Adult education AoPS

[Kit]

Adult, home learner, almost 45 years since I last did any maths.

So.....my AoPS pre algebra got here, and I set off into chapter 1. I watched the relevant video, then did the questions and read the text. What an eye opener. I thought I understood all about the 4 processes....in fact it is about the only bit of maths I thought I really knew! 
I have battled through the chapter, re watching the videos as necessary but even so I only got about half the review questions correct. I have not attempted the challenge questions. I have re worked every single question I got wrong as I went along. I am excited about the different way of looking at maths rather than the plug and go method I learnt all those years ago, but I am finding it very challenging indeed. I am not yet ready to throw the towel in, but I do have to admit, it is tough going.

My plan is to wait a week and then redo the chapter. Reworking all the questions, rewatching all the videos etc etc, and see if I do any better a second time around. Assuming I do better I will then attempt the challenge questions. Maths people out there.....first question, is this a good idea? Just keep redoing the chapter until I get it? And...second question....is it really this hard? This is for kids! Bright kids I know, but even so, what is pre algebra? 12 year olds? Or am I missing something here.....doing something wrong.....

All help and suggestions gratefully received...

kit.

[Kit]

1 hour ago, square_25 said:

I think math is harder if you've already internalized math as something you don't have to think about and that doesn't always have reasons why. So kids actually have an easier time!! 

Are there any questions I can help you with? I'm good at explaining concepts and very familiar with conceptual issues that crop up. 

The biggest problem I seem to have is the commutative and associative properties. I have the basics down and then there is a highly mixed up problem and they move even subtraction and division around gaily, and I’m left going....I thought you couldn’t do that! I’m getting the hang of changing things from division to multiplication of the inverse, but it’s slow going. And sometimes they just seem to move them at random!

I think I will start again. I started this chapter with the attitude that I knew all this and it’s just revision. Only to find I didn’t. I need to slow down and make sure I understand everything before moving on. I think maybe I was a little too sure of myself!

Thank you for your offer. I will take you up on it if I don’t get the hang of it next time around. If I don’t understand something, I will ask!

[Kit]

1 hour ago, mms said:

I did this five or six years ago. I have a decent math background, having taken math through the 500 level in college but I gained a lot from the different approach. It wasn’t a walk in the park for me either, but it was fun (didn’t use the videos either, didn’t know there were videos).  I didn’t stick with aops because it was too expensive, but moved on to Dolciani and Gelfand.

Stick with it, your efforts will make you a far better teacher!

eta: sorry, I am no help on your actual questions. Just wanted to cheer you on!

Thank you for your support. I think I was a little too sure of myself, I thought I knew the stuff in chapter one and approached it with that attitude. I am going to start again with the assumption I know nothing. After all it’s not too far from the truth! 😅😅

[daijobu]

I think you'll find as you keep working through the book it will get easier.  

When you read the introductory problems and solutions, write them out as you are reading them.  Don't just read the solutions.  Write out all the solutions and draw all the diagrams, step by step.  Make sure you understand each step before continuing on to the next step.  

Even if you get the right answer, check their solution to see if the solved it a different way.  If so, then write out their solution and make sure you understand it.

Line up your equal signs.  Draw nice big clear diagrams.  

Good luck and have fun!  You won't find a better math textbook.  

[El...]

I've had to draw pictures of the questions.  A lot. 🙂

Also,  it seemed to me that chapter 2 was the hardest in the book, so don't get discouraged. 

[Jackie]

13 hours ago, Kit said:

The biggest problem I seem to have is the commutative and associative properties. I have the basics down and then there is a highly mixed up problem and they move even subtraction and division around gaily, and I’m left going....I thought you couldn’t do that! I’m getting the hang of changing things from division to multiplication of the inverse, but it’s slow going. And sometimes they just seem to move them at random!

I think I will start again. I started this chapter with the attitude that I knew all this and it’s just revision. Only to find I didn’t. I need to slow down and make sure I understand everything before moving on. I think maybe I was a little too sure of myself!

Thank you for your offer. I will take you up on it if I don’t get the hang of it next time around. If I don’t understand something, I will ask!

 

I remember finally understanding that subtraction is just the adding of the opposite, and that division is simply multiplying by the reciprocal. That made all the rules make so much more sense to me! So, if I see something as simple as 9-7, my brain knows it can restate that as 9+(-7), and then it is easy to move things around because I know I can move the numbers around with addition. If it helps with the problem, I can now rearrange things to be -7+9, because that is still obviously valid in terms of the addition rules.

I see division similarly. 12 divided by 3 is the same as 12 times 1/3. It is also the same as the fraction 12/3. If the rules for any way of thinking of that make more intuitive sense to me, I can think of the problem in that way.

It is hard for brains to learn things in new ways. Even working my way through the AOPS books, it is easier to let my brain default back to what it already knew instead of always implementing new approaches. 

I know for my kid, she much preferred watching the videos in advance, before there was any expectation of working problems. She watched some sections repeatedly. It let the concepts have time to rattle around in her brain and sink in, and that works much better for her than having to immediately apply new learning.

[Kit]

Thank you for the support everyone. 😀

I am setting out again this morning (Monday morning here) in the sprit of adventure. I am so lucky to have the gift of time....this takes as long as it takes.

The plan is to watch the video. Then do the section. The introductory exploratory questions I will do one at at time, and then check the answer. If my working is different from theirs I will work it again their way. I think this is where I went wrong. I got the right answer to the introductory questions in my old fashioned way, and was then flummoxed on the more complex questions.

This is actually quite fun in a weird sort of way. When I was at school (1960’s and 70’s) one did not ask questions. There was one right way and one right answer, and the students job was to learn it and repeat it when asked. As one of those kids who always wants to know ‘but why.....’ I did not do well.

So.....onwards! Once more unto the breach dear friends....

Kit.

[Kit]

2 hours ago, square_25 said:

 

May I see the more complex questions? 🙂 I don't own the book. 

One I got really stuck on was 

Simplify (-13)+(-13)divide(-13)x(-13)-(-13).   (I don’t have a divide symbol) The only bit I was sure of was the double negation at the end being positive.

Also....
what is the value of 123,123 divided by 1001.     Obviously I could do it by 1000 and I knew there had to be a ‘trick’ or shortcut, but had no idea how to seperate that 1 from the rest of it.

There are others, where the answer is simple and I got it right but I did it a long winded way and they just did some maths wizard stuff and got the answer in far fewer steps.

Thank you for you help.

Kit.

[regentrude]

4 hours ago, Kit said:

what is the value of 123,123 divided by 1001.     Obviously I could do it by 1000 and I knew there had to be a ‘trick’ or shortcut, but had no idea how to seperate that 1 from the rest of it.

Look carefully at the number and notice the recurring 123: 

123,123 = 123,000 + 123 =123*(1000+1)=123*1,001. So 123,123 /1001 = 123.

Edited February 17, 2020 by regentrude

[regentrude]

9 minutes ago, Kit said:

Simplify (-13)+(-13)divide(-13)x(-13)-(-13).   (I don’t have a divide symbol) The only bit I was sure of was the double negation at the end being positive.

Could you retype that and let us know what all is on the top (numerator) and what's in the denominator, i.e. what all you divide by? This looks like it might need more parentheses to represent the actual problem

[Kit]

1 hour ago, regentrude said:

Could you retype that and let us know what all is on the top (numerator) and what's in the denominator, i.e. what all you divide by? This looks like it might need more parentheses to represent the actual problem

It was all in a line and the divide word was the divide symbol....2 dots and a line between...

[Kit]

1 hour ago, regentrude said:

Look carefully at the number and notice the recurring 123: 

123,123 = 123,000 + 123 =123*(1000+1)=123*1,001. So 123,123 /1000 = 123.

Now this is where I get lost.....

I can see that you have split the 123,123 into 123000+123. .......and then you have totally lost me! Does * mean multiply or divide? And where did the 1 go?

Aggghhhh........

[regentrude]

36 minutes ago, Kit said:

Now this is where I get lost.....

I can see that you have split the 123,123 into 123000+123. .......and then you have totally lost me! Does * mean multiply or divide? And where did the 1 go?

* means multiply. 

123,123 = 123,000 + 123

Now you do what we call "factor out" a 123:   123,000 + 123 = 123* 1,000 + 123 * 1

Still with me?

Now you apply the distributive property:  123* 1,000 + 123 * 1  =123*(1000+1)

 

[regentrude]

44 minutes ago, Kit said:

It was all in a line and the divide word was the divide symbol....2 dots and a line between...

OK soif there were no parentheses and it wasn't written as an actual fraction, we have this:

(-13)   +    (-13)  divide(-13)  x(-13)  -  (-13)

Are you familiar with order of operations? Multiplication and division precede addition and subtraction. We first need to divide/multiply; then we add/subtract

So we first work out (-13) / (-13) which is 1

Let's rewrite

(-13)   +    1  x(-13)  -  (-13)

Now  the multiplication 1 x (-13) = - 13. Rewrite:

(-13) + (-13) + 13 = -13

 

 

[Kit]

I am so sorry to be so slow here.

i have worked it through and arrived at 

123(1000+1)/1001.

And then....?

123*1000 = 123,000 makes sense to me. Mainly because in the number the 1 is in the 100,000 place and 100*1000 is 100,000.....and the the same for the 2 and 3 in the 10,000 place and 1000 place respectively.

Im just going to leave this one for a moment and look at the 13’s.

Sorry to be so slow. I don’t know if I’m stupid or uneducated!

Kit.

[Kit]

1 hour ago, regentrude said:

OK soif there were no parentheses and it wasn't written as an actual fraction, we have this:

(-13)   +    (-13)  divide(-13)  x(-13)  -  (-13)

Are you familiar with order of operations? Multiplication and division precede addition and subtraction. We first need to divide/multiply; then we add/subtract

So we first work out (-13) / (-13) which is 1

Let's rewrite

(-13)   +    1  x(-13)  -  (-13)

Now  the multiplication 1 x (-13) = - 13. Rewrite:

(-13) + (-13) + 13 = -13

 

 

Ahhh! Light bulb moment . This now makes sense......just off to write it all out in steps for myself. I thought brackets came first so was bamboozled by all the -13’s all alone in their brackets! But the bracket is just holding the 13 and it’s own - sign together so they don’t get lost.

 

thank you so much 😀  Now... back to the 123’s......

[Kit]

Can we just cancel the 1001’s? And if so....why?

[annegables]

4 hours ago, regentrude said:

Look carefully at the number and notice the recurring 123: 

123,123 = 123,000 + 123 =123*(1000+1)=123*1,001. So 123,123 /1000 = 123.

I think you dropped a 1 here at the end. It should be 123,123/1001 = 123

[regentrude]

4 minutes ago, Kit said:

Can we just cancel the 1001’s? And if so....why?

 

9 minutes ago, Kit said:

I am so sorry to be so slow here.

i have worked it through and arrived at 

123(1000+1)/1001.

And then....?

123*1000 = 123,000 makes sense to me. Mainly because in the number the 1 is in the 100,000 place and 100*1000 is 100,000.....and the the same for the 2 and 3 in the 10,000 place and 1000 place respectively.

Im just going to leave this one for a moment and look at the 13’s.

Sorry to be so slow. I don’t know if I’m stupid or uneducated!

You're not stupid. You just had an inadequate math education. Do not apologize. It's awesome that you are working through this.

123(1000+1)/1001 = 123x1001/1001 = 123 x (1001/1001)= 123 x 1=123

So yes, basically we can cancel the 1001 because it's both in the denominator and the numerator

 

[regentrude]

2 minutes ago, annegables said:

I think you dropped a 1 here at the end. It should be 123,123/1001 = 123

thanks, fixed it!

[annegables]

1 minute ago, regentrude said:

thanks, fixed it!

I didn't want to be nitpicky, but I thought that was a rather crucial 1😜

[regentrude]

Just now, annegables said:

I didn't want to be nitpicky, but I thought that was a rather crucial 1😜

absolutely. Not nitpicky at all -thanks for catching that

[Kit]

7 minutes ago, regentrude said:

 

You're not stupid. You just had an inadequate math education. Do not apologize. It's awesome that you are working through this.

123(1000+1)/1001 = 123x1001/1001 = 123 x (1001/1001)= 123 x 1=123

So yes, basically we can cancel the 1001 because it's both in the denominator and the numerator

 

 So 1001/1001 is 1.....of course it is!!!!! Because 2/2 is 1. Ha! 🌞🌞🌞🌞🌞🌞🌞🌞🌞

[Kit]

1 minute ago, square_25 said:

Don’t apologize, please :-). However, I’d like to make sure. What does 1000*123 mean to you, in words?

Ummm......in words!

1000 lots of one hundred and twenty three. Is that what you mean? So if I had 123 bottles in a crate it would mean I had one thousand crates.....and 123,000 bottles altogether........?

[Kit]

7 minutes ago, square_25 said:

Perfect :-). And what does dividing by 1001 mean?

Well.....now we have those  123,000 bottles in a big heap and we have to divvy them up into 1001 crates. And if they don’t fit exactly some crates will be short if we put the extras in some crates, leaving others with one less, or..............we have to start smashing bottles and spreading the shards around evenly 🤣🤣🤣🤣🤣

This doesn’t sound like a very mathematical explanation......nor very useful if you want to use the bottles........

Edited February 17, 2020 by Kit

[Kit]

3 minutes ago, square_25 said:

Hahahaha, perfect!!! And yes, uhmm, let's use apples or something, so we can split them evenly if need be. 

OK, so you were asking a couple of questions. 

1) Why is 1000*123 + 123 = 1001*123. Do you see why that is now, if we think in apples? 

2) Why is 1001*123/1001 = 123. Do you see how that works out, if we're splitting 1001*123 between 1001 people? 

In my opinion, getting this kind of visual for the operations, and getting used to thinking of them this way, is precisely what takes you from fumbling about and using whatever the teacher tells you and actually understanding things. And I'm really impressed you're doing this!! I hope that one day I get the energy to do this for the stuff I didn't learn well in school... 

Light bulbs are pinging on all over the place. Thank you, and Regentrude as well for explaining all this. Just wait until we get to the challenge questions 🤣🤣🤣

it seems you had a great education, in maths at any rate. I am nearly 60, and semi retired. I thought learning maths would be a nice little hobby, 😉

[annegables]

1 hour ago, Kit said:

Light bulbs are pinging on all over the place. Thank you, and Regentrude as well for explaining all this. Just wait until we get to the challenge questions 🤣🤣🤣

it seems you had a great education, in maths at any rate. I am nearly 60, and semi retired. I thought learning maths would be a nice little hobby, 😉

If it makes you feel any better, I have an engineering degree, and I still have to say things in words to have it make sense. For instance, if I want to multiply 7*9 and I think, I will just multiply 7*10 and then subtract...9? or is it 7? I forget. Let's see. I have Ten groups of seven apples in each group, but I really just want nine groups of seven apples, so I need to subtract one group of seven apples to go from ten groups to nine groups. Now I get it. This was a relatively straightforward problem, but the idea stands with more challenging problems. Talking to yourself about apples is a legit way to think about numbers😜

[readinmom]

When you first posted I thought "I wonder where I would start with AoPS...maybe Beast Academy?"  

I also went through the early grades in the late 60s, early 70s...only one answer and don't ask why.   I also feel that I "missed something" when I started out. I used to be extremely math phobic, have improved a lot as I had professors in college that seemed to speak my language when it came to math.    

Keep posting your progress!  I look forward to reading about it.  You've got this!