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What does "mental math" mean to you?


Rosie
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When people talk about mental math I've always assumed they meant the way Singapore and Math Mammoth and RightStart and Math-U-See teach it - by making tens. (If you don't know what I'm talking about, there are video examples here - http://www.educationunboxed.com/mental_math.html) Recently, though, I've noticed on a few threads here that some people may be thinking of it as something else - something more like drilling math facts or something. One of the comments was made by someone who uses Saxon which makes me wonder if "mental math" in other programs means something totally different.

 

So, what do you think of when you hear or say "mental math?"

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When people talk about mental math I've always assumed they meant the way Singapore and Math Mammoth and RightStart and Math-U-See teach it - by making tens. (If you don't know what I'm talking about, there are video examples here - http://www.educationunboxed.com/mental_math.html) Recently, though, I've noticed on a few threads here that some people may be thinking of it as something else - something more like drilling math facts or something. One of the comments was made by someone who uses Saxon which makes me wonder if "mental math" in other programs means something totally different.

 

So, what do you think of when you hear or say "mental math?"

 

I mean the Singapore version of mental math, but that's what we use. It's more than making tens. That is just one of the methods taught in the program.

 

I read that thread too and had the same realization you did. I never thought of drilling or recalling memorized facts as mental math.

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I was introduced to the term 'mental math' through this book, which my sister recommended based on a recommendation to her from the ps where her children attended.

 

Because of this book, I first associated the term with math parlor tricks! Now that we've gone through 2 and a half levels of RightStart, I give it the same meaning you do.

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Well, making tens is the most common, but any thing you do to mentally add/subtract/multiply/divide without having to use paper is mental math. Another trick is to add multidigit numbers left to right...readjust if there's carrying, but adding them left to right in your brain is less work since you already say the digits in that order. Another trick is rounding...if you need to add prices in your head, you round to the nearest dollar. If you need an exact answer to 97cents plus $2.05, you add $1+$2, then you add the extra 5cents and subtract the missing 3 cents that were rounded off. When you learn the doubles facts, and use them to get doubles +1 facts...that's mental math. I think of it as anything that involves juggling the quanties in your head.

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I think of "mental math" as being able to calculate, say, 3290 * 472, or the square root of 456789, virtually instantly in one's head. Or to say what day of the week it was on some date hundreds of years ago. I know people who do this, and I have no real idea what they do, but I have also seen books on math tricks, such as that by Shakuntala Devi. Or generically as in being able to calculate tax or tip or do big sums in one's head instead of needing to write it down.

 

I think any math curriculum that doesn't involve use of the brain is, well, not on my radar. ;)

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I always thought it was re-grouping by making and breaking 10s too. What else is there to it?

 

:rant:My personal vent is though that mental math seems to have this pedestal of superiority. That frustrates me because my dyslexic ds who is incredible with manipulating numbers just cannot always do it in his head. He has to draw or see it spatially. I'm not sure mental math will ever be his thing, but he definitely uses the same techniques, albeit visually or on paper.

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I think of mental math as strategies for completing computations without having to write the problem/question down. Making tens is one of the strategies taught for mental math.

 

:iagree:This is how I think of mental math.

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I think of Mental Math as being able to employ the laws of mathematics with either an explicit or implicit understanding (and in Singapore it is often the latter) in order to (in mathematically valid ways) be able to manipulate numbers so one can more easily facilitate computations or other problem solving exercises in ones mind.

 

The potential strategies are myriad.

 

I do think of Mental Math as being distinct from memorization or fact-drill, even though both might be done orally.

 

Bill

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I think of it as just doing math in your head vs paper. That includes learning strategies to do math in your head.

 

I had the same twacher for 2nd and 3rd grade. Everyday, as we waited for the bell so we cohld go home, we did "mental math." She did a string of problems to see who could keep up and get the right answer. It was just practice of basic facts. 1+4*6/2-5*7+35.....etc.

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I do think of Mental Math as being distinct from memorization or fact-drill, even though both might be done orally.
Mental math is using the basic math facts (your definition of "basic" may differ from mine) to perform mental, as opposed to pen and paper, calculations. Some strategies are more efficient than others.
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Mental math is using the basic math facts (your definition of "basic" may differ from mine) to perform mental, as opposed to pen and paper, calculations. Some strategies are more efficient than others.

 

And indeed one point of drilling is to facilitate rapid mental calculations.

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I see it as the difference between doing it all in your head or having to write it down to keep it straight.

My DH, the engineer is amazing at the former. I usually have to write it to "see" it. He can listen to a word problem and have it in no time flat, and I would have to make a table, write the equations and do them.

 

I don't think it is the same as reciting facts you've memorized. I do drill, but usually it is in the form of questions that require the boys to try to hold the previous answer in their heads--4+6, now takeaway 5, now add 12, take away 4, etc. But I don't know if that would qualify as an example of mental math. I'd just rather they could do the math in their heads like their Dad and Grandfather, and not have to resort to having to grab a pencil.

Not that I don't allow them to use a pencil. There's probably a need for some people to visualize by writing it down first before being able to retain the numbers as you go along.

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Mental math is using the basic math facts (your definition of "basic" may differ from mine) to perform mental, as opposed to pen and paper, calculations. Some strategies are more efficient than others.

 

And indeed one point of drilling is to facilitate rapid mental calculations.

 

Still, don't you see a difference between rapid recall of math facts (which are unarguably good skills to have) and mental math (which might use math facts, but doesn't stop there)?

 

I see a distinction between math fact memory, which is a single-stage process, as in you either "know" the product of 8 and 7 or you don't, and the multi-stage processes that takes place when employing mental math stratagies. The latter involves manipulation numbers and employing the properties of mathematics to make problem solving easier. In contrast memorizing "math facts" requires no knowledge (either implicit or explicit) of the laws of mathematics.

 

To me these are different things. Yes, automaticity is helpful for performing mental math, but memorized math facts are not the same thing as having the sort of mathematical reasoning skills required for mental math.

 

So I see them as distinct but complimentary skills.

 

Bill

Edited by Spy Car
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Still, don't you see a difference between rapid recall of math facts (which are unarguably good skills to have) and mental math (which might use math facts, but doesn't stop there)?

 

Who doesn't see a difference?

 

I don't think my first answer to the OP's question in any way implied that I see mental math as knowing 2x5=10. Feel free to consult my original ideas of what mental math means to me. I don't see how saying that one goal of drilling is being able to calculate rapidly means I am dismissing the importance of higher level thinking skills. If one goal of learning the alphabet is to be able to write poetry, that hardly means the ABCs are the only requirements for being the next Shakespeare.

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Still, don't you see a difference between rapid recall of math facts (which are unarguably good skills to have) and mental math (which might use math facts, but doesn't stop there)?

 

I see a distinction between math fact memory, which is a single-stage process, as in you either "know" the product of 8 and 7 or you don't, and the multi-stage processes that take place when employing mental math stratagies. The latter involves manipulation numbers and employing the properties of mathematics to make problem solving easier. In contrast memorizing "math facts" requires no knowledge (either implicit or explicit) of the laws of mathematics.

 

To me these are different things. Yes, automaticity is helpful for performing mental math, but memorized math facts are not the same thing as having the sort of mathematical reasoning skills required for mental math.

 

So I see them as distinct but complimentary skills.

 

Bill

Isn't that what I said? :tongue_smilie:
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Who doesn't see a difference?

 

I don't think my first answer to the OP's question in any way implied that I see mental math as knowing 2x5=10. Feel free to consult my original ideas of what mental math means to me. I don't see how saying that one goal of drilling is being able to calculate rapidly means I am dismissing the importance of higher level thinking skills. If one goal of learning the alphabet is to be able to write poetry, that hardly means the ABCs are the only requirements for being the next Shakespeare.

 

Isn't that what I said? :tongue_smilie:

 

See, we all agree, but I'm just the last to see it :lol:

 

Bill

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It's more than making tens. That is just one of the methods taught in the program.

 

 

Making tens is one of the strategies taught for mental math.

 

Well, making tens is the most common, but any thing you do to mentally add/subtract/multiply/divide without having to use paper is mental math. Another trick is to add multidigit numbers left to right...readjust if there's carrying, but adding them left to right in your brain is less work since you already say the digits in that order. Another trick is rounding...if you need to add prices in your head, you round to the nearest dollar. If you need an exact answer to 97cents plus $2.05, you add $1+$2, then you add the extra 5cents and subtract the missing 3 cents that were rounded off. When you learn the doubles facts, and use them to get doubles +1 facts...that's mental math. I think of it as anything that involves juggling the quanties in your head.

 

See I think of it all as "making tens" (except for the doubles +1, I guess) because you are always thinking about tens... or hundred, thousands, etc.... always thinking about place value. If you add $1+$2 in the example above, you are thinking of tens/hundreds when you are rounding. Maybe there's a better term than "making tens?" Any suggestions? The term "mental math" doesn't seem like it will work since people seem to have varying definitions.

 

I just think it's important that we all know what we mean by certain words so that when we recommend "mental math strategies" it is clear to the person what we are implying.

 

 

:rant:My personal vent is though that mental math seems to have this pedestal of superiority. That frustrates me because my dyslexic ds who is incredible with manipulating numbers just cannot always do it in his head. He has to draw or see it spatially. I'm not sure mental math will ever be his thing, but he definitely uses the same techniques, albeit visually or on paper.

 

See, I wouldn't think of that as inferior to doing mental math. The importance of mental math, IMO, is that you are UNDERSTANDING numbers and how they work together. Having to draw or see it spatially is just a different learning style.

 

I think of it as just doing math in your head vs paper. That includes learning strategies to do math in your head.

 

I had the same twacher for 2nd and 3rd grade. Everyday, as we waited for the bell so we cohld go home, we did "mental math." She did a string of problems to see who could keep up and get the right answer. It was just practice of basic facts. 1+4*6/2-5*7+35.....etc.

 

I see it as the difference between doing it all in your head or having to write it down to keep it straight.

My DH, the engineer is amazing at the former. I usually have to write it to "see" it. He can listen to a word problem and have it in no time flat, and I would have to make a table, write the equations and do them.

 

I don't think it is the same as reciting facts you've memorized. I do drill, but usually it is in the form of questions that require the boys to try to hold the previous answer in their heads--4+6, now takeaway 5, now add 12, take away 4, etc. But I don't know if that would qualify as an example of mental math. I'd just rather they could do the math in their heads like their Dad and Grandfather, and not have to resort to having to grab a pencil.

Not that I don't allow them to use a pencil. There's probably a need for some people to visualize by writing it down first before being able to retain the numbers as you go along.

 

Yes, here's where the confusion comes in because, in both of the above examples, that IS doing math in your head, but you could just be relying on memorized facts without clear understanding of our decimal system. I could have done that kind of thing in school but could not have added/subtracted large numbers in my head. What Chepyl did in school sounds more like a working memory exercise to me. Of course, you do need to use your working memory when doing mental math the "making tens" way, but that's not the important skill, IMO. The understanding is more important.

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I see a distinction between math fact memory, which is a single-stage process, as in you either "know" the product of 8 and 7 or you don't, and the multi-stage processes that take place when employing mental math stratagies. The latter involves manipulation numbers and employing the properties of mathematics to make problem solving easier. In contrast memorizing "math facts" requires no knowledge (either implicit or explicit) of the laws of mathematics.

 

To me these are different things. Yes, automaticity is helpful for performing mental math, but memorized math facts are not the same thing as having the sort of mathematical reasoning skills required for mental math.

 

So I see them as distinct but complimentary skills.

 

Bill

 

 

This is such a clear way of explaining what I've been thinking but couldn't articulate! Thank you, Bill! I especially like the sentence I bolded above.

 

 

This reminds me of another current thread. Let's Play Math said something I really liked. I don't know how to do the quote feature from another thread so I'll just put it in regular quotation marks....

 

"Your goal at this level is NOT for your son to memorize a series of math facts and procedures, but to develop confidence in working with numbers. In fact, if parents stress fact memorization too much, we short-circuit the child’s learning process. Once children “know†an answer, they don’t bother to think about it — but it is in the “thinking about it†stage that they build a logical foundation for understanding all numbers. - Let's Play Math"

 

I strongly believe in the importance of not pushing memorization too early. I want my kids to have LOTS of practice in manipulating numbers and developing their mathematical reasoning skills before it becomes automatic recall. This may be more important to me than some others because my kids memorize very well so I've had to on purpose try not to give them little memorization tricks that I learned in school because I want them to use their noggins in a different way than that!

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The purpose of mental math is to calculate more quickly, right?

 

Hmmmm, interesting. I guess, yes, that would be one purpose.

 

Maybe because of my specific experience, though, I see a greater purpose of mental math is developing and honing mathematical and critical thinking skills. It was such an epiphany to me when I saw how to do math the "Singapore" way that I wanted that for my kids. I didn't want them to go through their life not truly understanding the world of numbers. I feel like all through school my teachers were holding their hands over my eyes and teaching me how to navigate the "numbers world" as though I were blind, and now it's like the hands were taken away and I can actually look around and see where to go and what's around me. Maybe it's because I'm a visual learner, but that's the best description I can come up with. I don't want to put blinders on my kids by just making them memorize facts. Sure, getting around fast is important, but isn't being able to "see" even more important?

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When I say "mental math" I mean that you are faced with a calculation that you do not instantly know the answer for. For a child, this might be something simple like 7+8=?, while for an adult it could be anything from the oft-forgotten 7*8=? to the "mathemagician" problems Stripe mentioned.

 

Anyway, you need to find the answer to this calculation, and you don't just know it. So you tweak the numbers somehow --- finding tens, or estimating and then correcting, or using the distributive property, or working from left to right, or taking the numbers apart and putting them together in new ways. You figure out some way to make the calculation easier, so that you can find the answer. There are a LOT of different strategies.

 

Someone doing mental math may jot down some intermediate numbers on paper or a white board to keep track of them, or they may count on their fingers as a memory aid (for instance, to make sure they really did count by 5's seven times, as they intended).

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My personal vent is though that mental math seems to have this pedestal of superiority. That frustrates me because my dyslexic ds who is incredible with manipulating numbers just cannot always do it in his head.

 

The superior thing is to be able to manipulate numbers BEYOND just following a recipe of steps for carrying/borrowing/regrouping or whatever. That is, we want our children to be thinking about the numbers, not just following a memorized rule.

 

If your son understands, for instance, that 768 can be thought of as...

 

  • 76 tens and 8 ones,
  • or 75 tens and 18 ones,
  • or 7 hundreds and 68 ones,
  • or even some weird combination like 5 hundreds and 22 tens and 48 ones

... then he has a pretty thorough understanding of numbers and is well on his way to mastering arithmetic.

 

The reason that I push mental math is because so many homeschooling (and traditional schooling) parents seem to think that the important thing is for their children to memorize math facts so they can follow the steps of pencil-and-paper arithmetic. Their goal is for their children to do what any $5 calculator can do, only to do it slower and with many opportunities for error. That is not what math is all about!

 

Children who learn to manipulate numbers in their heads are forced to master the basic principles of how numbers work --- especially the principles of place value, commutativity, and the distributive property --- and in that way they are better prepared for algebra and everything that follows. If your student cannot manipulate the numbers in his head because of his dyslexia, that's fine, as long as you make sure that he learns and can work with those important principles.

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Children who learn to manipulate numbers in their heads are forced to master the basic principles of how numbers work --- especially the principles of place value, commutativity, and the distributive property --- and in that way they are better prepared for algebra and everything that follows. If your student cannot manipulate the numbers in his head because of his dyslexia, that's fine, as long as you make sure that he learns and can work with those important principles.

 

Oh he can, he just has to have visual or tangible representation. He solved something this morning using the distributive property (or is that commutative?), choosing to use 4 9s instead of 9 4s to get the perimeter he needed. He just can't consistently remember that 9 + 9 = 18, without drawing dots or using c-rods to find equivalent lengths. Ok, that's not completely fair to him. He can do this mentally, it just takes him a long time to access that 8 +8 = 16 and then think up 2 because his processing speed is slow. He is lightening fast with visual/spatial so he usually chooses to go that route (which also cuts down on his dyslexic computation errors of flipping 1s and 10s or forgetting some of the intermediate numbers).

Edited by FairProspects
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Mental math is using the basic math facts (your definition of "basic" may differ from mine) to perform mental, as opposed to pen and paper, calculations. Some strategies are more efficient than others.

 

Yes, this.

 

My mom called it "doing it in your head".

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We use MM (and a little bit of SM) and I think of mental math as strategies for completing computations without having to write the problem/question down. Making tens is one of the strategies taught for mental math.

 

:iagree: It's doing the computations in your head, using strategies that are simpler and faster The BJU math series teaches several strategies of this type.

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When I say "mental math" I mean that you are faced with a calculation that you do not instantly know the answer for. For a child, this might be something simple like 7+8=?, while for an adult it could be anything from the oft-forgotten 7*8=? to the "mathemagician" problems Stripe mentioned.

 

Anyway, you need to find the answer to this calculation, and you don't just know it. So you tweak the numbers somehow --- finding tens, or estimating and then correcting, or using the distributive property, or working from left to right, or taking the numbers apart and putting them together in new ways. You figure out some way to make the calculation easier, so that you can find the answer. There are a LOT of different strategies.

 

Someone doing mental math may jot down some intermediate numbers on paper or a white board to keep track of them, or they may count on their fingers as a memory aid (for instance, to make sure they really did count by 5's seven times, as they intended).

 

The superior thing is to be able to manipulate numbers BEYOND just following a recipe of steps for carrying/borrowing/regrouping or whatever. That is, we want our children to be thinking about the numbers, not just following a memorized rule.

 

If your son understands, for instance, that 768 can be thought of as...

 

 

  • 76 tens and 8 ones,

  • or 75 tens and 18 ones,

  • or 7 hundreds and 68 ones,

  • or even some weird combination like 5 hundreds and 22 tens and 48 ones

 

... then he has a pretty thorough understanding of numbers and is well on his way to mastering arithmetic.

 

The reason that I push mental math is because so many homeschooling (and traditional schooling) parents seem to think that the important thing is for their children to memorize math facts so they can follow the steps of pencil-and-paper arithmetic. Their goal is for their children to do what any $5 calculator can do, only to do it slower and with many opportunities for error. That is not what math is all about!

 

Children who learn to manipulate numbers in their heads are forced to master the basic principles of how numbers work --- especially the principles of place value, commutativity, and the distributive property --- and in that way they are better prepared for algebra and everything that follows. If your student cannot manipulate the numbers in his head because of his dyslexia, that's fine, as long as you make sure that he learns and can work with those important principles.

 

Denise, thank you for your wisdom. It's helping to clarify some things for me....

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Oh he can, he just has to have visual or tangible representation. He solved something this morning using the distributive property (or is that commutative?), choosing to use 4 9s instead of 9 4s to get the perimeter he needed. He just can't consistently remember that 9 + 9 = 18, without drawing dots or using c-rods to find equivalent lengths. Ok, that's not completely fair to him. He can do this mentally, it just takes him a long time to access that 8 +8 = 16 and then think up 2 because his processing speed is slow. He is lightening fast with visual/spatial so he usually chooses to go that route (which also cuts down on his dyslexic computation errors of flipping 1s and 10s or forgetting some of the intermediate numbers).

 

4 nines = 9 fours. That's the commutative property, because you are switching the numbers around --- they are moving, or commuting. The distributive property would sound something like this:

 

 

  • "4 is 2 + 2, so 4 nines would be 2 nines and 2 more nines..."

  • or "9 is 10 - 1, so 9 fours would be 10 fours take away 1 four."

 

Over times, as your son works with the numbers, you might encourage him to come up with as many different strategies as he can to figure things out. That way, whenever he can't remember something, he will have plenty of options for how to think it through.

 

 

  • For instance, with 9 + 9 = ?, I find it easier to imagine one 9 donating a block to the other, so that 9 + 9 becomes 10 + 8 = 18.

 

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4 nines = 9 fours. That's the commutative property, because you are switching the numbers around --- they are moving, or commuting. The distributive property would sound something like this:

 

  • "4 is 2 + 2, so 4 nines would be 2 nines and 2 more nines..."
  • or "9 is 10 - 1, so 9 fours would be 10 fours take away 1 four."

Over times, as your son works with the numbers, you might encourage him to come up with as many different strategies as he can to figure things out. That way, whenever he can't remember something, he will have plenty of options for how to think it through.

 

  • For instance, with 9 + 9 = ?, I find it easier to imagine one 9 donating a block to the other, so that 9 + 9 becomes 10 + 8 = 18.

 

Yeah, he uses all of these strategies too, he just can't do them mentally because it takes him so long to remember his math facts. He is actually great at making 10s. He just does it more easily with rods or by drawing a picture. I don't think I'm explaining it well because this is not at all I how I think. But based on your definition of mental math, he is doing just fine. :D

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Can someone show me mental math with multiplication that isn't x9? I know 7x8 is 7 groups of 8 or vice-versa. They still have to memorize the answer right?

 

X10 is easy

X5=x10 /2. !

There's a trick for x11

 

More here

 

http://easycalculation.com/funny/tricks/6-10-finger-multiplication.php

http://www.abcteach.com/free/m/mulitplication_quicktricks_elem.pdf (yes, the URL has a misspelling)

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The reason that I push mental math is because so many homeschooling (and traditional schooling) parents seem to think that the important thing is for their children to memorize math facts so they can follow the steps of pencil-and-paper arithmetic. Their goal is for their children to do what any $5 calculator can do, only to do it slower and with many opportunities for error. That is not what math is all about!

.

 

I like that MEP has kids estimate to get a sense of what two 3digit numbers will add up to, so they have one potential way to identify mistakes.

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X10 is easy

X5=x10 /2. !

There's a trick for x11

 

More here

 

http://easycalculation.com/funny/tricks/6-10-finger-multiplication.php

http://www.abcteach.com/free/m/mulitplication_quicktricks_elem.pdf (yes, the URL has a misspelling)

 

And after the 2's and 3's are mastered (and 3's are double plus one more), the rest are a snap:

 

4's are double, double

8's are double, double, double

6's are triple, double or x5 plus one more

 

That leaves 7x7=49, a fact I teach first and ask daily and repeatedly in goofy voices while the rest are being learned.

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Can someone show me mental math with multiplication that isn't x9? I know 7x8 is 7 groups of 8 or vice-versa. They still have to memorize the answer right?

 

 

Try this -

 

Doing it this way brings about understanding. It's not just memorizing unconnected pieces of information. The child in this video has JUST been introduced to the concept of multiplication but she discovered the distributive property on her own with the rods!

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And after the 2's and 3's are mastered (and 3's are double plus one more), the rest are a snap:

 

4's are double, double

8's are double, double, double

6's are triple, double or x5 plus one more

 

That leaves 7x7=49, a fact I teach first and ask daily and repeatedly in goofy voices while the rest are being learned.

 

So 7X8 is 14, 28, and the answer is 56. If my kids get the meaning of what they are doing when they multiply, what is the difference to just memorize 56 for speed's sake. I guess I don't get the superiority of these mental math tricks. :confused:

Edited by LNC
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So 7X8 is 14, 28, and the answer is 56. If my kids get the meaning of what they are doing when they multiply, what is the difference to just memorize 56 for speed's sake. I guess I don't get the superiority of these mental math tricks. :confused:

 

There is no superiority. My child has a language based learning disability with a weaker than normal working memory. Skip counting and pictures and straight up memorizing make no sense to him. Mental math and the attending strategies with the distributive property of multiplication enable him to solve these problems. Whatever works for your family is great.

Edited by Heathermomster
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Mental Math = calculating in your head. It is not necessarily one algorithm or another though the mental math STRATEGIES taught in the programs like RightStart or Singapore are not the standard algorithm, and (at least for us) are more useful than trying to do the standard algorithm in your head.

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I agree that it is the ability to manipulate numbers in your head to solve a problem. Making tens, doubles +1, etc. I noticed these patterns on my own. My oldest picked them up fairly easily. She's only had a taste of multiplication, but just today her brother asked her what four 9s was and she reasoned that it would be 18+18, which is 36.

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So 7X8 is 14, 28, and the answer is 56. If my kids get the meaning of what they are doing when they multiply, what is the difference to just memorize 56 for speed's sake. I guess I don't get the superiority of these mental math tricks. :confused:
Well, this is just a "trick" to use while the basic facts are being mastered. But it's not really a "trick" because 8 is 2*2*2. :001_smile:

 

Very simple mental math would be more like:

4*98 = 4*100 - 4*2 = 392

Edited by nmoira
clarity
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That's funny.

 

I've got a square number lover in my house, so 7x7 was a cause for celebration. ;)

For some reason my youngest decided to fall in love with 6*7. She *says* "sigh" (in a dreamy, romantic way) whenever it comes up. I think she's destined to be a Douglas Adams fangirl. :D

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So 7X8 is 14, 28, and the answer is 56. If my kids get the meaning of what they are doing when they multiply, what is the difference to just memorize 56 for speed's sake. I guess I don't get the superiority of these mental math tricks. :confused:

 

Of course, the goal is that they eventually have it memorized to the point of automaticity, but if they do that by rote through drill, flashcards, etc. without the understanding first, then those facts will be forgotten eventually. (Ask a few adults around you what 7x8 is and you'll most likely find that to be true!) If they understand the mathematical properties/laws they can actually FIGURE it out when they can't seem to grab it from long term memory.

 

Plus, using the distributive and commutative properties over and over again to figure out multiplication facts when you are just learning will build and solidify number sense. Straight memorization will NOT do that. I know this from experience!

 

Oh, and I don't think of that as mental math "tricks." To me, tricks are ways to calculate that almost seem like magic because you DO NOT UNDERSTAND why it works. I want my kids to understand why it works!

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I agree that it is the ability to manipulate numbers in your head to solve a problem. Making tens, doubles +1, etc. I noticed these patterns on my own. My oldest picked them up fairly easily. She's only had a taste of multiplication, but just today her brother asked her what four 9s was and she reasoned that it would be 18+18, which is 36.

 

Maybe, but a lot of people who are not mathematically inclined don't. I'm not sure I was ever given manipulatives to play with when I learned math and I probably wouldn't recognize patterns like that anyway on my own. I don't think I ever heard the word play and math in the same sentence. I can memorize absolutely anything, but I had no idea what a number meant until I tried to teach my kids math, and I went through Calculus in college. Math was taught as a series of puzzles to memorize when I grew up and since I am a fast processor, I could keep up. To this day though, I have no idea how to do anything but plug and chug. (To be fair, I also had zero interest once I hit algebra so I didn't put in a lot of effort.)

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Maybe, but a lot of people who are not mathematically inclined don't. I'm not sure I was ever given manipulatives to play with when I learned math and I probably wouldn't recognize patterns like that anyway on my own. I don't think I ever heard the word play and math in the same sentence. I can memorize absolutely anything, but I had no idea what a number meant until I tried to teach my kids math, and I went through Calculus in college. Math was taught as a series of puzzles to memorize when I grew up and since I am a fast processor, I could keep up. To this day though, I have no idea how to do anything but plug and chug. (To be fair, I also had zero interest once I hit algebra so I didn't put in a lot of effort.)

 

Oh, I completely agree that not all will notice those patterns. In fact, I dare say most will not if it's not pointed out. I've always like math and have been conceptually strong since I was very young. Part of that is due to the way I process numbers and another part is thanks to my older brother "playing math" with me when we were supposed to be listening to the speakers at church.

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Yeah, he uses all of these strategies too, he just can't do them mentally because it takes him so long to remember his math facts. He is actually great at making 10s. He just does it more easily with rods or by drawing a picture. I don't think I'm explaining it well because this is not at all I how I think. But based on your definition of mental math, he is doing just fine. :D

 

He's young, though. I've noticed with my VSL that he's gotten much better at doing mental math as he's gotten older and he's become less reliant on drawing what Right Start calls "part-whole circles" and what Singapore calls "number bonds".

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... any thing you do to mentally add/subtract/multiply/divide without having to use paper is mental math. Another trick is to add multidigit numbers left to right...readjust if there's carrying, but adding them left to right in your brain is less work since you already say the digits in that order. Another trick is rounding...if you need to add prices in your head, you round to the nearest dollar. If you need an exact answer to 97cents plus $2.05, you add $1+$2, then you add the extra 5cents and subtract the missing 3 cents that were rounded off. When you learn the doubles facts, and use them to get doubles +1 facts...that's mental math. I think of it as anything that involves juggling the quanties in your head.

:iagree:

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Mental math is doing math mentally. Using your brain.

 

 

I agree that it is not merely memorizing math facts, and that mere drill can actually kill the child's drive to figure it out for themselves. Thus the phrase, "Drill and Kill." It's the act of figuring it out for yourself over and over again that makes it stick permanently. It's also what builds a problem solving foundation.

 

 

The child who always knows the answer to the problem, but hates writing out the work is a natural at mental math. The child who likes to memorize the algorithm and when to use it may look good on paper, but is not really a "mathy" person...they've learned a process, not how to think math.

 

 

That leaves 7x7=49, a fact I teach first and ask daily and repeatedly in goofy voices while the rest are being learned.

 

 

I teach 6x4=24, 6x6=36, 6x8=48. Then I ask "What is 6x7?" (They think 6x8-6=42.) Then I ask "What is 7x7? (They think 42+7.) It's a mental maze, but the acrobats to them good...and they get to automacity quickly...and they use the same technique of going from known to unknown in other problems.

 

 

I think mental math includes everything from the basic 2+3=5 and beyond. It's simply using your brain, your own problem solving skills.

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