Final Fantasy Papercraft - Bahamut

Now that I'm back, I want to make it up to my subscribers for all the silent months that went by without a single post. Here's a paper model of Bahamut, my favorite Final Fantasy summon of all. I don't have an assembly instruction for this one, but it's pretty simple to figure out, I've posted a lot of high quality pictures on Flickr so that you can visualize how to put him together. Good luck and enjoy!

Final Fantasy Papercraft - Bahamut [via Mediafire]
Bahamut Photos [via Flickr]


Update - I've received a lot of email complaints from other people about not being able to open the file, if you see an error message from your PDF reader indicating they can't open it, most probably it's because you don't have your Asian/European font pack installed for your PDF reader. Here are the links for Adobe Acrobat and FoxIt readers' language packs.

FoxIt Reader - Download
Adobe Acrobat - Download

A little contest

In an effort to promote my newer site www.MathMammoth.com, I'm announcing this little "contest", in which it is quite easy to win a Math Mammoth book - or even two - for free, if you have a blog or website. The rules are as follows:

1. You write a little about MathMammoth.com site on your blog or website, and place within that text a link to the page of the book or books that you'd like to win!
   
   For example, if you'd like to win the Geometry 1 book, then you'd link to the page http://www.mathmammoth.com/geometry_1.php
   
   Instead of the individual book pages, you may link to the home page (www.mathmammoth.com), if that fits better your write up or your website. But I greatly appreciate the links pointing to the 'inside' pages.



2. Your link must be normal link, and not a scripted one or with nofollow tag. 



3. The link must point to www.mathmammoth.com or some subpage.



4. No adult, hate, or other weird or offensive blogs/websites/forums. 



5. No spammy sites, scraper sites, free-for-all pages, directory listings, or any such next-to-worthless sites. 



6. A blog comment on someone else's blog won't do either. 



7. Also I will not accept a brand new blog or site that you start because of this. It has to be an existing blog or site. 



8. Don't forget to let me know! My contact info is here or here. Remember to include your email address or I can't contact you. 



9. I will check your writeup and link, and reward you with one or two (maybe even three) books!

Questions and answers

1. How much do I have to write?

However much you feel like writing. I'm not setting any lenghts as such. It should be just something that appears normal to your readers, or fits well within your website. If it is a school website with a link list, then just put MathMammoth.com to the link list. If it is a blog, then a separate blogpost is in order, or a link in the "blogroll".


2. What should I write?

Just about anything... well, as long as it's not 'lashing' or mocking or being negative. I'm doing this for promotion purposes, so if you go against that, you're disqualified from the contest!

You can write something you liked about the site, or about the free sample worksheets, or even link directly to a PDF worksheet (but you ALSO need to link to one of the normal .php pages).

Or write about your math troubles (if any), and just put a link to the site somewhere within that rant. Or write about various math curricula, and include Math Mammoth as one. If it appears natural in a context, then that's great!


3. I don't have a website or blog.

You can do a posting on some public forum that you frequent already. It has to be a public access forum, and not a private Yahoo group or email list. The forum must be decent.. no 'adult' or 'hate' or other weird offensive forums.



These rules may be modified as need be.

Happy linking!

Maria Miller

A little contest

In an effort to promote my newer site www.MathMammoth.com, I'm announcing this little "contest", in which it is quite easy to win a Math Mammoth book - or even two - for free, if you have a blog or website. The rules are as follows:

1. You write a little about MathMammoth.com site on your blog or website, and place within that text a link to the page of the book or books that you'd like to win!
   
   For example, if you'd like to win the Geometry 1 book, then you'd link to the page http://www.mathmammoth.com/geometry_1.php
   
   Instead of the individual book pages, you may link to the home page (www.mathmammoth.com), if that fits better your write up or your website. But I greatly appreciate the links pointing to the 'inside' pages.



2. Your link must be normal link, and not a scripted one or with nofollow tag. 



3. The link must point to www.mathmammoth.com or some subpage.



4. No adult, hate, or other weird or offensive blogs/websites/forums. 



5. No spammy sites, scraper sites, free-for-all pages, directory listings, or any such next-to-worthless sites. 



6. A blog comment on someone else's blog won't do either. 



7. Also I will not accept a brand new blog or site that you start because of this. It has to be an existing blog or site. 



8. Don't forget to let me know! My contact info is here or here. Remember to include your email address or I can't contact you. 



9. I will check your writeup and link, and reward you with one or two (maybe even three) books!

Questions and answers

1. How much do I have to write?

However much you feel like writing. I'm not setting any lenghts as such. It should be just something that appears normal to your readers, or fits well within your website. If it is a school website with a link list, then just put MathMammoth.com to the link list. If it is a blog, then a separate blogpost is in order, or a link in the "blogroll".


2. What should I write?

Just about anything... well, as long as it's not 'lashing' or mocking or being negative. I'm doing this for promotion purposes, so if you go against that, you're disqualified from the contest!

You can write something you liked about the site, or about the free sample worksheets, or even link directly to a PDF worksheet (but you ALSO need to link to one of the normal .php pages).

Or write about your math troubles (if any), and just put a link to the site somewhere within that rant. Or write about various math curricula, and include Math Mammoth as one. If it appears natural in a context, then that's great!


3. I don't have a website or blog.

You can do a posting on some public forum that you frequent already. It has to be a public access forum, and not a private Yahoo group or email list. The forum must be decent.. no 'adult' or 'hate' or other weird offensive forums.



These rules may be modified as need be.

Happy linking!

Maria Miller

Geometric patterns in islamic art

This was in the news recently so you might have seen it...

Physics student Peter J. Lu has discovered, after his trip to Uzbekistan, that the geometric patterns shown on the walls of old islamic shrines depicted a very complex mathematical pattern, one that was only found in the west during last century.

I feel the article at ScienceNews.org explains it all very well, and shows plenty of pictures of the patterns AND of the underlying tilings, plus has references and links for further study, so I'll just refer you there:

Ancient Islamic Penrose Tiles

Geometric patterns in islamic art

This was in the news recently so you might have seen it...

Physics student Peter J. Lu has discovered, after his trip to Uzbekistan, that the geometric patterns shown on the walls of old islamic shrines depicted a very complex mathematical pattern, one that was only found in the west during last century.

I feel the article at ScienceNews.org explains it all very well, and shows plenty of pictures of the patterns AND of the underlying tilings, plus has references and links for further study, so I'll just refer you there:

Ancient Islamic Penrose Tiles

Some links

First of all, the carnival of homeschooling #60 is online this week at Homeschool Hacks.

---

Secondly, recently I've learned of Squidoo lenses created by Rebecca Newburn on topics such as integers and fractions. She's collected some of the best resources about these topics, including games, articles, explanations, books and even free video clips.

---

Then I also wanted to include a link to some magic square addition worksheets that my daughter has greatly enjoyed recently (scroll down to item #5 on the page). We've been practicing adding single-digit numbers where the sum goes over 10, such as 7 + 7 and 9 + 5.

I always tell her that when it's 9 and other number, then one (dot) of the other number "jumps" to go with the nine and makes ten. That way she has learned to add 9 + 6 or any other such sum.

And if it's 6 + 7, I tell her to think of 6 + 6 which she knows by heart, and figure it out from there.

I hope she gets to remembering these by heart eventually, but at this point I'm just glad she is able to figure them out by using these 'helping' ideas. And of course using those also teaches her important principles of mathematics.

---

And lastly, this one was interesting: What's special about this number? - a list of numbers from 0 to 9999, and there's something special about almost all of them!

Some links

First of all, the carnival of homeschooling #60 is online this week at Homeschool Hacks.

---

Secondly, recently I've learned of Squidoo lenses created by Rebecca Newburn on topics such as integers and fractions. She's collected some of the best resources about these topics, including games, articles, explanations, books and even free video clips.

---

Then I also wanted to include a link to some magic square addition worksheets that my daughter has greatly enjoyed recently (scroll down to item #5 on the page). We've been practicing adding single-digit numbers where the sum goes over 10, such as 7 + 7 and 9 + 5.

I always tell her that when it's 9 and other number, then one (dot) of the other number "jumps" to go with the nine and makes ten. That way she has learned to add 9 + 6 or any other such sum.

And if it's 6 + 7, I tell her to think of 6 + 6 which she knows by heart, and figure it out from there.

I hope she gets to remembering these by heart eventually, but at this point I'm just glad she is able to figure them out by using these 'helping' ideas. And of course using those also teaches her important principles of mathematics.

---

And lastly, this one was interesting: What's special about this number? - a list of numbers from 0 to 9999, and there's something special about almost all of them!

Printed (hardcopy) versions of Math Mammoth books

You can now buy a hardcopy (already printed version) of any of the 16 "BLUE" series Math Mammoth books.

These are found at Lulu.com - at the address www.lulu.com/mathmammoth/

The drawback is, they're black-and-white (or grayscale to be exact). This is because the price per page for color printing is quite high and I figured no one would buy the books then. As it is, the b&w versions cost around $8-$11 each.

Lulu.com provides on-demand printing for self-publishing folks (such as me). They have a nice service; you can even print calendars and other stuff and not just books.

Printed (hardcopy) versions of Math Mammoth books

You can now buy a hardcopy (already printed version) of any of the 16 "BLUE" series Math Mammoth books.

These are found at Lulu.com - at the address www.lulu.com/mathmammoth/

The drawback is, they're black-and-white (or grayscale to be exact). This is because the price per page for color printing is quite high and I figured no one would buy the books then. As it is, the b&w versions cost around $8-$11 each.

Lulu.com provides on-demand printing for self-publishing folks (such as me). They have a nice service; you can even print calendars and other stuff and not just books.

How the four operations become two

Awhile back I posted about fraction division; that post made me think of reciprocals, algebra, and even properties of addition and multiplication.

You see, when you study abstract algebra, you get to concept of a group, or a ring, or a field - and all of those consist of a set of elements and one (group) or TWO (ring or field) operations.

Yet, in school math, we always study the FOUR basic operations.

Why are two operations enough in higher-level algebra? Where did they throw subtraction and division?

Addition and multiplication

Did you ever notice these similarities between addition and multiplication?

• Both addition and multiplication are commutative:
  a + b = b + a and ab = ba for all real numbers a, b.
  Division and subtraction are not.


• Boht addition and multiplication are associative:
  (a + b) + c = a + (b + c).
  Division and subtraction are not.

• There exists an element so that adding it to a number doesn't change it (ADDITIVE IDENTITY). We call it zero:
  0 + x and x + 0 both end up being just x.


• There also exists an element so that multiplying by it doesn't change the number (MULTIPLICATIVE IDENTITY). This is number one:
  1x and x1 both are just x.

You might a lready know these well... but how about the INVERSE:

• For any real number x, there is a real number so that when you add the two, you will get the additive identity element or zero.(THE ADDITIVE INVERSE, or OPPOSITE)


• For any real number x (excepting zero), there is another real number so that when you multiply those, you will get the multiplicative identity element or one. (THE MULTIPLICATIVE INVERSE, or RECIPROCAL)

For example, for 8 we have −8: if you add them, you get zero.
For -5.4 you have 5.4; if you add them, you get zero.

And with multiplication:
For 8 we have 1/8. If you multiply them, you get 1.
For -5.4 or -54/10, we have -10/54: if you multiply them, you get 1.

Of course we normally call those the opposite (with addition) and the reciprocal (with multiplication).

But notice how both are based on the similar principle: for each number you can find a "pair" so that when you do the operation (either add or multiply), you get that operation's identity element (zero or one).

Nothing like that is true at all about subtraction and division.

Getting "rid" of subtraction and division

Now, you don't really have to get rid of these two operations, but just observe how it CAN be done:

Simply define "division" as multiplying by the reciprocal... (doesn't that sound a bell?)

...and...

define "subtraction" as adding the opposite (that should sound a bell as well).

You are actually using these in elementary math as well. Just remember the fraction division; you're told to "invert and MULTIPLY" - to change the division to multiplication by the reciprocal.

Or, remember the rule about subtracting negative numbers; you're told that 8 − (−9) is actually adding 9... You're changing subtraction into addition of the opposite (opposite of −9 is 9).

But did you ever consider; that same thing is TRUE of EVERY DIVISION problem and EVERY SUBTRACTION problem!

For example:

24 ÷ 4 is the same as 24 × 1/4.
0.08 ÷ 0.02 is the same as 0.08 × 100/2.

10 − 3 is the same as 10 + (−3).
−1/5 − (−1/2) is the same as −1/5 + 1/2.

In both cases, the change is made in a similar way: the operation is changed to mult/add, and the number is changed to its multiplicative or additive inverse.

So instead of four operations, we can get by with two. And the two that are left have some nice properties (like we saw above).

I think all this is really neat!

(Don't worry though; just keep using subtraction and division in your everyday life and teach them to students, too. I just feel the TEACHER should know about all this, and then bring little bits and pieces of this underlying algebra structure in his/her teaching when appropriate.)

How the four operations become two

Awhile back I posted about fraction division; that post made me think of reciprocals, algebra, and even properties of addition and multiplication.

You see, when you study abstract algebra, you get to concept of a group, or a ring, or a field - and all of those consist of a set of elements and one (group) or TWO (ring or field) operations.

Yet, in school math, we always study the FOUR basic operations.

Why are two operations enough in higher-level algebra? Where did they throw subtraction and division?

Addition and multiplication

Did you ever notice these similarities between addition and multiplication?

• Both addition and multiplication are commutative:
  a + b = b + a and ab = ba for all real numbers a, b.
  Division and subtraction are not.


• Boht addition and multiplication are associative:
  (a + b) + c = a + (b + c).
  Division and subtraction are not.

• There exists an element so that adding it to a number doesn't change it (ADDITIVE IDENTITY). We call it zero:
  0 + x and x + 0 both end up being just x.


• There also exists an element so that multiplying by it doesn't change the number (MULTIPLICATIVE IDENTITY). This is number one:
  1x and x1 both are just x.

You might a lready know these well... but how about the INVERSE:

• For any real number x, there is a real number so that when you add the two, you will get the additive identity element or zero.(THE ADDITIVE INVERSE, or OPPOSITE)


• For any real number x (excepting zero), there is another real number so that when you multiply those, you will get the multiplicative identity element or one. (THE MULTIPLICATIVE INVERSE, or RECIPROCAL)

For example, for 8 we have −8: if you add them, you get zero.
For -5.4 you have 5.4; if you add them, you get zero.

And with multiplication:
For 8 we have 1/8. If you multiply them, you get 1.
For -5.4 or -54/10, we have -10/54: if you multiply them, you get 1.

Of course we normally call those the opposite (with addition) and the reciprocal (with multiplication).

But notice how both are based on the similar principle: for each number you can find a "pair" so that when you do the operation (either add or multiply), you get that operation's identity element (zero or one).

Nothing like that is true at all about subtraction and division.

Getting "rid" of subtraction and division

Now, you don't really have to get rid of these two operations, but just observe how it CAN be done:

Simply define "division" as multiplying by the reciprocal... (doesn't that sound a bell?)

...and...

define "subtraction" as adding the opposite (that should sound a bell as well).

You are actually using these in elementary math as well. Just remember the fraction division; you're told to "invert and MULTIPLY" - to change the division to multiplication by the reciprocal.

Or, remember the rule about subtracting negative numbers; you're told that 8 − (−9) is actually adding 9... You're changing subtraction into addition of the opposite (opposite of −9 is 9).

But did you ever consider; that same thing is TRUE of EVERY DIVISION problem and EVERY SUBTRACTION problem!

For example:

24 ÷ 4 is the same as 24 × 1/4.
0.08 ÷ 0.02 is the same as 0.08 × 100/2.

10 − 3 is the same as 10 + (−3).
−1/5 − (−1/2) is the same as −1/5 + 1/2.

In both cases, the change is made in a similar way: the operation is changed to mult/add, and the number is changed to its multiplicative or additive inverse.

So instead of four operations, we can get by with two. And the two that are left have some nice properties (like we saw above).

I think all this is really neat!

(Don't worry though; just keep using subtraction and division in your everyday life and teach them to students, too. I just feel the TEACHER should know about all this, and then bring little bits and pieces of this underlying algebra structure in his/her teaching when appropriate.)

How to divide irrational numbers?

Many times students just accept what they are told in math class without much questions; it's perhaps boring, they don't want to take time to investigate and delve deeper, or whatever the reasons.

The teacher feeds them with knowledge and they take it in: "Oh, okay, there's such a thing as Pi. Oh, okay, some numbers are irrational."

But a question such as this shows that the person is wondering, wanting to understand the material more. So it's a good question!

I will divide the 'division' question to two parts.

1) When exact answer is desirable, often we do cannot technically divide. For example, let's take Pi ÷ 2. Well, we just write it as Pi/2 or π/2 and leave it at that.

If we can simplify the answer, we do that. For example, √15/√5 can be simplified to √3.

We could simplify √2/2 this way: since 2 = √2 √2, then √2/2 = √2/(√2 √2) = 1/√2.

But often the expression √2/2 is left as is, since there is a convention that we should consider a root in the numerator to be "prettier" than a root in the denominator. Or I don't know why it is; I just remember this little rule from school math.

2) When you need a numerical answer, then you use the decimal approximation of the irrational number, and divide normally as you would decimals.

For example, to find Pi/2, you take the decimal approximation to Pi as, say, 3.14159, and go 3.14159 ÷ 2 = 1.570795.

How to divide irrational numbers?

Many times students just accept what they are told in math class without much questions; it's perhaps boring, they don't want to take time to investigate and delve deeper, or whatever the reasons.

The teacher feeds them with knowledge and they take it in: "Oh, okay, there's such a thing as Pi. Oh, okay, some numbers are irrational."

But a question such as this shows that the person is wondering, wanting to understand the material more. So it's a good question!

I will divide the 'division' question to two parts.

1) When exact answer is desirable, often we do cannot technically divide. For example, let's take Pi ÷ 2. Well, we just write it as Pi/2 or π/2 and leave it at that.

If we can simplify the answer, we do that. For example, √15/√5 can be simplified to √3.

We could simplify √2/2 this way: since 2 = √2 √2, then √2/2 = √2/(√2 √2) = 1/√2.

But often the expression √2/2 is left as is, since there is a convention that we should consider a root in the numerator to be "prettier" than a root in the denominator. Or I don't know why it is; I just remember this little rule from school math.

2) When you need a numerical answer, then you use the decimal approximation of the irrational number, and divide normally as you would decimals.

For example, to find Pi/2, you take the decimal approximation to Pi as, say, 3.14159, and go 3.14159 ÷ 2 = 1.570795.

Back From The Grave

I'm looking for people who would like to join Paperkraft.net as a contributor/blogger. If you have crazy skills in paper crafting (paper models, origami, etc.) send me an email describing why you want to join and some sample photos of your work. You don't have to design your very own paper craft, you can create whatever model you can find on the net and then take pictures of it like you would if you where going to post it on a blog (so I could see your photography skills). Obviously you will also need to have your own digital camera or whatever it is you use to take pictures (having a video camera would be nice but not necessary). There will be no financial gain in this endeavor(coz' there isn't any), only a chance to show the world how great you are with paper crafts.

An 8-year old boy finds error in science exhibit

Parker Garrison is an 8-year old math prodigy who recently discovered an error in the equations of a candy exhibit in Discovery Place.

The problem there asked to find the number of jellybeans in a half-pyramid, using the formula for the volume of the pyramid, and the known volume of one jellybean. The problem asked to multiply the volume of pyramid by 0.9 to account for the spaces between the jellybeans.

The boy's mom took the numbers home and there he calculated and found an error in the given measurements. They had already given dimensions of the half-pyramid in the problem, but still the problem asked to divide by 2...

(You can read more details here)

It's just an interesting example how adults can make simple mathematical mistakes and those can go unnoticed for years...

An 8-year old boy finds error in science exhibit

Parker Garrison is an 8-year old math prodigy who recently discovered an error in the equations of a candy exhibit in Discovery Place.

The problem there asked to find the number of jellybeans in a half-pyramid, using the formula for the volume of the pyramid, and the known volume of one jellybean. The problem asked to multiply the volume of pyramid by 0.9 to account for the spaces between the jellybeans.

The boy's mom took the numbers home and there he calculated and found an error in the given measurements. They had already given dimensions of the half-pyramid in the problem, but still the problem asked to divide by 2...

(You can read more details here)

It's just an interesting example how adults can make simple mathematical mistakes and those can go unnoticed for years...

MathTV.com's special offer

I got word a little while ago that MathTV.com has a very special offer: you can get online access to their entire library of math videos for only $25 (for a full year).

That is indeed a very low price for such a service!

On their site, you can also buy those videos on a CD.

The videos cover basic mathematics, prealgebra, algebra 1, 2, and 3, word problems, and trigonometry

I have some of their CDs and found them very good. They show a math teacher solving problems on a board, step by step.

MathTV.com's special offer

I got word a little while ago that MathTV.com has a very special offer: you can get online access to their entire library of math videos for only $25 (for a full year).

That is indeed a very low price for such a service!

On their site, you can also buy those videos on a CD.

The videos cover basic mathematics, prealgebra, algebra 1, 2, and 3, word problems, and trigonometry

I have some of their CDs and found them very good. They show a math teacher solving problems on a board, step by step.

Multiplying decimals

Someone asked,

how can you use models to multiply decimals?

Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions.

So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that.

Multiply a decimal by a whole number

Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem

2 × 0.34

You can use little hundredths cubes, or draw something that's divided to 100 parts.



BUT you can also just use fractions, and justify the calculation that way:

2 × 0.34 = 2 × 34/100 = 68/100 = 0.68.

OR you can explain it as repeated addition:

2 × 0.34 = 0.34 + 0.34 = 0.68.

I employ that idea in these lessons:

Multiply mentally decimals that have tenths and
Multiply decimals that have hundredths

Multiply a decimal by a decimal

When students are learning to multiply a decimal by a decimal, they're on 5th or 6th grade perhaps. One of the most obvious ways to teach this is to use fractions:

0.5	×	0.3	=
5 10	×	3 10	=	3 × 5 10 × 10	=	15 100

= 0.15.

With such calculations the student is made to notice that when both factors had one decimal digit (and thus had ten as denominator when written as fraction), then the result has two decimal digits (because you multiplied 10 × 10 to get 100 as denominator).

I do not know if there is any obvious pictorial model or manipulative that could be used to multiply a decimal by a decimal.

That is because it's no longer multiplying by a whole number.
It's no longer fitting the simple idea of "repeated addition".

It is instead best understood as 'taking part of' (just like multiplying a fraction by a fraction).

For example 0.1 × 45.9 is the same as taking 10th part of 45.9.

Or, 0.25 × 45.9 is taking 25/100 part of 45.9

And so on.

Understanding this is also crucial so that students can fathom how the answer number can sometimes be smaller than any of the factors... It's no longer adding where everything gets 'bigger' - it's taking a part of something.

See for example:

0.2 × 0.5 = 0.10

0.3 × 0.4 = 0.12

0.6 × 0.05 = 0.030

Here's one worksheet I've made that lets students practice the intricacies of decimal multiplication (a PDF file from my 5th grade worksheets collection).

Multiplying decimals

Someone asked,

how can you use models to multiply decimals?

Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions.

So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that.

Multiply a decimal by a whole number

Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem

2 × 0.34

You can use little hundredths cubes, or draw something that's divided to 100 parts.



BUT you can also just use fractions, and justify the calculation that way:

2 × 0.34 = 2 × 34/100 = 68/100 = 0.68.

OR you can explain it as repeated addition:

2 × 0.34 = 0.34 + 0.34 = 0.68.

I employ that idea in these lessons:

Multiply mentally decimals that have tenths and
Multiply decimals that have hundredths

Multiply a decimal by a decimal

When students are learning to multiply a decimal by a decimal, they're on 5th or 6th grade perhaps. One of the most obvious ways to teach this is to use fractions:

0.5	×	0.3	=
5 10	×	3 10	=	3 × 5 10 × 10	=	15 100

= 0.15.

With such calculations the student is made to notice that when both factors had one decimal digit (and thus had ten as denominator when written as fraction), then the result has two decimal digits (because you multiplied 10 × 10 to get 100 as denominator).

I do not know if there is any obvious pictorial model or manipulative that could be used to multiply a decimal by a decimal.

That is because it's no longer multiplying by a whole number.
It's no longer fitting the simple idea of "repeated addition".

It is instead best understood as 'taking part of' (just like multiplying a fraction by a fraction).

For example 0.1 × 45.9 is the same as taking 10th part of 45.9.

Or, 0.25 × 45.9 is taking 25/100 part of 45.9

And so on.

Understanding this is also crucial so that students can fathom how the answer number can sometimes be smaller than any of the factors... It's no longer adding where everything gets 'bigger' - it's taking a part of something.

See for example:

0.2 × 0.5 = 0.10

0.3 × 0.4 = 0.12

0.6 × 0.05 = 0.030

Here's one worksheet I've made that lets students practice the intricacies of decimal multiplication (a PDF file from my 5th grade worksheets collection).

"Conversations with bird girl" Part I & "Darling You're so Square"

One thing I've noticed is that, although I may have swapped my paint brush for a sewing machine, I still love language. I like the potential in clothes to be more of an art, as opposed to something you just put on. A title takes it to another level.

Maybe this is more personal and evident in the creative process, and not so much something you think of when you reach into your wardrobe... hmmmm, will I wear "Mr Smith Takes Auntie Jackie Out" or "Dorothy Lost the Can of Tomatoes on the Way Home" or will I wear "Conversations with Bird Girl Part I?"

For most people, it's not really like that at all! Who can remember all the titles anyway?! It's more like ' hmmmm, red dress, bird dress, green dress...." Nevertheless, it's the process that matters to me, and so I decided to start a a series of two new projects. Yes, you guessed it... "Conversations with Bird Girl" and "Darling You're So Square". Each is a celebration of different elements. Bird Girl is more whimsical and fairytale like for me, it embodies what I like about folk art, amongst other things, and "Darling You're So Square" is a celebration of garish, and not so garish vintage fabrics made between the 1950's and late 1960's.

Math Mammoth Grade 7 (prealgebra) worksheets available

Just as of today, I got online and available the new Math Mammoth Grade 7-A and 7-B worksheets collections.

These pre-algebra sheets contain problems for beginning algebra topics such as expressions, linear equations, and slope - plus your typical 7th grade math topics such as integers, fractions, decimals, geometry, statistics, and probability.

You will find the problems are very varying as these worksheets have been created one by one (not script made).

Go download and enjoy the free sample sheets! - including a Percent Fact Sheet.

Math Mammoth Grade 7 (prealgebra) worksheets available

Just as of today, I got online and available the new Math Mammoth Grade 7-A and 7-B worksheets collections.

These pre-algebra sheets contain problems for beginning algebra topics such as expressions, linear equations, and slope - plus your typical 7th grade math topics such as integers, fractions, decimals, geometry, statistics, and probability.

You will find the problems are very varying as these worksheets have been created one by one (not script made).

Go download and enjoy the free sample sheets! - including a Percent Fact Sheet.