Specific algebra question

[Calizzy]

I’m evaluating algebra programs for my daughter and I would really like one with teaching videos. I am drawn to videoText but I have a question about the way they solve the problems with variables. They always want to add or multiply to solve for X and never subtract or divide. So if the question is 8x + 3= 42 the first step is to add -3. Ok, I would just subtract 3, but I can deal with that. But then when you have 8x= 39 instead of dividing by 8 they multiply both sides by 1/8. I understand that it’s the same thing, but it seems more complicated and harder for a child to grasp. Is there a reason for this method? Do other programs do this as well? When I look around it seems like most programs just “divide by 8” instead of “multiply by 1/8”. If you have done it this way, do kids catch on or is it the bane of you algebra existence?

[importswim]

I think the reason why they do this in VideoText (and other places) it to show that you're making 8/8 which reduces into 1/1 or 1 so then you have 1x = 39 * 1/8.

For math I go for further understanding rather than easy. I want my kiddos to understand why. If they don't understand equivalent fractions then I'd go back and do more of that so that they can grasp it.

 

I'm not sure if this is what you're asking, though!

Edited November 25, 2020 by importswim

[Monica_in_Switzerland]

That does seem a bit weird to me.  A kid who is ready for algebra should not stumble over this, but I would consider that to be a bit odd.  

 

 

[SoCal_Bear]

This actually makes a lot of sense to me because in higher math, there is no division sign in algebraic expressions only fractions which are the expression of division. She needs to see that 1/8 is the reciprocal of 8 or 1/a is the reciprocal of a. This is a really important concept to grasp and why reciprocals are important for manipulating an algebraic expression. Thinking about reciprocals is very important in math.

Also subtraction shouldn't be used in algebra because that is not what is happening. Students must grasp the concept of positive and negative terms and that whenever you see - it is not an operation. An expression that say 2x - 4 = 12 really means + 2x + (-4) = 12. A student who doesn't grasp that this is the addition of postive and negative terms will struggle in algebra because it will be intuitive to them to always keep the positive or negative with the term.

[EKS]

I don't remember this specifically about VideoText, but what I do remember is that they tended to overcomplicate things.  We jumped ship to Jacobs and it was like a breath of fresh air.  If you want video lessons, take a look at Derek Owens or Math Without Borders.  Neither uses Jacobs, but both are excellent.

[Not_a_Number]

2 hours ago, calbear said:

This actually makes a lot of sense to me because in higher math, there is no division sign in algebraic expressions only fractions which are the expression of division. She needs to see that 1/8 is the reciprocal of 8 or 1/a is the reciprocal of a. This is a really important concept to grasp and why reciprocals are important for manipulating an algebraic expression. Thinking about reciprocals is very important in math.

Sure there's division in higher math. You write it with the / symbol but it doesn't stop being division for all that. 

I am not really sure that thinking about reciprocals is particularly important in higher math. A reciprocal is 1 over a number. It's not any more or less important than other numbers, although sometimes it's convenient. 

If you get into sufficiently high math, you do wind up using the inverse instead of the reciprocal, but that's quite a long way away. 

 

Quote

Also subtraction shouldn't be used in algebra because that is not what is happening. Students must grasp the concept of positive and negative terms and that whenever you see - it is not an operation. An expression that say 2x - 4 = 12 really means + 2x + (-4) = 12. A student who doesn't grasp that this is the addition of postive and negative terms will struggle in algebra because it will be intuitive to them to always keep the positive or negative with the term.

Subtraction can be used in algebra. Subtraction is a valid operation. You can say that 

2x - 4 = 12 

means that "if you take away 4 from 2x, you get 12," and that's a perfectly FINE and valid interpretation of the equation. In my experience, kids understand "taking away" far better than they understand "add the negative," so I see no reason to insist on that interpretation. 

I'm not sure what you mean about "keeping the positive or negative with the term," to be honest. What's the context? The negative or positive simply represents the operation being performed on the term... 

 

Edited November 25, 2020 by Not_a_Number

[almondbutterandjelly]

If I had to guess, I would say it is so that they can still work the problem if instead of 8 as a coefficient, it's something like 2/3.  Then you multiply by the reciprocal.  Same procedure they already learned from the 1/8 scenario.  So maybe it's just to teach one rule that works for everything?

Edited to add:  Lest it not be clear, I am explaining this for kid-speak.  I have a degree in math and know exactly all the intricacies.

Edited November 25, 2020 by perky

[Not_a_Number]

31 minutes ago, perky said:

If I had to guess, I would say it is so that they can still work the problem if instead of 8 as a coefficient, it's something like 2/3.  Then you multiply by the reciprocal.  Same procedure they already learned from the 1/8 scenario.  So maybe it's just to teach one rule that works for everything?

Edited to add:  Lest it not be clear, I am explaining this for kid-speak.  I have a degree in math and know exactly all the intricacies.

You could divide by 2/3 in the second scenario, though? So the division rule works for everything, right?

(Although I think my kiddo hasn’t become comfortable enough with that manipulation yet and would probably multiply by 3 first.)

[wendyroo]

2 hours ago, Not_a_Number said:

I'm not sure what you mean about "keeping the positive or negative with the term," to be honest. What's the context? The negative or positive simply represents the operation being performed on the term... 

 

I know that at this house I have my kids change all subtraction to addition of the negative (once they know negatives) because their accuracy is much much higher.

For me, the context of "keeping the positive or negative with the term," is 7 - 5 (2x - y). Left in that form, dollars to doughnuts they are going to simplify to 7 - 10x - 5y. However, if I get rid of the 7, suddenly -5 (2x - y) will be correctly converted to -10x + 5y. It all hinges on whether they are viewing the negative as attached to the 5 or as a subtraction sign floating between the 7 and the 5. So if I just force them to rewrite subtractions as addition of the negative, it becomes 7 + -5 (2x + -y) which they can accurately deal with.

At this point, my oldest does it voluntarily because he recognizes that it helps him keep track of the signs. Just like I recognized the same thing for myself long ago and continued to use that strategy throughout my time at MIT. Obviously it is not essential, but it works for us.

[Not_a_Number]

22 minutes ago, wendyroo said:

I know that at this house I have my kids change all subtraction to addition of the negative (once they know negatives) because their accuracy is much much higher.

For me, the context of "keeping the positive or negative with the term," is 7 - 5 (2x - y). Left in that form, dollars to doughnuts they are going to simplify to 7 - 10x - 5y. However, if I get rid of the 7, suddenly -5 (2x - y) will be correctly converted to -10x + 5y. It all hinges on whether they are viewing the negative as attached to the 5 or as a subtraction sign floating between the 7 and the 5. So if I just force them to rewrite subtractions as addition of the negative, it becomes 7 + -5 (2x + -y) which they can accurately deal with.

At this point, my oldest does it voluntarily because he recognizes that it helps him keep track of the signs. Just like I recognized the same thing for myself long ago and continued to use that strategy throughout my time at MIT. Obviously it is not essential, but it works for us.

Interesting. I guess I’ve never had trouble with that, so I don’t do it. DD8 occasionally has this issue but I found her understanding of what she’s doing better if she doesn’t switch to negatives, so while I let her switch if she likes, she doesn’t usually.

ETA: I just asked DH, and he doesn't do this either. Apparently it's a matter of taste! 

Edited November 26, 2020 by Not_a_Number

[Calizzy]

This is a really interesting and helpful. Personally I struggle with multiplying by the inverse instead of just dividing, since dividing just seems easier to me. But probably since she’s learning from scratch and doesn’t have the baggage of already learning a different way dd can catch on. But I do wonder about what @EKS said about videotext over complicating (or over explaining) things. Dd is very straight forwards when it comes to math, she doesn’t want to know why, she just wants to memorize the formulas. I know that’s not the current approach in mathematics, but she is what she is. 

[Not_a_Number]

1 minute ago, Calizzy said:

This is a really interesting and helpful. Personally I struggle with multiplying by the inverse instead of just dividing, since dividing just seems easier to me. But probably since she’s learning from scratch and doesn’t have the baggage of already learning a different way dd can catch on. But I do wonder about what @EKS said about videotext over complicating (or over explaining) things. Dd is very straight forwards when it comes to math, she doesn’t want to know why, she just wants to memorize the formulas. I know that’s not the current approach in mathematics, but she is what she is. 

I personally don't actually let kids memorize the formulas. I think it makes algebra harder to memorize formulas, not easier. 

If you're going to go with memorizing formulas, though, multiplying by the reciprocal is as easy as dividing 🤷‍♀️.

[EKS]

1 hour ago, Calizzy said:

But I do wonder about what @EKS said about videotext over complicating (or over explaining) things. Dd is very straight forwards when it comes to math, she doesn’t want to know why, she just wants to memorize the formulas. I know that’s not the current approach in mathematics, but she is what she is. 

Just to clarify--I am all about kids understanding why.  My memory of videotext is that it seemed to overcomplicate things for the sake of overcomplicating them.  If she just wants to memorize formulas, I'd make it my life's mission to undo that impulse.

Edited November 26, 2020 by EKS

[SoCal_Bear]

2 hours ago, wendyroo said:

I know that at this house I have my kids change all subtraction to addition of the negative (once they know negatives) because their accuracy is much much higher.

For me, the context of "keeping the positive or negative with the term," is 7 - 5 (2x - y). Left in that form, dollars to doughnuts they are going to simplify to 7 - 10x - 5y. However, if I get rid of the 7, suddenly -5 (2x - y) will be correctly converted to -10x + 5y. It all hinges on whether they are viewing the negative as attached to the 5 or as a subtraction sign floating between the 7 and the 5. So if I just force them to rewrite subtractions as addition of the negative, it becomes 7 + -5 (2x + -y) which they can accurately deal with.

At this point, my oldest does it voluntarily because he recognizes that it helps him keep track of the signs. Just like I recognized the same thing for myself long ago and continued to use that strategy throughout my time at MIT. Obviously it is not essential, but it works for us.

This is exactly the point I was trying to get at...your illustration with the pitfalls of not understanding this when you have to work with the distributive property. 
 

 

3 hours ago, perky said:

If I had to guess, I would say it is so that they can still work the problem if instead of 8 as a coefficient, it's something like 2/3.  Then you multiply by the reciprocal.  Same procedure they already learned from the 1/8 scenario.  So maybe it's just to teach one rule that works for everything?

Edited to add:  Lest it not be clear, I am explaining this for kid-speak.  I have a degree in math and know exactly all the intricacies.

Yes, one rule that works for everything. I don't like teaching someting and having to unteach or modify it when it not longer works as easily.

[Not_a_Number]

2 minutes ago, calbear said:

This is exactly the point I was trying to get at...your illustration with the pitfalls of not understanding this when you have to work with the distributive property. 

That's interesting, because I think my family thinks about this in a different way. Neither I, DH, nor DD8 do this at all and we don't have any sign issues. I think we all verbalize the distributing instead of doing it symbolically, so maybe that's why? 

Edited November 26, 2020 by Not_a_Number

[SoCal_Bear]

@Not_a_Number I think when you already think this way automatically (math adept and math gifted students), then it is harder to see how this is a pitfall for students who aren't oriented this year. Sign issues are very common problem for students. I am painfully anal about this because I've seen students easily lose track of signs because they don't habitually associate the sign with the number in their thinking. They associate signs with the operation in their minds and the numbers somehow exist separately from that.

I think another way this is problematic is that when they start thinking about factorization, they will look at something like -12 and not know what to do with the negative. They don't easily grasp that it could be +1, -1, +2, -2, +3, -3, +4. -4, +6, -6, +12, -12 if they are still thinking about the negative as a operation. The reason why I think this is really important when it gets you start to have to work quickly with quadratic equations.

 

[Not_a_Number]

37 minutes ago, calbear said:

@Not_a_Number I think when you already think this way automatically (math adept and math gifted students), then it is harder to see how this is a pitfall for students who aren't oriented this year. Sign issues are very common problem for students. I am painfully anal about this because I've seen students easily lose track of signs because they don't habitually associate the sign with the number in their thinking. They associate signs with the operation in their minds and the numbers somehow exist separately from that.

I think another way this is problematic is that when they start thinking about factorization, they will look at something like -12 and not know what to do with the negative. They don't easily grasp that it could be +1, -1, +2, -2, +3, -3, +4. -4, +6, -6, +12, -12 if they are still thinking about the negative as a operation. The reason why I think this is really important when it gets you start to have to work quickly with quadratic equations.

No, I do see the issue a fair amount with kids who aren't in my family, so I know what you mean. I just know that I personally tackle this issue by verbalizing it and not by thinking of it as multiplying by the negative. I did see DD8 have this issue once or twice a few months ago but we talked it out and she now no longer makes the mistake. 

We did negatives with DD8 a looong time ago (I think she was 5), so she does understand that adding a negative is the same as subtracting a positive and she's fluent at multiplying them. I just know that's not really how she models something like 3 - 5(x-2) -- she definitely thinks of that as "5 copies of x-2 taken away from 3." She'd be able to explain to me why it's the same thing as 3 + (-5)(x + (-2)), but that's a later stage understanding for her, not her intuitive way of grappling with the expression.

[frogger]

On 11/25/2020 at 3:58 PM, Calizzy said:

This is a really interesting and helpful. Personally I struggle with multiplying by the inverse instead of just dividing, since dividing just seems easier to me. But probably since she’s learning from scratch and doesn’t have the baggage of already learning a different way dd can catch on. But I do wonder about what @EKS said about videotext over complicating (or over explaining) things. Dd is very straight forwards when it comes to math, she doesn’t want to know why, she just wants to memorize the formulas. I know that’s not the current approach in mathematics, but she is what she is. 

If she can remember that multiplying by 1/8 is basically dividing it will actually help when she has fraction coefficients. That hung up one of my children. I did explain we were dividing by the numerator to get rid of it and multiplying by the denominator but doing both steps at once when we multiplied by the reciprocal.

 

It might actually be easier (for newbies without baggage) to think of it fractionally to begin with. If I have 8x, I only need 1/8th the total to get down to one x. Especially if they understand fractions properly. Moving to 3/4x should be easy then. 🤷

[frogger]

On 11/25/2020 at 3:58 PM, Calizzy said:

This is a really interesting and helpful. Personally I struggle with multiplying by the inverse instead of just dividing, since dividing just seems easier to me. But probably since she’s learning from scratch and doesn’t have the baggage of already learning a different way dd can catch on. But I do wonder about what @EKS said about videotext over complicating (or over explaining) things. Dd is very straight forwards when it comes to math, she doesn’t want to know why, she just wants to memorize the formulas. I know that’s not the current approach in mathematics, but she is what she is. 

Ok, back to respond to the actual curriculum. Videotext worked great for my oldest but I didn't ever help him. He did it by himself. The youngest likes it because the videos and problem sets are very short and very incremental BUT they do teach differently then I learned and sometimes I have to really take time to figure out the point of the lesson etc now that I'm actually helping someone get through. 

Forester is very good at sneaking in the why while "doing". I'm very impressed with the textbook layout. There are videos at mathwithoutborders.com to go with Foresters since you are looking at videos. 

[Not_a_Number]

2 hours ago, frogger said:

It might actually be easier (for newbies without baggage) to think of it fractionally to begin with. If I have 8x, I only need 1/8th the total to get down to one x. Especially if they understand fractions properly. Moving to 3/4x should be easy then. 🤷

I think it's the "understanding fractions properly" thing that would trip me up, lol. In my experience, kids are far better into dividing by 8 than at multiplying at fractions. 

[frogger]

1 minute ago, Not_a_Number said:

I think it's the "understanding fractions properly" thing that would trip me up, lol. In my experience, kids are far better into dividing by 8 than at multiplying at fractions. 

True, but long term it is important to really understand fractions/ratios because you are way more flexible with your math and understanding concepts further down the line then if you don't understand them but then again I have a tendency to over explain to my children, so there is that. 😁

[Not_a_Number]

Just now, frogger said:

True, but long term it is important to really understand fractions/ratios because you are way more flexible with your math and understanding concepts further down the line then if you don't understand them but then again I have a tendency to over explain to my children, so there is that. 😁

Oooh, I can't say I disagree there. It's just that in my experience, putting together TWO ideas kids are shaky on is a recipe for disaster. You already have a variable in the equation, which makes things much harder for kids -- now we have to reason about EVERY number at once, not just one number. Plus you have to multiply by a fraction, and maybe you're still iffy on that, too. So then doing the two together can be overwhelming for kids. 

[Lecka]

Skipping to the bottom..... I have told my oldest son it’s easier to always multiply by a reciprocal, because to me it makes it easier when working with fractions.  By A LOT.

But for very beginning — I think it’s fine and could be needed for understanding what is going on.

But when getting into wanting to do stuff — I think it’s much easier.

So — here is the thing to me — haven’t many kids been introduced to this in pre-Algebra?  And then maybe in Algebra I they are expected to be working with fractions, not just understand the concept?

To me — I think it would be appropriate for Algebra if there had already been an introduction in pre-Algebra or an introduction in general and it was already introduced previously.

I think if someone is being introduced — introduce them to dividing and subtracting.

If someone already has been introduced — then I think there is a time to focus on adding and multiplying, because I think it is easier and more straightforward at that point.

To me — I think it would be appropriate for Algebra.  
 

But if it is going to be a stumbling block then I would not be in favor.

This is all based just on me. 
 

I definitely was saying this to my oldest son when he was going further into Algebra I but I wasn’t saying it to him when he was first learning how to manipulate equations and might not have understood what was going on.  
 

Edit:  to me, bottom line, it just prevents confusion with fractions.  It makes it a lot easier to keep from making mistakes with  fractions.  It is something that helped me personally to do better in Algebra.

I have also told him to always think of negative signs as “the opposite of” — that is just something that has always helped me so much, and if I am confused, or forget something, I can just think of “the opposite of.”  “Multiply by the reciprocal” is like that for me also.  

Edited December 16, 2020 by Lecka

[Not_a_Number]

3 minutes ago, Lecka said:

If someone already has been introduced — then I think there is a time to focus on adding and multiplying, because I think it is easier and more straightforward at that point.

To me — I think it would be appropriate for Algebra.  

I have to say, I have never really thought of adding and multiplying being more appropriate for algebra than subtracting and dividing. If you understand what you're doing, they all make equal sense. You might want to keeping to adding and multiplying to keep things in order, but I think practicing doing all the operations on both sides is just as important. 

[Lecka]

Well obviously they make equal sense.

I am just saying — it is easier for me to always “multiply by the reciprocal” because then I am always doing one thing and can do it properly.  I don’t have to “sit and think” and get distracted and lose my place in what I am doing.

Of course it is the same as dividing, and *I do just divide if there are no fractions.*  But a lot of the time there are fractions, and it adds a step to divide by a fraction — or, well, it is the same to multiply by the reciprocal.  But I definitely introduce a place where I am likely to make a mistake if I do that.

 

[Lecka]

I think it’s important to know the concept, but then it’s also important to be able to solve a problem without making a stupid mistake.

I think it can take away an opportunity to make a stupid mistake. 

[Not_a_Number]

Just now, Lecka said:

Well obviously they make equal sense.

I am just saying — it is easier for me to always “multiply by the reciprocal” because then I am always doing one thing and can do it properly.  I don’t have to “sit and think” and get distracted and lose my place in what I am doing.

Of course it is the same as dividing, and *I do just divide if there are no fractions.*  But a lot of the time there are fractions, and it adds a step to divide by a fraction — or, well, it is the same to multiply by the reciprocal.  But I definitely introduce a place where I am likely to make a mistake if I do that.

I guess I think that at the BEGINNING, it's probably most important to encourage flexibility with equations and operations. Kids get the oddest ideas if you start out by making things "convenient." For instance, I'm trying to patiently teach kids that you can do ANYTHING to both sides of the equation (as long as it's defined), absolutely everything, and the textbooks that were trying to make things more compact by telling them you can't square root both sides are really getting in the way 😉 . 

So I think that at an early point in algebra, I'd really encourage all operations that make sense to the kid, to encourage logical and sensible variable and equation work. Later on, when you're having issues with keeping signs straight, you can introduce shortcuts. 

[frogger]

1 hour ago, Not_a_Number said:

Oooh, I can't say I disagree there. It's just that in my experience, putting together TWO ideas kids are shaky on is a recipe for disaster. You already have a variable in the equation, which makes things much harder for kids -- now we have to reason about EVERY number at once, not just one number. Plus you have to multiply by a fraction, and maybe you're still iffy on that, too. So then doing the two together can be overwhelming for kids. 

Which is why taking 1/8 of something (you can even draw physical X's on the board and show you are taking 1/8th of them makes sense. Really, it doesn't matter. I, as a teacher, like to have multiple tools under my belt to explain a concept in case one way doesn't work. 

 

Dealing with reciprocals would, of course, be a second step. You do explain one thing at a time. 

Try whichever way you like and switch if the kid isn't getting it. 👍

[Not_a_Number]

Just now, frogger said:

Which is why taking 1/8 of something (you can even draw physical X's on the board and show you are taking 1/8th of them makes sense. Really, it doesn't matter. I, as a teacher, like to have multiple tools under my belt to explain a concept in case one way doesn't work. 

I know, I know, but in my experience variables make kids skittish. I had tons of kids in prealgebra who could do "What is the supplementary angle to (fill in any number)?" very well but weren't quite sure what the supplementary angle to x was. 

I do agree with you about having more than one way to explain it!! I would just not hesitate to stick to division if multiplying by a fraction was confusing for now, though 🙂 . 

[mom31257]

I have found my students understand solving equations better when they are "undoing" the operations that were done to the variable.

When you have have 2x - 4, 4 was subtracted from 2x. They add 4 back. 2 was multiplied by x, so they divide by 2. As far as when 2/3 is a coefficient, they should have learned long ago that dividing by a fraction is multiplying by a reciprocal. If they don't have that down, they need more review in fractions before moving on anyway. 

As to the original question, I have several algebra books. Most do not teach the method Video Text does. One presented the idea of multiplying by a reciprocal along side dividing but continued with dividing in the rest of the book. 

 

 

[Not_a_Number]

27 minutes ago, mom31257 said:

I have found my students understand solving equations better when they are "undoing" the operations that were done to the variable.

When you have have 2x - 4, 4 was subtracted from 2x. They add 4 back. 2 was multiplied by x, so they divide by 2. As far as when 2/3 is a coefficient, they should have learned long ago that dividing by a fraction is multiplying by a reciprocal. If they don't have that down, they need more review in fractions before moving on anyway. 

As to the original question, I have several algebra books. Most do not teach the method Video Text does. One presented the idea of multiplying by a reciprocal along side dividing but continued with dividing in the rest of the book. 

Yeah, this matches my experience.

[Lecka]

Somehow I have just remembered why I liked making the “dash” mean “the opposite of” and for a long time (2 years?) I did do only “add the opposite” and not subtraction when I did written work.

Because sometimes I would, I don’t know, be distracted for a moment, and then just get this feeling like I didn’t know if a “dash” meant  to subtract or to work with a negative number.  I might have to start all over.  I think I had some understanding — but sometimes I would just get confused and it would be frustrating. 
 

Then if I knew that “the dash” would always mean “the opposite of” because I was “adding the negative” for subtraction — which would have a “plus” sign and be very obvious — I just didn’t have to worry about that.

 

I would never say “you CAN’T use subtraction” or “don’t teach subtraction” — but honestly I think it was helpful to me to just use “add the opposite,” and never have to wonder what “the dash” was.

It could keep from getting to a step and having to go back and look at the previous line or lines to go back and see “is this negative or subtraction” because — if I got distracted, or had to do some other steps and had lost track of things, I could get to that place of not remembering what I was doing with the dash.

Anyway — I think it’s easier.

And for the fractions thing — to me if you would write out a line showing “divide by the fraction” and then another line “here we are multiplying by the reciprocal because that is how you divide” — why not just skip that line?  Or would that be done mentally by people?  I think it’s just — a relief to skip writing another line and introduce another place to make a mistake or become confused.  
 

To me saying an Algerba student should understand that conceptually is more of an argument for showing them it is easier that way.

I am not — like — to the point I would argue with someone who had a way that worked better for them.  Or insist it was the best way.  
 

I do like it, though.  Maybe I was taught more this way, because to me it is obviously easier.
 

Today I don’t even really see how I would ever have confused the dash for minus or a negative number, but I remember that I did.   

Edited December 18, 2020 by Lecka