sort of picture model to help understand what was going on. I remember getting scolded by my 5th grade teacher for drawing little diagrams in the corner of my test papers to help figure out problems. "No more drawings! That's for babies!" 🙄 

I just asked DS12 if he knew that fractions were actually division and he was like "Yes...of course?", like I was nuts for asking the question, lol. That makes me feel good! 

[Not_a_Number]

1 minute ago, MissLemon said:

Cutting down on the number of concepts to internalize would have been considered "cheating" or "taking shortcuts" by my elementary and middle school math teachers. 

And of course, they didn't know much math, probably. Because that's who we need teaching our kids math -- people who didn't like it themselves!

That's, like, a major cornerstone of my teaching philosophy -- the fewer concepts, the better. 

 

1 minute ago, MissLemon said:

How Lecka thinks of fractions is how I was expected to think of them. They were specific points on a number line and *not* division. We were also discouraged from drawing the actual number line or any sort of picture model to help understand what was going on. I remember getting scolded by my 5th grade teacher for drawing little diagrams in the corner of my test papers to help figure out problems. "No more drawings! That's for babies!" 🙄 

Oooh, that reminds me of discouraging kids from counting on their fingers!! I saw someone do that (a homeschooler, no less!) just a few years ago. The little girl who she scolded hasn't done it since, which is a crying shame, since she could use some help with her number sense, and fingers are great manipulatives... 

 

1 minute ago, MissLemon said:

I just asked DS12 if he knew that fractions were actually division and he was like "Yes...of course?", like I was nuts for asking the question, lol. That makes me feel good! 

Good! It sounds like he's doing great. 

[Wheres Toto]

3 hours ago, Not_a_Number said:

You say "Where's Toto?", I say "Where's the apostrophe?".... 😉 

The latest board change evidently took away the ability to use punctuation in names.  I couldn't get it back at all for the longest time, when it finally worked, I wasn't going to quibble.  😁

[Clemsondana]

I read over this thread and put it out of my mind, but on Friday I had a strange experience with my younger that was directly relevant to how kid thinks about fractions and how thinking of it as division and not a specific number was, in this case, leading to unexpected confusion.  We are doing pre-A and had progressed to the 2x + 6 = 3X + 4 stage and were doing fine.  Then the book threw in problems like 2/3 x + 5 = 7/8  x + 1 .  Kiddo was fine dealing with the constants and then hit a roadblock at what to do with the variables...because if you have division then you can't just add and subtract the numbers from each other, and then there was some order of operations thinking where we'd need to divide first but how do you divide x by 8, but if you leave it as a fraction then it's a division problem.  We talked and used simple examples like 2x = 1/2 x + 6 and kid could easily see that you can do 2x - 1/2 x, and actually had done problems like 3/4 x -1/6 x before and was fine, but somehow combining it all was confusing.  I pointed out that 1/2, or 5/6, was just a number, and a number times x could, of course, be added and subtracted (8x - 5x, doesn't confuse kid).  Anyway, it was on point for this conversation!  

[Not_a_Number]

2 minutes ago, Clemsondana said:

I read over this thread and put it out of my mind, but on Friday I had a strange experience with my younger that was directly relevant to how kid thinks about fractions and how thinking of it as division and not a specific number was, in this case, leading to unexpected confusion.  We are doing pre-A and had progressed to the 2x + 6 = 3X + 4 stage and were doing fine.  Then the book threw in problems like 2/3 x + 5 = 7/8  x + 1 .  Kiddo was fine dealing with the constants and then hit a roadblock at what to do with the variables...because if you have division then you can't just add and subtract the numbers from each other, and then there was some order of operations thinking where we'd need to divide first but how do you divide x by 8, but if you leave it as a fraction then it's a division problem.  We talked and used simple examples like 2x = 1/2 x + 6 and kid could easily see that you can do 2x - 1/2 x, and actually had done problems like 3/4 x -1/6 x before and was fine, but somehow combining it all was confusing.  I pointed out that 1/2, or 5/6, was just a number, and a number times x could, of course, be added and subtracted (8x - 5x, doesn't confuse kid).  Anyway, it was on point for this conversation!  

Oh, that’s super interesting. We’ve always done fractions as division (that’s how I introduced them), but we haven’t had this issue.

I’m curious, why does thinking of it as division mean you don’t think of it as a number? Doesn’t one get a number as the outcome of any operation?

[Clemsondana]

I don't know - I mean, I'd kind of expect it to be thought of as a number not yet calculated maybe, and as a point on a number line would be OK, too, since it's a number, but this kid has always been weird with fractions even after taking an entire semester off to work on them.  But, part may be a personality thing, because as soon as kid sees something that doesn't immediately make sense kid becomes angry, which makes it harder to learn anything...for this kid, fractions may be caught in a loop of not fully learning because too angry to process it..But, kid also doesn't want to write 2/3 x - they want to write 2x/3, and although I've talked about them being the same thing and we can sub in values for x and show that it's the same, I think that takes some of the 'it's just a coeffecient' out of it - it looks like a complex problem that kid doesn't yet know how to solve.  I mean, they know that you could then multiply both sides and such, but then you're in the same boat once you multiply the constant and get another fraction that has a variable in it.  🙂   

[Not_a_Number]

5 minutes ago, Clemsondana said:

I don't know - I mean, I'd kind of expect it to be thought of as a number not yet calculated maybe, and as a point on a number line would be OK, too, since it's a number, but this kid has always been weird with fractions even after taking an entire semester off to work on them.  But, part may be a personality thing, because as soon as kid sees something that doesn't immediately make sense kid becomes angry, which makes it harder to learn anything...for this kid, fractions may be caught in a loop of not fully learning because too angry to process it..But, kid also doesn't want to write 2/3 x - they want to write 2x/3, and although I've talked about them being the same thing and we can sub in values for x and show that it's the same, I think that takes some of the 'it's just a coeffecient' out of it - it looks like a complex problem that kid doesn't yet know how to solve.  I mean, they know that you could then multiply both sides and such, but then you're in the same boat once you multiply the constant and get another fraction that has a variable in it.  🙂   

Oooh. I see how writing 2x/3 would make it much harder, actually. I don’t know if DD8 would think of that as a coefficient, either, to be honest... she might very well multiply both sides at that point, to clear the denominators. I would probably need to remind her that it’s the same thing, or she’d do a common denominator? 

Maybe I’ll give her this tomorrow, to see what she does! 

Was your younger kiddo introduced to fractions as division, or is that a later understanding?

ETA: this reminds me that I need to work on the associative property with division... well, I should show her how multiplying by the reciprocal lets us regroup things as multiplication.

Edited December 7, 2020 by Not_a_Number

[Clemsondana]

And today after some start of math drama, I wrote a page of problems with a mix of 2/3 x and 2x/3 notation and kid proceeded to covert everything to 2x/3 format and then solve everything correctly (well, there were a few careless mistakes, but no confusion).  And, when older heard us discussing why younger prefers 2x/3 notation older said that it was also their preferred method most of the time.  They don't use it when factoring, but do most of the rest of the time, or at least switch back and forth.  I've said before that I understand math at least through algebra well enough to be very comfortable teaching it, but my neither of my kids see it the same way that I do, or the same as each other.  

Oh, and to answer your earlier question, in the beginning I know that Singapore math taught 1/2 as one divided into 2 pieces, etc.  We spent some time drawing 3 pies divided into 4 parts and seeing that everybody got 3/4 of a pie, so the division problem yielded a fixed quantity.  We probably did it with blocks, too...it's hard to remember because it was a long time ago.  

[Not_a_Number]

9 minutes ago, Clemsondana said:

Oh, and to answer your earlier question, in the beginning I know that Singapore math taught 1/2 as one divided into 2 pieces, etc.  We spent some time drawing 3 pies divided into 4 parts and seeing that everybody got 3/4 of a pie, so the division problem yielded a fixed quantity.  We probably did it with blocks, too...it's hard to remember because it was a long time ago.  

That does sound like it's a division from the start, then. Interesting. 

 

10 minutes ago, Clemsondana said:

And today after some start of math drama, I wrote a page of problems with a mix of 2/3 x and 2x/3 notation and kid proceeded to covert everything to 2x/3 format and then solve everything correctly (well, there were a few careless mistakes, but no confusion).  And, when older heard us discussing why younger prefers 2x/3 notation older said that it was also their preferred method most of the time.  They don't use it when factoring, but do most of the rest of the time, or at least switch back and forth.  I've said before that I understand math at least through algebra well enough to be very comfortable teaching it, but my neither of my kids see it the same way that I do, or the same as each other.  

Interesting. I definitely have days with DD8 where it seems like she's forgotten a whole year's worth of material that day, lol... and then it just comes back. 

I use both notations interchangeably, I think. I really will give a problem in that format to DD8 tomorrow and see what happens 😄 . 

How do your kids see math differently from you and each other? My little one is too little to compare to the big one so far. They both seem mathy, and that's all I know so far. 

[Not_a_Number]

OK, @Clemsondana, I got impatient and just asked DD8 today 🙂 . 

I gave her 

1/2*x = 2x/5 + 3. 

She was a little bit confused by the two different formats using x on the two sides and multiplied by 2 first, getting 

x = 4x/5 + 6, 

then multiplied by 5, getting 

5x = 4x + 30,

then got x = 30.

I asked whether 2x/5 was the same thing as 2/5*x and she thought about it and said yes and explained why. (She said she was getting a fifth of one x from each x in 2x/5, and that wound up as 2/5*x.) Then I rewrote it as 

1/2*x = 2/5*x + 3

and asked her if she could do anything differently without first multiplying through and she had to think but said, yes, she could subtract 2/5x from both sides and that she had to use tenths, getting 

1/10*x = 3, 

and again, x = 30. 

So, she can do it either way, but I don't think she's yet internalized that you can treat 2x/5 as something times x. I'll have to give her some more problems like this, so she's flexible with the notation.