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Algebra Textbook for Younger Students

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7 minutes ago, mathmarm said:

Does you DD like manipulatives in math?

No, she never has. Visuals, yes, but not manipulatives. (And with fractions, she spurned visuals, too, and decided to talk every calculation out out loud. It was a little excruciating to listen to, to be honest, but it worked for her.) 

How about your son? 

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i have no real suggestions to offer.  I just wanted to join in and say I think you all are doing amazing things. My kid did Khan as his first pass through algebra because that’s what was available and has continued with that and added Life of Fred and AOPS but since I’ve never been his math teacher I have no idea if those were the right choices.

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22 hours ago, square_25 said:

No, she never has. Visuals, yes, but not manipulatives. (And with fractions, she spurned visuals, too, and decided to talk every calculation out out loud. It was a little excruciating to listen to, to be honest, but it worked for her.) 

How about your son? 

Yes, Jr. is very handsy. He typically likes to use manipulatives, but I have to be very conscientious about the manipulatives when we use them during a lesson or they just become distractions.

In general, I will teach with manipulatives and lots of discussion. We will work with the manipulatives a lot, talking through what's happening and why. Making observations. Writing down what we noticed. Posing questions and using the manipulatives to investigate them.

Even though we use manipulatives, I think it's the discussion that's key, because as he becomes "fluent" in the discussion of a topic, he's mastering the concept and internally dominating the ways it can be processed. Once Jrs reached the point that he's able to tell me why something happened in a previous step, or tell me what's going to happen after some step, then the visual will be mostly redundant. For some topics, I let him create the visuals when we make doodle-notes for the math. But we often go from manipulatives + disucssion, to written work.

Some times, as we go deeper into a concept or if he begins to stumble at some point doing the written work, I will use visuals in place of manipulatives. We draw it and discuss it and he's back on track quickly then. Some times if I've moved him to written work too quickly, then while he's still processing a concept, I will toggle between manipulatives and visuals to help talk him through what he's doing and eventually just go to only/mostly visuals.

When he is satisfied that he can get it right without the manipulatives--like, really, really sure that he knows it--he doesn't want to use the manipulatives anymore. 🤷‍♀️

We used manipulatives a lot in the early stages of graphing, but he's resisting the manipulatives for the graphs more now, even if he's stuck, he'll screw his eyes shot and imagine it.

Jr. didn't use visuals for fractions either. I just extended the concept of units, and the operations. We talked through them.
JR, understood the prerequisite concepts well enough that fractions took like...4 days once we begin formally covering them.

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1 hour ago, mathmarm said:

Yes, Jr. is very handsy. He typically likes to use manipulatives, but I have to be very conscientious about the manipulatives when we use them during a lesson or they just become distractions.

In general, I will teach with manipulatives and lots of discussion. We will work with the manipulatives a lot, talking through what's happening and why. Making observations. Writing down what we noticed. Posing questions and using the manipulatives to investigate them.

Even though we use manipulatives, I think it's the discussion that's key, because as he becomes "fluent" in the discussion of a topic, he's mastering the concept and internally dominating the ways it can be processed. Once Jrs reached the point that he's able to tell me why something happened in a previous step, or tell me what's going to happen after some step, then the visual will be mostly redundant. For some topics, I let him create the visuals when we make doodle-notes for the math. But we often go from manipulatives + disucssion, to written work.

Some times, as we go deeper into a concept or if he begins to stumble at some point doing the written work, I will use visuals in place of manipulatives. We draw it and discuss it and he's back on track quickly then. Some times if I've moved him to written work too quickly, then while he's still processing a concept, I will toggle between manipulatives and visuals to help talk him through what he's doing and eventually just go to only/mostly visuals.

When he is satisfied that he can get it right without the manipulatives--like, really, really sure that he knows it--he doesn't want to use the manipulatives anymore. 🤷‍♀️

We used manipulatives a lot in the early stages of graphing, but he's resisting the manipulatives for the graphs more now, even if he's stuck, he'll screw his eyes shot and imagine it.

Jr. didn't use visuals for fractions either. I just extended the concept of units, and the operations. We talked through them.
JR, understood the prerequisite concepts well enough that fractions took like...4 days once we begin formally covering them.

 

I also think the discussion is the most important part. That, and being patient while a kid integrates the knowledge in a good way. We always do written work along the way, though -- it's a format that comes easily to DD7. 

Fraction operations definitely took us a while. Or at least, it took DD7 a while to start applying shortcuts to the operations -- she could do the operations immediately, but she'd do it in a roundabout, slow way. 

I did try manipulatives when DD7 was younger, but she wasn't a fan. I've been using manipulatives (mostly poker chips, but also cards and other things one can count) in my homeschool math classes, so I'm not sure if I'll use them with DD3.75 or not... it'll probably depend a lot on what she wants. 

What kinds of manipulatives do you use for graphing? 

 

Edited by square_25

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On 3/29/2020 at 11:01 PM, square_25 said:

What kinds of manipulatives do you use for graphing? 

 

😞 I had a massive reply with color coding, indented lists and I lost it. I'm completely demoralized...

😔

The short answer is I used a wooden board and toys, then transitioned to a home made template on a whiteboard which was too slippery, so I finally landed on using a big XY-plane on cardboard. I got a lot of tips and suggestions from @Gil when I was starting out. If you want more detailed advice, you could reach out to him directly.

We played first with placing the toy and reading the coordinates.
Then placing the toy, reading the coordinates and then I would move the toy to a different place on the plane and JR would read the new coordinates and work out how the coordinates had changed and tell me what path(s) I took or could've taken to get there.
Another exercise was to place the toy, then I'd give him the new coordinates of where the toy should be and he'd work out how far left/right or up/down he needed to move the toy.

We talked about the "somersault shadow" also, and where the "somersault shadow" would be (reflections of the graph across an axis) and when we'd do the the What Was My Path version of the game,  I mixed in "somersault shadows" from time to time.

Later I transitioned to using cutouts of functions and we did the same thing. We just worked our way up to it.

In our pen-based math, we were working on functions a lot at that and I had him doing 1 or 2 graphs per day by transforming in stages. We use the pages with 4 seperate XYplanes on a single side. He'd pick a graph card and then we'd read the graph and say what kind of graph it was, what it looked like and how we knew. Even though we'd talked through the entire graph, I made him graph it in stages.

1st the parent function. On the 2nd XY plane, we'd say if it had wiggled away from "home base" left/right and graph that. Then we'd say if it had shimmied up or down from the main-line, and finally we'd say if it was fatter or thinner than usual based on the 4th XY-plane, we'd have the completed graph.

Flip the sheet of paper and do a second one.

From the math play and the pen-math exercises, when it was time to merge the steps, formally, the transition was very smooth. He deduced logically that the x +- h, notation would need to be different from the vertical shift notation.

It gave him a fantastic sense of spatial awareness for 2D graphs.

When he was comfortable with the basic functions, I introduced the math curves and as he was getting consistent with circles, I introduced the Trig functions. He loved graphing trig functions. Something about unwrapping a circle, and then "phase shifts" just absolutely tickles him. I think it sounds "cool". In hindsight, I should've introduced the word Transformations to him sooner.

This isn't nearly as good as my Lost Reply...

😪

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@square_25. We had our first day with Foersters.

We read through and discussed the first 3 lessons, because he didn't want to stop. When he picked what was important to remember, I had him do notes for 2 lessons, and apparently he did notes for the  3rd lesson all on his own. He's so proud of his math notes. Algebra 1 CH1 - JR.pdf

He understood but missed the very last problem that he did in 1-3. The instructions were to write the amount as a power and he didn't. I won't mention it directly, but I will highlight the habit responding to the question asked or the directions given if this pattern continues.

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2 minutes ago, mathmarm said:

@square_25. We had our first day with Foersters.

We read through and discussed the first 3 lessons, because he didn't want to stop. When he picked what was important to remember, I had him do notes for 2 lessons, and apparently he did notes for the  3rd lesson all on his own. He's so proud of his math notes. Algebra 1 CH1 - JR.pdf

He understood but missed the very last problem that he did in 1-3. The instructions were to write the amount as a power and he didn't. I won't mention it directly, but I will highlight the habit responding to the question asked or the directions given if this pattern continues.

Is that his handwriting? It's lovely! 

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1 minute ago, square_25 said:

Is that his handwriting? It's lovely! 

Thank you! Yes it is.

I wouldn't have termed it lovely, 😊 !

He was a real wiggle worm today so it is wobbly-er than I like it to be.

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#45 for 1-3 got out of control. He was writing all over the page--he knows that's a no-no!  😏

He told me "It's sloppy but I checked it!" How can you be mad at a kid who checks their math? Answer: You can't be.

The answer was supposed to be written as a power so he should've written just 2^18, but he missed that and worked it out long hand.

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35 minutes ago, mathmarm said:

Thank you! Yes it is.

I wouldn't have termed it lovely, 😊 !

He was a real wiggle worm today so it is wobbly-er than I like it to be.

No, I think it's awesome for his age :-). Very clear and legible. 

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If you have the energy, I'd LOVE to see you post the notes from the lessons, at least occasionally. Was this stuff he knew already or new stuff? (I'm guessing he knew most of it.) 

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33 minutes ago, mathmarm said:

#45 for 1-3 got out of control. He was writing all over the page--he knows that's a no-no!  😏

He told me "It's sloppy but I checked it!" How can you be mad at a kid who checks their math? Answer: You can't be.

The answer was supposed to be written as a power so he should've written just 2^18, but he missed that and worked it out long hand.

 

Hah, you don't want to see my notes to self from my math olympiad (and frankly, my Ph.D) days. They would go ALL OVER the page. Lots would be sideways. A few would be upside down ;-). 

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32 minutes ago, square_25 said:

If you have the energy, I'd LOVE to see you post the notes from the lessons, at least occasionally. Was this stuff he knew already or new stuff? (I'm guessing he knew most of it.) 

Well, I'll probably be home a lot more for the next few months so I will try and update the notes every so often.

Yes, this was material that he has seen similar to before (even though its been a while for some things) and we read the book together and discussed the lessons we did many problems orally just snuggled on the couch together.

 

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16 minutes ago, mathmarm said:

Well, I'll probably be home a lot more for the next few months so I will try and update the notes every so often.

Yes, this was material that he has seen similar to before (even though its been a while for some things) and we read the book together and discussed the lessons we did many problems orally just snuggled on the couch together.

 

Weirdly enough, we were doing exponents today, too... 

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17 minutes ago, square_25 said:

Weirdly enough, we were doing exponents today, too... 

Oh? What was the lesson?

How'd it go?

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3 minutes ago, mathmarm said:

Oh? What was the lesson?

How'd it go?

It went well! We’re working on writing short proofs, so that’s what we focused on. She doesn’t know properties of exponents yet, so we’re exploring that at the same time.

19E026A9-68F8-49A5-BEA9-F5D78C78C758.jpeg

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I think that the exponent properties make for good first proofs. They are easy for young kids to experience by using the definitions themselves.

Exponentiation is often less familiar than addition/multiplication, so the properties aren't so obvious you can't tell the point of what you're doing by manipulating them.

Jr. had a lot of fun with exponents back when we did them.

Today, when he saw the lesson on exponents he grinned and said "yay! Exponents!"

 

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1 minute ago, mathmarm said:

I think that the exponent properties make for good first proofs. They are easy for young kids to experience by using the definitions themselves.

Exponentiation is often less familiar than addition/multiplication, so the properties aren't so obvious you can't tell the point of what you're doing by manipulating them.

Jr. had a lot of fun with exponents back when we did them.

Today, when he saw the lesson on exponents he grinned and said "yay! Exponents!"

 

 

Yeah, I agree :-). Our first proofs were really solving factorable quadratics, but this seems like a good avenue, given that I do think that proving things like the distributive property can be kind of unsatisfying. It just kind of... is :-P. Either you find it really intuitive or you shouldn't be proving it. 

She's been able to explain things for a while in words, but her writing only got fast and efficient enough recently to be able to write proofs (she lost serious writing ground when I sent her to public kindergarten, grr.) So I'm excited to work on them with her! They still take a while for her to phrase correctly... I think that page took like an hour for her to put together, with input! But I was happy with her overall. 

We've also been taking notes on viruses: that's our note-taking exercise right now. That's been really fun, too. (And surprisingly enough, something we've been meaning to do for ages, not something that she became interested in due to not being able to leave the house...) 

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18 minutes ago, square_25 said:

(she lost serious writing ground when I sent her to public kindergarten, grr.)

Fortunately we weren't able to find a suitable academic setting for Jr.

Hubby was telling me recently that he believes that us not finding a school for him was one of the main reasons that JRs core skills are so strong--that their development wasn't disrupted by school. He wasn't taught or allowed psuedo-alternatives or destructive practices like sight words.

He learnes--and practices--writing, reading and drawing without the disruption of disjointed/contrary instruction.

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26 minutes ago, square_25 said:

So I'm excited to work on them with her! They still take a while for her to phrase correctly... I think that page took like an hour for her to put together, with input! But I was happy with her overall.

That's impressive that she was able to focus for an hour. Do you trained and developed her focus intentionally, or is that a personality thing?

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5 minutes ago, mathmarm said:

Fortunately we weren't able to find a suitable academic setting for Jr.

Hubby was telling me recently that he believes that us not finding a school for him was one of the main reasons that JRs core skills are so strong--that their development wasn't disrupted by school. He wasn't taught or allowed psuedo-alternatives or destructive practices like sight words.

He learnes--and practices--writing, reading and drawing without the disruption of disjointed/contrary instruction.

 

She was luckily already a very fluent reader and good at math, so those didn't get disrupted. But her writing was still starting out, and that got seriously set back. She came into kindergarten forming letters quite well and came out writing some of them backwards, mixing uppercase and lowercase letters, and also hating to write. They did a TON of writing, which burned her out, and refused to check her work at all, which both stressed her out and led to the deterioration of her handwriting. 

I was pretty mad. But we knew about mid-year we were going to pull her out, so at least I didn't feel desperate. 

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1 minute ago, mathmarm said:

That's impressive that she was able to focus for an hour. Do you trained and developed her focus intentionally, or is that a personality thing?

 

She's an extremely focused child naturally. She did have to come and talk to me a few times, but she CAN focus for an hour. We've done practice Math Kangaroos this year, and she's totally willing to write one for 75 minutes by herself in the corner of the classroom in which I teach other kids without bothering me. 

I'd like to take credit but I really can't ;-). 

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So, I have a quandary: What to do about the quadratic equation?

It's in Ch. 6 of my text, but I don't like the way it's introduced.
The book does guide the students to explore it and explain it more through out the chapter. But I just don't like the texts coverage of it and their introduction is just so...uninspired.

I will be reading Ch6 of the text more closely over the next few days. I don't know that I want to use Ch6 as it's written. I may write some alternative lessons for that chapter.

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8 minutes ago, mathmarm said:

So, I have a quandary: What to do about the quadratic equation?

It's in Ch. 6 of my text, but I don't like the way it's introduced.
The book does guide the students to explore it and explain it more through out the chapter. But I just don't like the texts coverage of it and their introduction is just so...uninspired.

I will be reading Ch6 of the text more closely over the next few days. I don't know that I want to use Ch6 as it's written. I may write some alternative lessons for that chapter.

I know nothing about math other than what I have taught my kids over the yrs, but I do like the way the quadratic equation is introduced.  IIRC, the equation is introduced as a formula at the beginning of the chpt so that they start memorizing it as a formula, but they do not use it at all.  He spends the rest of the chpt introducing the concepts that lead to being able to derive the equation.  My kids have all been able to write a proof deriving the equation by the end of the chpt, so for my kids it has worked well in solidifying their understanding of it.

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31 minutes ago, mathmarm said:

So, I have a quandary: What to do about the quadratic equation?

It's in Ch. 6 of my text, but I don't like the way it's introduced.
The book does guide the students to explore it and explain it more through out the chapter. But I just don't like the texts coverage of it and their introduction is just so...uninspired.

I will be reading Ch6 of the text more closely over the next few days. I don't know that I want to use Ch6 as it's written. I may write some alternative lessons for that chapter.

What don’t you like about it? I’ll give you suggestions if you like!

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3 hours ago, 8FillTheHeart said:

I know nothing about math other than what I have taught my kids over the yrs, but I do like the way the quadratic equation is introduced.  IIRC, the equation is introduced as a formula at the beginning of the chpt so that they start memorizing it as a formula, but they do not use it at all.  He spends the rest of the chpt introducing the concepts that lead to being able to derive the equation.  My kids have all been able to write a proof deriving the equation by the end of the chpt, so for my kids it has worked well in solidifying their understanding of it.

Yep. That's about it. I don't like that it's just plopped down for them to memorize without wider context, but I do like that the book spends the next several lessons developing the ideas needed to derive it/understand the derivation of it and that the book highlights each connection as it crops up. I am not sure how I feel that they just keep asking the kids to regurgitate the QF from memory (it's in the exercises for each lesson in Ch6  But that could just me being picky or hard to please.

6-1 is the shortest lesson so far. About 1/2 a page.
It's pretty much just "This is the Quadratic Formula. Learn to read it and say it. Here is a strategy to help you memorize it."

6-2 is all about evaluating radicals and exploring the root properties for + and -, * and /.
They lean really heavily on the calculator for this lesson, but the exercises are well thought out. We can get around this by reasoning with properties and factoring the radicands then using a reference chart to calculate the square roots to two decimal places as requested. They call on the student to produce the QF from memory.

6-3 they formalize the whole concept of absolute value. They introduce the +- notation and note it's appearance in the QF.
They call on the student to produce the QF from memory. I do like the problem sets. I already know which ones are "can't skip" for us.

6-4 They connect absolute value and squares. They make the connection between absolute value and principle or positive square roots of expressions.
They now expect solution sets, instead of just "a solution" so that is good. (Note to self: Teach some introductory set theory before this lesson). They call on the student to produce the QF from memory. The problem sets are solid, and the final question is a discussion question.

6-5 This lesson is all about making the connection between trinomials (extensively covered in Ch5)  and quadratics. This is absolutely a stepping stone lesson because it bridges very smoothly to the next lesson. They call on the student to produce the QF from memory.

6-6 They work with their base in trinomials and begin completing the square. The exercises for this section are very sound and I like them. They call on the student to produce the QF from memory.

6-7  They solve quadratic equations by completing the square. I like how they've built up to this through 3 lessons and this section see's  the return of interesting problems to discuss. They call on the student to produce the QF from memory.

6-8 By this point they've explored all of the fundamental ideas used in proving the QF, so they are told that they can now learn what it means and why it works so that they can use the formula to solve quadratics, the final exercise in this section is deriving the quadratic formula from ax^2 + bx + c = 0. By now the student should have a good idea of what they're trying to prove and should have both some intuition and some insight to guide them through the whole process.

6-9 They get to use the QF to work out expressions, equations and solutions for 18 multipart scenarios/word problems using vertical motion. 

6-10 Guides the student to consider the discriminant of a QE and what the discriminant can tell you about the QE itself

6-11 is the Chapter Review and Test.

6-12 is a cumulative review for Ch1-6.

This is NOT a weak chapter. The development of the  QF taken as a whole is good--or even very good.
But there is *something* about it, that I can't quite put my finger on--that I just don't love.

This is a really good text. But there is something about this chapter that just doesn't...pop for me.

Edited by mathmarm

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36 minutes ago, mathmarm said:

Yep. That's about it. I don't like that it's just plopped down for them to memorize without wider context, but I do like that the book spends the next several lessons developing the ideas needed to derive it/understand the derivation of it and that the book highlights each connection as it crops up. I am not sure how I feel that they just keep asking the kids to regurgitate the QF from memory (it's in the exercises for each lesson in Ch6  But that could just me being picky or hard to please.

6-1 is the shortest lesson so far. About 1/2 a page.
It's pretty much just "This is the Quadratic Formula. Learn to read it and say it. Here is a strategy to help you memorize it."

6-2 is all about evaluating radicals and exploring the root properties for + and -, * and /.
They lean really heavily on the calculator for this lesson, but the exercises are well thought out. We can get around this by reasoning with properties and factoring the radicands then using a reference chart to calculate the square roots to two decimal places as requested. They call on the student to produce the QF from memory.

6-3 they formalize the whole concept of absolute value. They introduce the +- notation and not it's appearance in the QF.
They call on the student to produce the QF from memory. I do like the problem sets. I already know which ones are "can't skip" for us.

6-4 They connect absolute value and squares. They make the connection between absolute value and principle or positive square roots of expressions.
They now expect solution sets, instead of just "a solution" so that is good. (Note to self: Teach some introductory set theory before this lesson). They call on the student to produce the QF from memory. The problem sets are solid, and the final question is a discussion question.

6-5 This lesson is all about making the connection between trinomials (extensively covered in Ch5)  and quadratics. This is absolutely a stepping stone lesson because it bridges very smoothly to the next lesson. They call on the student to produce the QF from memory.

6-6 They work with their base in trinomials and begin completing the square. The exercises for this section are very sound and I like them. They call on the student to produce the QF from memory.

6-7  They solve quadratic equations by completing the square. I like how they've built up to this through 3 lessons and this section see's  the return of interesting problems to discuss. They call on the student to produce the QF from memory.

6-8 By this point they've explored all of the fundamental ideas used in proving the QF, so they are told that they can now learn what it means and why it works so that they can use the formula to solve quadratics, the final exercise in this section is deriving the quadratic formula from ax^2 + bx + c = 0. By now the student should have a good idea of what they're trying to prove and should have both some intuition and some insight to guide them through the whole process.

6-9 They get to use the QF to work out expressions, equations and solutions for 18 multipart scenarios/word problems using vertical motion. 

6-10 Guides the student to consider the discriminant of a QE and what the discriminant can tell you about the QE itself

6-11 is the Chapter Review and Test.

6-12 is a cumulative review for Ch1-6.

This is NOT a weak chapter. The development of the  QF taken as a whole is good--or even very good.
But there is *something* about it, that I can't quite put my finger on--that I just don't love.

This is a really good text. But there is something about this chapter that just doesn't...pop for me.

 

I have never, ever gotten good results out of teaching a formula first and then explaining it to kids. I've experimented with this at AoPS -- if you start with the formula, you run a much higher chance of someone treating something like a black box they not only will not understand, but will not really feel motivated to understand. 

I would probably work up to this in the opposite order -- I'd start with solving equations like (x+2)^2 = 9 (weirdly enough, that's what I did today with DD7, lol), and then I'd move to completing the square and finally to the quadratic formula. Setting things up so that things are motivated will lead to, yes, slower memorization of the quadratic formula, but a very minimal chance that someone will be unable to rederive it. Starting with the formula first will lead to a higher chance that you will need to seriously remediate something that has become a black box. 

(I found these results fascinating when I saw them in my classes, by the way. I always thought it'd be fine to rigorously introduce the formula, use the formula, and remind kids where it came from along the way to keep them functioning in a logical way. However, it turned out that once most kids thought of something as a formula, their brains shut off to other understandings of it for a good long while. Somehow, the formula became primary. Probably once the formula becomes fully internalized and part of long-term memory, one should be able to go back and really, honest-to-goodness explain it. But I'm skeptical that you'd be able to switch from a procedural to a logical orientation by the end of the chapter, even with an explanation.) 

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1 hour ago, square_25 said:

 

I have never, ever gotten good results out of teaching a formula first and then explaining it to kids. I've experimented with this at AoPS -- if you start with the formula, you run a much higher chance of someone treating something like a black box they not only will not understand, but will not really feel motivated to understand. 

I would probably work up to this in the opposite order -- I'd start with solving equations like (x+2)^2 = 9 (weirdly enough, that's what I did today with DD7, lol), and then I'd move to completing the square and finally to the quadratic formula. Setting things up so that things are motivated will lead to, yes, slower memorization of the quadratic formula, but a very minimal chance that someone will be unable to rederive it. Starting with the formula first will lead to a higher chance that you will need to seriously remediate something that has become a black box. 

(I found these results fascinating when I saw them in my classes, by the way. I always thought it'd be fine to rigorously introduce the formula, use the formula, and remind kids where it came from along the way to keep them functioning in a logical way. However, it turned out that once most kids thought of something as a formula, their brains shut off to other understandings of it for a good long while. Somehow, the formula became primary. Probably once the formula becomes fully internalized and part of long-term memory, one should be able to go back and really, honest-to-goodness explain it. But I'm skeptical that you'd be able to switch from a procedural to a logical orientation by the end of the chapter, even with an explanation.) 

I have had excellent outcomes with mathematical understanding in terms using  nothing but Foerster's methodology with my 6 kids that have moved on from alg to cal+. (7th is in alg 1 right now.) So, I know it teaches well bc my kids have had nothing but Foerster's since I dont have the ability to teach it any other way. So, it is hardly a completely flawed approach.

Kids do nothing with the formula other than write it down until they know how to derive it. It isnt memorized and followed by simply explaining how it works. The formula is simply presented as a formula to memorize and then students spend 6 sections working through step by step processes until they encounter a lesson that explains now they are ready to understand what it is and why it works.

It would be very easy to skip the 1 question in the lesson that tells them to write it from memory until after they have learned to derive it. But while they dont use the equation until the later lesson bc he does explain how the step by step processes look like parts of the equation.

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1 hour ago, square_25 said:

I have never, ever gotten good results out of teaching a formula first and then explaining it to kids. I've experimented with this at AoPS -- if you start with the formula, you run a much higher chance of someone treating something like a black box they not only will not understand, but will not really feel motivated to understand.

I'm not super-worried about this. The thing about teaching 1-1 day in and day out, is that I can do both to whatever degree is needed, simultaneously.

I'm not about to pretend I've never presented a formula first, then worked on the derivation of it. I've done it dozens of times.

No, I can't quite put my finger on what it is about Ch.6 that's bothering me. But it's not strictly their approach to the QF.

 

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5 minutes ago, mathmarm said:

I'm not super-worried about this. The thing about teaching 1-1 day in and day out, is that I can do both to whatever degree is needed, simultaneously.

I'm not about to pretend I've never presented a formula first, then worked on the derivation of it. I've done it dozens of times.

No, I can't quite put my finger on what it is about Ch.6 that's bothering me. But it's not strictly their approach to the QF.

 

I'm sure if you're working one-on-one, it matters a lot less :-). It was just a surprising observation of a classroom experience. It definitely explained some things that had nagged at me from earlier teaching experience -- namely, feeling like I explained things very clearly and thoroughly, and yet the kids would still in some fundamental way not be engaging with the ideas. But the ability to seriously redirect and explain is, of course, orders of magnitude bigger one-on-one, and your son sounds very mathematically fluent! So I'm sure it's fine. 

Any idea what bothered you about it? Again, I can only comment on my experience: for me, memorizing the formula would be a non-starter. I would also want to spend time on absolute value aside from square roots, as it's a concept and requires internalizing. Same with properties of radicals -- unless your son is already fluent with the properties of radicals, I'd let him play with them first, so the ideas feel integrated. (But it's possible he already understands this well. I know my daughter does not.) I would almost certainly spend time solving quadratics in a wide variety of ways BEFORE getting to the quadratic formula (completing the squares, spotting the answer, factoring, etc.) so that a kid wouldn't get too stuck on the formula as the one and only way to do it. 

Is any of this getting at the kind of thing that's bothering you? Or is it something entirely different? 

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21 hours ago, square_25 said:

 I would also want to spend time on absolute value aside from square roots, as it's a concept and requires internalizing. Same with properties of radicals -- unless your son is already fluent with the properties of radicals,

I only gave a rough over-view of Ch6, but keep in mind that I'm talking about the sixth chapter in this text.
In the text, Exponents are taught in Ch-1, Absolute value in Ch 2-2,  and Radicals are tuaght in Ch 5. So the students will have seen and worked with AV and radicals before being reminded of it in Ch6.
By the time that they see these things in Ch 6, students are recalling and extending something that they've already been exposed to and practiced.

In Ch6 they formalize the AV concept with a more precise definition and use radicals it in a way that will tie in with students ability to understand its use in the derivation of the QF by chapters end.

From what I've seen, the text has already introduced, explained and practiced the student with the prerequisite skills.

The book is well written and conceptually sturdy. It's scope and sequence might not be 100% perfect according to me, but it's a strong text or I wouldn't have ever bought it and I certainly wouldn't have started it with my Big Boy if it wasn't "good enough" for him 🙂

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14 minutes ago, mathmarm said:

I only gave a rough over-view of Ch6, but keep in mind that I'm talking about the sixth chapter in this text.
In the text, Exponents are taught in Ch-1, Absolute value in Ch 2-2,  and Radicals are tuaght in Ch 5. So the students will have seen and worked with AV and radicals before being reminded of it in Ch6.
By the time that they see these things in Ch 6, students are recalling and extending something that they've already been exposed to and practiced.

In Ch6 they formalize the AV concept with a more precise definition and use radicals it in a way that will tie in with students ability to understand its use in the derivation of the QF by chapters end.

From what I've seen, the text has already introduced, explained and practiced the student with the prerequisite skills.

The book is well written and conceptually sturdy. It's scope and sequence might not be 100% perfect according to me, but it's a strong text or I wouldn't have ever bought it and I certainly wouldn't have started it with my Big Boy if it wasn't "good enough" for him 🙂

Sorry, I’m just trying to go off of what you wrote — I don’t know the book!! I don’t know what is bugging you about the chapter, so I’m just taking shots in the dark. If you figure out what it is, I’d be happy to troubleshoot :-).

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28 minutes ago, square_25 said:

Sorry, I’m just trying to go off of what you wrote — I don’t know the book!! I don’t know what is bugging you about the chapter, so I’m just taking shots in the dark. If you figure out what it is, I’d be happy to troubleshoot :-).

I was just addressing that particular point, because I know other people will see these messages at some point. I am just writing these notes real time and on the fly, so they aren't coherent or comprehensive but since I'm making them public, I need to address (potential) concerns as they are raised or it won't be a good review of the book for others reference.

However, you are fine so don't apologize so much.

 

 

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1 minute ago, mathmarm said:

I was just addressing that particular point, because I know other people will see these messages at some point. I am just writing these notes real time and on the fly, so they aren't coherent or comprehensive but since I'm making them public, I need to address (potential) concerns as they are raised or it won't be a good review of the book for others reference.

However, you are fine so don't apologize so much.

Anyway, let me know if there's anything I can help troubleshoot. 

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We're only a few days in, but I"m so happy with our book choice thus far. Jr is still enthusiastic about it, which is very important.

Jr. loves to write and is loving having a real math book. These are his math notes for the last 3 lessons. Algebra 1 CH1 - JR_2.pdf
I'm going to ignore the doodle (which he knows definitely isn't supposed to be in this book 🙁 when he has tons of drawing papers).

For now our approach of buddy reading and discussing with a board near by is working. He's loving having to write the math notes--that might be his favorite part. I only allow him to write a few. For 1-6, I assigned 4, but he snuck in some extra because he wanted to.

So far, we buddy read and discuss the lesson, then we answer all of the lesson problems, but we discuss them and he does them orally. Today he asked that we not orally answer the problems he's going to write before he writes them. So moving forward we will give that a try.

 

 

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39 minutes ago, square_25 said:

What are you guys working on now, by the way? 🙂

He's loving everything about using the math book on the couch, then making notes.  I think that aside from giving him more "writing homework" it feels very "big kid" to him, He's asked for "writing homework" several times in the past but even though I gave him some, we've always kept his writing "homework" to a minimum. We want him to write in his own time but I guess making up something to write is a struggle for him still? He draws much more freely than he writes.

I'm torn between letting him go at his pace vs pacing him.

I think I will be stricter with pacing the rate at which he writes for Chapter 2.

Outside of the book, we have other math strands that we practice regularly. I'm going to introduce and begin teaching some set theory soonish.

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9 minutes ago, mathmarm said:

We want him to write in his own time but I guess making up something to write is a struggle for him still? He draws much more freely than he writes.

We've never had much luck with writing on our time, but then I have a very linear kid. She writes much better for me with assignments, and she found "free writing" in kindergarten akin to torture. 

I imagine he'll slow down once he hits stuff he doesn't know yet? Are you still going over stuff he basically knows? 

10 minutes ago, mathmarm said:

Outside of the book, we have other math strands that we practice regularly. I'm going to introduce and begin teaching some set theory soonish

 

Set theory is fun! Thanks for the reminder -- we haven't done it, and we probably should at some point. 

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1 hour ago, mathmarm said:

We're only a few days in, but I"m so happy with our book choice thus far. Jr is still enthusiastic about it, which is very important.

Jr. loves to write and is loving having a real math book. These are his math notes for the last 3 lessons. Algebra 1 CH1 - JR_2.pdf
I'm going to ignore the doodle (which he knows definitely isn't supposed to be in this book 🙁 when he has tons of drawing papers).

For now our approach of buddy reading and discussing with a board near by is working. He's loving having to write the math notes--that might be his favorite part. I only allow him to write a few. For 1-6, I assigned 4, but he snuck in some extra because he wanted to.

So far, we buddy read and discuss the lesson, then we answer all of the lesson problems, but we discuss them and he does them orally. Today he asked that we not orally answer the problems he's going to write before he writes them. So moving forward we will give that a try.

 

Didn't see this before I commented :-). So I'd assume this is stuff he knows already, then, from stuff you've said. 

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WHOOPS NEVER MIND!

Edited by mathmarm
Nevermind! I was wrong, wrong, wrong!
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Ok! So I grabbed the book and looked again after posting my previous post and I was super wrong :blush:.

If you saw what the other post used to say, please ignore it.

I checked the book and Chapter 6 Forester Algebra 1 includes plenty of discussion worthy problems. Some kind of way, I was just missing them!

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1 minute ago, mathmarm said:

Ok! So I grabbed the book and looked again after posting my previous post and I was super wrong :blush:.

If you saw what the other post used to say, please ignore it.

I checked the book and Chapter 6 Forester Algebra 1 includes plenty of discussion worthy problems. Some kind of way, I was just missing them!

 

Hah, then I won't continue with the list I was typing! What kinds of problems look discussion-worthy to you? 🙂 I think we'll get into the quadratic formula relatively soon (we're about to start solving quadratics by completing the square), so as usual, I'm on the lookout for fun ideas. 

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1 minute ago, square_25 said:

 

Hah, then I won't continue with the list I was typing!

By all means, please do!

Edited by mathmarm

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Just now, mathmarm said:

By all means, please do!

 

Good thing I still had it copied, lol! 

I'm going to brainstorm bit by bit, so I don't type up long lists of stuff that don't actually work :-). I think problems involving graphing can be really nice for getting a holistic visual of quadratic functions, so that can be nice. So, say, studying what you know about the equation if, say, you only know the distance between roots on the x-axis, or if you only know something about the vertex could be interesting. 

In my experience, absolute values are actually kind of tricky even for mathy kids, so a fun thing to do there is to graph the absolute value function and also compositions of it with all sorts of other functions. Say, graph f(|x|) and |f(x)| and whatnot for a variety of functions f. You can also engage with the absolute value as the distance between numbers on the number line, although maybe that's something your son knows already :-). 

To me, personally, the formula itself is not super interesting, to be honest -- I have it memorized and that's about how I engage with it. However, I think the method of solving quadratics is interesting! Maybe some historical context here would be cool -- how the cubic formula was hard to find (you could show the derivation, if you feel ambitious!), the quartic formula was even harder, and the quintic formula turns out to be impossible! 

Is any of this interesting at all to you? I'd love some guidance about what kinds of questions you like! 

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4 hours ago, square_25 said:
 

Good thing I still had it copied, lol!  😄

 

Is any of this interesting at all to you? I'd love some guidance about what kinds of questions you like! 

Yes, keep going.

We've got a whiteboard going about finding zeros of a functions. It's an exploration that grew out of our daily graphing activities.

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1 minute ago, mathmarm said:

Yes, keep going.

We've got a whiteboard going about finding zeros of a functions. It's an exploration that grew out of our daily graphing activities.

Hahahah, I can go on forever! But are any of those more appealing than others, just so my internal machine learning algorithms can have some data? 😉

Edited by square_25

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4 hours ago, square_25 said:
I'd love some guidance about what kinds of questions you like! 

Questions that prompt a second-look or a deeper look at a concept. They don't have to be super-difficult questions--in fact, I don't want them to be hard questions, just thoughtful questions.

Questions that--when you think about and discuss them, they help you flesh out your understanding of a topic or they help you realize something else.

For example, in the lesson that introduces integers (2-1) there are the questions:

(32) Adding a negative number to another number is the same what other operation.
(33) What is meant by
- a negative number,  
- an integer, 
-a real number?

In 2-3 (subtracting signed numbers) they have the practice problems where you simply tell if the parenthesis are needed in each case as a part of the oral drill.
This is an intelligent exercise that I've not seen in many remedial or even college algebra textbooks. The thoughtfulness of the exercises inclusion is impressive, because I've had tons of students who never know if they're supposed to use parenthesis or not.

In 2-4 (multipltication of signed numbers) these are 2 of the questions:
(51) From the answers to Problems 47-50, you can see that -x and -1*x always stand for the same number. True or false: Does the expression -x always stand for a negative number? Explain.
(52) Explain the difference in meaning between -x^2 and (-x)^2.

 

All of these questions are simple, but thoughtfully included. Rather than just can you compute this it's inviting thought. It's requiring the student to pause, reflect on what they've been doing to pick up on something that they may or may not have even noticed in the first place.

Edited by mathmarm
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In lesson 3-3 they have:
(63) Tell the difference between an axiom and any other property.

In 5-2 they introduce a formal definition of polynomials have:
(57) The expression 7 / (x-5) is not a polynomial, since it involves division by an expression containing a variable.
For what value of x is the expression not equal to a real number? Explain.

These are not hard questions. But they encourage thoughtfulness.Their inclusion in the text helps me remember any points that I might want to make. Some of these questions are obviously designed to even preempt students from making bad turns and ending up in common pitfalls that students find themselves in.

I like to collect these types of thinking questions, because it could be the sort of thing that I might take for granted or forget to ask, but by asking and exploring it a few times with Jr. I'm able to help him grow or reinforce his understanding of the concept.

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Hmmmm, I assume they already talk about when the quadratic formula yields equal roots, when it yields different roots, and when it yields no roots? That's another fun and not too tricky thing to explore. And you could do some examples. 

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