Math Concept Progression Question

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I know I've been posting a lot about math lately. Sorry! But this came up today and I was wondering if any mathies would like to help me out here. :)

I'm trying to pin down why KB isn't working, so I know what to look for in our next math program. I'm fairly confident that we'll go with Saxon. I've been going through each lesson with ds and what was an easy unit, turned very difficult and this is a pattern with KB. They seem to want him to reverse engineer the concepts at the end of the unit and on the test. After giving him the basic concepts and providing some practice, to me, it seems like they make a huge leap with no other information. I can explain it to him, but he just runs into trouble every time.

Example

Exponents unit

The end of unit problems go from

(a/b)10

(x8)7

(x5y6)5

(x4/y10)11

(x7y-2)-3

up until here it is simple and easy enough to him that he could do most in his head.

This is where the reverse engineering comes in. There was no teaching of or problems like this until the end of the unit review and on the test.

Write 1256 as a power of 5. (answer 518)

Write 6359 in the form x3, where x is a number. (answer 7503)

He has some issues with distribution no matter how often we discuss it. :glare:

Do most kids "get" this kind of progression? Is this conceptual?

I'm glad they do this because it gets him thinking, but I think it's a bit unfair to throw it on him with no lesson or additional information. This seems to be how KB works since it has happened in many units. In the percentages unit they went through the basics of %'s and then in the test had him work the problem backwards and figure out the original price from the sale price. I will admit, if he is comfortable in the why, he should be able to do the how, but he's not making that leap. :(

So, for someone who is new to teaching math, but feels fairly strong in it, am I making much ado about nothing? We used MM and this our first year away from that. Is this normal for math curriculum? What type would you say KB is? I feel like a huge newbie right now, so try to go easy on me. :nopity:

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They seem to want him to reverse engineer the concepts at the end of the unit and on the test. After giving him the basic concepts and providing some practice, to me, it seems like they make a huge leap with no other information. I can explain it to him, but he just runs into trouble every time.

Example

Exponents unit

The end of unit problems go from

(a/b)10

(x8)7

(x5y6)5

(x4/y10)11

(x7y-2)-3

up until here it is simple and easy enough to him that he could do most in his head.

This is where the reverse engineering comes in. There was no teaching of or problems like this until the end of the unit review and on the test.

Write 1256 as a power of 5. (answer 518)

Write 6359 in the form x3, where x is a number. (answer 7503)

I assume the laws of exponents have been discussed in the lesson, because the first set of problems requires them. I am not entirely sure what you mean by "reverse engineering", because the two later problems are straightforward applications of the laws of exponents as well. There is nothing reverse about them.

Is your student familiar with squares and cubes? Those should have been taught alongside the multiplication tables. The problem requires the student to express 125 as a power of 5 (which is very easy if the student recognizes that 125=5^3; if the student does not have this fact recall, finding the power by repeated division always works). The problem is then reduced to a problem of exactly the same kind as your "first" set: (5^3)^6=5^(3*6)=5^18. The second problem, again, applies laws of exponent and requires the student to express 9 as 3*3. So (6^3)(5^3)^3=(6*125)^3. x=125*6=750/

I am not sure what explicit teaching you would expect beyond the laws of exponents; I am obviously not grasping the issue here.

At some point, most likely early in the introduction of the concept of exponents, the student should have been given problems of the type: write 125 as a power of 5. Write 9 as a power of 3 or similar. I would be surprised if your curriculum had not done this, before laws of exponents are introduced. From what you describe, it seems as if your student can not transfer this earlier acquired skill.

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This is where the reverse engineering comes in. There was no teaching of or problems like this until the end of the unit review and on the test.

Write 1256 as a power of 5. (answer 518)

Write 6359 in the form x3, where x is a number. (answer 7503)

Write 1256 as a power of 5. (answer 518)

125 = 53

(Am)n = A mn

1256=53.6

= 518

Write 6359 in the form x3, where x is a number. (answer 7503)

We have to first write

59 = (53)3 = 1253

6359 = 631253

= (6.125)3

= 7503

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I assume the laws of exponents have been discussed in the lesson, because the first set of problems requires them. I am not entirely sure what you mean by "reverse engineering", because the two later problems are straightforward applications of the laws of exponents as well. There is nothing reverse about them.

I mean they want him to work the problem backwards since all that was covered in the lesson was (53)6 = 518. Just as Arcadia did.

Is your student familiar with squares and cubes? Those should have been taught alongside the multiplication tables. The problem requires the student to express 125 as a power of 5 (which is very easy if the student recognizes that 125=5^3; if the student does not have this fact recall, finding the power by repeated division always works).

He did not automatically recognize that, but he did eventually see that it was a power of 5 and then that led to repeated division after some prompting from me.

The problem is then reduced to a problem of exactly the same kind as your "first" set. The second problem, again, applies laws of exponent and requires the student to express 9 as 3*3.

I am not sure what explicit teaching you would expect beyond the laws of exponents; I am obviously not grasping the issue here.

At some point, most likely early in the introduction of the concept of exponents, the student should have been given problems of the type: write 125 as a power of 5. Write 9 as a power of 3 or similar. I would be surprised if your curriculum had not done this, before laws of exponents are introduced. From what you describe, it seems as if your student can not transfer this earlier acquired skill.

See, that's the problem I'm having because they did not have any problems like that until the end of unit problems. I went through every problem in this unit with him. Then I double checked to avoid embarrassment. He did have basic exponent concepts lessons for this along with when he did MM. Naturally, I figured it out, but he basically froze up, sat there for awhile and got upset that he didn't know how to do it.

The unit progression was

Exponents

- Basic Rules

- Negative numbers raised to a power

- Exponents and order of operations

- First power

- Product of Powers

- 3332 to a single power of 3.

- Quotient of Powers

- Multiplying and Dividing Powers

- Zero as an Exponent

- Power of a Product Rule

- (2x)4 = 16x4

- (xy)n = xnyn

- Negative Exponent

- Power of a Quotient

- (w/5)5

- Probability Application

- What is the probability of rolling a six-sided die three times, and getting a 1 each time? Assume that the probability of rolling a 1 is 1/6.

- Power of a power

- multiple problems of Simpify (z10)8

- End of Unit problems with

-Write 1256 as a power of 5 and 646 as a power of 4.

-Write 6359 in the form of x3, where x is a number and 64416 in the form of x4, where x is a number.

- Quiz

I just think there is either something missing in the curriculum or in his skill set or both. This isn't the first time something like this has happened. Am I off here?

Write 1256 as a power of 5. (answer 518)

125 = 53

(Am)n = A mn

1256=53.6

= 518

Write 6359 in the form x3, where x is a number. (answer 7503)

We have to first write

59 = (53)3 = 1253

6359 = 631253

= (6.125)3

= 7503

Hubby rushing me for library. Hope no typos.

Thanks for taking the time to answer that. All of those exponents probably took awhile. :blushing: I knew how to work out the problem, but he didn't. :(

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We quite KB before figuring out what the problem was. But yes, it seemed to throw the kids to the wolves. We switched to TT and he is getting an A now.

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Lol. It really does.

I couldn't figure out why he was doing so well on the homework and quick checks and then would break down on the end of unit problems and tests.

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I just think there is either something missing in the curriculum or in his skill set or both. This isn't the first time something like this has happened. Am I off here?

I think your child needs more worked examples going for easy to hard. You could get examples from another book/text and keep using KB, or switch to another curriculum if it is going to be more math frustration down the road.

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I think your child needs more worked examples going for easy to hard. You could get examples from another book/text and keep using KB, or switch to another curriculum if it is going to be more math frustration down the road.

I'm at the library :)

Well I was planning to switch to Saxon. I hear that has loads of review and problems. I'm hoping it will be a good fit.

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Well I was planning to switch to Saxon. I hear that has loads of review and problems. I'm hoping it will be a good fit.

Saxob has plenty. I have done a quick read of the books. Saxon geometry teachers guide is pretty good. Saxon Algebra 1 teacher guide sample.

Whatever works :)

ETA:

Maybe get your son to do his own math cheat sheet for quick revision. Something like this

http://www.regentspr...heetAlgebra.pdf

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Lol. It really does.

I couldn't figure out why he was doing so well on the homework and quick checks and then would break down on the end of unit problems and tests.

Same here. Mine needed explicit instruction. Not half the instruction, with them assuming he'd figure out the rest of it. TT has explicit instruction on how to do everything.

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Saxob has plenty. I have done a quick read of the books. Saxon geometry teachers guide is pretty good. Saxon Algebra 1 teacher guide sample (http://msmilesmath.f...alg1-sect04.pdf)

Whatever works :)

ETA:

Maybe get your son to do his own math cheat sheet for quick revision. Something like this

http://www.regentspr...heetAlgebra.pdf

I'm feeling better already. :) Ahhhh....solutions guides. That would be nice. :D

Also, thanks for the cheat sheet. He made his own when he was doing volumes of solids. I printed out the book that Tablet Class gave out for signing up to their email. A 2 page sheet is much easier to handle so I'll post that on his bulletin board above his desk.

I have MM Gold for 7A/B and Pre-Algebra and I'll have him do those if I feel he needs to. MM didn't have that kind of question either. I also have him work on Khan separately at his own pace.

I think once we are done with KB, I'll spend some extra time shoring up the little things before we move on. I do think he should have gotten those questions, he must have some gaps still.

I feel bad because I was thinking I might have to repeat pre-algebra because he was so hot and cold this year. KB seems to be for more mathy kids that can pick it up quick.

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I assume the laws of exponents have been discussed in the lesson, because the first set of problems requires them. I am not entirely sure what you mean by "reverse engineering", because the two later problems are straightforward applications of the laws of exponents as well. There is nothing reverse about them.

Is your student familiar with squares and cubes? Those should have been taught alongside the multiplication tables. The problem requires the student to express 125 as a power of 5 (which is very easy if the student recognizes that 125=5^3; if the student does not have this fact recall, finding the power by repeated division always works). The problem is then reduced to a problem of exactly the same kind as your "first" set: (5^3)^6=5^(3*6)=5^18. The second problem, again, applies laws of exponent and requires the student to express 9 as 3*3. So (6^3)(5^3)^3=(6*125)^3. x=125*6=750/

I am not sure what explicit teaching you would expect beyond the laws of exponents; I am obviously not grasping the issue here.

At some point, most likely early in the introduction of the concept of exponents, the student should have been given problems of the type: write 125 as a power of 5. Write 9 as a power of 3 or similar. I would be surprised if your curriculum had not done this, before laws of exponents are introduced. From what you describe, it seems as if your student can not transfer this earlier acquired skill.

I let that one sink in overnight. He has done them, but I would say he's rusty with cubes specifically. I looked it up on the KB Table of Contents and square roots and cube roots are actually covered in the next unit under Radicals.

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I let that one sink in overnight. He has done them, but I would say he's rusty with cubes specifically. I looked it up on the KB Table of Contents and square roots and cube roots are actually covered in the next unit under Radicals.

Yes, but what I mean is a familiarity with, and recognition of, cubes and squares of integers. The next unit may teach the definitions and formal dealings with the concept of a root, but none of those were required for the previous problem.

I notice that many students are lacking familiarity with numbers: they do not see a prime, a square, a cube, what factors a number has. These are skills that would have been developed along with elementary math, through lots and lots of arithmetic practice and mental math - before concepts like "root" and "prime factors" and "exponents" have even been introduced.

I am talking about practice to the extent that the student can never see the number 196 without remembering that this is 14*14, that seeing 81 immediately connects to 9*9 and 3*3*3*3 - things like this. This has nothing to do with algebra, but solely with arithmetic. And it is not a conceptual issue, but a very basic skill that only comes from time on task.

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Jen - My oldest would have sat there staring at those problems like a deer in the headlights. The resulting crash would have been terribly messy.

Makes me glad I decided not to go with KB as was my original pre-algebra plan. I don't see any issue with that type of problem in a problem set as an extension of the lesson, but to just throw it at you at the end on a test ... Not for my type of kid (at least dd#1 & dd#2)!

Neither of my older kids is STEM-minded and has no desire to think of numbers beyond what they are required to. I think of this as a combination of Desire & Aptitude. Oldest would not see 169 & think 13*13. She would probably think it looked like a great candidate to divide by 3. :glare: At least until she got a remainder & thought to do a digit sum to see if it was divisible by 3. :crying:

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Neither of my older kids is STEM-minded and has no desire to think of numbers beyond what they are required to. I think of this as a combination of Desire & Aptitude. Oldest would not see 169 & think 13*13. She would probably think it looked like a great candidate to divide by 3. :glare: At least until she got a remainder & thought to do a digit sum to see if it was divisible by 3.

I have a question for you (and I do not mean that snarky at all, just very curious): how often has she been required to see things like 169=13*13?

I believe we are too quick to blame intrinsic aptitude when it is frequently a lack of practice. I know that I did not magically grow up to see squares and cubes because I have some intrinsic ability - I do so because we had daily hard drill in math class where this kind of fact recall, mixed in with rapid fire mental math, was done, and were never allowed to use calculators until very late in high school.

It is like playing an instrument: of course not everybody is becoming a world class musician, but to get to some medium level, practice of simple skills is all that is required, just lots of it.

So, often when I hear stories like this, I wonder to what extent what we excuse with lack of aptitude may be simply a lack of training.

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.

I notice that many students are lacking familiarity with numbers: they do not see a prime, a square, a cube, what factors a number has. These are skills that would have been developed along with elementary math, through lots and lots of arithmetic practice and mental math - before concepts like "root" and "prime factors" and "exponents" have even been introduced.

I am talking about practice to the extent that the student can never see the number 196 without remembering that this is 14*14, that seeing 81 immediately connects to 9*9 and 3*3*3*3 - things like this. This has nothing to do with algebra, but solely with arithmetic. And it is not a conceptual issue, but a very basic skill that only comes from time on task.

That really help both my boys when they were doing HCF, LCM, and factorisation in general. They did not purposely memorise but knew the squares and some cubes from off the top of their heads. It has help my older do his exponents chapter comfortably. AoPS preaglebra book has a list of perfect squares to 29 x 29 (page 54).

Another thing I notice about public school math in general is that while multipliction drills are in 3rd grade, very few teachers highlight that the diagonal is the squares. Or when doing the Sieve of Eratosthenes exercise for prime numbers, kids come away thinking that they just did some circling/coloring exercise. The coherence is just not there.

The multiplication table with the diagonals highlighted as perfect squares.

http://www.sinclair.edu/centers/tlc/pub/handouts_worksheets/mathematics/DEV085/1_whole_numbers/multiplication_table.pdf

The typical worksheet on prime numbers

http://www.mathgoodies.com/webquests/number_theory/PDF/unit3_wks2.pdf

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Another thing I notice about public school math in general is that while multipliction drills are in 3rd grade, very few teachers highlight that the diagonal is the squares. Or when doing the Sieve of Eratosthenes exercise for prime numbers, kids come away thinking that they just did some circling/coloring exercise. The coherence is just not there.

This is actually the first thing we did when learning the multiplication tables! We made a poster, realized that we need only a triangle of problems (because multiplication is commutative) and made the diagonal a special color, to emphasize the squares.

I have the sad suspicion that the Sieve of Eratosthenes remains a mystery to some elementary math teachers themselves.

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Thanks for the links. I saved and printed them. I think it will be a good summer drill along with other sheets and anything else you all think would be helpful. :)

Sadly, he missed a lot of math in ps because they would pull him out for GT during math instruction. :glare: They said it was the only time slot available. I tried to fill in as much as I could and spent 6th grade with MM dark blue and lots of drill sheets. I can see cubes and squares getting touched on, but not drilled. I'm thinking that I might want to try the pre-algebra diagnostic test that HSBC has. I'm hoping that will help me see what I should take some time to focus on before the next school year.

He wants to be a computer programmer which means strong math and logic skills, so figuring this mess out before he gets into high school is really important to us.

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I have a question for you (and I do not mean that snarky at all, just very curious): how often has she been required to see things like 169=13*13?

I believe we are too quick to blame intrinsic aptitude when it is frequently a lack of practice.

Squares of everything up to 12*12 (cubes of 3, 4, & 5, and up to 2^5), oldest dd saw/sees repeatedly ... several times a week for several grades (that repetition of elementary math leading up to pre-algebra that some kids apparently don't need but that my older two did/do). For my specific example (13*13), she hasn't seen it much at all, because we haven't yet done a LOT of squares-past-12. (That is something we have in the plans for this, her "pre-algebra" year. I know that's when I memorized the squares up to 25*25, but I did it for math competitions.)

I know my kids pretty darn well. Oldest is actually _good_ at arithmetic concepts, but she dislikes "Math" (the work involved for the actual class). She makes careless mistakes and in everything, dislikes having to actually THINK. She'd much rather do pages of arithmetic drill than six story problems. Story problems require her to work (mentally). Drill just asks her to recall. That's why that type of a problem would throw her into a spin. She'd actually have to use her brain.

Dd#2 has talents in areas other than arithmetic. I think she might like geometry when we get to it. Her mind works in completely different ways than most of the rest of the family's. I could write pages on what this means and how she's still trying to make the jump from concrete to abstract when it comes to multiplication & division.

Dd#3 has so far been very "mathy." But she's still young, so I try not to extrapolate much or I would have said that dd#3's eyes light up when you throw something (numbers-wise) at her that she hasn't seen before and she gets to puzzle it out. She brings her math book to bed with her some nights.

DH & I are both engineers. Math, for us, is FUN. It is exciting and intriguing to continue to explore topics and applications of mathematical ideas. (Spelling - not so much!)

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I'm positive training is an issue. He definitely needs lots of problems that vary in difficulty and lots of drill.

The good news is that he is now doing the op problems quick. I'll give him a problem tomorrow just to see how he does. ;)

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I'm positive training is an issue. He definitely needs lots of problems that vary in difficulty and lots of drill.

The good news is that he is now doing the op problems quick. I'll give him a problem tomorrow just to see how he does. ;)

Check out the math workbook thread I started. There are quite a lot of free to use resources there and you can browse through those with copyright to check for suitability before buying.

ETA:

If you don't mind internet games, Lure of the Labyrinth is a fun, free prealgebra game for de-stressing.

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That really help both my boys when they were doing HCF, LCM, and factorisation in general. They did not purposely memorise but knew the squares and some cubes from off the top of their heads. It has help my older do his exponents chapter comfortably. AoPS preaglebra book has a list of perfect squares to 29 x 29 (page 54).

Another thing I notice about public school math in general is that while multipliction drills are in 3rd grade, very few teachers highlight that the diagonal is the squares. Or when doing the Sieve of Eratosthenes exercise for prime numbers, kids come away thinking that they just did some circling/coloring exercise. The coherence is just not there.

The multiplication table with the diagonals highlighted as perfect squares.

http://www.sinclair.edu/centers/tlc/pub/handouts_worksheets/mathematics/DEV085/1_whole_numbers/multiplication_table.pdf

The typical worksheet on prime numbers

http://www.mathgoodies.com/webquests/number_theory/PDF/unit3_wks2.pdf

SM 5 does the Sieve without calling it that, although the formal name could be in the HIG (don't have it).

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• 2 years later...

Yes, but what I mean is a familiarity with, and recognition of, cubes and squares of integers. The next unit may teach the definitions and formal dealings with the concept of a root, but none of those were required for the previous problem.

I notice that many students are lacking familiarity with numbers: they do not see a prime, a square, a cube, what factors a number has. These are skills that would have been developed along with elementary math, through lots and lots of arithmetic practice and mental math - before concepts like "root" and "prime factors" and "exponents" have even been introduced.

I am talking about practice to the extent that the student can never see the number 196 without remembering that this is 14*14, that seeing 81 immediately connects to 9*9 and 3*3*3*3 - things like this. This has nothing to do with algebra, but solely with arithmetic. And it is not a conceptual issue, but a very basic skill that only comes from time on task.

I have a question for you (and I do not mean that snarky at all, just very curious): how often has she been required to see things like 169=13*13?

I believe we are too quick to blame intrinsic aptitude when it is frequently a lack of practice. I know that I did not magically grow up to see squares and cubes because I have some intrinsic ability - I do so because we had daily hard drill in math class where this kind of fact recall, mixed in with rapid fire mental math, was done, and were never allowed to use calculators until very late in high school.

It is like playing an instrument: of course not everybody is becoming a world class musician, but to get to some medium level, practice of simple skills is all that is required, just lots of it.

So, often when I hear stories like this, I wonder to what extent what we excuse with lack of aptitude may be simply a lack of training.

That really help both my boys when they were doing HCF, LCM, and factorisation in general. They did not purposely memorise but knew the squares and some cubes from off the top of their heads. It has help my older do his exponents chapter comfortably. AoPS preaglebra book has a list of perfect squares to 29 x 29 (page 54).

Another thing I notice about public school math in general is that while multipliction drills are in 3rd grade, very few teachers highlight that the diagonal is the squares. Or when doing the Sieve of Eratosthenes exercise for prime numbers, kids come away thinking that they just did some circling/coloring exercise. The coherence is just not there.

The multiplication table with the diagonals highlighted as perfect squares.

http://www.sinclair.edu/centers/tlc/pub/handouts_worksheets/mathematics/DEV085/1_whole_numbers/multiplication_table.pdf

The typical worksheet on prime numbers

http://www.mathgoodies.com/webquests/number_theory/PDF/unit3_wks2.pdf

This is actually the first thing we did when learning the multiplication tables! We made a poster, realized that we need only a triangle of problems (because multiplication is commutative) and made the diagonal a special color, to emphasize the squares.

I have the sad suspicion that the Sieve of Eratosthenes remains a mystery to some elementary math teachers themselves.

This old thread really hit the nail on the head for me today.

DD is home for the first time this year, as a 7th grader.  Conceptually, she grasps math easily and likes math.

With Everyday Math in ps(!), she didn't get the "training" she needed, which would make things easier now.

Now that she's home, we supplement AoPS with Singapore/MEP, depending on the topic.

Could we start a list of any basic skills that you find lacking in many kids?  I want to make sure that we don't miss anything.  Ideas on how to efficiently do this is welcomed!

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Lisa, I would recommend starting a new thread because otherwise this is going to get filled up with people answering the OP's question.

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