Rapidly catching up in math (advice?)

[Eilonwy]

2 minutes ago, pinball said:

With that in mind, I’d like to suggest triangle flash cards, done daily for both addition/subtraction and then multi/division.

How do triangle flash cards work? Or, how do you use them?

[Not_a_Number]

1 minute ago, Eilonwy said:

How do triangle flash cards work? Or, how do you use them?

I'd assume it's the usual "number bond" thing. Personally, I tend to spend quite a lot of time on WHY those are the same thing (like, why 12 - 5 = 7 is the same bit of information as 5+7 = 12) before I drill it, and then I only drill the additions and use them to have kids CHECK their subtractions. 

Edited April 17, 2021 by Not_a_Number

[pinball]

3 minutes ago, Eilonwy said:

How do triangle flash cards work? Or, how do you use them?

I’m linkin’ cuz I’m lazy, LOL

 

[Eilonwy]

Just now, Not_a_Number said:

I'd assume it's the usual "number bond" thing. Personally, I tend to spend quite a lot of time on WHY those are the same thing (like, why 12 - 5 = 7 is the same bit of information as 5+7 = 12) before I drill it, and then I only drill the additions and use them to have kids CHECK their subtractions. 

Yes, looks like you’re right.  I looked them up, and I think you cover up a corner, and then you’re using number bonds.  I think this would work better if they understand why, like you said.  I am asking my kids to “undo”operations and explain why it works, and their explanations are getting better. 

 

2 minutes ago, pinball said:

I’m linkin’ cuz I’m lazy, LOL

Thanks, a video is worth 1005 words, at least!

[Not_a_Number]

1 minute ago, Eilonwy said:

Yes, looks like you’re right.  I looked them up, and I think you cover up a corner, and then you’re using number bonds.  I think this would work better if they understand why, like you said.  I am asking my kids to “undo”operations and explain why it works, and their explanations are getting better. 

Exactly. Otherwise, kids tend to forget "which way" to use the operation. As is, half the kids I work with think that 8-12 = 4, because why not? 

[Sk8ermaiden]

Those triangles look like a flashcard form of the "number families" Singapore teaches for +/-, and I teach again for multiplication and division. We just draw them on the paper though. I did plan to teach him number families, especially as a problem solving strategy (even though his basic arithmetic is strong, he's missing questions like 852-?=262, and 864/?=8 so he's never been taught the relationship between addition/subtraction and multiplication/division or how to use it to solve problems) and I plan to go all the way back to unit cubes to teach it. Everyone loves unit cubes, right? I even pull them out with my 6th grader sometimes when I think she's not getting the relationship between the algorithm and real life. 

I am not going to just shove him along to pass the test though. He is adamant that he wants to be a marine biologist when he grows up, and that of course will require upper level math and sciences with math. I'm going to tell mom that we will go exactly as fast as he can understand. If that doesn't get him to Saxon 7/6, then I will commit to continuing this next year to get him there. With a whole year there's no question he can get there, and in fact, DD and I find the placement questions showing mastery of 8/7 WAY easier than the ones that show mastery of 7/6. I don't know why, but if he does as well, he could potentially test into Algebra 1/2 (Saxon pre-a) after another year. I don't know. I can't see passing the 7/6 portion of the test and not the 8/7, because it's so much more basic.

[Not_a_Number]

4 minutes ago, Sk8ermaiden said:

Those triangles look like a flashcard form of the "number families" Singapore teaches for +/-, and I teach again for multiplication and division. We just draw them on the paper though. I did plan to teach him number families, especially as a problem solving strategy (even though his basic arithmetic is strong, he's missing questions like 852-?=262, and 864/?=8 so he's never been taught the relationship between addition/subtraction and multiplication/division or how to use it to solve problems) and I plan to go all the way back to unit cubes to teach it. Everyone loves unit cubes, right? I even pull them out with my 6th grader sometimes when I think she's not getting the relationship between the algorithm and real life. 

I love unit cubes, although I love place value poker chips even more 😉 . I'd do questions like that without the "inverse relationship" to start with, so he understands how they work. Then I'd introduce what you're talking about. 

 

4 minutes ago, Sk8ermaiden said:

I am not going to just shove him along to pass the test though. He is adamant that he wants to be a marine biologist when he grows up, and that of course will require upper level math and sciences with math. I'm going to tell mom that we will go exactly as fast as he can understand. If that doesn't get him to Saxon 7/6, then I will commit to continuing this next year to get him there. With a whole year there's no question he can get there, and in fact, DD and I find the placement questions showing mastery of 8/7 WAY easier than the ones that show mastery of 7/6. I don't know why, but if he does as well, he could potentially test into Algebra 1/2 (Saxon pre-a) after another year. I don't know. I can't see passing the 7/6 portion of the test and not the 8/7, because it's so much more basic.

I think you're exactly right. And you're doing a wonderful thing for this family. 

[Eilonwy]

On 4/17/2021 at 3:58 PM, Not_a_Number said:

Exactly. Otherwise, kids tend to forget "which way" to use the operation. As is, half the kids I work with think that 8-12 = 4, because why not? 

Yes, it could actually reinforce this issue if they didn’t have a good understanding to start with.

23 hours ago, Sk8ermaiden said:

I am not going to just shove him along to pass the test though. He is adamant that he wants to be a marine biologist when he grows up, and that of course will require upper level math and sciences with math.

That’s great, because it will give him a reason to work hard at this. He’s lucky to have you there teaching him!

Edited April 18, 2021 by Eilonwy

[Not_a_Number]

1 minute ago, Eilonwy said:

hat’s great, because it will give him a reason to work hard at this. He’s lucky to have you there teaching him!

He really is!! I'm impressed that you're doing this for them, @Sk8ermaiden. 

[cougarmom4]

I just wanted to add that you may find some helpful supplemental videos on MathAntics.com. These would be great for the student to watch on his own...they are enjoyable to watch and teach things in a pretty visual way. 

[Carol in Cal.]

On 4/17/2021 at 1:16 PM, Sk8ermaiden said:

Those triangles look like a flashcard form of the "number families" Singapore teaches for +/-, and I teach again for multiplication and division. We just draw them on the paper though. I did plan to teach him number families, especially as a problem solving strategy (even though his basic arithmetic is strong, he's missing questions like 852-?=262, and 864/?=8 so he's never been taught the relationship between addition/subtraction and multiplication/division or how to use it to solve problems) and I plan to go all the way back to unit cubes to teach it. Everyone loves unit cubes, right? I even pull them out with my 6th grader sometimes when I think she's not getting the relationship between the algorithm and real life. 

I am not going to just shove him along to pass the test though. He is adamant that he wants to be a marine biologist when he grows up, and that of course will require upper level math and sciences with math. I'm going to tell mom that we will go exactly as fast as he can understand. If that doesn't get him to Saxon 7/6, then I will commit to continuing this next year to get him there. With a whole year there's no question he can get there, and in fact, DD and I find the placement questions showing mastery of 8/7 WAY easier than the ones that show mastery of 7/6. I don't know why, but if he does as well, he could potentially test into Algebra 1/2 (Saxon pre-a) after another year. I don't know. I can't see passing the 7/6 portion of the test and not the 8/7, because it's so much more basic.

Off topic, but there is a fantastic set of summer camps for future marine biologists locally that might be worth a Santa Cruz vacation trip.  Check out the Seymour Center for details.  They are great.

https://seymourcenter.ucsc.edu/learn/youth-teen-programs/ocean-explorers-summer-camp/descriptions/

Edited April 20, 2021 by Carol in Cal.

[Sk8ermaiden]

I will pass that along!

We started Monday. I get the feeling that he has *seen* a lot of these concepts before, but not done them enough or had them explained enough to be cemented. So The things I planned to cover for this first week, I touched on and reviewed, and then (even though he is still practicing those concepts this week - mostly place value), I spent most of the time teaching geometry, because he's had basically none. This lets me spend 3 full face to face sessions on geometry if he needs it before he gets to those workbook pages. Geometry and fractions are going to be the two big units that need to have full attention devoted to them, so the more time I have for them the better. He did all the days work I assigned in the book in about the amount of time I was hoping it would take and only missed 1 question out of many, so that's good too. 

I brought the book and white board and explained to my friend what he was working on so she could be the first line of questioning if he had an issue. She said several of the things I talked about she was never taught at any age and she's upset how badly her math education failed her. 😕

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

I brought the book and white board and explained to my friend what he was working on so she could be the first line of questioning if he had an issue. She said several of the things I talked about she was never taught at any age and she's upset how badly her math education failed her. 😕

Oh no 😞 . What are some of the things she was never taught? 

[Sk8ermaiden]

There were a lot, but especially why carrying works the way it does (you are filling in the ones place but you have created a new ten and so however many tens you created gets added to the tens place.) She could carry but she didn't know why. That to get the area of a rectangle with a piece cut out, you you find the area of the rectangle and the area of the missing piece and subtract. Oh, the little tick marks geometry books use to show that two lines are equal in length? She had not seen them before or known what they were. 

Does anyone know when Math Mammoth does parallel and perpendicular lines? I need to figure out if I need to go back and teach it or if it will come up later. 

My only real complaint about MM is that they don't mark their right angles or equal sides. I did it myself, lol. 

[Not_a_Number]

1 hour ago, Sk8ermaiden said:

There were a lot, but especially why carrying works the way it does (you are filling in the ones place but you have created a new ten and so however many tens you created gets added to the tens place.) She could carry but she didn't know why. That to get the area of a rectangle with a piece cut out, you you find the area of the rectangle and the area of the missing piece and subtract. Oh, the little tick marks geometry books use to show that two lines are equal in length? She had not seen them before or known what they were. 

Oh, wow. So she really had a rotten math education 😞 . To know that about area, you just need to know what area IS... sounds like she doesn't even have a rudimentary mental model of either place value or area. 

 

1 hour ago, Sk8ermaiden said:

My only real complaint about MM is that they don't mark their right angles or equal sides. I did it myself, lol. 

That would annoy me! 

[Sk8ermaiden]

I think she does know what area is and how to calculate it, I think she was just really shorted on problem solving and critical thinking in math.

She says he's doing well on his work and really seems to understand. This is mostly place value, so yay, because that's so important. I'll see him Thurs.

[Not_a_Number]

3 minutes ago, Sk8ermaiden said:

I think she does know what area is and how to calculate it, I think she was just really shorted on problem solving and critical thinking in math.

In my experience, people who don’t know things that are one step away from the definition have very fuzzy mental models. I’m not blaming her for that, of course! 
 

3 minutes ago, Sk8ermaiden said:

She says he's doing well on his work and really seems to understand. This is mostly place value, so yay, because that's so important. I'll see him Thurs.

Yay!! Place value is such a big stumbling block.

[Sk8ermaiden]

He's doing all right, but he got all but one of the assigned word problems wrong. When I asked him to do them again, he did them right with zero assistance, so it's either that his work is too messy and numbers get lost, or that he was going to fast. 

Geometry continues to be a big issue. There were problems like, using only straight lines, divide this rectangle into a square and a rectangle, or two triangles, or 4 triangles. I feel like these are things small children pick up playing blocks (the first time they fit two triangles together and realize they made a square) or Lego (I need another piece that's 8 long, but I only have a 6 long and a 2 long). And none of that geometric awareness is there. Even after I explained and modeled solutions on near identical problems, he couldn't do it. I asked mom to download him a tangram app, because I know that won't feel too babyish. 

Part of me doesn't want to spend to terribly long trying to drive shapes home, but there is so much in high school geometry that is breaking shapes down into smaller shapes so you can calculate lengths and angles...
 

[Not_a_Number]

6 minutes ago, Sk8ermaiden said:

He's doing all right, but he got all but one of the assigned word problems wrong. When I asked him to do them again, he did them right with zero assistance, so it's either that his work is too messy and numbers get lost, or that he was going to fast. 

At least it's a relief he got them by himself the second time around.

 

6 minutes ago, Sk8ermaiden said:

Geometry continues to be a big issue. There were problems like, using only straight lines, divide this rectangle into a square and a rectangle, or two triangles, or 4 triangles. I feel like these are things small children pick up playing blocks (the first time they fit two triangles together and realize they made a square) or Lego (I need another piece that's 8 long, but I only have a 6 long and a 2 long). And none of that geometric awareness is there. Even after I explained and modeled solutions on near identical problems, he couldn't do it. I asked mom to download him a tangram app, because I know that won't feel too babyish. 

Wow. What happens if he tries? That sounds pretty unusual to me. A tangram app sounds like a good idea. 

 

6 minutes ago, Sk8ermaiden said:

Part of me doesn't want to spend to terribly long trying to drive shapes home, but there is so much in high school geometry that is breaking shapes down into smaller shapes so you can calculate lengths and angles...

Yeah, I think he'll need to be able to do this. 

[Sk8ermaiden]

Most things are going well! He seems to have area fairly well down and be doing well in the other topics we've been over, but shapes and perimeter are still a big stumbling block. After working with him, it doesn't seem like he doesn't understand perimeter, it seems like he is not used to being slow and careful in his work. I showed him three different ways to do perimeter of an oddly shaped object and said he needed to figure out which one worked best for him, but he will count three times and get three different answers. We will keep working on it and hopefully with more practice he will get better. 

Neatness is a big issue and I gave him graph paper and told him to work his longer problems on it.  He said, "I don't do so well with graph paper." I was like, because you're used to being messy in your work! If you use it, one number per square, you will have less silly mistakes. 

I am going to make him some shape flashcards, just for quick recognition of basic shapes and identification of the 5-10 sided shapes. Everything I can find online is super babyish. 

And in great news, the online tutor he has has spent the whole month on fractions and he seems to have a decent understanding of what they are and even equivalent fractions now. Since that was the other big topic he needed in the 3B book, it allows me to spend more time on geometry without worry. 🙂

[Not_a_Number]

12 minutes ago, Sk8ermaiden said:

Most things are going well! He seems to have area fairly well down and be doing well in the other topics we've been over, but shapes and perimeter are still a big stumbling block. After working with him, it doesn't seem like he doesn't understand perimeter, it seems like he is not used to being slow and careful in his work. I showed him three different ways to do perimeter of an oddly shaped object and said he needed to figure out which one worked best for him, but he will count three times and get three different answers. We will keep working on it and hopefully with more practice he will get better. 
 

What would be 3 ways to do perimeter? 😄 I feel like I only know one!!

 

12 minutes ago, Sk8ermaiden said:

Neatness is a big issue and I gave him graph paper and told him to work his longer problems on it.  He said, "I don't do so well with graph paper." I was like, because you're used to being messy in your work! If you use it, one number per square, you will have less silly mistakes. 

I am going to make him some shape flashcards, just for quick recognition of basic shapes and identification of the 5-10 sided shapes. Everything I can find online is super babyish.

Makes sense! Too bad he doesn’t have that experience from life.
 

12 minutes ago, Sk8ermaiden said:

And in great news, the online tutor he has has spent the whole month on fractions and he seems to have a decent understanding of what they are and even equivalent fractions now. Since that was the other big topic he needed in the 3B book, it allows me to spend more time on geometry without worry. 🙂

Oh good!! Sounds like there’s a lot of progress.

[Sk8ermaiden]

I realized after I wrote it that might be confusing. 😀 It is still beginning perimeter when shapes are drawn on graph paper. I mark my starting point and trace around the outside, counting as I go, until I reach the beginning again. My friend has her kids put a tick mark in every "unit" as they count, and my daughter prefers to write the length of each side and then add them. This is for unusual shapes with many sides of course, that look like capital "H" or a right triangle with stairstep sides, etc.

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

I realized after I wrote it that might be confusing. 😀 It is still beginning perimeter when shapes are drawn on graph paper. I mark my starting point and trace around the outside, counting as I go, until I reach the beginning again. My friend has her kids put a tick mark in every "unit" as they count, and my daughter prefers to write the length of each side and then add them. This is for unusual shapes with many sides of course, that look like capital "H" or a right triangle with stairstep sides, etc.

So, this is all due to my focus on "mental models," but I really prefer just saying "perimeter is total length around the figure" and going from there and letting kids figure it out using their own ideas of how to work with length... it's always interesting and surprising what kids can come up with when you give them the key to the question! Even not very mathy kids. 

DD8 had the WEIRDEST notion of perimeter ever after kindergarten. I had to actually unteach her!! It's always fascinating what kids can accidentally pick up. 

[Sk8ermaiden]

That was how I started in the first week. The issue isn't in the idea, but the execution. He got them all wrong. Sometimes twice. He knew what he needed to do, but he always skipped or missed sides, even when marking them. I do believe it comes from not having practice being precise in doing something. So I gave him some example methods for counting them, but he's just going to have to practice until he gets so bored of doing them that he focuses and makes sure to get every last side because he wants to move on.

When it is a simple shape like a quadrilateral or regular octagon or something where the length of the sides is written, he generally has no problem calculating the perimeter.

[Sk8ermaiden]

I think I am going to bring a tape measure Monday and make him calculate the perimeter of and the draw to scale the small room we work in. 

[Not_a_Number]

7 minutes ago, Sk8ermaiden said:

That was how I started in the first week. The issue isn't in the idea, but the execution. He got them all wrong. Sometimes twice. He knew what he needed to do, but he always skipped or missed sides, even when marking them. I do believe it comes from not having practice being precise in doing something. So I gave him some example methods for counting them, but he's just going to have to practice until he gets so bored of doing them that he focuses and makes sure to get every last side because he wants to move on.

When it is a simple shape like a quadrilateral or regular octagon or something where the length of the sides is written, he generally has no problem calculating the perimeter.

Yeah, it sounds like he needs practice. I'd just be careful that he remembers WHAT he's calculating. Personally, what I do with situations like this is let them TELL me what it is they are calculating first conceptually, then we scheme about the best way to keep track. But keeping track of the main idea is actually not as easy as it sounds -- I've known kids who "forgot" the difference between area and perimeter, believe it or not. 

I know you're under time pressure, so you're going to have to do the best you can -- I'm just saying that this approach has always led to better retention for me. 

[Not_a_Number]

It also sounds like maybe part of the issue is that he isn't comfortable adding lots of numbers? 

[Sk8ermaiden]

This is more just counting. These perimeters are mostly under 20 units and he is just going around counting the units on the graph paper. But when there are many small sides in a row he is always accidentally skipping some. The problems where he has to add many numbers he usually does fine on. As long as you give him the numbers.

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

This is more just counting. These perimeters are mostly under 20 units and he is just going around counting the units on the graph paper. But when there are many small sides in a row he is always accidentally skipping some. The problems where he has to add many numbers he usually does fine on. As long as you give him the numbers.

That sounds highly tedious, lol. Marking the sides he's gotten already sounds like a good idea! 

Would any of these be faster by adding, though? Because the thought of counting around a figure like that makes me want to stab myself with a fork... 

Edited April 29, 2021 by Not_a_Number

[Sk8ermaiden]

4 hours ago, Not_a_Number said:

That sounds highly tedious, lol. Marking the sides he's gotten already sounds like a good idea! 

Would any of these be faster by adding, though? Because the thought of counting around a figure like that makes me want to stab myself with a fork... 

No, adding takes at least three times as long, and you would still have to count all the lengths of all the sides anyways. And since he can't manage to get every side without forgetting one I have absolutely no confidence that he would pull the 6+ numbers from around the shape and not miss one. I did offer it as an option though.

Truly it should not be long or tedious, I mean how long does it take you to count to 10 or 15 or 20? That's what's so frustrating, is it's such an easy thing to do - my first grader is doing it without much hand holding needed from me. And obviously counting isn't an issue for him, but paying close attention is and it shows up all through his work.

I do not believe he's really had any kind of school work that was checked for correctness, or was held to any kind of standard, so when he has occasionally had some kind of class or workbook, he's just used to doing whatever and then moving on, good enough. I can imagine it's a difficult transition at 12 years old to suddenly have to pay close attention when you haven't really had to thus far.

But the placement test, and really any test in general is get 16 out of 20 right to place where you need to. Not get 10 right and I would have gotten the other 10 right if I'd been paying attention. 

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

Truly it should not be long or tedious, I mean how long does it take you to count to 10 or 15 or 20? That's what's so frustrating, is it's such an easy thing to do - my first grader is doing it without much hand holding needed from me. And obviously counting isn't an issue for him, but paying close attention is and it shows up all through his work.

Not very long, but if I did 20 of these, I'd get bored and I'd get sloppy. I just know myself 😛 . 

This sounds useful for him, unfortunately. It's too bad he's so unused to having to do things correctly 😞 . 

[Sk8ermaiden]

We did a several together and then there were...4? on his homework, three days later. I think he got one right. But when asked to try again, he only got one of the three incorrect ones right. So then we did a bunch more today, three days after that. And he can explain what he needs to do...

In things where the practice is crucial, like place value, I assigned a lot of problems, but for most days, I am picking and choosing select problems, with the idea that if he gets them right, we don't have to spend a lot of time drilling them over and over. And for the most part he gets them right, or when I check and it's wrong and I let him try again, he gets it right. But something about counting every one. It reminds me of when my kids were first graders and they had to sort pictures or manipulatives into groups of 10 and then leftover 1s and sometimes they would have to count SOOOO many times because they would think they counted one that they didn't, or count one twice, etc.

[stripe]

Just wanted to throw this out there. Schofield & Sims’ Understanding Maths books from the UK have explicit, scaffolded coverage of various topics, e.g. addition and subtraction, fractions and decimals. They are available from bookstores like Blackwells, Wordery, and Book Depository to be shipped worldwide.

[Eilonwy]

On 4/29/2021 at 10:38 PM, Sk8ermaiden said:

We did a several together and then there were...4? on his homework, three days later. I think he got one right. But when asked to try again, he only got one of the three incorrect ones right. So then we did a bunch more today, three days after that. And he can explain what he needs to do...

I’m wondering if it could be a confidence issue, and maybe working up slowly in difficulty might help?  For example, set him a shape that is quite simple, but no side lengths given, and increase his confidence that he can get these right, to encourage him to count carefully. Then, slowly increase the shape complexity.  Maybe you are already doing this. 

[Sk8ermaiden]

Just to give a quick update - he's finished the MM 3B book and we're almost done with 4A. He does very well with arithmetic and seems to be catching on to most of the conceptual ways MM teaches. Order of operations was no problem and neither is multi-digit multiplication. I'm having him do all the review at the end of 4A because he's had less-than-ideal amounts of practice and I want him to have that review time. But in the week or so he's working though it, I've been teaching the first new lessons of 4B - namely long division and averages. No problems and with practice he will be fine. 

Geometry still a sticking point. My mom (who was an elem. teacher for decades) expressed a thought. What if the spatial reasoning part of the brain mostly closes at a certain point - much like language/learning foreign language. And a child who has had no exposure to basic spatial concepts may have a much harder time grasping them when introduced at a later age. It would certainly explain a lot about how a child who can grasp long division on the first go is still struggling with area of rectangles months after its introduction (though it's getting better - slowly.)

But he's doing great - gets his work done. It's slowly getting neat enough to read (I don't know if I mentioned that earlier.) Cross your fingers for 4B - there's a decent amount of geometry in it!

Edited June 17, 2021 by Sk8ermaiden

[Not_a_Number]

37 minutes ago, Sk8ermaiden said:

It would certainly explain a lot about how a child who can grasp long division on the first go is still struggling with area of rectangles months after its introduction (though it's getting better - slowly.)

Question for you: do you think he understands why the rectangle area formula works? 

[Sk8ermaiden]

I do believe so. He understands the concept of multiplication, and we started, and spent a long time on area on graph paper, where you could see the square units. We talked about what area is (for example the square footage of a room - how much flooring one would need to buy) vs length or volume. We've talked about how (on the graph paper) you can see that the 5x3 rectangle has five groups of 3 sq units. 

And he can still get there on his own. It's just never intuitive. For example in the last set of problems it asked him to find the perimeter and area of a 5x7 rectangle. And he gave me perimeter written (5x2)+(7x2) and then "area" as 5+5+7+7. When I asked him how we find the area, he said, "length times width, Oh I did perimeter again." And then he did it correctly. Later gave him two review problems for area and perimeter, the first one was a regular rectangle, which he did correctly. The second one was a 7x2 rectangle attached to a 3x3 square. He got the perimeter (YAY! That was a huge issue for a while) but didn't know how to get started finding the area and needed help. And yes, we spent a ton of time on that when we originally did it. But breaking shapes apart into other shapes - even a rectangle or square into two triangles or this figure I drew that was clearly a square and rectangle sharing a 3 unit edge into its component parts is super, super hard for him. If you do something like draw a 10x2 rectangle and ask him to divide it into 2 rectangles he really struggles. 

 

Edited June 17, 2021 by Sk8ermaiden

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

I do believe so. He understands the concept of multiplication, and we started, and spent a long time on area on graph paper, where you could see the square units. We talked about what area is (for example the square footage of a room - how much flooring one would need to buy) vs length or volume. We've talked about how (on the graph paper) you can see that the 5x3 rectangle has five groups of 3 sq units. 

And he can still get there on his own. It's just never intuitive. For example in the last set of problems it asked him to find the perimeter and area of a 5x7 rectangle. And he gave me perimeter written (5x2)+(7x2) and then "area" as 5+5+7+7. When I asked him how we find the area, he said, "length times width, Oh I did perimeter again." And then he did it correctly. Later gave him two review problems for area and perimeter, the first one was a regular rectangle, which he did correctly. The second one was a 7x2 rectangle attached to a 3x3 square. He got the perimeter (YAY! That was a huge issue for a while) but didn't know how to get started finding the area and needed help. And yes, we spent a ton of time on that when we originally did it. But breaking shapes apart into other shapes - even a rectangle or square into two triangles or this figure I drew that was clearly a square and rectangle sharing a 3 unit edge into its component parts is super, super hard for him. If you so something like draw a 10x2 rectangle and ask him to divide it into 2 rectangles he really struggles. 

Draw him more pictures. With the unit squares. He's missing the mental models. I know it seems like he's too old for that, but as you say, he doesn't have enough practice. Don't give him questions without the unit squares. He'll get there. 

[Sk8ermaiden]

This one was drawn on graph paper. 😬 It had the unit squares, and he knows he could find the area by counting them, but he also knows that after the first week of practice I didn't really want him just counting them anymore, because 1) there's more room for mistakes and 2) it doesn't help you when your dimensions are something like 17x252. He can not take for example an L shape and turn it into two rectangles to be able to calculate the area of both. I feel like what he really needs is more practice with shapes, rather than more practice with area - because he can do the area of a basic shape - just not one that is more complicated. I am a little stumped about how to bring that into lessons though. I tried with pencil and paper, but it didn't help. 

Edited June 17, 2021 by Sk8ermaiden

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

This one was drawn on graph paper. 😬 It had the unit squares, and he knows he could find the area by counting them, but he also knows that after the first week of practice I didn't really want him just counting them anymore, because 1) there's more room for mistakes and 2) it doesn't help you when your dimensions are something like 17x252. He literally can not take for example an L shape and turn it into to rectangles to be able to calculate the area of both. I feel like what he really needs is more practice with shapes, rather than more practice with area - because he can do the area of a basic shape - just not one that is more complicated. I am a little stumped about how to bring that into lessons though. I tried with pencil and paper, but it didn't help. 

Backtrack to rectangles and things very like rectangles and let him count until he's comfortable? 

[Not_a_Number]

3 minutes ago, Sk8ermaiden said:

This one was drawn on graph paper. 😬 It had the unit squares, and he knows he could find the area by counting them, but he also knows that after the first week of practice I didn't really want him just counting them anymore, because 1) there's more room for mistakes and 2) it doesn't help you when your dimensions are something like 17x252. He literally can not take for example an L shape and turn it into to rectangles to be able to calculate the area of both. I feel like what he really needs is more practice with shapes, rather than more practice with area - because he can do the area of a basic shape - just not one that is more complicated. I am a little stumped about how to bring that into lessons though. I tried with pencil and paper, but it didn't help. 

I'm going to tell you that from my experience a week is nowhere near enough time to form a mental model. I know you're not used to kids his age without an area mental model (ugh, that's a real failure), but he's clearly there, so he may very well need a while just counting. After he's comfortable counting, have him count in 4s or whatever. Then eventually have that combine with area. I promise he'll remember MUCH better if you do that. 

[Sk8ermaiden]

I really don't think that will help. He can do a rectangle, and he can do a square. But he can not do a rectangle+square. I could have him count until he was blue in the face and it wouldn't get us any closer to the next step, because the breakdown is not with what area is or how to calculate it - it comes when he has to recognize how two shapes come together to make a new one. I'm going to put my creative hat on and come up with some activities. Maybe even something like tangrams but much more basic. And I've been asking him for a month to measure a real life room and calculate its area and perimeter, but he hasn't, so maybe we will do that on Friday as well. 

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

I really don't think that will help. He can do a rectangle, and he can do a square. But he can not do a rectangle+square. I could have him count until he was blue in the face and it wouldn't get us any closer to the next step, because the breakdown is not with what area is or how to calculate it - it comes when he has to recognize how two shapes come together to make a new one. I'm going to put my creative hat on and come up with some activities. Maybe even something like tangrams but much more basic. And I've been asking him for a month to measure a real life room and calculate its area and perimeter, but he hasn't, so maybe we will do that on Friday as well. 

Maybe he also doesn't have a good model of shape? But back up. Do both. 

[Not_a_Number]

Like, don't put them together. Do area as counting and do "putting together shapes" as "putting together shapes." I bet he's missing both. Work on both. Don't combine -- it's too much. 

[Not_a_Number]

4 minutes ago, Sk8ermaiden said:

I really don't think that will help. He can do a rectangle, and he can do a square. But he can not do a rectangle+square. I could have him count until he was blue in the face and it wouldn't get us any closer to the next step, because the breakdown is not with what area is or how to calculate it - it comes when he has to recognize how two shapes come together to make a new one. I'm going to put my creative hat on and come up with some activities. Maybe even something like tangrams but much more basic. And I've been asking him for a month to measure a real life room and calculate its area and perimeter, but he hasn't, so maybe we will do that on Friday as well. 

And if you're sure he understands the area of a rectangle, have him explain it. If you're right and he totally gets it, he'll be able to walk you through the unit square argument. But I would bet that he can't. He doesn't know why anymore because all of this is very overwhelming for him. 

[Sk8ermaiden]

Without question he does not have a good mental model of shape. 6 weeks ago he could not differentiate a square from a rectangle and I know that is something most kids learn as toddlers and spend years cementing. My big hurdle is how to spend some serious time on it without him thinking I'm treating him like a baby. 

Edited June 17, 2021 by Sk8ermaiden

[Sk8ermaiden]

He has explained it to me, basically every time we go over it. 

[Not_a_Number]

Just now, Sk8ermaiden said:

He has explained it to me, basically every time we go over it. 

But somehow he's not pulling it up, right? So something's not working. 

[Sk8ermaiden]

I feel like I'm a bit of a broken record at this point. He can't figure out how two shapes form a new shape. Once you take a marker and draw a single line for him that separates a strange shape into a square and rectangle, he can do the area with no issues or prompting. Understanding shapes is what's not working. 

[Not_a_Number]

1 minute ago, Sk8ermaiden said:

I feel like I'm a bit of a broken record at this point. He can't figure out how two shapes form a new shape. Once you take a marker and draw a single line for him that separates a strange shape into a square and rectangle, he can do the area with no issues or prompting. Understanding shapes is what's not working. 

But are you sure he totally understands WHY the area is additive? The perimeter isn't. That's weird, you know? One thing you can add, the other you can't. Why?