A mom recently wrote in to ask this question about standard and expanded notation.
- “How do you know when you are writing in standard form, expanded form? For example, is the expanded for of 30,048
30000 + 40 + 8 ?
Or for 29,486, the expanded form = 20000 + 9000 + 400 + 80 + 6 ?”
Professor Homunculus replies:
Precisely! Oddly enough, though, there seems to be no “standard” for “expanded.” What I mean is, for 30,048 the expanded form could also be considered:
- 3(10,000) + 4(10) + 8(1)
as well as
3(104) + 4(101) + 8(100).
To make it even more confusing, some teachers might require you to include the empty columns. That would mean they might require you to write:
- 30000 + 0 + 0 + 40 + 8
or
3(10,000) + 0(1000) + 0(100) +4 (10) + 8(1)
or
3(104) + 0(103) + 0(102) + 4(101) + 8(100)
Even though it seems silly to be that “complete” (I can’t imagine having to include the empty columns in “the real world”) it isn’t bad to teach it like that in order for students to get the feeling for the whole concept. It helps solidify it in one’s mind. It’s just when kids get tested on it and writing out the whole thing is required, that it gets a little ridiculous.
There may be other ways to express “expanded notation.” Of course, it depends on what one means by being “expanded.”
One of the problems with the way schools teach is that they usually teach you one way and imply that is the “official” way. Most often, the teachers are not even aware that they could be confusing students by not at least making them aware that there are other ways to do “standard” things.
Schools do that with many things in math, and in other subjects as well. I think that is an educational crime, because it encourages students to stop thinking and accept the narrow information they are given as the “absolute truth,” which, of course, it never is.
One thing you may want to keep in mind, is that although school lessons on “standard and expanded notation” tend to be boring and leave you with the feeling that those things are not important, and just basically fodder for tests, that is not necessarily so. To find out more about expanded notation, and why you should understand it, check out:
http://mathmojo.com/interestinglessons/expanded_notation/expanded_notation.html
There is also something called scientific notation which is often confused with expanded notation. They are not the same. For a good explanation of scientific notation, refer to:
http://www.nyu.edu/pages/mathmol/textbook/scinot.html
“Mom” also asked:
- “3205; is this number written in standard form?”
Prof. Hunc sez:
Yes. You could also have written it with the comma, as in 3,205. (You don’t have to write the comma in the real world, but sometimes schools require it.)
Thanks for writing in. Reader input is what drives Math Mojo. Keep it coming!
this was very helpful to me, i had the same question while working with my daughter’s homework! I can’t believe this site is here, I have bookmarked this site and will use it often!
i always confued in expanded notation ………..this site helped me a lot.
“Very informative stuff – thank you for taking the time to share it!
I’m also interested, which theme have you installed on this blog? Its a great design and I would like to know if it’s custom or not.”
Professor Homunculus sez:
Thanks for your kind words. I’m surprised that you like the theme, though. It’s old, and out of date. Doesn’t allow widgets, etc. I think the creator has an update. You can find it at: 3C SEO by Guido W. Stiehle
I did a little goofing around with the sidebars and css, and changed the header, obviously, but not much else. At Guido’s website, he has some free videos about how to install and tweak the theme. It was my first WordPress attempt, and I learned a lot from his info and his theme.
OmG.. THis site iszx the bomb. i like it …. It helped me with my homework
Thank you helped me so much~!
(is studying for maths assessment)
OMG! this is such a good site..it helped me with my studyguide homework!
I am studying for my final math assesment.thankz
I am not very good at math and this website helped me so much. Exspecially the mixed fractions , turning them into percents.
thanks
I have a related question:
My son 10-year-old son is having some difficulty converting expanded value expressions to standard notation. For a given standard notation number, he can tell me without hesitation what each digit represents (e.g., hundreds, ten thousands, etc.) He can go from standard notation to expanded value reliably. At this point the major stumbling block is the sort of number in which one or more of the digits should be 0. For example, this is one of the problems he had today:
40,000,000 + 10,000 + 3,000 + 60 + 1
The correct answer would be 40,013,061. Here is what my son put:
40,10,361
Misplaced commas are common in his answers. When we go over the missed problems he can see where he made a mistake, but he doesn’t seem to be getting more proficient at this type of question.
Any advice you can offer on how to teach this concept is much appreciated.
Thanks.
Hi, Maggie,
No problem! This is exactly the kind of problem I had a child myself, and still have a bit of trouble with. The system I’ve “invented” for myself is to make dots for the amount of the highest number. For example, in the problem above, I’d count the digits in the 40,000,000 and make that many dots, with the commas in the same place.
I’d write . . , . . . , . . . (making sure the commas are in every third place).
Then I’d erase the first dot and replace it with the digit of the largest number (4). Then I’d tell myself what place value it has (“forty million”).
Next, I’d look at the next highest number (10,000) and see where the 1 is in relation to the commas and zeros, erase the dot in the corresponding position and replace it with that 1, and once again tell myself what place value it has (“ten thousand”).
Next, I’d look at the next highest number (3,000) and see where the 3 is in relation to the commas and zeros, erase the dot in the corresponding position and replace it with that 3, and once again tell myself what place value it has (“three thousand”).
At this point I’d usually recognize that the ten thousand and the three thousand make thirteen thousand. If your son doesn’t right away, it’s no problem. When the problem is finished, have him read the entire number out loud, he will recognize it by then.
When that happens often enough, he’ll begin to see how to “chunk” the numbers in -illions and thousands as he goes along.
Then I’d continue on with the 60, and the 1, in the same fashion.
It worked for me. I hope it can have some good effect for you and your son.
One of the things I struggle with is a mild case of dysgraphia, which is sort of like dyslexia. I transpose numbers and letters occasionally. It is not extreme, and has never become a big handicap, but it is annoying when it happens. I’ve written about it before. One thing I do a lot is transpose numbers when I’m doing the crossword puzzle. I may read clue 41 across and try to put the answer in 14 across, or 61 across, or 41 down.
Nowadays I immediately self-correct. It was worse when I was a kid and felt a bit of pressure to “get the puzzle right,” but giving myself as much time as I need to solve the puzzle without pressure has almost eliminated the problem.
Now that I’ve realized that most pressure to solve these things is artificial (bogus standardized tests, etc.) I can take lots of pressure without goofing up like that. And I’m faster than ever.
One of the biggest hints I can give you is that until your son “gets” this and becomes proficient at it, you should make sure he is under no time pressure to get the answer write.
You might want to give him an example, then leave the room and get a cup of tea or whatever, and ask him to come to you when he thinks he has it absolutely correct. This way he develops the habit and confidence of self-correction.
I hope this helped.
Please try it and write in and let us know if this was of any use for you guys.
Thanks for reading (and writing to) the Math Mojo Chronicles.
Brian (a.k.a. Professor Homunculus)
Dear Professor Homunculus,
First, many thanks for your very speedy reply to my post.
Your method is an OUTSTANDING SUCCESS! I explained the method to my son and we walked through three examples together. Then I gave him additional problems to work on his own.
The result? He only missed one problem out of 17 (94%)! On a similar assignment two days ago he scored 33%. HUGE IMPROVEMENT IN UNDERSTANDING! I can’t thank you enough!
We did make one small change. My son has a mild form of cerebral palsy, and it’s a small difficulty for him to hold the paper down to erase. So instead of dots I had him draw a blank for each digit and simply fill them in as he worked the problem.
Thank you so much! I am officially bookmarking your site!
how do we do something in math like 600+50+4+0.2+0.7+0.009=
Jenna,
I’m not sure of what you mean by “do.” Do you mean how do you add those numbers? It looks like a simple addition problem to me. Do you understand place value (where the decimal point goes, for example?)
Can you write in again and be more specific? Let us know exactly what it is that you are getting at, and I’ll try to help.
- Brian
you helped me with my homework! thanks a bunch!
will you help me with the following math problem?: 123×29
Well, this isn’t the post for it (It’s about standard notation) but that one is a piece of cake, so let’s go:
- 29 is one less than thirty. So multiply 123 by 30 and subtract 123 from the answer.
- Multiplying 123 by 30 is like multiplying it by 3 and tacking a zero on the end.
- To multiply it by 3, you can notice that none of those digits will go over 9 when multiplied by 3, so there will be no carries. That makes it easy to multiply it from left to right.
- So start multiplying the digits 1, 2, and 3 from left to right. You’ll bet 369. Tack a zero on the end and get 3,690.
Now take 123 away from it.
- An easy way to do that in your head is to first take 100 away and get 3,590. Subtract 20 from that and get 3,570.
- Lastly, take 3 from that and get 3,567.
Do not be fooled by the long explanation. A monkey could almost do this. It just seems complicated because most schools do not teach how to look at numbers and understand what you can do with them. They teach the easiest ways to teach and grade, not the best ways to learn.
Also, don’t be tempted to use pencil and paper. None of those operations are something that you can’t do in your head. Get used to using your brain. It will thank you for the party!
On my test with expanded notation I got problems wrong that looked like this.
e.g. 20,466 I wrote: 6+60+400+20000 Why is this wrong? It still adds up to the right number.
Brittany
Hi, Brittany,
Wrong, in this case, is a subjective word. Yes, you did expand the number. But I don’t know which of the above types of expansion your teacher taught you and expected you to deliver.
Here’s the main problem: Schools are almost never about actual knowledge. They are about some curriculum they expect you to follow – sort of like hoops you are expected to jump through. At some point in life you have to realize this, and also realize that you can’t fight it. Schools are helpless about this. It sucks, but there it is.
So if your teachers want something, and something else makes sense as well, it generally doesn’t matter. You have to give them what they want in order to get the grade you want. It is like some weird contract.
So your answer could be considered right under some circumstances, but wrong under others.
Now here’s the bad news: The more you deliver, and the more you understand, the better it is for you. By your answer, you delivered the minimum. Look again at the post. There were three ways. Your was was the simplest, delivering the least amount of detail. Then there would have been the way using parentheses, and then there’s the way with parentheses and powers of ten. The last way shows that you understand not only how to expand, but how to represent the expansion in a concise way, using exponents.
If the teacher was specific about the way he or she wanted the answer, and you delivered less than he or she taught and asked about, the teacher would be absolutely right to mark your answer incorrect. On the other hand, if he never taught you any way other than the one you showed, you could have a leg to stand on.
But there’s more bad news – you didn’t simply expand it – you reversed it. I can’t imagine that you were taught to do that. I would be like if someone simply asked you what number 2 1/2 was, and you said, “Two and five tenths.” Technically you might be correct, but you’d have to admit that would be less than useful or efficient. It would not It would be “right,” but it could legitimately be marked “incorrect.”
School is a funny balance between what’s right, and what the teacher wants. Sometimes the teacher is right, sometimes you are right. Generally one is “righter” than the other. In this case, unless there are issues you didn’t include (like if the teacher never taught you any method but the one you demonstrated, or if he specifically required you, for some odd reason, to reverse the answer) the teacher seems to be “righter.”
Try to get over it, and learn the lesson that although some stuff in school is BS, there is a lot that isn’t. Try to learn the most you can. Actually, if you want to develop your real knowledge and maturity, learn more than is required, from sources beyond school, like the library. If you do that, you’ll be writing me again with similar problems, but the situation will be reversed. You will be “righter” than the teacher.
Then I’ll help you rip him a new one, OK?
- Professor Homunculus
I am 11yrs old and am having problems understanding writing expanded numbers as single numbers.
eg. 50 000 + 50 + 400 000 + 7 + 3 000 000 + 9 000 + 70 000 000
Most of the problems are in the millions
PLEASE HELP!!!!!!
Vidyuta,
Somebody’s been throwing you curve balls. What I mean by that is that those numbers should have been in order from highest to lowest. Otherwise it is confusing.
There is no actual “law” that it has to be that way, but it really is the only way that makes sense. After all, we don’t normally say a number as, “fifty thousand fifty four hundred thousand seven three million nine thousand seventy million,” do we? That would sort of be like calling 462 “Two sixty four hundred.” It could work, but it would be a little idiotic in most cases.
But let’s tackle it anyway. There are several ways to do it, but the most straightforward would be to put those numbers in order from highest to lowest. The highest number, of course (and by highest I really mean greatest) would be the seventy million. Obviously, the best way to identify which number is greatest is to notice which number has the most zeros after the initial digit.
Trying to rearrange them yourself before reading any further.
Did you get 70 000 000 + 3 000 000 + 400 000 + 50 000 + 9 000 + 50 + 7 ?
Now, as you speak them or write them, take them in groups of threes, or “illions,” first. The highest “illion “is the millions. You’ll see 70 million, and 3 million. Write or say that as “73 million “.
The next group of three would be the thousands. You’ll see 400 thousand, 50 thousand, and 9 thousand. So you would say or write, “459 thousand.”
In the last group of three, there is nothing in the hundreds place. And what’s left are fifty and seven, so you would simply say or write, “fifty seven.”
So, 70 000 000 + 3 000 000 + 400 000 + 50 000 + 9 000 + 50 + 7 becomes “seventy-three million, four hundred fifty-nine thousand, fifty-seven,” or 73,459,057.
I hope this answered your question.
If you are still not clear on it, write back, but make sure you have read the entire post, and this answer again, and then be very specific about what might be confusing to you, so we can zero in on exactly what the problem might be.
Brian (a.k.a. Professor Homunculus)
im 11 years old and i used this site to help me it really helped alot thanx
this website is the most wicked site in the world
help me. 34.98 expand notation?
I hate 5th grade math?!?! UGGGGG.
Daughter came home needing how to write out decimals in standard form but in fractions?! Help.
.97 + 11.07
[but like I said - has to be written out as fractions]
Emily,
Do you know how to add the two as decimals? 97 hundredths plus 7 hundredths equals 104 hundredths (or one “whole” and four hundredths, which is written as 1.04). Add that to 11 and get 12.04, which is read as
“twelve and four hundredths.”
From their it should be clear: twelve and four hundredths is the same as 12 4/100. Reduce it to simplest terms if you like, )but no law says you must – regardless of what schools say.)
Marie,
Make sure you’ve read the post at http://mathmojo.com/interestin…..ation.html
That would make sure you know that 34 in expanded notation is (3 x 10) + (4 x 1). So far so good?
.98 is only slightly trickier. The 9 signifies 9/10, which would be (9 x .1) because 1/10 is written as .1 as a decimal.
Can you guess how we’d write 8/100? 1/100 is written as .01 as a decimal, so 8/100 would be 8 x .01.
Therefore, 34.98 could be expressed in expanded form as (3 x 10) + (4 x 1) + (9 x .1) + (8 x .01)
Please let me know if that helped.
Brian (a.k.a. Professor Homunculus)
what a lovely site my daughter has found teacher….
3rd grade homework…..how is a zero used in standard form, expanded notaion, and word form…(help)
Dina,
I have a sneaking suspicion that this is the kind of vague question that means nothing, but seems like it needs a specific answer. I’m not sure what your teacher is getting at, unless s/he wants some trite answer like, “it is used to represent the fact that there is nothing in a particular column” or something like that. Please write back in a comment and let me know what the teacher wanted for an answer. I may be wrong, but it seems like the question is meant to be “busywork.” I am really curious, and hope that I am wrong, and we’ll all have something to learn from the question and the answer.
im doing 8th grade homework i need help on how to write 467,149 in expanded notation.
I am curious about this…
take the number 47.
I can write it in expanded notation as 40+7 or 4(10) + 7(1)
what if I decompose the number to
30+17 or 3(10)+17(1) – can I still call this expanded notation?
Basically, you can call it anything you want. But if you want to communicate with other people, it’s best to stick with standard terminology. There may be esoteric reasons for individual situations that mathematician’s may do that, but I don’t think looking for exceptions when you are just learning is a good strategy.
http://www.mathmojo.com/chroni.....ments-form
Of course, I could be wrong (that’s my default) so take this all with a grain of salt. But unless you know specific reasons for not doing so, I’d stick with the standard methods and terminology.