s/o what is casting off 9s?

[johnandtinagilbert]

I wonder if I'm doing this and don't know it.

[Capt_Uhura]

Siloam write it up in the thread where it was mentioned. If you go to Wikipedia, it gives the historical background. Casting out 9s is taught in Rightstart Math and taught in Singapore Math in one of it's accessory books. Basically, you make 9s and caste them out. Whatever is left over, is the check number. You assign numbers 1-9, check numbers of 1-0. So 11 has a check number of 2, 12 has a check number of 3, 13 has a check number of 4.

 

891 (0)

+637 (7)

_____

1528 (7)

 

The numbers in parentheses are the check numbers. You then add the check numbers 0+7=7, it checks so more than likely your computation is correct. There are instances where check numbers matched, such as when numbers are transposed (1528, 1258 have the same check number) and the answer is incorrect but it doesn't happen often.

 

it works for subtraction but you add the check number of the difference to the check number for the subtrahend which must equal the check number for the minuend.

 

it works for multiplication and division as well.

 

And for checking divisibility as Siloam outlined.

[johnandtinagilbert]

Thank you. That is interesting. I knew I could do that for 9s, but not for whole problems. I'm going to teach this tomorrow. I think my mathy kids will LOVE it! I think the other three will think I'm speaking Cambodian!

[siloam]

Thank you. That is interesting. I knew I could do that for 9s, but not for whole problems. I'm going to teach this tomorrow. I think my mathy kids will LOVE it! I think the other three will think I'm speaking Cambodian!

Here and here are examples of my oldest using it on her miltivides (if you can read her handwriting).

 

There are two slightly different ways to teach it. You can do it by casting out the 9's, so for a number like 98345 you can think 9=9 and 4+5=0 that leaves you taking 1 from the 3 and adding to the 8 to make 9=0. Your grand total is 2, what is left from the 3. If you prefer to just add and not cast out the 9's you can do it that way too: 9+9-17+3=20+4=24+5=29. Now add 2+9=11. Now add 1+1=2. Both get you to the same answer, it is just a matter of which works for you at the moment.

 

Heather

[johnandtinagilbert]

Thank you, Heather. How fun!

[SonshineLearner]

I did it for A Beka when I was homeschooled. I never totally understood it... but did well enough I could pass...

[Gooblink]

:lol: I thought you were talking about knitting - casting off 9 stitches.

[Joker]

My girls do it with CLE but they call it finding the digit sums. The girls think its a lot of fun.

[kiana]

Another interesting fact: If we worked in base 8, casting out sevens would work the same way.

Edited January 25, 2011 by kiana

[choirfarm]

(if anyone cares)

 

Because 9 is 1 less than 10, in our base 10 number system, the sum of the digits (or the repeated sum of the digits) is the remainder when you divide by 9.

 

If you are adding two numbers, and the first one has a remainder of 5 (all division is by 9 until further notice) and the second one has a remainder of 2, the sum will have a remainder of 7.

 

If the first one has a remainder of 5 and the second a remainder of 8, the sum will have a remainder of ... well, 13 is 9+4, so the remainder is really four = 1+3.

 

For subtraction, if you are subtracting two numbers and the first has a remainder of five, and the second has a remainder of 2, the sum will have a remainder of 5-2 = 3.

 

If the first has a remainder of five, and the second has a remainder of seven, well ... -2 is a silly remainder (if we were dividing, it would mean our quotient was one too high), so the remainder has to be 7.

 

For multiplication, if you are multiplying two numbers, and the first has a remainder of five, and the second has a remainder of two, the sum will have a remainder of 5*2 = 10, but since that's greater than 9, the remainder will really be 1.

 

It doesn't work directly for division other than for divisibility tests. However, if you want to check your work for division, use the multiplication procedure with the divisor and the quotient, add the remainder (from the original divisor), and check work that way.

 

For example -- if I take 6536/89, and get 73 with a remainder of 39, that means (assuming my calculations are correct) that 73*89 + 39 = 6536. Computing check numbers for 73*89 gives me 1*8, adding the check number for 39 gives me 3, 8+3 = 11 which reduces to 2. So the check number for the left-hand side is 2. The check number for the right-hand side is 6+5+3+6=20, which reduces to 2. Since the check numbers for the left-hand and right-hand side are equal, there is a good probability the work was correct.

 

Quite honestly, I only use this when I don't have access to a calculator but I frequently *do* use it to double-check off-the-cuff calculations.

 

Another interesting fact: If we worked in base 8, casting out sevens would work the same way.

 

You completely lost me.. I don't understand this at all. I thought I was good in math. I always made A's, but this is like a foreign language. So you just use casting nines when you are checking long division??? I just don't get it.

[A.J. at J.A.]

You completely lost me.. I don't understand this at all. I thought I was good in math. I always made A's, but this is like a foreign language. So you just use casting nines when you are checking long division??? I just don't get it.

 

:iagree:

[TechWife]

I. just. don't. get. it.

 

I'm totally lost as to the how to, but I'm also lost as to the why I would want to part.

[Capt_Uhura]

I only use check numbers or casting off nines the way I posted above. It's a quick check for addition, subtraction, multiplication and division. Granted, addition and subtraction are easy to check but I've seen DS whip through a page of 25 4-digit-4-digit subtraction problems with check numbers in a matter of minutes. I use it mostly to check multi-digit multiplication and division.

 

43 (7)

x24 (6)

____

172

860

____

1032 (6)

 

So in this case, the check numbers are 7x6=42, and the check number for 42 is 6 which is exactly the check number for 1032 therefore we are confident we have the correct answer. If we got the wrong check number, say we got 932 (failure to trade the tens), the check number would be 5 and therefore the check numbers wouldn't check 7x6=42 (6), 6 doesn't equal 5 so we know for certain our answer is incorrect.

[choirfarm]

Ok, so you add the numbers together for each line.. Then you multiply the numbers together. If the numbers you add together end up to the numbers you added in your answer, then you are ok. example

 

1032 divided by 24 ( 6 times 6 equals 36) The answer is 43 so it should be ???? I get it in your example, but it doesn't work when you divide.

 

Christine

 

I only use check numbers or casting off nines the way I posted above. It's a quick check for addition, subtraction, multiplication and division. Granted, addition and subtraction are easy to check but I've seen DS whip through a page of 25 4-digit-4-digit subtraction problems with check numbers in a matter of minutes. I use it mostly to check multi-digit multiplication and division.

 

43 (7)

x24 (6)

____

172

860

____

1032 (6)

 

So in this case, the check numbers are 7x6=42, and the check number for 42 is 6 which is exactly the check number for 1032 therefore we are confident we have the correct answer. If we got the wrong check number, say we got 932 (failure to trade the tens), the check number would be 5 and therefore the check numbers wouldn't check 7x6=42 (6), 6 doesn't equal 5 so we know for certain our answer is incorrect.

[kiana]

You completely lost me.. I don't understand this at all. I thought I was good in math. I always made A's, but this is like a foreign language. So you just use casting nines when you are checking long division??? I just don't get it.

 

Meh, sorry. I had separate parts 'for addition, for multiplication, for subtraction, for division' -- I just only listed an example for division because I'd already seen good examples for addition, subtraction, multiplication from everyone else. Division is more complicated and doesn't work directly.

 

If what I wrote seems nonsensical, don't worry. I probably should go back and delete it anyway.

[Capt_Uhura]

Ok, so you add the numbers together for each line.. Then you multiply the numbers together. If the numbers you add together end up to the numbers you added in your answer, then you are ok. example

 

1032 divided by 24 ( 6 times 6 equals 36) The answer is 43 so it should be ???? I get it in your example, but it doesn't work when you divide.

 

Christine

 

1032/24=43

(6)/(6)=(7) so just like you check subtraction by adding the check numbers for the difference and the subtrahend to get the check number for the minuend (that avoids negative check numbers) you check division by multiplying the check numbers. So you'd take (7)x(6)=42 (6) are the check numbers for 43x24=1032 and the check number for 1032 is 6 so it checks!

 

For addition, you add the check numbers.

For subtraction, you add the check numbers (diff+sub=minuend)

For multiplication, you multiply the check numbers

for division, you multiply the check numbers (quotientxdivisor=dividend)

[hooahwife]

:confused::blink:I'm beginning to wish I'd never brought the subject up.

 

The great part about homeschooling with Beka is you can skip this stuff!

[ELaurie]

After thinking it through, your post made sense to me, and I found it helpful :001_smile:

 

The part that I found confusing was this part:

 

For example -- if I take 6536/89, and get 73 with a remainder of 39, that means (assuming my calculations are correct) that 73*89 + 39 = 6536. Computing check numbers for 73*89 gives me 1*8, adding the check number for 39 gives me 3, 8+3 = 11 which reduces to 2. So the check number for the left-hand side is 2. The check number for the right-hand side is 6+5+3+6=20, which reduces to 2. Since the check numbers for the left-hand and right-hand side are equal, there is a good probability the work was correct.

 

Would you mind walking me through that again? I'm not sure what you meant when you referred to the right hand side and the left hand side?

 

ETA: Now I get it! Thanks for the great explanation! I plan to print your post so I can refer to it and teach it to my dc.

 

Thanks!

Edited January 26, 2011 by ELaurie
I just figured this out : )

[mycalling]

Oooo, I'd never heard of this. Fun! I know what we're learning tomorrow!

[AngelBee]

I wonder if I'm doing this and don't know it.

After saying I did not know how to diagram or what a printer that duplexes means......I wasn't gonna ask! :D :lol:

[kiana]

After thinking it through, your post made sense to me, and I found it helpful :001_smile:

 

The part that I found confusing was this part:

 

For example -- if I take 6536/89, and get 73 with a remainder of 39, that means (assuming my calculations are correct) that 73*89 + 39 = 6536. Computing check numbers for 73*89 gives me 1*8, adding the check number for 39 gives me 3, 8+3 = 11 which reduces to 2. So the check number for the left-hand side is 2. The check number for the right-hand side is 6+5+3+6=20, which reduces to 2. Since the check numbers for the left-hand and right-hand side are equal, there is a good probability the work was correct.

 

Would you mind walking me through that again? I'm not sure what you meant when you referred to the right hand side and the left hand side?

 

Thanks!

 

The right-hand side and left-hand side were referring to 73*89 + 39 = 6536, where the right-hand side is 6536 and the left-hand side is 73*89 + 39.

 

Checking that 73*89 + 39 = 6536 is true is the same as checking that 6536/89 is 73 with a remainder of 39 -- they are just different ways of saying the same thing. But it is much easier to verify the multiplication.

 

Finding the check number for 73*89 + 39 is a two-step process -- first, find the check number for 73*89 (it's 8), then add the check number for 39 (which is 3, so together they make 11 -- but since 11 is bigger than 9, the check number reduces to 2).

[ELaurie]

Here and here are examples of my oldest using it on her miltivides (if you can read her handwriting).

 

There are two slightly different ways to teach it. You can do it by casting out the 9's, so for a number like 98345 you can think 9=9 and 4+5=0 that leaves you taking 1 from the 3 and adding to the 8 to make 9=0. Your grand total is 2, what is left from the 3. If you prefer to just add and not cast out the 9's you can do it that way too: 9+9-17+3=20+4=24+5=29. Now add 2+9=11. Now add 1+1=2. Both get you to the same answer, it is just a matter of which works for you at the moment.

 

Heather

 

Hi Heather!

 

I figured out kiana's post, and now I'm working through yours :001_smile:

 

I don't get why 9=9 but 4+5=0 but I think the rest of the first part makes sense. I don't understand the second way to do this :confused:

 

What are multivides?

 

Thank you!

[kiana]

Hi Heather!

 

I figured out kiana's post, and now I'm working through yours :001_smile:

 

I don't get why 9=9 but 4+5=0 but I think the rest of the first part makes sense. I don't understand the second way to do this :confused:

 

What are multivides?

 

Thank you!

 

It's a typo, it should read 9+8=17+3=20+4=24+5=29

She is adding up all the digits to get 29.

[LoveBaby]

I. just. don't. get. it.

 

I'm totally lost as to the how to, but I'm also lost as to the why I would want to part.

 

:iagree:

[ELaurie]

It's a typo, it should read 9+8=17+3=20+4=24+5=29

She is adding up all the digits to get 29.

 

For both of your replies to my questions :001_smile:

[Annabel Lee]

Hey, Abeka math teaches this. I did it in grade school. I forgot it, too! Thanks for the great ideas to help me solidify this w/ ds9.

[Storm Bay]

Siloam write it up in the thread where it was mentioned. If you go to Wikipedia, it gives the historical background. Casting out 9s is taught in Rightstart Math and taught in Singapore Math in one of it's accessory books. .

I learned it from a math teacher when I was in grade 6 and still use it when checking dc's work in a hurry (I don't usually pull out answer keys for anything before Algebra since it doesn't take that long to figure it out). I'm off to see the history of it now.

 

eta it is true that you can get false positives, but you can't get false negatives--this is why I use it when in a hurry and not all of the time.

Edited January 26, 2011 by Karin

[siloam]

Hi Heather!

 

I figured out kiana's post, and now I'm working through yours :001_smile:

 

I don't get why 9=9 but 4+5=0 but I think the rest of the first part makes sense. I don't understand the second way to do this :confused:

 

What are multivides?

 

Thank you!

 

Sorry about that! I get excited and think faster than I can type. Because I know 9=0 it didn't catch that I had typed it wrong when I proofed it.

 

Basically the point is all 9's equal 0, so if you have a 9 you, "cast it out," as being 0 or if you can take two numbers and add them up to 9 they too are cast out. Or you can do the RS method and keep adding the numbers, but again if your final number is 9, the answer is 0.

 

Honestly I haven't figured out how to use it with division yet. I think I get it, but I haven't had the time to really sit down and make sure I own it. Nor is that likely to happen soon. :D If life would only slow down a tad. Nope that wouldn't work either, I would just sit around and do nothing. :smilielol5: Oh well, you just have to keep moving forward, and if I really need to make sure I understand it I know where to go.

 

Multivides are something used in RS as well. You start with a number, say 9, then multiply it by 2, 3, 4, 5, 6, 7, 8 and 9 in that order. Then you go backwards and divide it by 2, 3, 4, 5, 6, 7, 8, 9 using short division. The end result is you come back to the number 9 when you are all finished. I posted some samples of my dd's multivides earlier on the thread.

 

Heather

[ELaurie]

Now I get it :tongue_smilie:

 

It will be fun to show this to my "mathy" ds 8!