Who wants to help me with algebra tonight?

[Critterfixer]

Got a theory problem that came up and it's driving me a little nuts.

 

The equation is 12=18 + x/6 

I solved it fine, but how come I can't do the multiplication principle first? I absolutely must do the Addition Principle first or the answer is wrong. Just do the simple thing first isn't cutting it for me...is it because the entire x/6 is the variable component, and I have to isolate it first to solve it? Is there a name for this that I should learn?

I seem to remember this used to make my hair curl when I did algebra in the past, and I'm simply not feeling like swallowing the procedural bit without some explanation.

 

So....

 

ETA: DH says it has something to do with not splitting up expressions? I multiplied the 6 by the entire thing (6 times 12 on one side and 6 times 18 + x/6 on the other and that works fine.

[desertflower]

I get the same answer both ways.

 

Multiplication principle first:  72 = 108 + x; 

                                    72 - 108 = 108 -108 + x;

                                           -36 = x

 

Addition principle first:    12 - 18 = (18 - 18) + x/6;

                                            - 6    =  x/6;

                                            -36 = x

 

I hope I'm right. 

[8filltheheart]

I would help, but I don't actually understand what you did originally. What do you mean by doing the multiplication principle first if it wasn't multiplying the entire expression by 6?

[Arcadia]

ETA: DH says it has something to do with not splitting up expressions? I multiplied the 6 by the entire thing (6 times 12 on one side and 6 times 18 + x/6 on the other and that works fine.

Whatever you do to the left hand side (LHS) you have to do to the right hand side (RHS)

 

So you could either subtract 18 on both sides or multiply by 6 on both sides to start.

 

ETA:

Easier if you see as

 

12 = 18 + x/6

(12) - 18 = (18 + x/6) - 18

-6 = 18 -18 + x/6

-6 = x/6

x = -36

 

or

12 = 18 + x/6

(12)(6) = (18 + x/6)(6)

72 = 108 + x

x = -36

[foxbridgeacademy]

So cool, I understood what she was asking and I was able to do the problem both ways and I got the right answer (I didn't make it past Pre-Algebra in H.S.).  I love HSing, it's taught me so much.

[Critterfixer]

I was trying to do it by using the multiplication principle on the variable component only. I did not do the entire expression. When you multiply the entire expression, it works.

So, is there a name for that rule? 

 

This was my error:

12=18 + x/6 

 (6)12=18+ x/6(6)

Obviously (to everybody in the world but me) you can't do that. It has to be the entire expression, and I'd like to know the name for that. The way I am rationalizing the rule in my mind is to picture x/6 as the entire variable, and since I don't know what it is, I can't multiply the other side by the variable that I do not know. But I'm sure there is a better explanation of why my bone-headed way didn't work. (Yes, I figured out how to do it, but that's not the same as having a name and a reason for why I have to do it that way.)

[wapiti]

Obviously (to everybody in the world but me) you can't do that. It has to be the entire expression, and I'd like to know the name for that. The way I am rationalizing the rule in my mind is to picture x/6 as the entire variable, and since I don't know what it is, I can't multiply the other side by the variable that I do not know. But I'm sure there is a better explanation of why my bone-headed way didn't work. (Yes, I figured out how to do it, but that's not the same as having a name and a reason for why I have to do it that way.)

 

I would worry less about the name for it (transformation?) than understanding the very important concept that whatever you do to one side of the equation, you must do to the other entire side.  If you multiply one side by six, but multiply only part of the other side by six, then you have not performed the same operation to both sides.  (If you do not perform the same operation to both sides, then the sides, which are supposed to be equal, will no longer be equal.)

 

As a practical matter, visualize each side of the equation as having parentheses around it before multiplying, as in Arcadia's second example above.  Where you have more than one term in the parentheses, as in (18 +x/6), then you multiply both terms times six according to the distributive rule.  

[Pegs]

x is the variable, but (18 + x/6) is the expression. The equation is a statement which tells you that the expression (18 + x/6) is equal to a term: 12.

 

You're applying an operation to terms on both sides of the equation, when you should be applying it to both the *expression* on one side, and the term on the other.

 

I think.

 

Eta: cross-posted!

[Critterfixer]

The DH knew it had something to do with the expression, but he couldn't remember the name for it. 

I suspect that in the recesses of my clearly dim intellect that I figure I am simplifying before solving. If it had a parentheses around it I would not have even thought about trying to dink with the variable in that way. But obviously they didn't come with parentheses. 

[wapiti]

I seem to remember this used to make my hair curl when I did algebra in the past, and I'm simply not feeling like swallowing the procedural bit without some explanation.

 

FWIW, you are very wise to not simply swallow a procedure without an explanation of the concept.

[Critterfixer]

I'd just like to not have it be a stumbling block when I get to teaching it... :tongue_smilie:

I do think that I am really having an issue seeing the entire expression on one side as equal to the entire expression on the other side, and therein lies the issue.

[Arcadia]

I do think that I am really having an issue seeing the entire expression on one side as equal to the entire expression on the other side, and therein lies the issue.

When you are adding or subtracting to both sides, think of it as balancing a balance scale (beam balance). The balance scale has to be at equilibrium

 

When multiplying or dividing both sides, think of it as baking cookies. If you double the quantity of flour, you have to double the quantity of sugar, milk and eggs.

[wapiti]

I see from your sig that you have MM - try the bar models on page 108 of MM 5A (new version). 

 

I like Arcadia's recipe example - I'm supposed to be mixing cookie dough right now  :tongue_smilie:

[Critterfixer]

I have some MM workbooks, not the whole series. Just topic stuff. But I'll see what I have. It's quite possible I do have some things on that. I think for me, it will be better for me to simply put things in the parentheses to remind myself to think of it as a whole. I have no problem seeing it when it is nicely enclosed. I never violate parentheses. :laugh:

 

 

[RootAnn]

Perhaps it would help you would be to put together something that looked structurally like the equation down but with real numbers.

So, perhaps something like:

 

6 = 4 + 12/6

 

Then, I'd do to it whatever I planned to do to the other and see if it worked. While Order of Operations says to divide first, you decide to multiply by the 6 to get rid of the fraction.

 

As soon as you do that the way you originally thought to, you realize it is no longer a true statement.

6*6 ?=? 4 + (12/6)*6

36 ?=? 4 + 12 * 6/6

36 not equal to 4 + 12

 

Then, you go back and figure what you could have done to make it true and find....

6*6 ?=? [4 + (12/6)]*6

36 ?=? 4*6 + (12/6)*6

36 ?=? 24 + 12*(6/6)

36 = 24 + 12

[Critterfixer]

Yep, I did that, RootAnn. It isn't hard to prove it to yourself, that's sure. 

Oddly enough, it is the whole order of operations things that rears it's ugly head with this kind of problem, because I see that division bar and want to take care of multiplying first (except I don't think of the expression as a whole and did only part of it). The idea of removing the whole number by subtracting it from both sides first just flies in the face of my desire to multiply before adding or subtracting. I think that is why it helps me to think of the x/6 in my problem as being a variable with two parts-the x and whatever the division by the 6 does to that number. I just don't know if that is wrong-headed. When I think of it that way (as a fraction that is unknown and not a simple unknown) I don't seem to feel the need to multiply and get rid of it. 

 

 

When multiplying or dividing both sides, think of it as baking cookies. If you double the quantity of flour, you have to double the quantity of sugar, milk and eggs. 

 

 

 

If I was a reasonable baker I would do that. I'm kind of well known for not completely adhering to that principle, however. :D I'm constantly playing with quantities. Always tinkering...

[Pegs]

I'm constantly playing with quantities. Always tinkering...

So, tinker! Try coming at a few more questions from both directions, as a PP demonstrated, upthread. :)

[Critterfixer]

Okay. Taking my test tonight before moving on to Polynomials. I've got this problem, and here is my thinking process.

 

x/3 - 1=20  

First I ponder the answer. What number take away 1 will give me 20?

I think x/3 has to be equal to 21, because that's all that it could be.

Then I think to myself, if x/3 is equal to 21, then x must be (21)3=63. 

 

I still think that thinking of the x/3 as a compound unit is helping me not mess with it before it's time to crack into it...

 

ETA: Got a 90%, and only really missed one calculation. Stupid errors naturally. Got to work on those.

[wapiti]

Two suggestions that may or may not help you:  try thinking of x/3 as (1/3)x.  When you want to solve for x, think, "how do I get x by itself," i.e. what can I get rid of first.

[Btervet]

It may help to realize that when you do *anything* to an equation, there are always parenthesis around each side, we just don't always show them to save time. 

 

So for

3 = x + 1

I would subtract 1 from each side, the whole side

(3) - 1 = (x+1) - 1

then drop unnecessary parenthesis

3 - 1 = x + 1 - 1

2 = x

 

The same is true if there is multiplication

Doing addition first

2 = x/2 + 1

(2) - 1 = (x/2 + 1) - 1

2-1 = x/2 + 1 - 1

1 = x/2

2 = x

 

Same problem doing multiplication first

2 = x/2 + 1

2*(2) = (x/2 +1)*2

4 = (x/2)*2 + 1(*2)       <-- distribute the 2 

4 = x +2

(4) - 2 = (x+2) -2

2 = x

 

We normally ignore the parenthesis because they don't make a difference with addition/subtraction. However, with multiplication, we distribute, so they do matter. I hope this helps. 

[Critterfixer]

 

We normally ignore the parenthesis because they don't make a difference with addition/subtraction. However, with multiplication, we distribute, so they do matter. I hope this helps. 

 

Actually, that does help. Because if I can always talk myself into seeing the fences (parentheses) I will remember to distribute. That's evidently solid in my mind.

 

I tried my thinking with the original equation, and it helped a lot to think of what on earth could make 18+x/6 equal to 12, realizing that first, it has to be a negative number and second that the number must be -6. Therefore, for x/6 to equal a -6 the x has to be -36. 

[Arcadia]

Okay. Taking my test tonight before moving on to Polynomials

For polynomials, group like terms together first. It would help reduce careless errors.

[8filltheheart]

I actually think Hands On Equations might be helpful for you. The problems would help you visually see the exact type of operations you are doing and help you apply the concepts to word problems. These concepts are fundamental to all things moving forward, so good for you for wanting to make sure you understand.

[Critterfixer]

We're talking really stupid errors. Like not writing the problem correctly (saw 10x=0 instead of 10x=10), and on one problem I seriously wrote 5.100 and then proceeded to divide it. On the word problem I missed, all my calculations were correct and I totally missed the part where they said they only wanted the length of the rectangle.

Tell you though, it makes me more patient with my boys and their math errors that I ever was before. Their dear old mum is every bit as careless as they are, even when trying to be very, very careful. I swear, it's like carrying a big full pot of soup to the table and saying "I'm not going to spill this. I'm going to be very careful to not tip it as I walk, or set it down on the edge of the table." You just know what's going to happen. :closedeyes:

[Critterfixer]

 

I actually think Hands On Equations might be helpful for you. The problems would help you visually see the exact type of operations you are doing and help you apply the concepts to word problems. These concepts are fundamental to all things moving forward, so good for you for wanting to make sure you understand. 

 

 

I've had that on my wish list for the last two years. Just never felt justified in getting it. Now may be a good time. And if my boys are as visually oriented as their mum, I could probably use it with them as well. They are already leaning over my shoulder to watch me battle my way through equations. So they are interested... :D

[Arcadia]

I've had that on my wish list for the last two years. Just never felt justified in getting it. Now may be a good time. And if my boys are as visually oriented as their mum, I could probably use it with them as well.

If you have an iPad, there is hands on equation level 1 lite which is free. Good for seeing what it's about before buying.

 

It's on android and kindle too

http://www.borenson.com/tabid/1581/Default.aspx

[Pegs]

I bought HoE worksheets and a manual as PDFs so I could use them with DS, using manipulatives that I've crafted or sourced myself. That might be a cheaper way for you to try the physical system out?

[Critterfixer]

I'll probably just give myself an early Christmas present. And the boys will probably love it for some fun math. The great thing about self-educating is that you get to try several things and figure out what works and what doesn't and have fun doing it. Looks like I can probably pick it up at a good price, and the extra student kits are pretty inexpensive, so I think I may just pick it up to play with. 

[Critterfixer]

I was also thinking of getting Zaccaro's Real World Algebra for myself as well. I'm finding that the bane of my existence, the dreaded word problem, still has the ability to make me want to tear my hair out.  Might be fun for me to work through as well.

[RootAnn]

My kids like DragonBox 5+ and it also drills in the "do to one side what you do to the other side" principal. It won't let you bend that rule - and you have to distribute your multiplication/division stuff - but you can get yourself into a heck of a mess if you do things in the wrong order. The only problem is you & the boys might be done with it in 30-45 minutes and ready for Dragonbox 12+ . . More money out the door.  :laugh:

They had DragonBox 5+ for free one day on Amazon last year. I think....

[Incognito]

I was going to suggest Dragonbox also.  It's fun and shows the principles nicely.

[Critterfixer]

Basically, if it requires a) a popular device (ipod, etc), an internet connection that functions at something other than snail-pace, or a computer that isn't this laptop missing one key that I use to access the forum, it ain't gonna happen. But thanks for the suggestions anyway, guys. 

 

[RootAnn]

FYI - I have dragonbox on our computer. We don't have any "popular devices" in this house. ;-) It doesn't require the internet connection except for the initial buy & download part. Even our one cell phone is dumb.

I waited patiently until they had it for Windows so we could use it. 

[Critterfixer]

I couldn't download it, RootAnn. I can't download a song. It's really that bad, and not getting any better. I've checked.

 

The good thing in my case, I guess, is that I really dislike screens when it comes to reading anything, and I never was much of a game person. I won't play them on the computer at all. 

When it comes right down to it, I'm kind of a "hand me the book, and stand back" kind of person. If I get stuck on a math problem, I'm likely to save it and hash it out with the DH on the whiteboard, but most of the time I'd rather draw it out and mess with it. I looked again at the Hand's On Equations, and remembered why I have always kicked it off the list before I hit buy...I don't like little pieces that have to be bagged or get lost. I'll probably still get it though, because the boys will enjoy it, and I can probably tolerate something with manipulatives if I plan on teaching with it.

 

Moved on to Polynomials yesterday and am enjoying the idea of all these first, second up to fifth degree equations. Makes me want to get a staff and a magic sword or something. :tongue_smilie:

[kiana]

Critterfixer, sorry, I was out of town and didn't see this in time. Two things that I didn't see mentioned:

 

Multiplying by 6 is just fine but you must apply the distributive law (is this the name of the thing you were missing?) to multiply the entire right side by 6.

 

In general, though, with equations such as this it is frequently simplest to work in *reverse* order of operations. That is, undo adding and subtracting first, then multiplying and dividing, then exponents if they exist, then parentheses if they exist -- the exact opposite of how you would do an evaluation.