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mathmarm

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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