Algebra Question

[happyhome]

If dd can solve an equation by writing out each step of her work properly and gets the right answer, does she also need to know which property was used where?

 

We are using Foerster's Algebra 1 as an algebra review over the summer after having done most of the AoPS Intro book (we'll finish it in the next couple of months.)  In Chapter 3, Foresters has very basic equations that are completely solved in the text but next to each step of the solution, they have a blank alphabetical list asking the student to list each axiom that was used in each step.  When I wrote down the equation on the white board, dd solved it correctly (without looking in the textbook.)  But when I asked her which "axiom" she used for each step....deer in headlights.  Should I be worried?

[Mike in SA]

Not unless explicitly requested. There are only a handful, but the concept is what matters, not so much the name of the property (for now).

[Gil]

Nothing to worry about. There are pretty much 5 basic axioms so you can easily learn them by name, I made my boys learn them because thats the way we roll, we did a lot of oral quizzes and played trivia games so they had to be able to be precise.

 

It is my opinion and preference that by the time you finish Algebra you know them so that when a book mentions it as a part of an explanation you can follow without missing a beat. My suggestion: don't make a big deal about it but do have her learn the names.

[creekland]

I wouldn't be worried - knowing the math is more important, but I would start teaching the names as understanding "mathese" is fairly important IMO.

 

Students often pick up on the terminology fairly quickly if exposed to it correctly.  I wouldn't have her try to memorize all the names at once.  Pick one to start with.  Once she recognizes that one, add another.  She knows the steps.  All she's doing is putting terminology with it to make (math) communication easier.

[Julie of KY]

I wouldn't worry about it.

[happyhome]

Forester's doesn't focus on just the big five.  In Chapter 3 alone.....Commutative for Addition, Commutative for Multiplication, Associative for Addition, Associative for Multiplication, Distributive over Addition and Subtraction, Additive Identity, Multiplicative Identity, Additive Inverse, Multiplicative Inverse, Reflexive, Transitive and Symmetrical.  

 

I taught her that Commutative, Distributive and Associative were properties.  Forester's calls them axioms.  I thought those were two different things? I was always taught axioms are assumed to be correct, no proof needed.  Properties can be proven for each operation.  He's almost using the terms interchangeably and all this new vocabulary is confusing her.  After AoPS, I thought her core understanding was basically solid but now I feel like we're totally out of sync.   What am I missing?

 

How do I simplify this and what does she really NEED to memorize....just the big 5:  Reflexive, Symmetric, Transitive, Additive and Multiplicative?  

[go_go_gadget]

It's not a common use, but it's not incorrect: http://en.wikipedia.org/wiki/Axiom#Non-logical_axioms

 

When she starts geometry, she's going to be citing axioms to justify every step of her work, all the time. She should have commutativity, distributivity, and associativity of both addition and multiplication cold, and transitivity comes up frequently as well. For the others, I wouldn't worry about memorization but would expect that given a worked problem and a list of axioms/properties, the steps could be correctly identified. Reflexivity is the hardest for me to remember because the others are pretty much apparent from their names.

[Twigs]

Nothing to worry about. There are pretty much 5 basic axioms so you can easily learn them by name, I made my boys learn them because thats the way we roll, we did a lot of oral quizzes and played trivia games so they had to be able to be precise.

…

Would you mind listing (or providing a link for) the 5 basic axioms?

I am sure I was taught them, but not as a list of 5.

Thanks

[regentrude]

I cannot imagine where it would ever be necessary to list the axioms/properties when doing algebra - unless for some stupid state testing.

She has to be able to use the properties and manipulate her equations, but making her justify every step? NEVER heard of that.

 

ETA: I use algebra on a daily basis. I have not the foggiest idea what the "property" is called that I am using in each step.

What matters is that she conceptually understands what she does and can narrate her solution.

[regentrude]

Forester's doesn't focus on just the big five.  In Chapter 3 alone.....Commutative for Addition, Commutative for Multiplication, Associative for Addition, Associative for Multiplication, Distributive over Addition and Subtraction, Additive Identity, Multiplicative Identity, Additive Inverse, Multiplicative Inverse, Reflexive, Transitive and Symmetrical.  

 

I taught her that Commutative, Distributive and Associative were properties.  Forester's calls them axioms.  I thought those were two different things? I was always taught axioms are assumed to be correct, no proof needed.  Properties can be proven for each operation.  He's almost using the terms interchangeably and all this new vocabulary is confusing her.  After AoPS, I thought her core understanding was basically solid but now I feel like we're totally out of sync.   What am I missing?

 

How do I simplify this and what does she really NEED to memorize....just the big 5:  Reflexive, Symmetric, Transitive, Additive and Multiplicative?  

 

They seem to be making something simple unnecessarily complicated.

She needs to know that multiplication and addition are commutative, and how to use the distributive property.

 

All this other stuff - I have never come across the terms, and knowing them is not necessary for excellent performance in math.

 

The reflexive and symmetric properties are common sense and a big deal only for mathematicians and logicians. There is no need to memorize that there is a name for "a=b means b=a". The transitive property is pure logic. If she understands that a=b and a=c means b=c, she's fine.

There is no need to know that x+0=x has a special name.

 

I find it unnecessary to make the student memorize any of these terms.

[g1234]

I told my daughter to skip the naming-axioms part in Foerster. It never came back to haunt her, even in later parts of Foerster. So, agreeing with others here.

[creekland]

They seem to be making something simple unnecessarily complicated.

She needs to know that multiplication and addition are commutative, and how to use the distributive property.

 

All this other stuff - I have never come across the terms, and knowing them is not necessary for excellent performance in math.

 

The reflexive and symmetric properties are common sense and a big deal only for mathematicians and logicians. There is no need to memorize that there is a name for "a=b means b=a". The transitive property is pure logic. If she understands that a=b and a=c means b=c, she's fine.

There is no need to know that x+0=x has a special name.

 

I find it unnecessary to make the student memorize any of these terms.

 

I'd argue (based upon what I see in our high school) that knowing how the associative property works is beneficial too.  Many students do not realize they can multiply or add more quickly by doing things in a different order than written (2+ 6 + 8  doesn't have to be re-written - commutative property - to add 8+2 first and do the mental math more quickly if one understands the associative property).

 

And names for all of these can be picked up quickly and easily just by using the terminology itself.  It'll be useful when communicating in math ("ok, now distribute your constant across your polynomial") and for some proofs in geometry.  It's definitely not necessary for just performance in math, but reading isn't necessary for public speaking either.  It's just useful in certain circumstances.  There's no real reason NOT to learn it in context IMO.  It's just not worth beating to death as a super important big deal.

 

[Mike in SA]

I have to add, it'll drive your college prof crazy to call the arithmetic properties "axioms."  Axioms are fundamental, whereas properties are verifiable behaviors.  It's how a mathematician determines whether a set and an operation form a group, et al.  The fundamental five must exist to form an algebraic ring.

 

The fundamental properties of a ring are:

 

Associativity  (an operation % is associative if a%(b%c) = (a%b)%c )

Commutativity  (an operation % is commutative if a%b = b%a )

Distribution  ( one commutative, associative operation % and another associative operation & exist such that a%(b&c) = a%b & a%c )

Identity  ( an operation % has an identity if there is a number a such that for any number b, a%b = b%a = b )  

Inverse  ( for the operation % and any number a, there is always an inverse b, such that a%b = b%a = the identity )

 

There is an additive and multiplicative version of each, except for distributive.  Reflection, symmetry, and transitivity are also properties of sets, though they are not normally considered among the basic arithmetic properties, as they have more to do with relationships, whereas arithmetic has to do with operations.

 

To give an exercise for clarity, real numbers and rational numbers are both rings.  However, natural (counting) numbers are not, because there are no negative natural numbers.

 

Edit: to reiterate, I do not think it necessary to know all this.

[regentrude]

I'd argue (based upon what I see in our high school) that knowing how the associative property works is beneficial too.  Many students do not realize they can multiply or add more quickly by doing things in a different order than written (2+ 6 + 8  doesn't have to be re-written - commutative property - to add 8+2 first and do the mental math more quickly if one understands the associative property).

 

Oh, sure, I forgot to mention this one. It is important to know how it works. It is not, IMO, important to know what it is called.

[go_go_gadget]

Oh, sure, I forgot to mention this one. It is important to know how it works. It is not, IMO, important to know what it is called.

 

I disagree. It's not uncommon for commutativity to be referred to without an accompanying example, so if a student doesn't remember what it is means they can't comprehend. I don't think most of them need to be memorized, as I said, but commutativity, associativity, and distributivity do. And again, most of them have such intuitive names that memorization is basically moot.

[Gil]

Would you mind listing (or providing a link for) the 5 basic axioms?

I am sure I was taught them, but not as a list of 5.

Thanks

Axioms may be the wrong word but what I meant were the following axioms/properties.

 

Associativity (of addition and multiplication)

Commutativity (of addition and multiplication)

Identity (additive and multiplicative)

Inverse (additive and multiplicative)

Distribution of multiplication over addition.

 

[happyhome]

Thank you all. What I'm getting here is as long as she's solving correctly and can tell/show me what she did, we're good. And along the way, it will help to ask now and again, "Do you remember what that's called?" She's a quick study so pointing things out to her in context won't be a problem. Forester's pile of terminology definitely threw us for a loop today. I like Gil's idea of trivia games. That would work. I could get the other two in on it too. Gil, could you explain what you did?

 

Thanks everyone for chiming in. It helps to hear from those with experience. I just truly didn't know. Foerster's is a respected text and frankly, I thought it would be a breezy review after AoPS. I'm finding the different perspective/emphasis to be both interesting and frustrating. Expect me to be back with more questions. We're only in Chapter 4 now.

 

Makes me wonder, is there a better way to review/check retention after AoPS? Are these methodologies just that different? SHE was worried that she didn't remember as much as she needed to. This seemed like a good plan to run a check. Now I'm wondering......hmmmm.

[go_go_gadget]

I think this is a perfectly great way to review after AoPS. It sounds like your DD remembered the concepts well and just not the names of things, which makes me think she could do with a little more dialogue while AoPSing in future. I'm in grad school for math so pedant has become my middle name, and I speak the names of the properties as we use them all the time. My kids get irritated with me for asking why b/a is the multiplicative inverse of a/b, but by doing so they remember that it's because 1 is the multiplicative identity in multiplication for the reals, which is because 1 is the element of the reals that can be multiplied by any other element to get the same element back (and the analogous argument for addition of reals). I start getting nervous when people can't explain things and just remember them, but that may just be that pedant in me.

[regentrude]

I'm in grad school for math so pedant has become my middle name, and I speak the names of the properties as we use them all the time. My kids get irritated with me for asking why b/a is the multiplicative inverse of a/b, but by doing so they remember that it's because 1 is the multiplicative identity in multiplication for the reals, which is because 1 is the element of the reals that can be multiplied by any other element to get the same element back (and the analogous argument for addition of reals). I start getting nervous when people can't explain things and just remember them, but that may just be that pedant in me.

 

Of course they should be able to explain the concept - but that can be done without throwing around terms like "multiplicative identity".

The only people who use these terms in their dealings with math are the mathematicians :001_smile:

In other fields, this terminology never even comes up. We physicists just take the math and use it and make lovely theories ;-)

[g1234]

BTW, Foerster does not continue doing that. It was only for a few sections. We thought those were dumb, but overall we LOVED the text and did not find it fixating on trivia like that.

[Twigs]

Axioms may be the wrong word but what I meant were the following axioms/properties.

 

Associativity (of addition and multiplication)

Commutativity (of addition and multiplication)

Identity (additive and multiplicative)

Inverse (additive and multiplicative)

Distribution of multiplication over addition.

 

:hurray: Thank you!  :hurray:

[go_go_gadget]

Of course they should be able to explain the concept - but that can be done without throwing around terms like "multiplicative identity".

The only people who use these terms in their dealings with math are the mathematicians :001_smile:

In other fields, this terminology never even comes up. We physicists just take the math and use it and make lovely theories ;-)

 

Yeah, well, some kids end up with more education on dairy cows than they might need in their adults lives because of their parents' expertise, and my kids know the multiplicative identity. ;)

[creekland]

Of course they should be able to explain the concept - but that can be done without throwing around terms like "multiplicative identity".

The only people who use these terms in their dealings with math are the mathematicians :001_smile:

In other fields, this terminology never even comes up. We physicists just take the math and use it and make lovely theories ;-)

 

And when kids get out on the job they also will mostly end up retaining what they actually use.  It may not even be basic algebra.

 

When they are still in school and learning, there's absolutely no reason they shouldn't be exposed to far, far more.  Perhaps they'll want to be a mathematician! 

 

At school I often get asked that age old question, "Why do we need to learn _____?  We'll never use it!"

 

I remind them that there are two very good reasons even if they never use it on a job.

 

1)  The more they learn at their age the more brain cells they turn on (vs getting pruned around age 25 due to not being used).  They will have those brain cells and can write over them, or be thankful when they get memory issues later on that they have more paths to work with.

 

2)  They might need to help their own kids with their homework a few years down the road.

 

I don't get arguments afterward.  ;)

 

Kids in high school are exposed to oodles of things.  They won't actually use many of them in detail later on in their jobs, but the knowledge learned is never wasted IMO.

[Momling]

We did that lesson... That section is about algebraic proofs and the idea that that algebra has universal principles which can be laid out to prove an answer. I think it helps to have a name for what you're doing. It's not as important as the *doing* part, but it has its use.

 

It's the same in geometry... Or knowing the names of things like "indefinite article" and "prepositional phrase" when learning another language. You can certainly learn without the terminology, but it gives you some worthwhile ways of thinking and talking about the subject.

[happyhome]

And when kids get out on the job they also will mostly end up retaining what they actually use. It may not even be basic algebra.

 

When they are still in school and learning, there's absolutely no reason they shouldn't be exposed to far, far more. Perhaps they'll want to be a mathematician!

 

At school I often get asked that age old question, "Why do we need to learn _____? We'll never use it!"

 

I remind them that there are two very good reasons even if they never use it on a job.

 

1) The more they learn at their age the more brain cells they turn on (vs getting pruned around age 25 due to not being used). They will have those brain cells and can write over them, or be thankful when they get memory issues later on that they have more paths to work with.

 

2) They might need to help their own kids with their homework a few years down the road.

 

I don't get arguments afterward. ;)

 

Kids in high school are exposed to oodles of things. They won't actually use many of them in detail later on in their jobs, but the knowledge learned is never wasted IMO.

Sorry to get back here so late...we took a much needed break before our summer studies begin. Creekland, I like your approach. I do expect a lot from my kids, probably much more than they'll ever "need," but I love to teach and they love to learn so onward we go.

 

This kid loves to teach so I had her write out all the properties on notecards, teach them to her younger brother and sister and create a match game to play when we're in the car. The properties are rather simple to understand so this has been fun for all and my 10 year old loves telling people that he knows algebra!

 

I am a little surprised at some of the other things she didn't remember/learn in AoPS though. She blanked on radicals yesterday. Yikes! I thought this would be a quick and easy review. We're definitely going to slow down and work each chapter this summer. Geometry won't start right away.

[happyhome]

We did that lesson... That section is about algebraic proofs and the idea that that algebra has universal principles which can be laid out to prove an answer. I think it helps to have a name for what you're doing. It's not as important as the *doing* part, but it has its use.

 

It's the same in geometry... Or knowing the names of things like "indefinite article" and "prepositional phrase" when learning another language. You can certainly learn without the terminology, but it gives you some worthwhile ways of thinking and talking about the subject.

That's exactly how I explained it to dd...through the grammar analogy.

 

We went to a conference this past weekend and I had the pleasure of talking with Dana Mosley from Chalkdust. He confirmed what most have said here, that nomenclature is interesting and helpful but by no means necessary to succeed in algebra. Understanding and being able to clearly explain the concepts, however, is critical. I think with a proper algebra review this summer, we'll have both.....I hope.