The break-apart method

Published Date: 23 Mar 2015 Last Modified: 24 Apr 2017

What is an example of a calculation using one operation (addition, subtraction, multiplication, or division) that cannot be easily computed mentally? What are some strategies you can use to mentally solve this expression? Which mathematical properties does your strategy require? Respond to classmates that may have chosen a different operation. Identify any similarities or differences from the way you solved your operation. Think about another way your classmate might have solved the problem mentally.

When you have more than a few numbers and you have to multiply them together it is difficult to multiply mentally. Say, if you have more than 4+ numbers, it gets tricky to mentally multiply them all together. For example if you have (34)(65)(56)(76)(87) = then many get confused on even where to begin on how to solve the problem. If you round the problems then you can get close to your exact number. (30)(70)(60)(80)(90) = 907,200,000. My problem requires estimation and rounding because you have to use both in order to solve this problem in order to get close to the exact number without using a calculator. If you use a calculator you would get an exact number of 818,301,120.

Posted: 1/31/10 8:56 PM, by: Amy Arnold ([email protected]) A calculation that might be difficult to compute mentally is 642 + 815 + 171. Since this calculation has many numbers that are not compatible when computing mentally, I would use find the break-apart method the easiest one to use for this problem. Using this method, I would break the first number into 600, 40, and 2; the second number into 800, 10, and 5; and the third number into 100, 70, and 1. I would then add 600 + 800 + 100, which equals 1500. Then I'd add 40 + 10 + 70, equaling 120. Finally, I'd add 2 + 5 + 1, which equals 8. Adding these results together gives me 1628, the final sum of the problem.

The break-apart method requires both the associative and the commutative properties to work. The commutative property states that two numbers can be added to each other in either order and the sum is still the same, and the associative property states that any grouping of three numbers can be added together with the sum staying the same. These properties explain why breaking numbers apart in this way still equals the same number when all of the numbers are added together.

Amy Arnold:

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Posted: 2/1/10 5:39 AM, by: Kei Gibbs ([email protected]) One example of a calculation using addition cannot simply be computed mentally is 248+30-18. The approach best used to solve this expression is the count on and count back method. For instance, you can start with the biggest number of 248, and then count on by tens such as 248,258, 268, and 278. Therefore, we know 248+30=278, and then we can use the count back method to continue to find the sum of this problem. Such as, you can start with the sum of 278 then count back by tens 278, 268, then just subtract 8 to get 260. So by using the count on and count back method for addends of 1,2,3 or 100, 200, 300 or so on we can easily count on or count back, just as with this problem by tens to compute mentally. 248+30-18=260

Kei Gibbs:

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Posted: 2/1/10 8:53 AM, by: Misty Mullendore ([email protected]) Subtracting is not always easily computed mentally. For instance I chose 74-38. For this operation the technique I chose is using the equal addition technique for subtraction. Subtracting a multiple of 10 is much easier, so for this operation you add two to both numbers: 76-40=36 and 74-38=36. This technique is used when one of the numbers in a subtraction calculation can be changed so that it results in a number that is easy to do mentally. In this case the number is a multiple of 10. Selecting the best number to add is important. This strategy requires the associative property of addition. Furthermore, once you have the operation you can work with, you could also using the counting technique. So once you have 76-40 then you could easily count down in multiples of 10: 76, 66, 56, 46, until you get to your answer of 36.

Misty Mullendore:

Posted: 2/1/10 11:06 AM, by: Susana Novella ([email protected]) One of the operations that cannot be easily calculated in the computer mentally would be adding. For example add 1 plus 2 plus 3 plus 4 http://www.uscg.mil/hr/psc/ps/whgdata/whlstf2.htm" o:spid="_x0000_i1025">� 100+101+102+103+104+........+200. Basically, is adding the first number to the last number which is 1+200=201, and then add 2+199=201 and so on. The last sum is 100+101=201, so the total of the sum is 201 with 100 times equal 201100. The associative property is use to join the term as 201+201_201....+201(100 terms) and then distribute property and 201, which is 1=1+1+....+1= 201x100=20100. Another operation would be, (2 x 8) x (5 x 7). One way of doing it 2 times 5 equals 10 and multiplying by 10 is easy. Then, 8 times 7 are 56 and 56 times 10 are 560. The product is 560. Second different way of doing it is 8 times 5 is 40. Then, 40 times 2 are 80 and 80 times 7 is 560. The product is 560.

Susana Novella:

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Posted: 2/1/10 11:32 AM, by: Amy Wood ([email protected]) Do you feel this way would work better than rounding or estimation? Do you like this way better and why? To me estimation works better for me than breaking a number down, how do you feel?

Posted: 2/1/10 11:37 AM, by: Amy Wood ([email protected]) I find this method a little more challenging, can you break it down and try to explain it a little better for me. I just cant seem to grasp the way this way is done. Can you please help?

Posted: 2/1/10 3:05 PM, by: Kei Gibbs ([email protected]) Hi Amy,

Ok let's start with the expression 248+30-18.The count on and count back method works like this you start off with 248 and count on tens to get the sum so 248, 258,268,278. I added 10 to each number that gets me to 278(this is what you will get when 248+30 is added) afterwards I did 278-18 which I needed up with 260.

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Posted: 2/1/10 4:57 PM, by: Amy Arnold ([email protected]) I suppose it depends on the type of answer needed. If I needed an exact answer, then rounding or estimation wouldn't be feasible, so I personally think that being able to use other methods is beneficial. I think the break apart method is really rather simple for addition, because it simplifies all the numbers into 100s and 10s, which are much easier to add together than the actual numbers.

Amy Arnold:

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Posted: 2/1/10 5:03 PM, by: Amy Arnold ([email protected]) Hi Amy,

What Kei did to solve her equation is break down the number 30 into three 10s. Therefore, 248+10=258+10=268+10=278. So 248+30(three 10s)=278. Then, she did the same with 18, breaking it down into one 10, and eight 1s. So 278-10=268-8(leftover from the 10 taken out of 18)=260. Using 10s and 100s to solve equations mentally is much easier than using less compatible numbers. Hope that helps :)

Amy Arnold:

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Posted: 2/1/10 5:58 PM, by: Cate Russell ([email protected]) Adding sales tax to a purchase, at least in Michigan where the sales tax is 6% is an example of a calculation using an operation that cannot be easily computed mentally. If the sales tax was 5% or 10% the math would be much easier to do mentally. To solve a 6% sales tax calculation the easiest option would be to break down the problem into smaller more manageable problems. Calculating 5% first and then adding a percent is a strategy to help solve this expression mentally. This strategy requires both multiplication and addition.

Cate Russell:

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Posted: 2/1/10 7:37 PM, by: Tricia Sauer ([email protected]) Unfortunately, I have always had a rough time doing even the simplest of math problems in my head. for instance, 64+49=.113..Until I began working with elementary math as an adult in the teachers role, I never learned or considered to round the 49 up to 50, count up by tens from 64 five times, and then subtract one. The elementary school my kids go to have started this math program this year that involves higher level thinking and lots of mental work. When I sub in the lower grades especially I am totally amazed at what the kids are able to do. They are better than I am!

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Posted: 2/1/10 7:41 PM, by: Tricia Sauer ([email protected]) That's a great example of a difficult computation to do in the head. I'm not even sure that rounding and multiplying that many numbers would be easy for me. I think I would have a hard time keeping track of how many 0's to add...I would immediately get a pen and paper with those numbers. I'm so used to using paper that it may be a crutch--trying to do it in my head overwhelms me. However, there are people who cannot function without a calculator, so at least I can use paper!

Tricia Sauer:

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Posted: 2/1/10 7:44 PM, by: Tricia Sauer ([email protected]) you have also given a great example. Mine was so much easier than that!!! I'm not sure I could keep the broken apart numbers straight in my brain without using paper... However, the method is great and I know so many kids will benefit from learning how to break apart math problems in his way. I wonder if I had learned to think this way as a child if I would have been able to do it well...

Tricia Sauer:

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Posted: 2/1/10 7:46 PM, by: Tricia Sauer ([email protected]) breaking the number down gives the exact answer and estimating does not, right? So, it depends on the results needed which method you would choose. Estimating may be easier to do int he head since it's fewer numbers to remember--even that is tricky for some if it's a lot of 0's.

Tricia Sauer:

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Posted: 2/1/10 8:17 PM, by: Krystal Brewer ([email protected]) When I go to the grocery store or any type of shopping I like to try and keep track of my spending through out the trip. It can be quite difficult keeping track of every price and knowing exactly what you have spent. $12.99 on these pants, $5.49 on this shirt, $19.95 on these shoes, etc. Using the Break Apart Numbers Technique you can round these prices up to the nearest dollar or fifty cents and it is much easier to add $13 + $5.50 + $20 = $38.50. This is an estimated price, to get the finished result you would need to then subtract the extra $.05 from the estimate on the shoes, and the extra $.01 on each the shirt and the pants. $38.50 - $.07 = $38.43 which is the exact amount, that is before taxes.

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Posted: 2/1/10 8:42 PM, by: Aimee Seymore ([email protected]) Do you think for a problem this large it would be better to round down for all numbers? I haven't worked it out yet, it just seems that doing it that way gives you a large margin of error.

Aimee Seymore:

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Posted: 2/1/10 8:54 PM, by: Aimee Seymore ([email protected]) If an equation holds to many numbers it will always be an obstacle to compute in your head, until you are able to use the numbers to your advantage. 8+19+25+6+48.

None of these numbers will work together to equal a simple number ending in '0' so you will have to find another way to get the numbers to work for you. While it is hard to accurately carry numbers when doing a large addition problem, one way to solve this in your head is to add all the numbers in the one's place first and then carry and add the numbers in the ten's place second, just as you would if you were using a pen and paper. Like so; 8+8+9+5+6=36, take the '3' and add it to 1+2+4+(3)= 10. Even if this isn't the quickest or easiest route to get the answer, it is the way that my brain sees it and is able to work the problem.

This problem used addition and the break apart method.

Aimee Seymore:

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Posted: 2/1/10 9:01 PM, by: Aimee Seymore ([email protected]) This was one of the first techniques discussed in Chapter 3 if you haven't read it yet. With some practice it will make more sense then just reading it. Once I work with it a few times I am sure it will be a very beneficial tool to use when looking at a problem.

Aimee Seymore:

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Posted: 2/1/10 9:15 PM, by: Gloria Schaffers ([email protected])

I think multiplication will be the most difficult to multiply mentally. Say you trying to multiply 897�46 in your head that can be the most difficult and confusing thing to do. The stratedy I would use is rounding. I would round the 896 to 900 and 46 to 50. Rounding your number to me will make the problem much easier to answer mentally.My problem requires estimation and rounding because you have to use both of them in order to solve the problem. in order to get an estimation number without a calculator you have to round it to the next closes number. If you use a calculator you would get an exact number of the problem you trying to solve. I always round my number to the nearest number especially when I am shopping. I truly believe when you round your numbers to nearest dollar or two when shopping, you will include your taxes.

Gloria Schaffers:

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Posted: 2/1/10 9:30 PM, by: Gloria Schaffers ([email protected]) Hi Kei,

That's a good example of the count on and count back method. I never thought of doing a problem like that. you made it very simple and easy to solve to me. I will be trying that method. Thank you a lot for a good example.Have a great week!!!!

Gloria Schaffers:

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Posted: 2/1/10 9:38 PM, by: Gloria Schaffers ([email protected]) Hi Amy,

That's a good example of a break down method. I haven't never thought about breaking a problem down like that to get the answer to a problem. I will definately try this method it seem very is to do mentally. Have a great week!!!!!

Gloria Schaffers:

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Posted: 2/1/10 11:24 PM, by: Kristen Cross ([email protected]) What is an example of a calculation using one operation (addition, subtraction, multiplication, or division) that cannot be easily computed mentally?

An example of a calculation that uses addition and subtraction and cannot be computed mentally is when I do my banking. I have so many different things to keep track of and the numbers are usually on the high side. I have to sit down with a calculator and go through every single purchase or deposit.

What are some strategies you can use to mentally solve this expression?

I will occasionally estimate when it comes to my banking just so I have an idea. I usually do this when I don't have the time to figure out the exact amount in my account. I don't like to do it all the time, but it is a short term fix.

Which mathematical properties does your strategy require?

I don't feel like there is an obvious property but the associative property would play a role. It feels like the transactions are just little groups.

Posted: 2/1/10 11:44 PM, by: Danielle Peters ([email protected]). One an example of a calculation using multiplication that can not easily be computed mentally would be 73 x 15.

A strategy you could use to mentally solve this expression would be to estimate. If you estimate 73 to 70 and estimate 15 to 10 you come up with 70 x 10 which can easily be done mentally to equal out to 700. The answer to the problem would be somewhere between 700 and 800.

The mathematical properties that my strategy requires is to multiply whole number and also to estimate or round whole numbers to the nearest tenth place.

Danielle Peters:

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Posted: 2/2/10 6:14 AM, by: Kei Gibbs ([email protected]) Hi Krystal,

I do the same thing, I often find myself in a store and need to figure the percentage of something. For example I was recently shopping for my son some school clothes and there were so many items marked 10%, 25%, and even 50% off. I find figuring percentages of tens to be easy to do mentally, but when I come across one that is say 32% off that is not so easily done mentally.

So what I do is always take the original price and times it by the percentage and then subtract that from the original cost to get the total sales price. I even round or use compatible numbers when appropriate to make mentally computation easier.

For example, boy�s shirts are on sale for 32% off the original price this week and the original price is $19.99. I would round $19.99 to $20.00 and times that by .30 to get the sum of $6.00 so I know now that the shirts will roughly be on sale for about $6.00 off for a sales price of about $14.00. This estimation is very close to the original price, but a lot easier to do mentally. For example my estimation of $20.00 * .30=$6.00 then $20.00-$6.00=14 is very close to the original sales price of $19.99*.32=6.40 then $19.99-$6.40=$13.49.

Kei Gibbs:

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Posted: 2/2/10 6:16 AM, by: Kei Gibbs ([email protected]) Hi Gloria,

I guess that is why everyone has their own different strategies for working problems. Whether it is with pen and paper or mentally. I find it easy to do problems especially mentally where I have to use strategies that break down the problem into proportions, that way I can see each step as I go along. For example using the count on techniques by tens I can easily see the solution as I progress through the problem.

Kei Gibbs:

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Posted: 2/2/10 7:23 AM, by: Amy Arnold ([email protected]) Tricia,

I actually was never taught this method early on either, and I wish I had been! My dad taught it to me in high school, and was surprised to know that I was never taught it in elementary school because he was. I think it's a very simple method to use when dealing with larger, non-compatible numbers, such as in my example. I'll be teaching this to my daughter once she is ready to learn basic math! What types of mental computation strategies were you taught when you were younger?

Amy Arnold:

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Posted: 2/2/10 7:28 AM, by: Amy Arnold ([email protected]) Aimee,

I agree about rounding - it seems that there can be such a large margin of error, that I usually try to avoid it unless I know I can estimate really close to what the exact answer would be. For Amy's example, for instance, I would probably not choose to round and estimate an answer, only because there are so many numbers in the equation that the final estimated product cannot come close to the exact answer, as shown in her example, which was off by close to 100,000,000. I think rounding might be a better strategy to use when dealing with only a few numbers and where the estimated answer can come somewhat close to the exact answer. Any thoughts?

Amy Arnold:

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Posted: 2/2/10 10:43 AM, by: Amy Wood ([email protected]) I think after looking at my example, it was a bad one and I agree with you. Rounding on that problem tended to be way off. Now looking back on it, I would chose either another way to solve the problem or another problem all together to use that method on.

Posted: 2/2/10 10:47 AM, by: Amy Wood ([email protected]) Thank you for that, I have to agree. When I have a lot of numbers to solve for, even sometimes I get myself confused. I have to break out the paper and solve it that way. I guess the only way to learn it easier and better is to constantly do it over and over again or repetition.

Posted: 2/2/10 11:25 AM, by: Vanessa Kempson ([email protected]) I am not the brightest when it comes to any kind of mathematics problems. An example of a math problem that I could not solve mentally would be any word problem. For myself word problems tend to confuse me, because I become overwhelmed very easily. When working on word problems I will break down each sentence, and gather all of the information. I then set my equations up from the information that is presented. After I have my equations I work them out, to find my answer. I also write everything down on paper when working on word problems. To solve a word problem it can require any type of mathematical properties.

Vanessa Kempson:

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Posted: 2/2/10 11:59 AM, by: Tricia Sauer ([email protected]) Honestly I don't remember being taught any mental math strategies as a child. Some of these methods mentioned in ch3 of our text are new to me and I'm irritated I never knew them. I've always depended on pencil and paper. I know that I've always struggled with math--I'm sure a negative attitude played a big role, but even with a good one math isn't my strength. Anyway, had I been using these strategies all my life mental math might not be so tough for me. As a future elementary school teacher, I'm concerned that I don't have the solid background necessary to give students the solid background they need. This class has been very informative so far.

Tricia Sauer:

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Posted: 2/2/10 12:02 PM, by: Tricia Sauer ([email protected]) To take her explanation one step further, I can't subtract 18 in my head easily, so I round 18 up to 20, subtract 20, and then add back 2--since 20 is 2 more than 18...

Tricia Sauer:

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Posted: 2/2/10 12:06 PM, by: Tricia Sauer ([email protected]) Amy A.,

Based on your post, you seem to have a good grasp on expanded notation. Do you understand that base 2, base 8, etc...stuff? My husband showed me the formula for figuring out what the numbers are in the different bases, but I have NO IDEA what it means or what I'm doing. The textbook on that part seems as foreign to me as the writing in Egyptian part...

Tricia Sauer:

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Posted: 2/2/10 12:10 PM, by: Tricia Sauer ([email protected]) that's a great strategy I've never used before. You're right, it's so much easier to work with 10's...

Tricia Sauer:

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Posted: 2/2/10 12:14 PM, by: Tricia Sauer ([email protected]) I round up to the nearest whole dollar when shopping also, but I don't ever worry about subtracting back out the extra change...I just assume it's close to the tax amount and go with that. If I think I'm coming close to my maximum that I can spend, I get a little worried with my estimations though!! To combat that, I may round up a few extra dollars just to be safe. I never want to get to the counter and not have enough $.

Tricia Sauer:

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Posted: 2/2/10 12:16 PM, by: Tricia Sauer ([email protected]) You make figuring out the sales price so easy...I'm gonna have to tell my husband I need to go shopping so I can practice!! :) Thanks.

Tricia Sauer:

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Posted: 2/2/10 6:31 PM, by: Chris-Anne Worrell ([email protected])

One calculation that cannot be easily computed mentally would be 645+30+15. One strategy that I would use is the break apart method. I would first add 30+15 which equals 45, and then use the Count On strategy and add 645+45, by doing 45+45+600 which would equal 690. The property that would be required would be the commutative property, because instead of adding 645+30 I would first add the 30+15 then, take the sum and add that to the 645 and come up with my answer.

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Posted: 2/2/10 6:39 PM, by: John Price ([email protected]) Amy, you certainly picked a difficult one! Your trick of rounding to estimate is excellent. How can you then take that the next step to get the exact answer? Would it help to do the multiplication two numbers at a time?

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"Amy Wood" wrote in message news:[email protected]

What is an example of a calculation using one operation (addition, subtraction, multiplication, or division) that cannot be easily computed mentally? What are some strategies you can use to mentally solve this expression? Which mathematical properties does your strategy require? Respond to classmates that may have chosen a different operation. Identify any similarities or differences from the way you solved your operation. Think about another way your classmate might have solved the problem mentally.

When you have more than a few numbers and you have to multiply them together it is difficult to multiply mentally. Say, if you have more than 4+ numbers, it gets tricky to mentally multiply them all together. For example if you have (34)(65)(56)(76)(87) = then many get confused on even where to begin on how to solve the problem. If you round the problems then you can get close to your exact number. (30)(70)(60)(80)(90) = 907,200,000. My problem requires estimation and rounding because you have to use both in order to solve this problem in order to get close to the exact number without using a calculator. If you use a calculator you would get an exact number of�818,301,120

Posted: 2/2/10 6:41 PM, by: John Price ([email protected]) Aimee, that is good thinking. Let's try it on something just a bit easier. Try 99 x 101. Would 90 x 100 or 100 x 100 give a better answer?

By the way, can you take it the next step and get the exact answer in your head? How?

John Price: FacultyUniversity of [email protected] [email protected] (570) 637-5875 Eastern Time Zone: Student Technical Support: 866-334-7332

"Aimee Seymore" wrote in message news:[email protected]

Do you think for a problem this large it would be better to round down for all numbers? I haven't worked it out yet, it just seems that doing it that way gives you a large margin of error.

Aimee Seymore:

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Posted: 2/2/10 6:48 PM, by: Chris-Anne Worrell ([email protected]) The method I think you should use is the compensation technique as oppose to the break apart method. Because the break apart is basically breaking apart certain numbers, for example, your problem was $12.99+$5.49+$19.95. If you use the break apart method you would break down the numbers like $12.00+$5.00+$19.00 which would equal $36.00 then you would add $.99+$.49+$.95 which would equal $2.43, then you would take $36.00 and add $2.43 which would then equal $38.43. The quickest way in my opinon would have been the compensation method.

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Posted: 2/2/10 6:50 PM, by: Misty Mullendore ([email protected]) Tricia,

Counting to tens and working with tens have always been the easiest for me. When you look at numbers and even bigger numbers, you can always break them down easily into tens and even hundreds. This makes for a quick way to mentally compute larger numbers or at least gives you a place to start.

Misty Mullendore:

Posted: 2/2/10 6:57 PM, by: Chris-Anne Worrell ([email protected]) I completely agree with what you did. I personally would have used the break apart method also, and then I would most likely use the commutative property.

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Posted: 2/2/10 6:57 PM, by: Misty Mullendore ([email protected]) Gloria,

I usually do not think of rounding in multiplication. Your example was a big number, but your method of rounding and estimating is useful for me. My question is, with such a big number, after rounding up, how do you get a more accurate answer without using the calculator?

Misty Mullendore:

Posted: 2/2/10 7:22 PM, by: Cate Russell ([email protected]) I have a hard time doing math completely mentally as well. My math classes never really taught me trick like they teach in elementary school now. Had I learned these tricks math would become a lot simpler. I can break the problems down but its something I really have to think about.

Cate Russell:

EST:

Posted: 2/2/10 7:24 PM, by: Cate Russell ([email protected]) I do that as well. Just like you, I round to make the numbers much more manageable. That trick really helps, otherwise I would need to keep a calculator in my purse.

Cate Russell:

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Posted: 2/2/10 7:32 PM, by: Gloria Schaffers ([email protected]) Hi Kristen,

I totally agree with you.I have to do the same.I make sure I keep every reciept after a purchase until it clears my account. Have a great week!!!!

Gloria Schaffers:

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Posted: 2/2/10 7:52 PM, by: Gloria Schaffers ([email protected]) Hi Danielle,

Good Example!!!!!!I do the same thing I round everything to the nearest number even when I shopping at stores I round it to the nearest dollars. I think by round numbers make mental math much easier. Have a great week!!!!!!!!!!!!

Gloria Schaffers:

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Posted: 2/2/10 8:25 PM, by: Susana Novella ([email protected]) This a very good point, especial that usually every women have to go through the mission of keep in track in how much money will be spending when it come to glossary. Every time I�m at the store the first thing I do is to wrap a pen and a paper and begging writing all each prize of the item I will be buying. The good thing about it is that these items usually run with a small category of praise which it give the facility to add up everything at the end and then multiplied the percentage to be able to find my estimate. Round number is the best way of how to find the correct estimate, which for me is like a habit that I do it every day and especially when I�m at the store.

Susana Novella , Miami, FL

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Posted: 2/2/10 8:37 PM, by: Susana Novella ([email protected]) I agree with you multiplication is not an easy work to handle especially with thought using calculate. When you need to find the estimate in the multiplication and there are more than two digits it get more complicated than anything. For instate, the other day I have to find the estimate of how much time I need to pay off my car, which I have to add and multiplied to be able to get my correct answers. This is why is very convenient to get familiar in doing estimation because once you get familiar the rest is easy.

Susana Novella , Miami, FL

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Posted: 2/3/10 6:26 AM, by: Amy Wood ([email protected]) I find that it is so much easier when multiplying more than 2 numbers at a time to add 2 at a time, a lot of times if I have 4 numbers, I will multiple them in groups of 2, then multiply the product of the two by each other. I find this works easier for me.

Posted: 2/3/10 8:13 AM, by: Amy Arnold ([email protected]) Hi Chris-Anne.

You picked a good example, but I just wanted to clarify that the count-on strategy was not used correctly. The count on strategy, in your example, would be able to be used if you take 645, and think of 30 as three tens. So 645 + 10 + 10 + 10 = 675. Then take 15, think of it as one ten and 5 ones, so 675 + 10 = 685. 685 + 5 = 690. What you did with the second part of your solution was actually use the break apart method again. Hope that helped clarify a bit :)

Amy Arnold:

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Posted: 2/3/10 8:21 AM, by: Amy Arnold ([email protected]) Hi Krystal,

I liked how you used shopping as an example for when to use mental computations. I tend to do it so often when shopping that I don't even realize it anymore! Rounding is perfect for shopping, because I hardly ever have a calculator available. I usually round all my prices up, that way I don't undercompensate for anything, and I can have a better idea of what I'll be spending after taxes get added on. Do you do this too, or do you usually just round to the nearest dollar, whether it's under or over?

Amy Arnold:

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Posted: 2/3/10 1:58 PM, by: John Price ([email protected]) Class, I am intrigued by this particular line of discussion. Please see the estimation DQ for a challenge that Ay and Krystal may find interesting.

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"Amy Arnold" wrote in message news:[email protected]

Hi Krystal,

I liked how you used shopping as an example for when to use mental computations. I tend to do it so often when shopping that I don't even realize it anymore! Rounding is perfect for shopping, because I hardly ever have a calculator available. I usually round all my prices up, that way I don't undercompensate for anything, and I can have a better idea of what I'll be spending after taxes get added on. Do you do this too, or do you usually just round to the nearest dollar, whether it's under or over?

Amy Arnold:

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Posted: 2/3/10 2:00 PM, by: John Price ([email protected]) Can anyone give me a list of steps, all that could be done mentally with little difficulty that will allow you to arrive at the exact answer for: 12 x 137 + 16?

It looks difficult at first, but I assure you that it can be done mentally. Look on this one as a puzzle. How can you use the techniques in this chapter to do this problem in your head?

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Posted: 2/3/10 3:52 PM, by: Susana Novella ([email protected]) yes, I do the same thing. I always rounding up because like that I would know how much money I can spend and know when to stop.

Susana Novella , Miami, FL

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Posted: 2/3/10 4:56 PM, by: Misty Mullendore ([email protected]) Here goes: 12x137+16

I made the 12 a 10 and multiplied that by 137. Easily I know the answer is 1370. To make up for the 2 I took away from the 12, I multiplied 2 times 137 and came up with 274. Multiplying 2 times 137 isn't too easy to do in my head so I just mentally added 130 + 130 together and then added 14 to that number. 1370 is a nice number so to add 274 to it I just did 1370 + 100+100+70+4 and came up with 1644. I then took 1644 +15 +1 to come up with the final answer of 1660. I pretty much used every technique available.

I am sure that there is an even simpler answer, but this was my technique.

Misty Mullendore:

Posted: 2/3/10 5:08 PM, by: Misty Mullendore ([email protected]) Vanessa,

Like you, I am not great at word problems, but I do appreciate what we are covering this week. Getting the information from your word problem and piecing it together is the hardest part. Once you have this information, then you can use the techniques in chapter 3 to help you mentally and quickly figure the problems out. The technique I rely on the most is counting in 10's, 100's, and so forth and rounding numbers.

Misty Mullendore:

Posted: 2/3/10 5:51 PM, by: Jennifer Ashley Giles ([email protected]) This problem would be hard to solve mentally: 3/18 + 4/12. What I would mentally do to solve this situation is to look at the 18 and the twelve, and find the common denominator, which is 6. So if 6 is the denominator, then 18 would be divided by 6 and would give you a 3, and if I divide twelve by 6, I get 2, which I would then add the three to the three getting six, and 4 added to two to get 6 , and then I would add 6 and 6 to get me 12/6, which, when simplified, gives me 6 as a whole number.

Once you write it on paper, its easier, but fractions can be kind of scary which is why I think it is hard to do mathematically.

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Posted: 2/3/10 6:47 PM, by: Jennifer Ashley Giles ([email protected]) I like they way that you computed how to do subtraction mentally. I also have a hard time subtracting mentally. I always like to resort to rounding when it comes to solving a subtraction problem. if I were to take the number s you used, 74-38, I would round 74 to 70, and 38 to 40, which would give me 110. It is not that much lower than the exact number.

Jennifer Giles:

Elementary Education

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Posted: 2/3/10 6:49 PM, by: Jennifer Ashley Giles ([email protected]) If I am not planning on buying a whole bunch of stuff, I usually like to round all of my numbers up, so it kind of includes the tax. If it is $ 5.49, I am going to round up to 6/ If it is 4.23, I am still going to round up. It helps so much because when I get to the check out, I have usually over calculated which is great because I stay within my budget. What do you think?

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Elementary Education

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Posted: 2/3/10 6:53 PM, by: Tricia Sauer ([email protected]) surely for the problem 99 x 101, rounding to 100 x 100 would be a better estimate than 90 x 100 because 99 is only 1 away from 100...please list the steps in doing 99 x 101 in your head. I would not know where to begin. I think mental math is much trickier with X. For addition, I would move both 99 and 101 to 100, and then know the answer is 200 since i both subtracted and added 1 in my rounding.

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Posted: 2/3/10 7:00 PM, by: Tricia Sauer ([email protected]) The break apart method in that example is not easy to me...I need paper to figure out the broken apart change!! I agree that the compensation method is best. Even with that, remembering how much over or under each amount is tricky I think.

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Posted: 2/3/10 7:02 PM, by: Tricia Sauer ([email protected]) I always round up to the next dollar to compensate for tax. It's also a nice surprise when the bill is lower than i estimated!!

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Posted: 2/3/10 7:04 PM, by: Tricia Sauer ([email protected]) I think it's funny you're intrigued. Is it because you are a man and don't do the shopping as much?? I think all shoppers round up maybe without even knowing it.. I'm totally curious about the estimation DQ now!!

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Posted: 2/4/10 9:17 AM, by: Vanessa Kempson ([email protected]) Misty,

I also agree that chapter 3 has great strategies to overcome word problems as well. Many times, in word problems, you have a variety of numbers. Majority of these numbers are not easy to mentally solve. So yes rounding would defiantly come into effect and help anyone out! I do like to round my numbers in ending with 0, it is just so much easier.

Vanessa Kempson:

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Posted: 2/4/10 9:28 AM, by: Tricia Sauer ([email protected]) oh my gosh--i can't believe you did all of that in your head. I'm really impressed. I had a hard time following the process reading it!! Do you use that length of mental math on a regular basis? I wonder who would win the race, me on paper or you in your head? if it were possible for me to win the race on paper, then why do I need to try to do that complicated stuff in my head? I can just keep paper and pen in my purse or car...

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Posted: 2/4/10 9:32 AM, by: Tricia Sauer ([email protected]) is this applicable to anything real in life other than math puzzles? It has never even crossed my mind to attempt anything like this in my head--I didn't know anybody did this! I usually have pen and paper handy..it would take me WAY WAY longer to do this in my head--and I would run out of memory to keep track of numbers...

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Posted: 2/4/10 9:34 AM, by: Vanessa Kempson ([email protected]) Jennifer,

I am constantly having problems with fractions. You have provided great strategies for anyone to easily solve this problem. Like you said, when reading this equation it is hard to understand, but I did in fact put the equation with the steps on paper. It was easy to solve with the steps you provided. Do you have any problems with the work in chapter 3 that you can provide a strategy for?

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Posted: 2/4/10 3:03 PM, by: John Price ([email protected]) Class, a variation on Misty's method is

subtract 74 - 40 = 34

Adjust by "giving back 2" 34 + 2 = 36, since we subtracted 2 more than the problem states.

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"Misty Mullendore" wrote in message news:[email protected]

Subtracting is not always easily computed mentally. For instance I chose 74-38. For this operation the technique I chose is using the equal addition technique for subtraction. Subtracting a multiple of 10 is much easier, so for this operation you add two to both numbers: 76-40=36 and 74-38=36. This technique is used when one of the numbers in a subtraction calculation can be changed so that it results in a number that is easy to do mentally. In this case the number is a multiple of 10. Selecting the best number to add is important. This strategy requires the associative property of addition. Furthermore, once you have the operation you can work with, you could also using the counting technique. So once you have 76-40 then you could easily count down in multiples of 10: 76, 66, 56, 46, until you get to your answer of 36.

Misty Mullendore:

Posted: 2/4/10 3:12 PM, by: Misty Mullendore ([email protected]) Tricia,

I am sure there was an easier way, but that is the only way I thought to do it. Of course, writing it all out sounded ridiculous. I do not usually do math unless I have a calculator placed firmly in my hand. I like this weeks topic because of all of the different mental techniques.

Misty Mullendore:

Posted: 2/4/10 3:17 PM, by: Misty Mullendore ([email protected]) Tricia,

I round in the store as well, but usually by the end of my shopping trip I loose where I am at. Like you, I always just round up and forget about getting a more accurate or exact amount. I assume that whatever I round up will carry over.

Misty Mullendore:

Posted: 2/4/10 7:38 PM, by: Krystal Brewer ([email protected]) Hello Kei,

The only way I go shopping for such things as clothes or electronics is when they are either on sale or clearance is even better. I love to figure out how much I am saving when they have rack of clothes that are a certain percentage off. Math has always been so natural to me and I do anything possible to save money so I guess it was just natural to learn to shop with sales and percentages.

I don't mentally do the multiplication. Instead I would take and divide the item. Like a $10 item was 25% off I would divide by 4 to figure out my savings of $2.50. In your example I would have done the same thing. I would have taken the $19.99 round up to $20 and divide by 3 to get an idea what about the 32% off would be since 33.33% is one third. So dividing by 3 would give me $6.66 give or take a few pennies. Then I would subtract the $6.66 away from the $20 to get about $13.40 as a price.

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University of Phoenix

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Posted: 2/4/10 7:50 PM, by: Krystal Brewer ([email protected]) Hello Tricia,

I usually round everything to the nearest dollar or fifty cents when actually shopping. I was just using this as an example to explain this procedu