Humor me please. Pre-algebra rec that is not AoPS?

[FairProspects]

What are some solid non-AoPS recommendations for pre-algebra for a younger student? If BA is any indication, ds does not currently have the emotional maturity to handle the frustration/change direction style of AoPS. We are working on this, and I do think he will eventually end up in AoPS materials with some more maturity (perhaps after another pre-algebra program), but if I was to go another direction for now what would I look at?

[Hoggirl]

My ds did Pre-Algebra in 4th grade. We used Chalkdust. In fact, we used Chalkdust until he enrolled in his charter school. Very solid program.

[Dmmetler]

LoF Pre-Algebra plus Key to Algebra (and various other resources used as needed-DD likes the Algebra Survival Guide and Painless Algebra as well, and I have some other problem solving and logic materials, too) seems to be working here. I don't think DD7 is ready for AOPS yet either-but is ready for more challenge. I'm thinking we'll either do Pre-algebra twice, or perhaps bridge over to AOPS later.

[mathwonk]

just a guess, but i would surmise that saxon, although it is not high on my list mathematically, might be good for a low stress level.

[In The Great White North]

He's how old? 7 or 8? I wouldn't use Saxon for him. How about Hands on Equations? MUS? LOF? Khan Academy?

[Crimson Wife]

Have you looked at Singapore Discovering Mathematics 7 (formerly DM 1)? It is challenging but doesn't have the "discovery" approach of AOPS. It is very similar to Singapore Primary Mathematics in presentation only the challenging problems are in the regular problem sets rather than being segregated into the IP books.

[FairProspects]

He just turned 8. I probably wouldn't jump him until the 2nd half of the year, but he's an interesting one. He is not a fast processor, so he doesn't whizz through lots of material or complete tons of problems, but he just jumps in terms of conceptual understanding. I have HOE and I'll admit I haven't dug into it yet, but he said today that he is bored with the BA variables chapter and I think he's right. Sometimes he just wakes up and knows things. So, if HOE is like BA 3C variables, he wants to do more interesting things than that (according to him).

 

There has to be a pre-algebra that is not super repetitive, but also not at the problem-solving level of AoPS. Ds sometimes feels like AoPS is designed to trick him, which makes him mad (again, immature), and I'd rather he practice getting used to that style with concepts he knows (like BA) while moving ahead with a more direct instructional approach conceptually at the same time. Isn't there something out there like that? What did people use before AoPS pre-algebra?

[wapiti]

Supposedly, the beginning chapters of Jacobs algebra are prealgebra. I just started a thread today asking about it, because my ds9 is not making the most of AoPS Prealgebra right now due to his age. My new plan is to have him do the chapter reviews for MM6 (except for geometry and probability & statistics, because I prefer the way they are handled in AoPS) and then start Jacobs.

 

At some point - I'm not sure when, perhaps halfway through Jacobs - I want him to do those geometry and probability chapters in the AoPS Prealgebra book, ch 9-15. My impression from this past year, when my dd did AoPS Prealgebra, was that those chapters were easier to get through than the first half of that book. Or, maybe I have him do those chapters after Jacobs algebra, and before AoPS Intro to Alg, so he'd be a bit older by then.

 

Other prealgebra options include Dolciani's Prealgebra, an Accelerated Course and Russian Math 6.

Edited September 26, 2012 by wapiti

[BlueMorpho]

My older son used Saxon Algebra 1/2 when he was 7-8 and it worked well for him -- the only time we've ever used a Saxon math book. He was also a strong reader and did it mostly on his own with me giving the homeschool packet tests after every 4th lesson. He corrected his own problems and figured out if his mistakes were conceptual or careless (usually they were "copied wrong"). We took breaks and did scattered chapters from Harold Jacobs's Mathematics: A Human Endeavor, and that way stretched pre-algebra out to 1 1/2 years.

 

AoPS didn't have algebra, much less prealgebra back then. He did VideoText Algebra next, then when he was 10 he started taking AoPS online courses.

 

If you want to introduce problem-solving along the way you might look at some of the MOEMS books -- they sell them on the AoPS website. They introduce strategies for solving various types of problems that come up in contests and AoPS courses.

 

I'm using the AoPS Prealgebra book with his younger brother. He's not as mathy as his older brother (he's almost 11) but he really likes it. Likes the videos, too.

[EKS]

We used Derek Owens for 4th grade prealgebra. It worked beautifully! By the end, my son was ready for AoPS prealgebra or a regular algebra program (like Jacobs).

[Beth in SW WA]

Dd9 started alg at age 8 with HoE, TT Alg 1, MUS Alg 1 (second half), Crewton Ramone and our online tutor, Rachna, who did SM cwp/ip with Abi. She didn't do prealg formally.

She did basic math and then algebra. It worked well. :)

 

ETA: Dd9 will be doing 'alg' for a few years. No rush.

 

Dd8 is playing with prealg concepts with Crewton Ramone. She is doing HoE while doing basic 4th grade math (just starting decimals this month).

Edited September 26, 2012 by Beth in SW WA

[JeanM]

My older ds used Singapore primary math 1 through 6, did LOF Fractions and Decimals & Percents, and then moved into algebra.

 

My younger ds is not as advanced as older ds was. He used Singapore 1 through 5, and is now working through AOPS prealgebra.

[FairProspects]

Thanks! So far I am liking the look of Derek Owens.

 

Is Dolciani really as bad as the first few Amazon reviews indicate? Some of them state that they wouldn't use it for a child even with LDs because info is so bad and it contains so little real mathematics.

[boscopup]

Dolciani definitely teaches math. It's very thorough. It's just DRY and has a gazillion problems. Very much traditional textbook style.

 

I'd suggest either doing SM through grade 6 and then jumping into Jacobs Algebra or doing something like MUS Algebra as prealgebra and then a more difficult algebra.

 

Have you looked at the sample of Jacobs Algebra on Google Books? It's pretty lengthy, so you get a good feel for the style of lessons. The only thing I didn't realize was how HUGE the book is. I don't know if that would matter to your son or not. There are plenty of words, but it's interesting and has some humor in it. Explanations are good, and they use the balance to illustrate linear equations (like MM does in grade 4+). The number of problems seems reasonable (I would only assign section I and III OR section II and III, not all 3... Sections I and II are the same types of problems, and Section III is review). It's not a whole page of one type of problem.

[Kathy in Richmond]

I'd suggest either doing SM through grade 6 and then jumping into Jacobs Algebra

 

:iagree:This is what I would suggest, too. Use a whiteboard, perhaps, if the size of the Jacobs text puts him off? I love Dolciani, (and it's definitely real mathematics), but I think that it would be too dry for a boy your son's age.

[FairProspects]

So, if I go the Singapore route, do I just let him test out of what he knows? We are still going to take a few weeks to finish up the BA we have, but I discovered last week while playing around with math with him that he had taught himself concepts in Singapore 4 & 5. Other than the placement tests, do I just give him some problems and move on if he gets them right? I just have a hard time with him jumping and figuring out where to place him because he is all over the place on concepts. We may be doing 4th grade fraction topics, but 5th grade decimals, etc.

[boscopup]

So, if I go the Singapore route, do I just let him test out of what he knows? We are still going to take a few weeks to finish up the BA we have, but I discovered last week while playing around with math with him that he had taught himself concepts in Singapore 4 & 5. Other than the placement tests, do I just give him some problems and move on if he gets them right? I just have a hard time with him jumping and figuring out where to place him because he is all over the place on concepts. We may be doing 4th grade fraction topics, but 5th grade decimals, etc.

 

Sometimes I have DS do some problems on the white board to demonstrate that he knows how to do them.

 

Since DS has been doing Life of Fred Fractions, he's already recently worked the concepts of fractions taught in 5A, so instead of making him do ALL those exercises, I had him do each "Practice" section. If he understands the concepts, those practice sections end up being reinforcement, without being a gazillion of the same type of problem that he already knows how to do. If he misses a certain type of problem consistently (hasn't happened yet), he'd need to go back and do the exercises associated with that type of problem. Oh, and I've been going over the type of problems from the textbook at the white board before having him do the practice section. So it's still teaching sort of, but more of a skimming through the teaching, if that makes sense. It's kind of like what I did with Math Mammoth much of the time.

 

I also have often been known to teach beyond what the text is teaching. For example, when introducing fractions back in MM3 or something like that, it had kids adding 2/6+2/6=4/6, and I had him reduce his answer. Reducing wasn't introduced at that grade level, but my son was ready for it and was actually naturally wanting to do it (noticing "Hey, 2/4 is the same as 1/2!"), so I taught him to do so and encouraged it. I often do stuff like that when teaching him, because I know he can move faster than the grade progression allows.

 

So yes, concept wise, we are also all over the place (well, he's solidly at 5A level or above in each concept), and when we get to a concept we've done to that depth already, we just practice it a bit as review and move on. We don't do ALL the exercises associated with that unit if we don't need to. Some concepts, I can tell he just needs more practice to be able to do it *quickly*, and we go through those exercises for that purpose (long division was a good example - he knows how to do it, but doing it quickly required more practice).

[FairProspects]

Sometimes I have DS do some problems on the white board to demonstrate that he knows how to do them.

 

Since DS has been doing Life of Fred Fractions, he's already recently worked the concepts of fractions taught in 5A, so instead of making him do ALL those exercises, I had him do each "Practice" section. If he understands the concepts, those practice sections end up being reinforcement, without being a gazillion of the same type of problem that he already knows how to do. If he misses a certain type of problem consistently (hasn't happened yet), he'd need to go back and do the exercises associated with that type of problem. Oh, and I've been going over the type of problems from the textbook at the white board before having him do the practice section. So it's still teaching sort of, but more of a skimming through the teaching, if that makes sense. It's kind of like what I did with Math Mammoth much of the time.

 

I'm pretty sure this is the wouldn't work for ds though. With his other difficulties, making him do additional problems he already knows how to do is not a review, but an exercise in exacerbating his LDs to the point of him giving up (and it would slow his speed to glacial when he his already bored). That is actually why I've chucked Singapore in the past, there are just too many problems. If I give selected problems to him orally though, he can frequently answer even the toughest ones. I'm going to have to carefully select problems for him and learn to not worry about completing a section as long as he gets it. I may start looking into that math software from AoPS so he can type math too.

 

And doing something "quickly" is never going to be a goal or an accomplishment here!

[Crimson Wife]

So, if I go the Singapore route, do I just let him test out of what he knows? We are still going to take a few weeks to finish up the BA we have, but I discovered last week while playing around with math with him that he had taught himself concepts in Singapore 4 & 5. Other than the placement tests, do I just give him some problems and move on if he gets them right? I just have a hard time with him jumping and figuring out where to place him because he is all over the place on concepts. We may be doing 4th grade fraction topics, but 5th grade decimals, etc.

 

If he knows the concepts, I would not use anything but the IP's.

[freesia]

Thanks! So far I am liking the look of Derek Owens.

 

Is Dolciani really as bad as the first few Amazon reviews indicate? Some of them state that they wouldn't use it for a child even with LDs because info is so bad and it contains so little real mathematics.

Oh, goodness, no. We love Dolciani here. It is very straightforward and clear. I didn't think it was drier than most math books. My ds was 10/11 when we used it. We are now doing the first chapters of Jacobs and he is flying through after a year of Dolciani Pre-Algebra.

 

You aren't suppose to do all the problems. We did odds of set A and B and occasionally did some set C.

[mathwonk]

one of my problems in answering is that i don't know what "pre-algebra: means, even though i have been told here before. it is not a mathematicians term. To a mathematician, the stuff before algebra is called arithmetic.

 

But having been reminded that Harold Jacobs includes what is thought of as pre algebra, I second that recommendation. It is both solid mathematically and un stressful. In fact it is downright fun.

 

If it is too heavy, you might consider cutting it up into 3 or 4 or more separate books. After all it's your book and you can do as you wish with it. Especially workable if you have a paperback, but they may not exist. (My 1600 page paperback Robert Parker guide to buying wine is on my shelf in two 800 page pieces.)

 

Or you could copy a chapter at a time for your child to work on, at some extra cost, and possibly losing the beautiful colors. I used to xerox one chapter at a time of our required calculus book when I taught the course, to avoid toting it all around.

Edited September 26, 2012 by mathwonk

[crazyforlatin]

I may be wrong, but Jacobs' text size is smaller than AOPS Pre-A, correct? I'm looking at the sample that boscopup linked above. What I see is smaller text than AOPS, but more white space. If the font size is just like MM, I could buy it, but it actually looks smaller. Has it bothered any younger kids here?

 

There's been so much discussion about younger kids and algebra that I'm thinking of a back-up plan. I have Dolciani Pre-A (comfortable text size) but....:blush:.... Just needed a back-up of the back-up plan.

[wapiti]

I may be wrong, but Jacobs' text size is smaller than AOPS Pre-A, correct? I'm looking at the sample that boscopup linked above. What I see is smaller text than AOPS, but more white space. If the font size is just like MM, I could buy it, but it actually looks smaller.

 

I'll compare text size when it arrives. It looks small to me. Fortunately, text size hasn't been a major issue for this ds generally (knock on wood!).

[boscopup]

I may be wrong, but Jacobs' text size is smaller than AOPS Pre-A, correct? I'm looking at the sample that boscopup linked above. What I see is smaller text than AOPS, but more white space. If the font size is just like MM, I could buy it, but it actually looks smaller. Has it bothered any younger kids here?

 

Text size is smaller than AoPS, and probably smaller than MM. I don't think my son will have any issues with it (text size doesn't bother him).

 

I got Dolciani out, and it has the same text size as Jacobs, exactly. So if Dolciani is ok, I would think Jacobs would be also.

[Dmmetler]

If text size is an issue, CK12 Beginning Algebra isn't a bad resource, and it's free. My DD still goes through phases where both book size and text size are an issue for her, which was why Key to Algebra was such a nice fit for pre-algebra, and I'm using CK12 at times because it includes more instruction, simply because, on an e-reader, book size and text size are non-issues.

[crazyforlatin]

I'll compare text size when it arrives. It looks small to me. Fortunately, text size hasn't been a major issue for this ds generally (knock on wood!).

 

Text size is smaller than AoPS, and probably smaller than MM. I don't think my son will have any issues with it (text size doesn't bother him).

 

I got Dolciani out, and it has the same text size as Jacobs, exactly. So if Dolciani is ok, I would think Jacobs would be also.

 

If text size is an issue, CK12 Beginning Algebra isn't a bad resource, and it's free. My DD still goes through phases where both book size and text size are an issue for her, which was why Key to Algebra was such a nice fit for pre-algebra, and I'm using CK12 at times because it includes more instruction, simply because, on an e-reader, book size and text size are non-issues.

 

Thanks, ladies! I just looked at Dolciani Pre-A again. I thought it was larger, but it doesn't matter since the type is fine for DD. I can't expect a Pre-A book to be like SM.

 

I've looked at CK, but I don't think DD will use an iPad for math. There are temptations in it and I haven't figured out a way to lock up everything and only allow iBook to open up. :tongue_smilie: And, she's dropped the iPad a couple of times. :glare:

 

I keep looking at Key to Algebra, ready to buy it, but I have made so many math purchases already. I do like the workbook format. I'm still trying to figure out what to do with the non-workbook feature of AOPS. DD has been writing in it!

[Mukmuk]

I'm late to the thread because I wonder if I have a very different type of child - he was/is highly manipulatives-based. Most posters' kids here are well past this stage. Still, I'll throw our experience out there as it may be helpful to some.

 

In context, my son is stealth dyslexic with working memory issues. I did think he was highly gifted in math because he had these marvelous insights about numbers. He had favorite numbers when he was young, they made shapes to him ("9 is so beautiful, it's like an even number to me", he said to my friend at a 5yo birthday party) and he had ideas of "dimensions" about them. Anyway, arithmetic proved to be something else, and I started thinking he *wasn't* that good after all! He's not sequential at all - he had trouble counting past 17 for the longest time. :001_huh:

 

In terms of curriculum, we started with Singapore Math till the grade 3 level. He hated it after awhile. Too much repetition. He loved just flying through curriculum and lapping up the ideas without actually doing the underlying problems. I have to say I was not cool about it! Anyway, we dabbled with Rightstart - he hated! At the time, he was not into manipulatives at all. Then we found the Zacarro books which were just fantastic. About this time (and this is perhaps just a year or two ago), I looked at Montessori Math and loved, but he was only okay with it. Then my approach became - if he doesn't know/cant recall a concept, I'd use manipulatives to help him out. I think having something to play with helped his recall skills. I found the Aimsedu website (every time I paste the link, I have to wait for approval before my post shows us, so I shall desist) which is very manipulatives driven, and he took to it well.

 

I hesitate to say we have success now, but he is totally loving AOPS. I'm very glad his neuropsych pushed me into it. It's still very early days, and a swallow does not a summer make (can you tell we've had a lot of U-turns in the past?)! He still loves patterns, weird connections, still has problems with recall (although a lot less) and we still use manipulatives for clarifying anything he doesn't understand. He hasn't gone through a formal pre-algebra program. But he's had no major problems so far (cross fingers).

 

I think I'm trying to say, there are alternative roads through pre-algebra!

[mommymilkies]

:lurk5: my eldest is good at math and a completely right brained thinker. However, there is too much wordiness in AoPS Pre-Algebra in my (probably crazy) opinion. She is getting frustrated with math for the first time in almost 2 years. So I was thinking of looking into something else for her.

[mathwonk]

as usual i advocate finding something that appeals to the child now, but not giving up on the possibility they will advance to appreciating the text that currently frustrates. i.e. words are useful, and eventually math is discussed in words. so get something else, but keep the AoPS and try to graduate to the level of discussing math in words.

[FairProspects]

as usual i advocate finding something that appeals to the child now, but not giving up on the possibility they will advance to appreciating the text that currently frustrates. i.e. words are useful, and eventually math is discussed in words. so get something else, but keep the AoPS and try to graduate to the level of discussing math in words.

 

Interesting that you say this, because just this week we have been making a "math words" chart, graphically organizing words by operation & meaning. It has made a huge difference in ds's ability to translate and simplify equations (as well as do word problems) and I am wondering if this might just have been one of those keys that enables him to open the door to more math understanding.

[slackermom]

Your DS might enjoy this lovely Usborne Illustrated Math Dictionary. I got it for DD when I realized the math terminology used at her school was leading to some real confusion and frustration.

[mommymilkies]

Interesting that you say this, because just this week we have been making a "math words" chart, graphically organizing words by operation & meaning. It has made a huge difference in ds's ability to translate and simplify equations (as well as do word problems) and I am wondering if this might just have been one of those keys that enables him to open the door to more math understanding.

 

I'm intrigued. Can you explain how you do this? My dd is very VSL and has weak verbal learning abilities, so she really needs to work on that. I had my dh look through what we were doing in AoPS last night and even he agreed that it was way too much. D they expect you to memorize all of those equations and formulae before moving on? It was overwhelming for both of us, and I've taken upper level maths and dh is a chemistry professor!

[mathwonk]

i am intrigued and puzzled. what equations and formulas are you asking about? I cannot think of any equations or formulas anyone needs until say the quadratic formula. even that can be derived.

[mommymilkies]

i am intrigued and puzzled. what equations and formulas are you asking about? I cannot think of any equations or formulas anyone needs until say the quadratic formula. even that can be derived.

 

1.2-1.3. Maybe more than that.

[boscopup]

i am intrigued and puzzled. what equations and formulas are you asking about? I cannot think of any equations or formulas anyone needs until say the quadratic formula. even that can be derived.

 

1.2-1.3. Maybe more than that.

 

She's talking about these:

 

Addition is commutative: a + b = b + a

Addition is associative: (a + b) + c = a + (b + c)

Multiplication is commutative: ab = ba

Multiplication is associative: (ab)c = a(bc)

Multiplication distributes over addition: a(b+c)=ab+ac

 

Frankly, these are things taught in elementary math. The child likely knows them already, but just hasn't used letters to represent the concepts. These properties are very important to know, but you shouldn't need to memorize "a+b=b+a". You should remember what commutative and associative mean and how the distributive property works. Those are foundational concepts. AoPS takes you through these and has you PROVE them. The whole book has you proving various concepts and properties before using them.

[FairProspects]

She's talking about these:

 

Addition is commutative: a + b = b + a

Addition is associative: (a + b) + c = a + (b + c)

Multiplication is commutative: ab = ba

Multiplication is associative: (ab)c = a(bc)

Multiplication distributes over addition: a(b+c)=ab+ac

 

Frankly, these are things taught in elementary math. The child likely knows them already, but just hasn't used letters to represent the concepts. These properties are very important to know, but you shouldn't need to memorize "a+b=b+a". You should remember what commutative and associative mean and how the distributive property works. Those are foundational concepts. AoPS takes you through these and has you PROVE them. The whole book has you proving various concepts and properties before using them.

 

Does AoPS Pre-A teach them just like that? That would be perfect for ds and would save us a lot of calculating re-call in BA. Maybe I should just order it, look through it, and re-think this whole thing.

[boscopup]

Does AoPS Pre-A teach them just like that? That would be perfect for ds and would save us a lot of calculating re-call in BA. Maybe I should just order it, look through it, and re-think this whole thing.

 

Yes, I copied the purple boxes from those sections. :D

 

Order it! Come to the dark side! We have cookies! :lol:

[FairProspects]

Order it! Come to the dark side! We have cookies! :lol:

 

Well maybe, but only if the cookies are vegan and gluten-free. :D

[mathwonk]

they should NOT be memorized. If i were teaching that basic stuff I would strive to make the principles memorable.

 

e.g. I would illustrate addition by lining up blocks. first 4 blocks then 2 more blocks. then i would ask why first lining up two and afterwards 4 gives the same thing? A simple rotation of the line of blocks shows that the length of the two is the same.

 

for AB = BA i would use the basic illustration of multiplication as area. i.e. multiplying 3 times 5 is illustrated by a rectangle with three rows and 5 columns, giving 15 blocks in all,. then again rotating the rectangle shows that it ahs the same size (area) as a rectangle with 5 rows and three columns.

 

distributivity is done by noticing that a rectangle with (6+2) rows and 3 columns can be split into two rectangles, one with 6 rows and 3 columns and another with 2 rows and 3 columns. then lots of practice on example problems. i believe harold jacobs illustrates all these with actual blocks instead of letters, which makes it much clearer.

 

associativity of multiplication is done by constructing a three dimensional block with edge lengths given by each factor. Then the volume is the same whether you think of it as multiplying the base times the height, or the side times the front edge.

 

memorizing is ok if it is fun or helpful, but requiring memorizing before proceeding further is a sure fire way to kill off progress, and discourage the student. when i was in college our algebra book had a list of 10 properties that made up a certain type of mathematical object (integral domain) on page one. i thought i had to memorize them before going further. what a disaster and joy kill. i hated that class. nobody had taught me how to read and study.

 

 

now i know for example that an integral domain is a number system where you can add and multiplky with the usual properties, but cannot necessarily divide. There is one other key point, it has the cancellation property, that AB = AC implies B = C, unless A = 0.

 

Equivalently, a product XY = 0 only when one of X or Y, or both, is zero.

 

ok? i still don't have them all memorized exactly but i know the two key facts, cannot always divide, and XY = 0 means either X=0 or Y=0.

 

actually what i remember first is that it is a number system that acts like the integers! ("integral domain" duh???)

 

so the point is to make the definitions memorable and meaningful. figure what really matters and key in on that.

 

in an integral domain you can solve equations by factoring, and setting the factors equal to zero separately. that's what matters.

 

 

another thing that matters is that an integral domain can always be enlarged to a number system where you CAN divide (except by zero)

 

e.g. the integers can be enlarged to the rationals. In fact this is equivalent. A number system with addition and multiplication can be enlarged to one where division is possible if and only if it has the cancellation property, i.e. iff it is an "integral domain".

Edited September 29, 2012 by mathwonk

[mathwonk]

by the way one of my heroes died last week (NY Times obituary today), the famous math author Irving Adler. He wrote scores of good books for all audiences especially children. I once spent a couple hours with him chatting delightfully about math games and other topics.

 

the houghton mifflin book level 5 that he apparently wrote, might be a good choice.

 

http://www.amazon.com/Houghton-Mifflin-Mathematics-Level-Student/dp/0618099794/ref=la_B001HOGD86_1_1?ie=UTF8&qid=1348958073&sr=1-1

 

 

heres his amazon page:

 

http://www.amazon.com/Irving-Adler/e/B001HOGD86/ref=ntt_dp_epwbk_0

[boscopup]

Well maybe, but only if the cookies are vegan and gluten-free. :D

 

You're on your own there! :lol:

 

they should NOT be memorized. If i were teaching that basic stuff I would strive to make the principles memorable.

 

e.g. I would illustrate addition by lining up blocks. first 4 blocks then 2 more blocks. then i would ask why first lining up two and afterwards 4 gives the same thing? A simple rotation of the line of blocks shows that the length of the two is the same.

<snip>

 

You can see the text we're talking about here, starting on page 4. Sounds like what you're describing. :D

[Kathy in Richmond]

they should NOT be memorized. If i were teaching that basic stuff I would strive to make the principles memorable.

 

e.g. I would illustrate addition by lining up blocks. first 4 blocks then 2 more blocks. then i would ask why first lining up two and afterwards 4 gives the same thing? A simple rotation of the line of blocks shows that the length of the two is the same.

 

for AB = BA i would use the basic illustration of multiplication as area. i.e. multiplying 3 times 5 is illustrated by a rectangle with three rows and 5 columns, giving 15 blocks in all,. then again rotating the rectangle shows that it ahs the same size (area) as a rectangle with 5 rows and three columns.

 

distributivity is done by noticing that a rectangle with (6+2) rows and 3 columns can be split into two rectangles, one with 6 rows and 3 columns and another with 2 rows and 3 columns. then lots of practice on example problems. i believe harold jacobs illustrates all these with actual blocks instead of letters, which makes it much clearer.

 

associativity of multiplication is done by constructing a three dimensional block with edge lengths given by each factor. Then the volume is the same whether you think of it as multiplying the base times the height, or the side times the front edge.

 

memorizing is ok if it is fun or helpful, but requiring memorizing before proceeding further is a sure fire way to kill off progress, and discourage the student.

 

Hi,Roy! Have you seen the AoPS prealgebra book? Here's an excerpt, including the lesson being discussed here. It actually does motivate the basic properties of arithmetic by drawing blocks and using concrete examples before summarizing them with variables (much in the same spirit as Harold Jacobs does:))..and btw, I find no mention of needing to memorize anywhere in the lesson. AoPS is one of the least memoriz-y curricula out there; if you do the lessons with thoughtfulness and understanding, there will be no need to memorize anything. JMHO.

Edited September 29, 2012 by Kathy in Richmond

[mommymilkies]

That was very a helpful walkthrough. Thank you. I think the problem is we went from curricula that didn't use many variables and did not really use the terms (commutative property, etc.) very often to now knowing what it *is* but having this huge daunting list that my overachiever feels compelled to memorize. Does that make sense?

[mathwonk]

thanks everyone. and hi Kathy! so the aops book does have the visual intuitive explanations, but as you say mommymilkies we obsessive types think we must memorize everything.

 

this is the next step to advanced learning i.e. when there is too much data, what should i concentrate on?

[Dmmetler]

We're not using AOPS (yet), but my DD7 LOVES the Algebra Survival Guide, which does a good job of pulling out some of these common formulas/concepts and explaining them in a light, humorous way. I think it's officially designed as a review/summary for people who have taken algebra and need a refresher, but it's working well to give her a little fleshing out of some of the topics/concepts. She also loves Painless Pre-Algebra and Painless Algebra.

[mommymilkies]

We're not using AOPS (yet), but my DD7 LOVES the Algebra Survival Guide, which does a good job of pulling out some of these common formulas/concepts and explaining them in a light, humorous way. I think it's officially designed as a review/summary for people who have taken algebra and need a refresher, but it's working well to give her a little fleshing out of some of the topics/concepts. She also loves Painless Pre-Algebra and Painless Algebra.

Thanks, I'll check my library!