"Show your work"

[UHP]

What challenges have you faced, what successes have you had, in getting your kids to "show their work"?

I have in mind the way the phrase is used in algebra, but maybe the concept applies in other parts of education too.

Even very young kids can learn to solve simple algebra problems like

Quote

10-A = 9

in their heads. "We don't know what A is, we have to find out. 10 minus what number equals 9?"

But training a kid to write down

Quote

10 - A = 9

10 - A + A = 9+A

10 = 9+A

10 - 9 = 9 + A - 9

1 = A

A = 1

must be harder. At the very least they need to know how to use lined paper. It requires a lot of writing, with what to a kid must feel like zero payoff. They have to accept criticism if they make a mistake on any one of those lines. And making a mistake in a deduction like this is more consequential than making a spelling mistake in a paragraph, they may have to redo every line that comes after their mistake.

I'm getting ready to start "Algebra 1" with my daughter, in a sort of formal way. What has it been like for you?

[EmilyGF]

I think it depends on the age of the kid. So, if your Algebra 1 kid is 11+, I might say, "Paper is much cheaper than time. Please use a lot of paper." I also emphasize the payoff - I make you redo the whole thing if it is wrong without work, but I can help you figure out your mistake if you write the whole thing down. Also, I emphasize they are less likely to make mistakes when they write down everything,

Also, I think it helps to have hard enough problems that the kid doesn't feel like t is an effort in futility. Even my most compliant kids with excellent fine motor skills would balk at writing out the example problem you gave. Whenever a silly mistake is made, I say, "You probably wouldn't have done that if you bothered to write your steps."

I encourage one equation per line and clear notes that show thinking.

Emily

[Brittany1116]

My 11yo is in the last bit of Saxon Alg 1. The habit training is important at 10-A=9 because it keeps them on track when they get to really convoluted stuff down the line and they 1) try to do it all in their head, or 2) skip 3 steps mentally and write down some work at the end, AND 3) ultimately get the wrong answer. Ask me how I know. 😉

[wendyroo]

I never require that level of showing their work.
For us it would look like:

10 - A = 9
     +A     +A
10 = 9 + A
-9     -9
1 = A    <---- They would box this answer, because I insist on all answers being boxed.

And very quickly it would evolve to:

10 - A = 9
-9   +A   -9  +A
1 = A

Well, actually, it would never look exactly like that here because I drill into my early algebra learners to change all subtractions into adding the negative. So 10 - A = 9 would first be changed to 10 + -A = 9 before we solved it.

For division they just draw a fraction line under each side of the equation and note what they are dividing by.

I guess, bottom line, I work with them to increase clarity while decreasing unnecessary writing.

Like Emily mentioned, I keep problems challenging enough that the writing isn't pointless. I do this by having the first two sweeps through simple equations be writing-free. First my kids go through Dragonbox to build up the concepts; then they go through the Hands on Equations app to put numbers to the concepts in a way that still doesn't require writing. Next, they work through the easiest level of Hands on Equations Verbal Problems using physical manipulatives on a paper balance beam (a cheap, homemade version of HOE manipulatives).

Once they are very confident solving the problems with the manipulatives, then I start "taking dictation" and translating what they are doing onto paper in exactly the format I want them to use (including numbering the problem, copying the starting equation, and boxing the answer). When I feel they are ready, we switch roles - I slowly solve problems using manipulatives, and they write down the steps for me. I always start them on a white board (that I added lines to with a permanent marker), because that is less taxing than paper and pencil. (So far, all the kids I have done this with have been writing-resistant 8 year old boys, so making it as painless as possible is crucial.)

By the time they hit HOE Verbal Problems levels 2 and 3, they are solving problems like, "At 10 am, two canoes are 16 miles apart on a river. They pass each other at noon headed in opposite directions. If the upstream canoe's rate of speed is 2 miles per hour slower than two-thirds of the rate of the canoe going downstream, find the distance the downstream canoe travels when the two canoes meet." and can't reliably solve them in their heads...which makes it slightly more likely they will give in and show their work. 🤪

One other resource I like is How To Write A Math Solution by Richard Rusczyk (founder of AOPS). It is very high level, so I do not go through all of it at once with a student, but I pick out sections as applicable to focus on.

[UHP]

7 minutes ago, wendyroo said:

Well, actually, it would never look exactly like that here because I drill into my early algebra learners to change all subtractions into adding the negative. So 10 - A = 9 would first be changed to 10 + -A = 9 before we solved it.

That's interesting, what are your reasons for it? How far does it go, when studying quadratics do they learn about b^2 + (-4ac) instead of b^2 - 4ac?

[UHP]

30 minutes ago, Brittany1116 said:

My 11yo is in the last bit of Saxon Alg 1. The habit training is important at 10-A=9 because it keeps them on track when they get to really convoluted stuff down the line and they 1) try to do it all in their head, or 2) skip 3 steps mentally and write down some work at the end, AND 3) ultimately get the wrong answer. Ask me how I know. 😉

My plan, with reservations, is also to use Saxon. Are you happy with it?

[Brittany1116]

2 minutes ago, UHP said:

My plan, with reservations, is also to use Saxon. Are you happy with it?

Yes, I am happy with it. It is easy to use and teach, and there is plenty of practice. My favorite part is the lesson number listed with each problem if you need a refresher. I am not a math fanatic, but it is solid and meets our purposes. 

[HomeAgain]

For us, it's a question of "can a person look at only your written work and follow your train of thought?"  If it doesn't read from initial problem to every step solved, then it's not good enough.

In early years Right Start's multistep problems and having to use check numbers at each step was a good training habit for this.  Being able to go back and find the mistake because every calculation is on the paper....it's an invaluable tool for middle schoolers.

[wendyroo]

10 minutes ago, UHP said:

That's interesting, what are your reasons for it? How far does it go, when studying quadratics do they learn about b^2 + (-4ac) instead of b^2 - 4ac?

I find subtraction introduces endless mistakes. 4x - 3(x + 6)…inevitably my early algebra learners are going to distribute 3 instead of -3 and get 4x - 3x + 18.

In quadratics, and all cases, they learn the standard equations, but I strongly encourage them to rewrite them right away to eliminate subtraction.  Obviously, when they get far enough along, I no longer care whether they do that or not as long as they get the right answer…but I will tease my precalc student if he makes that type of mistake because he thinks he is better than that now. Think again, young sir. I was still making that change at MIT to avoid careless mistakes. 

[Clarita]

38 minutes ago, UHP said:

That's interesting, what are your reasons for it? How far does it go, when studying quadratics do they learn about b^2 + (-4ac) instead of b^2 - 4ac?

FWIW I've never really stopped rewriting '- n' to '+(-n)'. I found it still useful through all of Calculus, and Stochastic Probability. It means instead of having 2 procedures one when you are subtracting and one when you are adding it's always the same procedure. It's actually very powerful tool. 

Edited January 31, 2023 by Clarita
typo

[wendyroo]

38 minutes ago, Clarita said:

FWIW I've never really stopped rewriting '- n' to '+(-n)'. I found it still useful through all of Calculus, and Stochastic Probability. It means instead of having 2 procedures one when you are subtracting and one when you are adding it's always the same procedure. It's actually very powerful tool. 

Exactly. Like when solving systems of equations with elimination. It isn’t a question of whether we are adding or subtracting one equation from the other. We always add, just first we multiply by whatever is necessary, even if that is just -1. That corrects the common mistake of subtracting the y terms, subtracting the x terms, and then getting to the number terms of, for example, 10 and -12 and “subtracting” them as 10 - 12 instead of 10 - (-12). If we had already multiplied through by -1, then we would just add 10 + 12 and avoid the problem. 

[Not_a_Number]

We've had a sort of circuitous path with it. DD10 was writing full-fledged proofs a few years ago and still knows how, but we used to butt heads over it, and now she's able to do it but feels nervous about it. So I'm currently not pushing her to do this, although we've had a lot of conversations about how she ought to work on it again.

For the time being, she's writing down the steps she feels she needs (which does involve writing down whatever algebra steps she finds useful -- she hasn't been having trouble with that) and we aren't focusing a ton on communicating to others. We've had conversations about changing that, but we aren't there yet.  

So I guess my suggestion will be to work with what your child feels is useful, @UHP, and try to run with that. Do you know whether she'd find this useful herself? 

[daijobu]

My rule of thumb is they are getting the correct answer, I don't change anything.  If they are making mistakes, then they need to include additional steps to prevent errors.  

[UHP]

1 hour ago, Not_a_Number said:

We've had a sort of circuitous path with it. DD10 was writing full-fledged proofs a few years ago and still knows how, but we used to butt heads over it, and now she's able to do it but feels nervous about it. So I'm currently not pushing her to do this, although we've had a lot of conversations about how she ought to work on it again.

I remember two years ago some of the impressive things she was doing, that you wrote about here. What is she up to now?

And you wrote also, a little bit, about butting heads. I also anticipate some amount of butting heads, that I wonder if I can mitigate with any kind of careful planning.

1 hour ago, Not_a_Number said:

So I guess my suggestion will be to work with what your child feels is useful, @UHP, and try to run with that. Do you know whether she'd find this useful herself? 

Right now, I wouldn't heed her own sense of what's useful and what's not useful. I usually think I know better. But I am a little bit uncertain of what the learning objectives of "high school algebra" should be. I took it in 8th grade, and have certainly mastered it, but don't remember how very well.

[Roadrunner]

If I even suggested my kids give me this level of detail, especially on a problem this simple, I would have a full fledge uprising on my hands. I mean there would be speeches given, walk outs staged…. So, whatever my kids could do in their heads got done in their heads. They wrote things down when math got complex enough. 

[FreyaO]

My kid is too young, but I never used that kind of detail myself. Seems a sheer torture.

[Jean in Newcastle]

49 minutes ago, Roadrunner said:

If I even suggested my kids give me this level of detail, especially on a problem this simple, I would have a full fledge uprising on my hands. I mean there would be speeches given, walk outs staged…. So, whatever my kids could do in their heads got done in their heads. They wrote things down when math got complex enough. 

I agree. Asking this much detail for something that simple is busy work in my opinion. 

[mathmarm]

   10-A = 9 would be solved and written out so that the equal signs are lined up, one equation per line, and with shorthand because I teach my kids to verbally, explain the answer so they'd tell me "add A on both sides", then "-9 from both sides" and "so you can see A is 1" so mathematically the working would look like this:
 10 - A = 9        | +A on b/s
       10 = 9 + A | -9 from b/s
         1 = A

They don't do write out the middle steps, but the note what that step is.

[Malam]

I'm going to mirror daijobu and say that if they can solve it in one step in their head, they shouldn't need to show their work. The need and usefulness of showing work comes when the problems reach the point where they can't be solved in a single step. At that point, having a more formal understanding of "do the same thing to both sides" becomes necessary, and showing work naturally follows - until, of course, these harder problems are "chunked" and able to be completed mentally, at which point you can move on to yet another level.

The higher level you go, the more you're expected to do in you head - when an algebra 1 teacher shows you that (x+3)(x-4)=0. they'll break it down into x+3=0 or x-3=0, and do one more step for each case. An algebra 2 teacher will get to (x+3)(x-4)=0 and expect the students to realize that the roots are -3 and 4. A somewhat terse calculus teacher might just say "by factoring, we can see that the roots of x^2-x-12 are -3 and 4"

Edited February 1, 2023 by Malam

[Clarita]

3 hours ago, UHP said:

Right now, I wouldn't heed her own sense of what's useful and what's not useful. I usually think I know better.

Why not heed her sense of usefulness? It'll be obvious to her and to you when she does need to write out all the excruciating details when she starts getting the answers wrong. 

IRL when you have to communicate your solutions you are going to clean up and re-write your "work" not just hand your colleagues your notebook with your calculation mistakes and bunny trails. There is an rare chance that on a patent dispute you'll need to give your personal notes to prove you came up with an idea first, but in those instances they are just looking for was some indication you had the idea on a particular date; it's not expected that it be that clear.

[Ellie]

11 hours ago, mathmarm said:

   10-A = 9 would be solved and written out so that the equal signs are lined up, one equation per line, and with shorthand because I teach my kids to verbally, explain the answer so they'd tell me "add A on both sides", then "-9 from both sides" and "so you can see A is 1" so mathematically the working would look like this:
 10 - A = 9        | +A on b/s
       10 = 9 + A | -9 from b/s
         1 = A

They don't do write out the middle steps, but the note what that step is.

And that's why algebra never made sense to me: people say stuff like that, but I don't understand why it is so.

As an arithmetic problem, I know that I can subtract the answer from the number given, and *that* answer will be the missing number. Easy peasy.

Sorry for the sidetrack. Carry on!

[8filltheheart]

6 minutes ago, Ellie said:

And that's why algebra never made sense to me: people say stuff like that, but I don't understand why it is so.

As an arithmetic problem, I know that I can subtract the answer from the number given, and *that* answer will be the missing number. Easy peasy.

Sorry for the sidetrack. Carry on!

Your answer doesn't really work this problem bc kids solving this problem would most likely not know negative numbers or multiplying by neg 1.

The sample 10 - A = 9   is obviously 1, but what if they numbers themselves weren't so simple?  What if the numbers were 261-A= 224?  How would have have the student solve the problem? 

The knowing A=1 in the first example is incredibly simple but also shouldn't be dismissed in terms of solving it so that they can understand why it is so.

Imagine a scale with the = sign in the middle.  You can only add or subtract from one side of the scale if you add or subtract the exact same number from the other side of the scale.   So......

2+3=5

2+3+4=5+4

2+3+4-4=5+4-4

All of those are equal statements.  So are

5=2+3

5+4=2+3+4

5+4-4=2+3+4-4, etc.  

So with

10-A=9

young kids don't know that subtracting 10 from both sides giving

-A=-1 is the same thing as A=1.  So you have to deal with the negative A in a way that makes sense to them.  The opposite of subtraction is addition.  So if you have -5 and want it to equal 0, you add 5.  If you have -A, you add A and then it equals 0 on that side.  If you add A on that side of the scale, you have to add A to the opposite side of the scale. 

10-A+A=9+A  (A is added to both sides of the scale.  A-A=0, so

10+0=9+A (Just the same as if I had added 2+3+4-4 to be 2+3+0 in my example above)

to solve for A you subtract 9 from both sides of the scale

10-9=9+A-9 (or commute the numbers so that you can easily see A+9-9)

1=A +0

(I can't really type it out in an easily visual way, but it should be clear enough by reading.)

My pt being that it shouldn't be dismissed as equalling 1, easy peasy.  Bc it is understanding why the above works that makes learning alg as no big deal bc they have been doing it from 2nd and 3rd grade.

 

[HomeAgain]

31 minutes ago, Ellie said:

And that's why algebra never made sense to me: people say stuff like that, but I don't understand why it is so.

As an arithmetic problem, I know that I can subtract the answer from the number given, and *that* answer will be the missing number. Easy peasy.

Sorry for the sidetrack. Carry on!

I think this is where Singapore bar models have an advantage.  A problem like this (or even more detailed ones) can be solved by drawing out a sketch of what is going on, how the number bonds work to give the answer.

While I am a fan of students showing all their work, I think they need flexibility in how to show it.  If a number bond triangle works for them, great.  If a bar model works, great.  If the area model works for them, awesome. If color coding works, I'm all for it.  Today my kid colored line segments to show supplementary angles.  Cool.  I'll take that as showing how his mind is working so that he can then move to the written step of 180 - x.  It doesn't all need to be algebraic, but it does need to be clear and easy to follow.

 

[UHP]

3 hours ago, Ellie said:

And that's why algebra never made sense to me: people say stuff like that, but I don't understand why it is so.

As an arithmetic problem, I know that I can subtract the answer from the number given, and *that* answer will be the missing number. Easy peasy.

I have an inkling now that this kind of exercise, of writing down all the intermediate steps between the question and the answer, perhaps could be called something more attractive than "showing your work." To me, the expression has a scolding tone to it: you hear it when you get points off of your homework even though the answer is right.

One famous thing that kids learn in algebra is to write a new equation that differs from an old equation in a simple way. Add to both sides the same quantity, multiply both sides by the same quantity, etc. They learn that if the old equation is true, the new equation is also true. And for some rarefied examples they learn that they can make these modifications repeatedly and judiciously until they reach an equation that simply tells the solution of the original equation.

This process isn't the only way a kid could get the right answer: maybe they drew a bar-model diagram, or another kind of diagram, or maybe they just had a hard-to-articulate inspiration. If that's the case, and you tell them they must now write lots of equations in between the question and the answer, then you aren't telling them to "show their work," because they didn't work the problem that way.

Maybe there's an opportunity to start fewer fights by calling it something else.

Edited February 1, 2023 by UHP

[8filltheheart]

9 minutes ago, UHP said:

 

One famous thing that kids learn in algebra is to write a new equation that differs from an old equation in a simple way. Add to both sides the same quantity, multiply both sides by the same quantity, etc. They learn that if the old equation is true, the new equation is also true. And for some rarefied examples they learn that they can make these modifications repeatedly and judiciously until they reach an equation that simply tells the solution of the original equation.

 

I don't believe that this is something that kids learn in algebra.  It is a basic elementary concept.  (I know it is in Horizons 3.  I can't remember if it is in Horizons 2.)  It is how Hands On Equations teaches how to solve their word problems.  

I personally don't think asking a student to show their work sounds like scolding.  It is no different than saying that paragraphs need to be written with complete sentences. Bc of this fact isn't an expository essay.  😉  Showing the operations performed is part of the answer.  The final answer is simply part of the whole once you get beyond basic math facts.  

[UHP]

1 hour ago, 8filltheheart said:

I don't believe that this is something that kids learn in algebra.  It is a basic elementary concept.

In your view or experience, what is a more prominent thing that kids learn in algebra?

[8filltheheart]

1 hour ago, UHP said:

In your view or experience, what is a more prominent thing that kids learn in algebra?

Understanding functions/the relationship between dependent and independent variables, graphing linear equations and inequalities, solving quadratic equations, etc.

What you are describing is solving for a variable that is a constant.  If a single variable is considered the key pt of algebra, than 1st grader filling in a blank, 1+ ___ =3 are essentially doing algebra since the ___ and a variable represent the same thing.  What makes the statement true?

Edited February 1, 2023 by 8filltheheart

[UHP]

1 hour ago, 8filltheheart said:

What you are describing is solving for a variable that is a constant.  If a single variable is considered the key pt of algebra, than 1st grader filling in a blank, 1+ ___ =3 are essentially doing algebra since the ___ and a variable represent the same thing.  What makes the statement true?

I would rather say "solving for an unknown" than "solving for a variable that is a constant." I'll contradict you and say that I do think that solving for an unknown is a very typical algebra problem. I agree that even very young children can be taught to understand and solve a limited variety of such problems. My own has had a procedure for solving 1+__ = 3 for as long as she's had one for solving 1+2 = __. But that procedure is not the equations-as-balancing-scales, simplify-a-little-at-a-time approach that I was taught in middle school.

I fully agree that learning about variables, functions of them and relations between them is an algebra skill of prime importance and one that is largely separate from "solve for the unknown." The contrast between them is interesting. I've seen lots of ideas around for getting 5-year-olds used to solving for the unknown, no ideas for getting them used to dependent and independent variables.

[Jean in Newcastle]

1 hour ago, 8filltheheart said:

Understanding functions/the relationship between dependent and independent variables, graphing linear equations and inequalities, solving quadratic equations, etc.

What you are describing is solving for a variable that is a constant.  If a single variable is considered the key pt of algebra, than 1st grader filling in a blank, 1+ ___ =3 are essentially doing algebra since the ___ and a variable represent the same thing.  What makes the statement true?

Understanding the relationship between addition and subtraction is something that grade schoolers can learn without making it into a formal algebraic equation. Even with bigger numbers. (Btw this gets back to the whole concept of teaching mathematics concepts that was discussed in another math thread.). 

[8filltheheart]

8 minutes ago, UHP said:

I would rather say "solving for an unknown" than "solving for a variable that is a constant." I'll contradict you and say that I do think that solving for an unknown is a very typical algebra problem. I agree that even very young children can be taught to understand and solve a limited variety of such problems. My own has had a procedure for solving 1+__ = 3 for as long as she's had one for solving 1+2 = __. But that procedure is not the equations-as-balancing-scales, simplify-a-little-at-a-time approach that I was taught in middle school.

I fully agree that learning about variables, functions of them and relations between them is an algebra skill of prime importance and one that is largely separate from "solve for the unknown." The contrast between them is interesting. I've seen lots of ideas around for getting 5-year-olds used to solving for the unknown, no ideas for getting them used to dependent and independent variables.

I have been using Horizons K-6 since 1994.  Solving for N in equations like you posted are pretty much standard problems starting somewhere around 3rd grade. They teach the concept by completing operations on both sides of the equation to isolate the variableon one side of the equation.  

Anyway, 1994 was a while ago, so this isn't a new concept for teaching elementary age kids to balance equations. And younger kids can understand the concept.  So, it obviously doesn't need to be a middle school concept.

[wendyroo]

29 minutes ago, 8filltheheart said:

They teach the concept by completing operations on both sides of the equation to isolate the variableon one side of the equation.  

I teach this concept pretty early, but mostly with shapes instead of numbers and letters. My 7 year old is working through Balance Benders, and knows how to maintain equality by adding or subtracting from both sides, or doubling or halving both sides. She can also manipulate the equations via substitution.

I consider those concepts pre-algebra level, but at that stage my kids are not ready for formalized notation; they don't have the fine motor skills, patience, or attention to detail to write the equations. My 7 year old wouldn't even know the notation for dividing each side in half...though she has no problem articulating that that is what she is doing and what the result it.

For me, it jumps to algebra level when 1) the equations (which include fractions, decimals, exponents, negatives and parentheses) require at least 3-4 steps to solve, 2) at least some of the equations are generated by the student a la HOE Verbal Problems rather than being handed to them, and 3) the student is developmentally ready to learn and implement a clear, accurate, sequential notation system to show their work.

[8filltheheart]

1 hour ago, Jean in Newcastle said:

Understanding the relationship between addition and subtraction is something that grade schoolers can learn without making it into a formal algebraic equation. Even with bigger numbers. (Btw this gets back to the whole concept of teaching mathematics concepts that was discussed in another math thread.). 

I guess my POV is simply, why not teach it with balancing equations?  It makes kids get used to the idea of how math works and that equal signs aren't some sympol with only the final answer on the right side.  (I have seen kids who are confused by 5=2+3 bc they think it is wrong and can't be written that way.)  From my perspective, it is teaching them the relationship between addition and subtraction and what it means to be equal.

[mathmarm]

11 hours ago, Ellie said:

And that's why algebra never made sense to me: people say stuff like that, but I don't understand why it is so.

As an arithmetic problem, I know that I can subtract the answer from the number given, and *that* answer will be the missing number. Easy peasy.

Sorry for the sidetrack. Carry on!

Well I'm more finnicky with the way things are verbalized, but I was just responding to the OP about "what does it mean to show work for a problem of this caliber" for young kids. By the time my students are in a position to write out the work to problems of this caliber algebraically, they've already learned to think through the problem and discuss the solutions.

I gradually teach an explicit understanding of equality and the equal symbols so there is a background of understanding before the kids get there.

You've clearly lead a long and full life without a particularly clear understanding of whatever it is that you missed in Arithmetic and/or Algebra, so you know how little it will impact you in the long-run if you do or don't get this concept while in school.
 

[Malam]

12 hours ago, wendyroo said:

My 7 year old is working through Balance Benders, and knows how to maintain equality by adding or subtracting from both sides, or doubling or halving both sides. She can also manipulate the equations via substitution.

When did she start with the beginning level (if that is where she started) and which of the 4 balance bender books is she doing now?

[wendyroo]

18 minutes ago, Malam said:

When did she start with the beginning level (if that is where she started) and which of the 4 balance bender books is she doing now?

She is at the very end of the Beginning book now, and will move to Level 1 in the next week or so.

She started at the beginning of the school year, but we worked very slowly at first. In fact, for quite a while we used an actual balance scale and random objects to test the solutions. I did not want her to take my word that A = B meant that A + C = B + C or that X = Y meant that 2X = 2Y. I wanted her to see that physically those were true.

Another caveat: My kids would not have been ready for Balance Benders as early as they were if I was not 100% confident in my teaching. Balance Benders offers very, very little help in teaching these abstract concepts to kids. If I had just handed the books to my kids, it would not have been pretty. The books were primarily problem sets...well designed and sequenced problem sets, but just problem sets nonetheless. I needed to scaffold my kids through them with a lot of exploration with manipulatives and Socratic dialogues.

[Brittany1116]

Love Balance Benders and Balance Math & More books. I agree they require instruction for at least the first book as they begin to understand how they work. 

[Malam]

10 minutes ago, wendyroo said:

The books were primarily problem sets...well designed and sequenced problem sets, but just problem sets nonetheless

Is this true for Hand-on Algebra as well? Which starts at a more accessible level, BB beginning, or HoA?

[8filltheheart]

20 minutes ago, Malam said:

Is this true for Hand-on Algebra as well? Which starts at a more accessible level, BB beginning, or HoA?

I have never seen BB.  I start HoE Verbal Problems Book with strong 3rd graders, avg 4th graders.  (I have never seen Hands On Algebra, only Hands On Equations.....which I would also not consider alg.  The content is the same as SM type word problems but solving with equations vs. bar models.)

[wendyroo]

2 hours ago, Malam said:

Is this true for Hand-on Algebra as well? Which starts at a more accessible level, BB beginning, or HoA?

Do you mean Hands on Equations? I don't know anything about a Hands-on Algebra.

We have only ever used the Hands on Equations apps and then the Verbal Problems book.

For us, the sequence looks like:
Dragonbox 5+ and then 12+
Balance Benders Beginnings and Level 1 (normally overlapping somewhat with Dragonbox 12+)
Hands on Equations apps (levels 1, 2, 3) (again, often overlapping somewhat with BB level 1)
Hands on Equations Verbal Problems book

All of this is done as supplements to Math Mammoth.
The Verbal Problems books typically coincides with doing the interesting parts of Math Mammoth 6 and 7.

Then a quick run through Zaccaro's Becoming a Problem Solving Genius: A Handbook of Math Strategies.

With my oldest we then went through AOPS Prealgebra...and hated it.
Spencer is finishing up Problem Solving Genius, and then we are going to slowly dip our toe straight into AOPS Algebra.

To answer your question - I don't know how much teaching there is in HoE because we have never actually used the curriculum. The apps are just well organized, self checking problem sets with digital manipulatives. I take my kids through BB first because it is 1) more fun, and 2) doesn't require a formal understanding of negatives numbers.

[Not_a_Number]

I think of early algebra as largely being about the concept of a variable as a generalization of number. I've found that this is the thing kids struggle with most when they work with variables (and I've seen this at the college level quite often) -- they don't reason about algebraic quantities using the same ideas as they do with actual numbers. 

Later on, there are functions and graphs. 

I cover the idea of an equals sign as meaning that two things are the same considerably before algebra, so I don't think of that as an essential algebra concept. I can believe it can be, though. 

We do negative numbers very early, but I don't insist that kids add negatives instead of subtracting. I would have hated that kind of rigidity in school and wouldn't want to dictate. (I can imagine giving that advice if a kid was making a lot of mistakes of this type, though. DD10 doesn't so far, however.) 

[Not_a_Number]

On 1/31/2023 at 9:39 PM, UHP said:

I remember two years ago some of the impressive things she was doing, that you wrote about here. What is she up to now?

She's doing well. She was self-teaching for a while so she's made less progress in the last year than she might have, but she's working through the various AoPS introductory books. She's still feeling nervous about proofs, so hasn't been writing them, although I can tell when she describes what she's thinking verbally that she ought to be able to pick it up again. Hopefully she'll get over the nerves soon. Fingers crossed.  

 

On 1/31/2023 at 9:39 PM, UHP said:

And you wrote also, a little bit, about butting heads. I also anticipate some amount of butting heads, that I wonder if I can mitigate with any kind of careful planning.

Right now, I wouldn't heed her own sense of what's useful and what's not useful. I usually think I know better. But I am a little bit uncertain of what the learning objectives of "high school algebra" should be. I took it in 8th grade, and have certainly mastered it, but don't remember how very well.

My two cents is that my kiddos learn more from my (doubtlessly superior) experience if they are allowed to make mistakes themselves first and see the point of my (usually excellent 😉 ) advice. 

[UHP]

1 hour ago, Not_a_Number said:

I cover the idea of an equals sign as meaning that two things are the same considerably before algebra, so I don't think of that as an essential algebra concept. I can believe it can be, though. 

I can't resist the opportunity to write that Robert Recorde invented the equals sign in 1557. You can tell that it's a marvelous invention by reading the algebra that Omar Khayyam was doing 500 years earlier. Khayyam states problems like

Quote

What is the amount of a square so that when six dirhams are added to it, it becomes equal to five roots of that square?

and reasons about them in the same kind of prose. To me, the fact that our good notation makes these things accessible to little kids shouldn't take them out of the universe of "algebra." It reminds me of punctuation and spaces, invented in very late antiquity, that made literacy accessible to little kids. We can dream of future inventions that will make the secret knowledge of today's geniuses accessible to little kids.

Here is a little bit of Recorde 1557, "The Whetstone of Witte...", quoted in a footnote on wikipedia:

Quote

Howbeit, for easie alteration of equations. I will propounde a fewe examples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in worke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: = , bicause noe 2 thynges, can be moare equalle.

"Gemowe lines" means "twin lines."

[flmom79]

Our rule of thumb has been that you need to show enough work so that someone who didn't know how to work the problem would be able to see how you solved it.  The level of detail you are indicating would (I think) be unnecessary.  I would only require that if they are *still* making careless errors due to those grains of minutiae not being demonstrated.  

[kiwik]

On 2/1/2023 at 4:16 PM, FreyaO said:

My kid is too young, but I never used that kind of detail myself. Seems a sheer torture.

Me too.

I would accept.

10-A =9 (or whatever it was)

-A = -1

A= 1