Wednesday, November 29, 2006

Square root problem

prove that

5√20 × √45 × √5 = 150√5

This is a quite easy problem. You use two basic "ideas" or properties relating to square roots:

* that √b × b is b - or that you can "pull out" a number times itself from under the root (this is just the definition of a square root of course).

* that √ab = √ab - or you can combine the radicands under the same root when they are multiplied.

So √20 * √45 is equal to √20*45 = √4 × 5 × 5 × 9
= √2 × 2 × 5 × 5 × 3 × 3 = 2 × 5 × 3 = 30.

So that's the crux of that problem.

Square root problem

prove that

5√20 × √45 × √5 = 150√5

This is a quite easy problem. You use two basic "ideas" or properties relating to square roots:

* that √b × b is b - or that you can "pull out" a number times itself from under the root (this is just the definition of a square root of course).

* that √ab = √ab - or you can combine the radicands under the same root when they are multiplied.

So √20 * √45 is equal to √20*45 = √4 × 5 × 5 × 9
= √2 × 2 × 5 × 5 × 3 × 3 = 2 × 5 × 3 = 30.

So that's the crux of that problem.

Math Mammoth Grade 5 Worksheets ready

I hope you're not tired of hearing this.... but this is what I've been busy with lately.

Just as of today, I got Math Mammoth Grade 5 worksheets ready and available for purchasing.

Like the others, there are two separate books, A and B, plus answer keys.

Price for the whole package is $10. And that includes 123 quality math worksheets all total.

Click the link to see sample worksheets.

And, I've also set up a volume discount for any of my math books:

For order totals at least $34 - a 20% discount.
For order totals at least $50 - a 25% discount.
For order totals at least $70 - a 30% discount.
Use coupon code 8A2301338 when ordering to get these discounts.

Math Mammoth Grade 5 Worksheets ready

I hope you're not tired of hearing this.... but this is what I've been busy with lately.

Just as of today, I got Math Mammoth Grade 5 worksheets ready and available for purchasing.

Like the others, there are two separate books, A and B, plus answer keys.

Price for the whole package is $10. And that includes 123 quality math worksheets all total.

Click the link to see sample worksheets.

And, I've also set up a volume discount for any of my math books:

For order totals at least $34 - a 20% discount.
For order totals at least $50 - a 25% discount.
For order totals at least $70 - a 30% discount.
Use coupon code 8A2301338 when ordering to get these discounts.

Tuesday, November 21, 2006

4th grade worksheets

I spent this last Saturday working on an idea that I probably should have done a long ago... making better use of my various worksheet generators.

I created a single page that has pre-made worksheets for 4th grade math. It's just long list of links, each of which generates a certain kind of worksheet. Should be handy for my visitors.

So these are still randomly generated, each time it's a new one. Just hit 'refresh from the browser and a new one is there.

And while building that page, I also fixed a few litte things in the generators, and added a few more options... For example, Fractions worksheet generator now has the option to make missing addend/factor/dividend sheets.

So hopefully with these changes they will better serve my visitors.

And yeah, I can hear some asking, "When are you going to do a similar page for 3rd or 5th grade?"

Well, I hope get such done before the year is over. We will see! I DO hope to get those done as well!

4th grade worksheets

I spent this last Saturday working on an idea that I probably should have done a long ago... making better use of my various worksheet generators.

I created a single page that has pre-made worksheets for 4th grade math. It's just long list of links, each of which generates a certain kind of worksheet. Should be handy for my visitors.

So these are still randomly generated, each time it's a new one. Just hit 'refresh from the browser and a new one is there.

And while building that page, I also fixed a few litte things in the generators, and added a few more options... For example, Fractions worksheet generator now has the option to make missing addend/factor/dividend sheets.

So hopefully with these changes they will better serve my visitors.

And yeah, I can hear some asking, "When are you going to do a similar page for 3rd or 5th grade?"

Well, I hope get such done before the year is over. We will see! I DO hope to get those done as well!

Friday, November 17, 2006

Multiplication trick

If you enjoy math 'tricks', here's one presented nicely in a slide show:


How to multiply numbers with 11-19 in less than 5 seconds


It explains how to multiply by 11, 12, etc.

And... here's a challenge (you can ask this of students, too):
After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm.

In other words, how do the standard multiplication algorithm (in columns) and this one compare?

Multiplication trick

If you enjoy math 'tricks', here's one presented nicely in a slide show:


How to multiply numbers with 11-19 in less than 5 seconds


It explains how to multiply by 11, 12, etc.

And... here's a challenge (you can ask this of students, too):
After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm.

In other words, how do the standard multiplication algorithm (in columns) and this one compare?

Thursday, November 16, 2006

Math Mammoth Grade 4 ready

Just tonight... I got my Math Mammoth grade 4-A and 4-B workbooks ready and available online. One of them is $5.50 and the other is $5...

I've truly put a lot of effort on those so I hope the worksheets serve well those who buy them.

They're mostly designed for teachers, but I've had homeschoolers buy the other grades so I guess they're for whoever needs a 4th grade math workbook or worksheets.

You can download sample sheets here.

Math Mammoth Grade 4 ready

Just tonight... I got my Math Mammoth grade 4-A and 4-B workbooks ready and available online. One of them is $5.50 and the other is $5...

I've truly put a lot of effort on those so I hope the worksheets serve well those who buy them.

They're mostly designed for teachers, but I've had homeschoolers buy the other grades so I guess they're for whoever needs a 4th grade math workbook or worksheets.

You can download sample sheets here.

Wednesday, November 15, 2006

All sorts of websites

I'm just going to drill through a few links people have sent me, or that I've taken notice of, lately. Hopefully you'll find something of interest!

1) JamesPoon.com is a teacher located in Singapore and his site has video tutorials (in English) and a forum getting started.

2) Weekly Math Problems from NASA - download free weekly math problems in the setting of space weather... quite interesting! You'll find topics such as radiation dosages at Mars, exposure calculations, background radiation on Earth... would be perfect for science-minded high school students.

3) This one you'll get to laugh at... Newton's laws revisited.

Some examples:
BATH THEOREM: When the body is immersed in water, the telephone rings.

LAW OF THE RESULT: When you try to prove to someone that a machine won’t work, it will!

LAW OF BIOMECHANICS: The severity of the itch is inversely proportional to the reach.

And so on... click on the link to read more.

4) Mathematical Moments - a series of one-page PDF posters or flyers... about the many various ways mathematics is used in modern society.

Teachers especially could use these to promote awareness of use of mathematics. It's just good to know in general how much mathematics is involved in various fields.. these flyers don't get into any deep stuff but are just real short 'snapshots'. Examples include:
Finding Oil
Solving Crimes
Unearthing Power Lines
Targeting Tumors
Beating Traffic

... and so on.

All sorts of websites

I'm just going to drill through a few links people have sent me, or that I've taken notice of, lately. Hopefully you'll find something of interest!

1) JamesPoon.com is a teacher located in Singapore and his site has video tutorials (in English) and a forum getting started.

2) Weekly Math Problems from NASA - download free weekly math problems in the setting of space weather... quite interesting! You'll find topics such as radiation dosages at Mars, exposure calculations, background radiation on Earth... would be perfect for science-minded high school students.

3) This one you'll get to laugh at... Newton's laws revisited.

Some examples:
BATH THEOREM: When the body is immersed in water, the telephone rings.

LAW OF THE RESULT: When you try to prove to someone that a machine won’t work, it will!

LAW OF BIOMECHANICS: The severity of the itch is inversely proportional to the reach.

And so on... click on the link to read more.

4) Mathematical Moments - a series of one-page PDF posters or flyers... about the many various ways mathematics is used in modern society.

Teachers especially could use these to promote awareness of use of mathematics. It's just good to know in general how much mathematics is involved in various fields.. these flyers don't get into any deep stuff but are just real short 'snapshots'. Examples include:
Finding Oil
Solving Crimes
Unearthing Power Lines
Targeting Tumors
Beating Traffic

... and so on.

Sunday, November 12, 2006

Worksheet generators

Someone pointed out to me how my addition worksheet generator had a bug: they couldn't make missing addend worksheets where the student is completing the next whole hundred.

Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.

These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.

I figure that's why on most websites, math worksheet generators don't have many options!

But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.

So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.

Here's a direct link for generating a random worksheet like the lady wanted:

Missing addend worksheet - complete the next whole hundred.

Refresh the page to get a new one and new one and so on.

Here's the link to the addition worksheet generator itself.

Worksheet generators

Someone pointed out to me how my addition worksheet generator had a bug: they couldn't make missing addend worksheets where the student is completing the next whole hundred.

Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.

These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.

I figure that's why on most websites, math worksheet generators don't have many options!

But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.

So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.

Here's a direct link for generating a random worksheet like the lady wanted:

Missing addend worksheet - complete the next whole hundred.

Refresh the page to get a new one and new one and so on.

Here's the link to the addition worksheet generator itself.

Thursday, November 9, 2006

Mental math tricks

Some children might be delighted to learn math 'tricks' - curious ways to do calculations such as multiplying 2-digit numbers.

The 'tricks' do not contain any magic but are based on solid mathematical principles.

For example, to subtract any number from 1000 or 10,000 or any power of ten... just subtract from 999 or 9999 etc. and add 1.

Subtracting 10,000 - 2,596 with the usual method in columns, you will get into lots of borrowing over zeros... and end up having a row of nines to subtract from - except in the ones column where you have 10.

So 10,000 - 2,596 is quickly done by looking at each digit's difference to 9 - except in the case of one's digit, when you will subtract from 10. The result: 7,404.



Another 'trick' is called vertically and crosswise and applies to multiplication.

It can easily be proven to work using simple algebra. But it is a nice little mental math method that can impress kids.

Read more about that trick here. You can also practice online.

(Please note: this so-called 'vedic' math probably is NOT from Indian vedas or anything. The original author of the book with that name just claimed that but it seems his claims have no base. It's just a system of mental math calculations. Learning a few tricks like that can be just some simple fun.)

Mental math tricks

Some children might be delighted to learn math 'tricks' - curious ways to do calculations such as multiplying 2-digit numbers.

The 'tricks' do not contain any magic but are based on solid mathematical principles.

For example, to subtract any number from 1000 or 10,000 or any power of ten... just subtract from 999 or 9999 etc. and add 1.

Subtracting 10,000 - 2,596 with the usual method in columns, you will get into lots of borrowing over zeros... and end up having a row of nines to subtract from - except in the ones column where you have 10.

So 10,000 - 2,596 is quickly done by looking at each digit's difference to 9 - except in the case of one's digit, when you will subtract from 10. The result: 7,404.



Another 'trick' is called vertically and crosswise and applies to multiplication.

It can easily be proven to work using simple algebra. But it is a nice little mental math method that can impress kids.

Read more about that trick here. You can also practice online.

(Please note: this so-called 'vedic' math probably is NOT from Indian vedas or anything. The original author of the book with that name just claimed that but it seems his claims have no base. It's just a system of mental math calculations. Learning a few tricks like that can be just some simple fun.)

Wednesday, November 8, 2006

Math Mammoth 3rd grade worksheets

... are ready as of today!

There are lots of sheets for multiplication concept and multiplication tables, division concept, geometry, measuring, for example.

Check out the sample sheets:

Math Mammoth Grade 3 Worksheets

Math Mammoth 3rd grade worksheets

... are ready as of today!

There are lots of sheets for multiplication concept and multiplication tables, division concept, geometry, measuring, for example.

Check out the sample sheets:

Math Mammoth Grade 3 Worksheets

Monday, November 6, 2006

Teaching integers

Recently I answered a question about teaching integers on an email list, and decided to post all that on my site as well.

Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.

The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.

Teaching integers
Enjoy!



Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.

how do you do this problems: elizabeth bought 3 1/3 pounds of tomatoes for $2.50. how much did she pay per pound?


Prices per pound are always given in [dollars per pound] or [dollars / pounds ].

This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.

$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.

Teaching integers

Recently I answered a question about teaching integers on an email list, and decided to post all that on my site as well.

Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.

The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.

Teaching integers
Enjoy!



Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.

how do you do this problems: elizabeth bought 3 1/3 pounds of tomatoes for $2.50. how much did she pay per pound?


Prices per pound are always given in [dollars per pound] or [dollars / pounds ].

This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.

$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.

Wednesday, November 1, 2006

Math learning and unhappiness

Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning.

The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.

The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.

American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!

Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.



The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.

I can almost hear my homeschooling readers' anger raising...!

But wait a minute.


Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."

"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'


This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.

Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.

In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.

In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.

But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.



Is there a solution?


Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.

But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.

The truth, as always, must be somewhere in the middle.

When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.

Maybe group work could be used with those, as well.

It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.

I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.


What are your thoughts?

Several other bloggers have gotten on to this too:

Gooseania

Luboš Motl's reference frame

Mathematics and (un)happiness by Alexandre Borovik

Mathematics & (un)happiness at NeverEndingBooks

See also:

CNN news: Confident students do worse in math; bad news for U.S.


The 2006 Brown Center Report on American Education:
How Well Are American Students Learning?
- With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores

Math learning and unhappiness

Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning.

The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.

The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.

American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!

Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.



The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.

I can almost hear my homeschooling readers' anger raising...!

But wait a minute.


Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."

"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'


This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.

Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.

In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.

In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.

But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.



Is there a solution?


Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.

But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.

The truth, as always, must be somewhere in the middle.

When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.

Maybe group work could be used with those, as well.

It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.

I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.


What are your thoughts?

Several other bloggers have gotten on to this too:

Gooseania

Luboš Motl's reference frame

Mathematics and (un)happiness by Alexandre Borovik

Mathematics & (un)happiness at NeverEndingBooks

See also:

CNN news: Confident students do worse in math; bad news for U.S.


The 2006 Brown Center Report on American Education:
How Well Are American Students Learning?
- With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores

Ejercicios de matemáticas

Recientemente he comenzado la tarea de traducir los generadores de ejercicios de matemáticas en español. Bueno, realmente es un amigo que hace la mayor parte de la traducción.

Ya tenemos algo de presentar: visita Mamutmatematicas.com y genera ejercicios de matemáticas gratis!

Los demás generadores de problemas siguen pronto...

Ejercicios de matemáticas

Recientemente he comenzado la tarea de traducir los generadores de ejercicios de matemáticas en español. Bueno, realmente es un amigo que hace la mayor parte de la traducción.

Ya tenemos algo de presentar: visita Mamutmatematicas.com y genera ejercicios de matemáticas gratis!

Los demás generadores de problemas siguen pronto...

Some math worksheets now in Spanish

I've had some of my math worksheet generators translated to Spanish, in case you're interested:

Math worksheets in Spanish. It's a new website... MamutMatematicas.com : )

Some math worksheets now in Spanish

I've had some of my math worksheet generators translated to Spanish, in case you're interested:

Math worksheets in Spanish. It's a new website... MamutMatematicas.com : )