Review of Math Rider software

If your children like horses, they might fall in love with Math Rider game!

Math Rider is a "math facts" software game; that is, it specifically trains and practices the basic addition, subtraction, multiplication, and division facts.

Some special features include:

• Each player has their own "account" in the game, and Math Rider is adaptive in that it trains those facts more that the child misses. I feel this is a great and important feature. More on that below...
  
  
• I also really liked the statistics. You can see a color-coded chart that shows what facts the child has mastered, what s/he has some trouble with, and what s/he has lots of trouble with. Also, after each ride, the game shows you a bar chart with a bar for each fact in that ride. The taller the bar, the longer the child took to solve the fact.

Here is the home screen. Click to enlarge.



The gameplay is centered around "quests". The player sets up a "quest" by choosing which operation to practice, and on which level. From there, the game first shows a storyline for a few screens.

Then starts the practice, in which a horse jumps a hurdle every time you solve a math fact correctly. If you answer wrong, the horse doesn't whinny but snorts, and you might have time to correct your answer. If you don't, or if you can't answer a problem at all, the horse stops at the hurdle and tells you the answer, and walks through the hurdle instead of jumping it.

The program adapts the problems as you go. The problems you miss or are slow at keep appearing more often than others. The speed adapts, too. If you are fast in answering, the problems appear faster and the horse rides faster, and vice versa.



These rides are 30 problems "long." After a ride of 30 problems, you can check your progress on the "quest" on a map. The backgrounds and hurdles on these rides vary as you proceed and travel through different landscapes. Completing the quest takes several of these 30-fact rides. Once you complete the quest, you are shown several more screens of the storyline to learn how it ends, and you get a special reward item.

The storylines include for example a sick mother, for whom you need to find a special flower, or you find a magical gem that a wizard tells you keeps darkness away, and you have to deliver the gem to the elven city. On the advanced level, you get to rescue a princess.

Personally I didn't care for the wizard/magic story, but you can easily fast forward through these short stories

You can read more about the game play on the Math Rider site.

The motivational reward items include points you get from riding, and the animated items you get from completing quests (flowers, flags, a castle, etc.). Additionally, the program tracks the "mastery" level of each operation and shows that in the animated items.

This is how the statistics looks like:


Both of my girls liked the game really well. They played and played and had to be dragged away from the computer!

My 5th grader already knows the basic facts pretty well, so for her the game was mostly fun with a little bit of review. She liked the changing backgrounds and the music. I really liked the music too (you can turn it off if you don't).

My 2nd grader just did addition and subtraction, and I feel the game did some good in helping her get speedier with her facts. They both liked best the different animated reward items you get when you finish the quests, such as flowers and flags.

It could be improved by adding more storylines. Right now, there is one story for any of the "easy" levels, no matter what the operation, and similarly for the other levels.


Also, I personally feel that children would benefit a lot from practicing the addition and multiplication facts using missing addends and factors (practicing "in reverse"). For example, in multiplication, the game could use 5 x ___ = 25, and in addition, it could use 6 + ___ = 13 types of problems.

I feel the game is well done and serves very well the purpose of practicing math facts and getting "up to speed" with them.  The adaptive features work well and really help the mastery to come to pass.

Of course there are so many different games and ways to practice math facts that it's hard to choose! I hope my review helps you in deciding if this game would be for you. I can definitely recommend it for what it does.

Website: MathRider.com Price: $37.


Disclosure: I was given a copy of Math Rider software for free but did not receive any other compensation.

Fraction of a fraction word problem

A problem about fraction of a fraction...

The sixth-graders have a fundraieser. They raise enough money to reach 7/8 of their goal. Nikki raises 3/4 of this money. What fraction of the goal does Nikki raise?

The picture below shows first of all 7/8. Nikki raises 3/4 of this goal. We need to find 3/4 of 7/8.

It's not easy to directly see what is 3/4 of 7/8. So to do that, I divide each 1/8 piece into four pieces, and then color three of the four. That way I color 3/4 of each of the seven eighths.

Of course, those tiny pieces are now 1/32 parts. I have colored 3 x 7 = 21 of them. So, the colored part represents the fraction 21/32.

This problem is also simple to solve without a picture, if you understand what is asked. To find 3/4 of 7/8, you simply multiply those two fractions. The word "of" translates into MULTIPLICATION in fraction math!

 3       7      21
---  x  --- =  ----
 4       8      32

A book on education crisis

I'm just passing on a link that some of you might enjoy (or be sad about). I found it interesting -- a math teacher (Larry Zafran) telling his story of teaching math in an inner city high school in New York in the early 2000s. It appears not much was able to be done but to try to survive and get through the day while presenting the official curriculum topics in class, at least on the board.

http://www.larryzafran.com/education-crisis-book.htm

Math Mammoth Thanksgiving Sale

Math Mammoth Thanksgiving sale is already in full swing... You'll get 20% off of all the downloads and CDs sold at Kagi store.

To take advantage of this offer, enter the coupon code THANK2010 at checkout.

Enter the coupon code on the shopping cart page (not the first order page). Coupon valid till November 29, 2010, only at Kagi store. Not applicable to printed copies.

Order pages:

• Blue series,
• Golden and Green Series;
• Light Blue complete curriculum products;
• Packages (CDs or downloads);
• Make It Real Learning workbooks;
• Versiones en español.

Notation for solving equations

I just found this via Math Teachers at Play...

 Carolin's Notation for Solving Equations

Carolin is a student from Germany. I just wanted to note that that is exactly how I was taught (in Finland) to note what is done to each side of the equation, and I really like the notation. I don't know if it's used in all Europe...

Basically, you note in "the right side margin" what you're going to do to both sides of the equation in your next step. The "margin" is made by writing a vertical line to the far right of your actual equation solving process.

6x - 5 = 2x      | -2x

4x - 5 = 0       | +5

    4x = 5       | ÷4

     x = 5/4

I just wanted to pass this on in case some of you who are teaching how to solve equations find it useful with students.

Math Mammoth Statistics & Probability

Math Mammoth Statistics & Probability is now available -- and it's the last book I'm intending for the Math Mammoth Blue Series. So the Blue Series is now complete!

Price: $5.30 (download), $10.30 (printed)

This book starts with the easiest topics, such as reading different graphs. These lessons are meant for 5th grade. Some are useful even in earlier grades. Then we study histograms and various kinds of line graphs, including how to draw them.

The next section deals with mean, median, and mode—the three measures of central tendency—how to use them, and how these concepts relate to line and bar graphs.

To create circle graphs, the student needs to calculate percentages, and it is assumed here that the student has already mastered how to calculate those. The lesson on data analysis presents various types of graphs for students to read, and reviews some other percent-related topics.

Then students learn how to make stem-and-leaf plots. Stem-and-leaf plots are simple plots that can be used with 15-100 data items. They are not often seen in media because you cannot use them with large amounts of data.

We study range as a simple measure of variance.

Probability is a topic that in the past was only taught in high school—for example, I personally encountered it the first time in 12th grade. However, in recent years it has “crept” down the grade levels and many states require probability topics even in elementary school.

We start with the concept of simple (classic) probability, and then expand into probability involving two events. This is all that is needful to master at this point (middle school). The exercises involve tree diagrams, dice, flipping coins, picking marbles, spinning spinners, and probability involving statistics, which are the usual types of situations in the study of probability.

See the free sample lessons (PDF) under the cover image on the book's page.

Two algebra 1 word problems (systems of linear equations)

Here are two problems for you to solve... OR to learn from me when I solve them. Both problems are for algebra 1, and use a system of 2 linear equations.

By the way, the comments have some wonderful ideas for solving these mentally, without using algebra. So please read them too!

---

Problem: John bought red pens for $4 apiece and blue pens for $2.80 apiece. If John purchased a total of 24 pens for $84, how many red pens did he purchase?

---

Solution: This is a typical problem that will have two variables and two equations.

Let r be the amount of red pens he buys, and b be the amount of blue pens he buys.

We get our first equation from this sentence: "He bought a total of 24 pens." So, r + b = 24.

We get the second equation from the fact that his total purchases were worth $84, and red pens cost $4, and blue pens cost $2.80

4r + 2.8b = 84

Now, just solve this system of two equations using your preferred method.

r + b = 24
4r + 2.8b = 84

I will multiply the top equation by -4, then add the two equations.

-4r - 4b = -96
4r + 2.8b = 84
-------------------
-1.2b = -12
From this, b = 10.

Then since r + b = 24, r must be 14.

---

Problem: In a group of 60 workers, the average salary is $80 a day per worker.  If some of the workers earn $75 a day and all the rest earn $100 a day, how many workers earn $75 a day?

---

Solution: To get started, first find out what are the unknowns. In this case, there are two. Some workers earn $75 a day, and some earn $100 (two quantities).

Let A = workers who earn $75 a day.
Let B = workers who earn $100 a day.

We need to have both variables when building the equations, even though the problem only asks for A.

THEN we need to somehow use the information given to build two equations. That is because to solve for two unknowns, you need two equations.

Well, there are 60 workers, so our first equation is pretty easy: A + B = 60.

Now, the first sentence may throw you off... but actually, the concept of average is not coming into the picture very much. The info in the first sentence is ONLY used to find out one useful fact: that the group of workers earns 60 x $80 = $4,800 in TOTAL each day. We use that to build our second equation, which has to do with the total earnings:

75A + 100B = 4,800

Now we have our two linear equations, and all that remains is to solve the system using any standard technique.

A + B = 60
75A + 100B = 4,800


Multiply the top equation by -100, and then add the two equations together:

-100A - 100B = -6000
75A + 100B = 4,800
-----------------------------
-25A = -1,200
A = 48

Then, B must be 12.

Check: 48 x $75 + 12 x $100 = $4,800. It checks.

Work & workers word problem

Here's another one of those job / workers word problems (inverse or direct variation). Try and see if you can solve it using the "table" method instead of equations:

A certain job can be done by 18 clerks in 26 days.  How many clerks are needed to perform the job in 12 days?


Again, we can set up a table and reason this out. Initially set it up like this:

jobs | clerks | days
--------------------------
  1  |  18    |  26
--------------------------
  1  |        |  
--------------------------
  1  |   ?    |  12

Then think of the "days" column. We want to "go" from 26 to 12. You could use a proportion here... or first figure out how many clerks are needed to do this job in 2 days, and then from that go to 12 days.

If 18 clerks do it in 26 days, then how many clerks would do it in 2 days... which is 1/13 the amount of time.... so we need 13 times as many clerks.
13 x 18 = 234 clerks are needed.

jobs | clerks | days
--------------------------
  1  |  18    |  26
--------------------------
  1  |  234  |   2  
--------------------------
  1  |   ?    |  12

Now, if 234 clerks do it in 2 days, how many clerks would do it in 12 days? Now, the time increases 6-fold, so we need only 1/6 as many workers.

234 / 6 = 39.

So 39 clerks are needed.

Simplify a ratio problem, for your entertainment :)

Simplify the ratio 186:403. The answer is 6:13. How do we get it?

To simplify ratios (or fractions), we need to find COMMON FACTORS of the two numbers. So, one way to do it is to first find the GCF (Greatest Common Factor) of 186 and 403. Then divide 186 and 403 by it.

Alternatively, just find ANY factor of 186 and 403, and divide both by it, to simplify the ratio somewhat, and to get started. Then repeat the process.

Okay, 403 is not divisible by 2, 3, 4, 5, 6. This I know by divisibility tests. Maybe it's divisible by 7... need to try (calculator). No, it isn't.

Maybe by 11? No.

Maybe by 13? YES. My calculator helps. 403 = 13 x 31. I happen to know both of these are primes, so therefore 403 doesn't have any other factors.

Then 186.... is it divisible by 13 or 31?
By 13, no.
By 31, YES!  186 = 31 x 6

So since 186 = 6 x 31 and 403 = 13 x 31, then the ratio 186:403 simplifies to 6:13. Clearly that's as far as we can get, as it's simplified to the lowest terms.

Free online quizzes & tests, middle & high school math

I just stumbled onto a pretty neat resource for all math teachers (parents too). It is Glencoe's Online Study Tools. What they have is online, multiple-choice tests, quizzes, and standardized test practices, for ALL middle and high school topics.

They are randomly generated so you can get as many different quizzes as you like.

Just choose any state you wish (it doesn't matter), then a book or course (such as algebra or geometry), and then you'll see the list of topics. In other words, you don't have to own the book in order to do the quizzes and tests.

Mathematically Interesting Buildings

Just yesterday I read this interesting article, so I thought I'd pass it on. It's a list of 9 most mathematically interesting buildings in the world... each one has something different that is interesting about them - from pi and symmetry to asymmetric hyperbolic paraboloids!

It includes the Great Pyramid, The Eden project, Sagrada Familia, the Gherkin, and some interesting-looking buildings I had never heard of.

Houses/workers/days problem (direct and inverse variation)

Today I want to feature a great video from a fellow blogger and Youtuber, Dave Marain. He solves this question, which at first can sound intimidating:

If 10 workers can build 3 houses in 60 days, how many workers are needed to build 5 houses in 40 days?

It sounds like you'd need algebra, proportions, inverse or direct variation, etc. And true, you could use those. But his method completely avoids all that and is based on setting up a simple TABLE, AND using common sense!

|  HOUSES  |  WORKERS  |  DAYS  |
----------------------------------
|    3     |    10     |  60    |
----------------------------------
|    1     |    10     |        |
----------------------------------
|    2     |    10     |        |
----------------------------------
|    5     |           |        |
----------------------------------

See the video below:




And here's a link to his other Youtube videos: www.youtube.com/user/MathNotationsVids.

Math Mammoth Integers is here!

Math Mammoth Integers worktext covers all important integer (signed numbers) topics for middle school (grades 5-8), with instructions written directly to the student. Thus, it requires very little teacher or parent involvement.

Check out a few free sample worksheets or lessons:

http://www.mathmammoth.com/preview/Integers_Adding_Integers1_Counters.pdf

http://www.mathmammoth.com/preview/Integers_Dividing_Integers.pdf

And see the contents.

More information and purchase info is here.

Special offer for printed Math Mammoth books at Lulu (coupon code)

Lulu.com has a special offer right now that you can use to purchase the printed copies of MathMammoth books.

Enter coupon code HOLIDAYVIP305 at checkout and receive $20 off your next order of $20 or more.

You can browse the printed versions of Math Mammoth books here.

This offer is only valid in US dollars, and cannot be applied to previous orders. You can only use this code once per account, and you can't use this coupon in combination with other coupon codes. The offer expires on November 30, 2010 at 11:59 PM. 

Offer posted at http://www.lulu.com/static/102610_HOLIDAYVIP305wv.html

The quickest path around the baseball bases

Some of you (sports fans?) might find this interesting: mathematicians have calculated the quickest path around the baseball diamond, through the four bases. And it's not in the shape that you'd think.

Please read the news article here: Winning the World Series with math
A nearly circular path could be the fastest way to home plate

Math Mammoth Australian Money

One more money-related book for Math Mammoth Blue Series... this time for Australian money. It covers counting coins, shopping, and making change, and is for grades or years 1-3.

Download $3.25
Printed copy $9.50

Carnival time

Please stop by the The Adventure Edition of Math Teachers at Play, posted at Homeschool Bytes. It's beautifully illustrated! Let the adventures begin!

Math Mammoth in other currencies

Did you know that Math Mammoth complete curriculum (Light Blue series) now comes with five different currencies?

When you purchase either grade 1, 2, or 3 downloadable curriculum, you will get the chapter on money in all of these currencies (as PDF files):

US money, Canadian money, European money (euros), British money, and Australian money

(The reason that higher grades don't have these is because I don't include instruction on coins and bills/notes after third grade... it is assumed as mastered.)

Of course, you only need to study through your own currency, but some of you might be curious to take a peek in the other monies as well - if nothing else, to see how their coins look like.

For example, Australian money does not use 1-cent coins at all. The smallest coin they have is 5 cents.  Or, British 50-pence coin is a heptagon in shape! Canadians use a 2-dollar coin nicknamed "toonie". It's pretty too.

Interested? You can find samples and purchase info from these links

Math Mammoth Grade 1 complete curriculum
Math Mammoth Grade 2 complete curriculum
Math Mammoth Grade 3 complete curriculum

Study shows females' math skills equal to males

This news comes from the university of Wisconsin-Madison.


Large study shows females are equal to males in math skills

The study author, Janet Hyde, says, "My message to parents is that they should have confidence in their daughter's math performance. They need to realize that women can do math just as well as men. These changes will encourage women to pursue occupations that require lots of math."

I agree; parents AND teachers should never in any way imply that girls can't do the math just because they are girls. That's just not true!

I also thought the mention of 48% of math majors being females was interesting. So, at least in pure math, genders seem to be quite equal. But I personally doubt that will happen in engineering because boys seem to be much more interested in that type of work.

Math teacher's error

It took Marie 10 minutes to saw a board into 2 pieces. If she works just as fast, how long will it take her to saw another board into 3 pieces?

The teacher marked this answer (20 minutes) wrong, and wrote down 15 minutes as the correct answer. Why is the teacher in error?

Geometry games online

I have updated the list of online geometry resources and games at HomeschoolMath.net. It became quite a long list of links! The links are categorized by topics, such as

Shapes & polygons 
Area & perimeter 
Angles 
Solids, volume, & surface area
Coordinate plane 
Congruent transformations 
Similar figures 
Circle / Pi 
Constructions 
Pythagorean Theorem 
General geometry websites 
Worksheets 
Books 
Advanced topics.

I hope this is helpful!

American Math Challenge

The American Math Challenge 2010 is around the corner... Registrations are now open at www.americanmathchallenge.com.

The actual event will take place over 48 hours, beginning on October 26, 2010 at 9 a.m. EST. This is a FREE online Challenge where U.S. students of all ages and skill levels compete against each other in a series of one-minute mental math games, as well as self-challenged curriculum-based activities.

October 18 will open a "practice competition week" open to all registrants.

Registration will close on October 22. 

Another guess my secret number game

Here's another one of those "guess my secret number" games where the computer tells what number you thought of originally.

http://myframeshoppe.ca/math/

First you need to choose your "secret" number between 1 and 10,000.

I thought some of you might enjoy trying to figure out how it works... I enjoyed it.

---

By popular demand I want to share some of why it works. So DON'T READ if you want to think about why it works!

---

In the LAST step, the computer guesses which digit you left out. For example, maybe you put in the number as 75x711. Now, the key to "guessing" or figuring out what the missing digit is, is the fact that this number is divisible by 9. Recall that before coming to this step, you had multiplied your number by 3, and again by 3. That means you multiplied it by 9, so it is now divisible by 9.

The numbers that are divisible by nine have a special property: the sum of their digits is also divisible by nine. OK, my number 75x711 IS divisible by nine. The sum of the digits I see is 7 + 5 + 7 + 1 + 1 = 21. The next bigger number after 21 that is divisible by 9 is 27. So the digit sum must be 27, and the missing digit is 6.

---

The game goes like this:
1) Choose a number between 1 and 10,000.
2) Multiply it by 4.
3) Add 5.
4) Multiply it by 75.
5) Choose any two digits from your number and add the number formed by those to your number.
6) Multiply it by 3.
7) Multiply it by 3.
8) Replace one of the digits by X and submit.
9) Computer guesses your number.

I already explained how the computer finds out the missing "secret" digit. Then of course, the program can easily go backwards any step that had adding or multiplying (by doing the opposite operation).

In step 5, where you choose two digits and add them, the program isn't trying to guess what number you added. The key idea is that no matter what number you add, it's less than 100, AND the number you got in the end of step 4 is divisible by 75 and ends in 5.

So, to undo step 5, the program ONLY looks for a number that is divisible by 75, ends in 5, and is no more than 100 less than the number from step 6.

Resources for multiplication tables

This time of year many students tackle multiplication tables. These resources can be of help:

1. This video explains my method for "structured drilling" of multiplication tables. We don't start with random drill (that comes later) but with drill that is using the structure of the tables. The tables are also practiced "backwards", which will facilitate the learning of basic division facts.
   
   Over the years I have hears from many individuals who have gotten their child to learn the tables using this method, but recently  I got word from a a principal of an English medium school in Pune, India, that they have already implemented my multiplication drilling method and it is working beautifully with their children! I feel flattered a whole SCHOOL is using it... and glad it is working.
   
   
2. Math Mammoth Multiplication 1 book. First it has a long section that concentrates on the concept of multiplication, and then the last part has lessons to faciliate the same drilling as explained in the video.
   
   
3. There is a place for random drill, and you can use flashcards, or the way shown in the video, and/or GAMES. I have compiled a list of online multiplication games here.
   
   This card game is also noteworthy  (the "Product War" version). Essentially, each player is dealt two cards face up, players multiply those, and the person with highest product captures all cards in that round.
   
   Or, try a dice game called Damult Dice.

I feel that learning the multiplication tables well is more important than even mastering addition and subtraction facts. Why? Because knowing the tables well facilitates the learning of basic division facts, multidigit multiplication, long division, most fraction math and factoring. Even in algebra you still need to be able to simplify rational expressions and factor polynomials, perhaps even multiply matrices.

Or, we can say it this way: if you child does not know the tables, he/she will have a terrible time mastering all those topics. I'm not saying children won't learn those topics conceptually -- I mean they will have hard time completing the problems and exercises quickly, and can instead get all "bogged down" just by the multiplications.

That is why I feel every teacher/parent should put forth a good effort for their students to learn the times tables. Spend 1-2 months on it. It can pay off!

That said, there is a BALANCE, as in everything. IF you've already expended a considerable amount of effort and the child is not retaining them, please don't make the multiplication tables to be the reason why your student hates math. You can back off and try again later. Some give their students a "crutch" -- tables written out -- and eventually the kid notices how slow it makes him/her having to check the answers in the table instead of knowing them, and decides to memorize them.

Math Mammoth Geometry 2 giveaway

Recently I offered one of my customers a free book for finding an error in the answer key... she declined and wished someone else could have it who is in need.

So I'll give one electronic copy of Math Mammoth Geometry 2 book (PDF file) to one randomly chosen lucky winner.

To participate, you need to do two things:
1) Leave a comment in this blogpost, explaining how and where you'd use this book, AND
2) Email me so I have your email address.

If you have no use for this geometry material, please don't participate, and let someone win who can use it.

If I choose a winner who didn't email me, then I cannot contact them. In that case I will have to choose another winner.

I'll first choose a few best entries based on how/where they'd use the book, and then choose randomly among those.

This 'contest' closes and winner will be chosen whenever I notice (in checking my mail) that there are at least 10 participants. So hurry up!

Free math tutoring

David Freeling from TutorTalk.org wants to let us know that as a part of a promotion to launch his site, he's offering FREE tutoring on every Friday.

Here is some information from David:

Free tutoring will apply to specific classes, labeled as Free. Students should visit www.TutorTalk.org in advance and register, because space is limited. Free lessons cover fundamental math, reading and writing skills and are suitable for a wide range of ages, but especially middle and high school. Curriculum focuses on problem-solving, and kids are encouraged to play along from home and be prepared with a calculator, pen and paper. Students will also be able to bring in their own homework problems for small group tutoring, in classes of up to 5 students.

The site also says, "TutorTalk Classes are dynamic, interactive small-group classrooms, led by a qualified teacher, and hosting up to five students. Students work together and chat to solve onscreen problems from the comfort of their own homes, with the help and hints of the instructor."

Homeschooling by the Numbers

My dh found an interesting "infographic" about homeschooling statistics. Click the image to enlarge.

Integer games online

I have updated the list of integer games online at my site. Now the games are organized into sections of:

• ordering integers, 
• addition & subtraction of integers, 
• multiplication & division of integers, and
• all operations with integers. 

Happy playtime!

How to use an abacus with Math Mammoth

Recently I've received several questions about abacus usage within Math Mammoth curriculum. Here is what I wrote and added to my FAQ on the site.

The only way the abacus is used in my books is where each bead counts as one. Nothing fancy. It is NOT used like Chinese, Russian, or any of the other abaci where one bead might count as 5, 10, or 100.

A 100-bead abacus or school abacus simply contains 10 beads on 10 rods, a total of 100. In the school abacus, each bead simply represents one. The 100-bead abacus lets children both "see" the numbers and use their touch while making them.

First and foremost, the abacus is used in the place value section in 1st grade where children learn about tens and ones (numbers up to 100). We use it to show clearly how 45 is made up of 4 tens and 5 ones, for example.

Secondly, you can use the abacus with addition and subtraction problems in 1st and 2nd grades. For example:

• Show the child additions and subtractions with whole tens. For example, to solve 50 + 20, first make 50 on the abacus. Then add 20 more.
• Add a two-digit number and a single-digit number. For example, to solve 23 + 5, first make 23 on the abacus. Then add five beads.
• Show some "shortcuts" in addition or subtraction. For example, to solve 34 + 20, first make 34 on the abacus. To add 20, add two whole rows of beads. Then the student checks how many whole tens and how many individual beads is the total.
  
  Or, to solve 85 − 20, first make 85. Then pull back two whole rows of beads.
  
  Or, to add 23 + 44. First make 23. Then make 44 on using the five lowest rows of the abacus. Have the child now count the whole tens (6), and the individual beads from the two rows (3 + 4). This shows adding the tens separately, and adding the ones separately. From this you can graduate to making first 23, then adding 4 full rows of beads for 40, and then adding 4 individual beads from the same row as the 3 beads.

The purpose is mainly to help children to visualize two-digit numbers, and to add and subtract two-digit numbers.

The goal in my books is to drop the abacus by 3rd grade. Even before that, students use visual models, and from those go on to the abstract. The quicker the child can use visual models, and then do the math problems without any models, the better.

At Amazon you can find Melissa & Doug Classic Wooden Abacus for around $12. An abacus where the beads alternate colors by fives is even more useful (but may be out of stock).

Browse Amazon's abacus selection here. Other stores carry abaci as well.

You can also use this virtual abacus. Or, make your own abacus. Just don't make it exactly like they show on that web page but instead use 10 bamboo skewer with 10 beads in each so you get a 10 x 10 abacus.

Order of operations / PEMDAS

PEMDAS does not cover matrices...
Photo courtesy of Stuartpilbrow

Someone asked me recently...

Could you please share with me your opinion of the "Please Excuse My Dear Aunt Sally" simplifying expressions. Any feedback could you give will be appreciated. Thank you.

This "PEMDAS" rule is a mnemonic for order of operations:

Please = Parenthesis
Excuse = Exponents
My = Multiplication
Dear = Division
Aunt = Addition
Sally = Subtraction

There's nothing wrong with using a mnemonic to remember the order of operations. However, one has to bear in mind that

1. This rule is not all-inclusive. It omits for example square roots. But the rule is good for all elementary grades. (Square roots would be on the same level or rank with exponents, by the way.)
    
2. The rule doesn't spell out the fact that in reality multiplication and division are "on the same level" or rank. This means that if you have several multiplications and divisions, you do them from left to right, and not "multiplication first, then division".
   
   For example: 60 ÷ 5 × 4. You go from left to right, and first do 60 ÷ 5 = 12. Then you multiply 12 × 4 = 48.
   
   If you want 5 × 4 to be done first, it needs to be in parenthesis: 60 ÷ (5 × 4). Here, first do 5 × 4 = 20, and then 60 ÷ 20 = 3.
   
   Similarly, addition and subtraction are on the same level: if both exist in an expression, they are to be done from left to right.
   
   An example: Simplify the expression 2 × 5 − 6 + 8.
   
   1: Multiply 2 × 5 = 10.
       The expression is now 10 − 6 + 8.
   
   2. Subtract. 10 − 6 = 4.
       The expression is now 4 + 8.
   
   3. Add. 4 + 8 = 12.
   
   So, perhaps it's more illustrative to lay out the PEMDAS rule like this:
   
   Please = Parenthesis
   Excuse = Exponents
   My Dear = Multiplication & Division
   Aunt Sally = Addition & Subtraction
   
   ...and say it with little pauses at the commas: Please, Excuse, My Dear, Aunt Sally.
   
   
   
3. Fraction line as a division symbol generates implicit groupings or parenthesis in the fact that anything written on top or bottom of the fraction line is to be done before the division. It's as if the whole numerator and denominator were inside parenthesis.
   
   So...
   
   
   
   

   4 × 32
   --------
   7 − 22
   

   
   means the same as (4 × 3²) ÷ (7 − 2²). Simplifying this is left as an exercise for the reader. The answer is 12.
   

See also a detailed lesson on order of operations at Purplemath and Wikipedia's note on mnemonics.

Estimation methods

A typical estimation problem:
guess how many paper clips are in a jar?
Photo by Dean Terry

Here recently I was asked a question about estimation methods:

My daughter's homework was to estimate the number of 953 divided by 18, using front-end estimation, using rounding to estimate, and using compatible numbers to estimate. What are the differences between these three methods? Do we get the same result? Thanks!

You definitely don't get the same results as these three methods are quite different!

1. Front-end estimation means you keep the "front" or first digit of each number, and make the other digits to be zeros.
   
   So, 953 ÷ 18 is estimated to 900 ÷ 10 = 90.
   
   Another example: 56 × 295 would be estimated as 50 × 200 = 10,000.
   
   
2. Rounding means you round the numbers, usually to their biggest place values, but sometimes you can round "creatively". In any case, the numbers you round to should be easy to work with mentally.
   
   So, in 953 divided by 18 we round 953 to nearest hundred, and 18 to nearest ten. 953 ÷ 18 becomes 1,000 ÷ 20 = 50.
   
   Another example: 56 × 295 would become 60 × 300 = 18,000.
   
   An example of rounding "creatively": with 24 × 32 you can round 24 to 25 (to the "middle five"), and 32 to 30. The estimated result is 25 × 30 = 750. The exact result in this case would be 768 so the estimation was fairly close.
   
   Another principle to keep in mind with when using rounding to estimate is that if you have an addition or multiplication problem, it's best to round one number down, the other up, in order to minimize the rounding error. If you have a division or subtraction, it's best to round both numbers "the same direction", either up or down.
   
   
3. Compatible numbers means finding numbers that are close to the numbers in the problems but such as are easily to work with mentally.
   
   So, 953 ÷ 18 could be estimated as either 960 ÷ 20 or 1000 ÷ 20, depending on your mental division skills.
   
   960 ÷ 20 = 96 ÷ 2 = 48.
   
   Another example: estimating 56 × 295 depends, again, on your mental multiplication skills. You could try to leave 56 as it is, and make 295 to be 300, to get 56 × 300 = 16,800. Or, you might make the numbers to be 60 and 300 (the same as in the rounding method) and get 18,000.

To compare all three methods, we check the exact result of our problem, which is 953 ÷ 18 = 52.944444444... From this we can see that the rounding method was most accurate in this case (it gave us 50), and front-end estimation did really bad (it gave us 90).

In the other example, the exact result is 56 × 295 = 16,520. Again, rounding (18,000) or compatible numbers (16,800 or 18,000) method did best, and front-end estimation the worst (10,000).

Factoring the time

I stumbled upon this comic that you might enjoy... factoring the time (from xkcd.com).

Carnival time: Math Teacher's at Play

Head on over to Math Teachers At Play #29 carnival to read a nice assortment of posts on math education!

Math tricks by a silly guy

A short youtube video where Kjartan Poskitt shows some math tricks. He's funny! Enjoy!

The equal sign problem

An interesting piece of research has just come out on the misconceptions with the equal sign (=).

Students' understanding of the equal sign not equal

According to the research, US students exhibit this misconception much more often than students in other countries. It has to do with thinking of the = sign as an operator. Kind of like thinking that = means "to do" the operation.

For example, a student with that misconception tends to solve the problem

7 + 6 = ____ + 2

by adding 7 + 6, and placing the answer on the empty line.

The correct way is of course to think of the equality: 7 + 6 equals 13, so the other side has to equal 13 too. 11 fulfills this little equation:

7 + 6 = 11 + 2

I have known of this problem for years, and have tried to include problems in my Math Mammoth books to help children NOT to develop this wrong idea. For example, children solve


200 + 50 + 6 = ____ + 200 + 50 in the place value section.

Or, I use problems where they have to put either <, >, or = in between (one the line here, but I like to use boxes in the books):

20 + 9  _______ 90 + 2

8 + 6   ________  7 + 7


Or, just simple missing addend problems from the very start (1st grade):

3 + _____ = 5.


Remember, students exhibiting the misconception would add 3 and 5, and put 8 on the empty line. But we can teach kids to think of this problem as "3 and how many more makes 5". It is a starting point in understanding the equal sign in the correct way.

Then they should also (later on) encounter the same problem this way:

5 = ____ + 3

And other variations.

Math Mammoth Geometry 2

I have just finished writing the material for Math Mammoth Geometry 2 book. The material in it is suitable for grades 6-7. Download price is $5.80.

The main topics in the book include:

* angle relationships
* classifying triangles and quadrilaterals
* angle sum of triangles and quadrilaterals
* congruent transformations, including some in the coordinate grid
* similar figures, including using ratios and proportions
* review of the area of all common polygons
* circumference of a circle (Pi)
* area of a circle
* conversions between units of area (both metric and customary)
* volume and surface area of common solids
* conversions between units of volume (both metric and customary)
* some common compass-and-ruler constructions.

I've included several complete lessons from the book as samples (PDF). Feel free to download these and use with your students!

Angles in Polygons
Review: Area of Polygons, 1
Surface Area

Besides those, there are two other sample pages:

Area and Perimeter Problems
Basic Compass and Ruler Constructions, 1

What is next?

• See more information
• Purchase a download at Kagi
• Purchase a download at Currclick

Review of Harold Jacobs Geometry

I have just posted a comprehensive review of this book that is popular among homeschoolers for a high school geometry course. Click to read:

Review of Harold Jacobs Geometry book

Math assessment and placement

My daughter just completed the end-of-year math test for 4th grade. She scored 82%. She missed doing one question (just forgot), made a few careless mistakes, and then missed several in the measuring unit section because she didn't remember how many ounces are in a pound or how many grams in a kilogram. So we need to review some of that.

But overall, I am happy with the results, because (and I knew this beforehand too) she understands the concepts well! That is the most important thing. It's always easy to review some forgotten facts (such as measuring units), but when child has not understood the main concepts, the "patch-up" work is more tedious.

I have provided these Math Mammoth end-of-year tests on my site as free downloads so that people can use them as placement tests or for general math assessment. I often have people email me about their kids' results, and based on that, I feel those tests are working fine.

I have also collected a few other math assessment resources on this page, notably the yearly tests from Texas, Florida, and California (which are free) and a few commercial resources. Note: in my opinion, California's math standards are a bit too advanced, so use care when using their tests.

Some parents are against testing, but I feel testing for assessment is often needful in order to know what the child knows and doesn't know. The tests are like a "diagnostic toolbox" for me when I need to guide parents in what Math Mammoth books they might need for their children. So the purpose is not to "put on the pressure", but just to find out.

Kenken game for summer fun

As most of you are taking a break from math or school work, here's a little (addictive) math game to play online or as printed version: Kenken.

In it, you have to place the numbers in the grid kind of like in Sudoku so that no number appears twice in the same row or column, BUT there's an additional twist: it gives you "cages", and the numbers within a particular cage have to work to a given answer with a given math operation (either addition, subtraction, multiplication, or division).

KenKen practices logical thinking and is, I think, more fun than Sudoku.

It's easier to understand if you go see and try it yourself. The smaller sizes, such as 4x4 game, are great for kids, and the larger ones are good for us adults or older students.

• Kenken.com has nice Kenken puzzles online plus instructions, but only one for each size.

• MathDoku.com has an unlimited number of Kenken puzzles to play online in three sizes and several difficulty levels.

My favorite music

This is a bit different entry from the norm, and if you're not interested in music (or oldies music), just ignore it. (I realize people's musical tastes vary tremendously.) I just wanted to share a bit of news of my favorite, 24K Gold Music Shows, and their new YouTube channel. They mainly play music from 50s and 60s; that is, oldies music.

(Well, I do also like classical music pretty well -- I played piano in my youth for 13 years, and still occasionally do at home.)

You can now see 24K Gold music videos at YouTube. For example (click on the song links to see these in high density):


Cara Mia (my favorite!):


Love Can Build a Bridge (a beautiful ballad with a strong emotion):


Or something faster such as Elvis' Burning Love:




Johnny B Goode



See more videos - or subscribe at their channel page.

Math Jokes 4 Math Folks book

I recently received Math Jokes 4 Mathy Folks from the author, Patrick Vennebush. This is a very comprehensive collection of math-related jokes that all "mathy" people will definitely enjoy, and math teachers could use this book to enliven their lessons.

I have seen several of the jokes on the Internet, but never such a large collection.

The jokes are categories into chapters such as

• One-Liners, ("A hungry clock goes back four seconds.")

• Graphic Jokes, (For example, "Cube Roots" picture you see on the cover of the book.)

• Three Dudes, (An engineer, a mathematician, and a chemist are...)

• Conversion Chart, (10⁹ antics = 1 gigantic, or 454 graham crackers = 1 pound cake)

• Professions,("The secretary of defense informed the president, 'Yesterday, three Brazilian soldiers were killed. 'Oh, no!' the president exclaimed. ... 'How many is a brazillion, anyway?' ")

And so on.

In each section, the jokes that require the least math knowledge to understand are first, and then the material "advances". People who have studied math in college will get the most out of it, but there are joke for every level.

Please note the author has included a few bathroom jokes and a few slightly dirty ones - if you don't like those, they can easily be marked over.

Algebra 1 curriculum advice

I have just finished writing a LONG article on homeschool algebra 1 recommendations and advice. It took me quite many hours to write and research.

Questions about what to do for algebra 1 have become one of the "frequently asked questions", so I decided to write down something that I can refer people to, from now on.

I realize you may have different opinions and even suggestions for algebra curriculum in homeschool, so if you let me know, I'm willing to look into other possibilities not mentioned in the article.

Common Core State Standards released

Common Core State Standards is a state-led initiative (not by the government) and designed by a diverse group of teachers, experts, parents, and school administrators.

They hope these standards will be adopted by various states. We shall see! It would definitely help if the math standards didn't vary so much among the various states. In some states, there are SO many standards and objectives per grade that it seems impossible students could learn it all well.

You can view the mathematics standards here. They also have language arts standards available.

Math Mammoth Percent

Math Mammoth Percent book is here! Part of the "Math Mammoth Blue Series", this is a topical worktext, which contains both the instruction and the exercises in the same book. Download price: $3.50.

I know that the concept of percent and related applications are often difficult for students. I wrote this text keeping this in mind, and I hope it can clear the confusion on percentages, percent of change, and such topics!

From the description:

Math Mammoth Percent teaches students the concept of percent, percentage of a number, discounts, sales tax, percent of change, circle graphs, and percent of comparison. It is suitable for 6th-8th grade (middle school).

The concept of percent builds on a student's understanding of fractions and decimals. Specifically, students should already be very familiar with the idea of finding a fractional part of a whole (such as finding 3/4 of $240). Students who have used Math Mammoth have been practicing that concept since 4th grade, and one reason why I have emphasized finding a fractional part of a whole so much in the division and fraction materials in the earlier grades is specifically to lay a groundwork for the concept of percent.

Assuming the student has mastered how to find a fractional part of a whole, and can easily convert fractions to decimals, studying percents in middle school should not be difficult.

Read more and see the free sample pages!

Online math curricula

I recently updated and revised my page listing various online math curricula. I tried to put the current pricing there so people could compare them easier, and also list them by level (either elementary/middle or middle/high school).

I feel using some online math program is an excellent choice for many parents, homeschooling or not. Sometimes they can be used as a supplement, and sometimes as a stand-alone program, especially with kids who like computer work because they get some extra motivation just because it's on computer. My own kids happen to like IXL for some additional math practice every now and then.

Of course many of these offer videos, and watching math videos online is a definite advantage of today's computer age.

Most of the curricula or programs are commercial but a few that I listed are actually free resources.

A study: U.S. Teachers Not Well Prepared to Teach Mathematics

I just found this news article at EducationNews.org and some of you might find it interesting:

U.S. Teachers Not Well Prepared to Teach Mathematics, Study Finds

It explains the findings of a new international study of future math teachers in 16 countries.

Math Mammoth sale

Math Mammoth end of year sale is here!

Get 20% off of all Math Mammoth downloadable books and CDs. Enter the coupon code MMSALE2010 on the shopping cart page at Kagi to receive 20% off. The sale is valid till June 1, 2010. At KAGI STORE ONLY.

Order pages at Kagi:

• Blue series
• Golden and Green Series
• Light Blue complete curriculum products
• Packages (CDs or downloads)
• Make It Real Learning workbooks
• Libros Mamut Matemáticas

Additionally, if you prefer printed versions, there is no percentage sale for those, BUT Lulu is having free summer shipping promotion where orders over $19.95 will receive free shipping (within the US). See a list of Math Mammoth books at Lulu. If your order is eligible, the coupon code FREESHIP is automatically applied.

How to teach long division

Long division is a hurdle to many students. I have written a long article about it where I approach teaching long division in several small steps. It is all also explained in this video:



How to teach long division video at YouTube


The video and the article do not delve into the "why" long division works, but suggest a specific sequence of teaching long division. It should also make the process of finding the remainder within long division more transparent to students.

Please read my article from this link: How to teach long division


What about teaching why long division works?

I feel the algorithm of long division AND why it works presents quite a complex thing for students to learn, so in this case I don't see a problem with first learning the algorithmic steps (the "how"), and later delving into the "why". Trying to do both simultaneously may prove to be too much to many students.

However, once your student has a basic mastery of how to do long division, it is time to also study what it is based on. To learn more about that, please see:

Long division as repeated subtraction
Why long division works (based on repeated subtraction)

Why long division works—a comparison with sharing money (bills) (video)

The Cookie Factory Guide to Long Division by Denise

Calculation Nation games

Calculation Nation is a new, free, math games website from the National Council of Teachers of Mathematics (NCTM).

It currently has 8 games, each of which you can play against the computer or against other players that are online at the same time as you. You will need to register first (free).

All of the games are educational and well made. They include:

Square Off
Drag a rectangle to cover as many spaceships as possible, calculating the perimeter.

Factor Dazzle
Find all the factors of a number to earn points. Then, choose a number for which your opponent must find the factors.


Dig It
Create and locate fractions on a number line to tell your shovel where to dig. Earn points for the amount of dirt you collect and the number of jewels you discover.

Click to enlarge

Times Square
Exercise your skill with factors and multiples! Try to get four squares in a row vertically, horizontally, or diagonally. I liked this one! I've played it before somewhere else as well.

Drop Zone
Make sums of 1, using different unlike fractions, and prevent your opponent from making sums of 1. I liked this one!

Click to enlarge

Slam Ball
Slam the ball into the sides of the game board, and use your knowledge of angles, symmetry, and reflections to choose the best path

Ker-Splash
A ball goes down a series of ramps and grabs either x, y, or plain number tokens. You need to figure out the best route for the ball, and combine terms. My kids really liked this one. I liked it alright as well. It's not very difficult - more for prealgebra level than algebra 1.

Click to enlarge

Fraction Feud
In each joust, earn points by creating a larger (or smaller) fraction than your opponent.

Click here to visit Calculation Nation games.

Teacher apprecation week is coming

I myself have enjoyed the downloadable resources from these Learning A-Z websites in the past, so I thought you might take a look, too. It's all going to be free for a week! Here's their blurb:

To thank the world's teachers during Teacher Appreciation Week from May 3-7, we're opening up all six of our PreK-6 Learning A-Z websites — for free. Sites include Reading A-Z, Raz-Kids, Writing A-Z, Science A-Z, Vocabulary A-Z, and Reading-Tutors.

Nearly 50,000 teachers took advantage of our Open House last year and pre-registration for this year is now open. Just go to http://www.learninga-z.com/openhouse — and then pass it on to your friends and colleagues!

Shopping City game hints

My daughter asked me to check out Shopping City game at HoodaMath.com, and so I did. Well, I became hooked... it's one of those "addictive" types of games. I spent nearly a week playing it every day, until I was able to finish all 9 levels.

I decided to write this blogpost about the gameplay because in the end, to be able to finish the last level, I had to do some serious analyzing of various factors in the game, which definitely reminds me of solving and analyzing math problems.

On each level, you are shown an empty street map, and your task is to build stores and to earn a certain amount of money in a given amount of days. As the levels progress, the street map gets more complex, you have a bigger goal to achieve and more days to do it. In the last level, you need to earn a million dollars in 29 days.

Then once you start the game, people start walking in from various directions, visiting the shops, and spending their money.

Click to enlarge

For the most part, I played the game kind of randomly, building different kinds of stores randomly here and there - toy stores, boutiques, pet stores, restaurants, jewelry stores, and so on. That strategy seemed to work alright in the beginning... but then the game got harder.

I was not able to finish the last level without thinking about and analyzing several things. I even wrote out how many percent I earned each day, and if I was falling behind my "best play", I would quit and start that level over (which you can do by pressing the "menu" button). I reached 96% or 97% several times before succeeding in reaching 100% in the 9th level.

I'm going to share some tips now for playing the Shopping City game. Warning: if you want to figure out these things yourself, don't read on! Just go play it. Of course, part of the fun IS when you figure out the strategy yourself. In fact, you could even use this game TO teach analyzing skills to youngsters.


Hints for the Shopping City game

The game itself gives you some instructions, such as the fact that the customers won't go to two of the same kind of stores, but they will visit a store and an upgraded version of the same store. Or, that corners are important locations.

1. On day 1, no matter what level, your shoppers have only $30 to spend, and they can spend it all in a pet store. On day 2, they have $60 to spend and they can still spend it all in a pet store. A pet store happens to be the best store to start with. On day 3 and 4, having a pet store and boutique is enough for them to spend all their money (as long as all the people can visit a boutique and a pet store on their walking path). If on any particular day, all your customers spend all their money, you get a $1000 bonus. This is important to get for the first few days on level 9.
   
   
2. The game tells you how much the customers spend in each particular store (before it's upgraded). Some stores are not nearly as cost-effective as others. For example, a restaurant costs $10,000 and a boutique costs $2,500, yet customers spend the same amount of money in both. Study carefully how much people spend in each store. Some stores look like fun to get, but they won't bring you in the most money.
   
   
3. When you upgrade a store the first time, customers will spend 1.5 times as much money there than before upgrading. When you upgrade a store the second time, customers will spend 2 times as much money there than before upgrading at all.
   
   
4. For best play, observe carefully the walking paths of the customers. They come from various directions and go off at several different exits. You want to choose the location of the best stores so that most people can visit them. Usually the corners work best for that. In level 9, I found out (eventually) which few locations worked best. If I put a pet store (for example) to two specific locations, I could be sure that everyone would visit one or the other. This is one of the most important things to analyze in the game.
   
   
5. As the days progress on any particular level, you will need to sell the smaller stores in the best locations in order to build the better-performing stores in those locations. For example, if you put a pet store in the best location at first, later on when you can buy a jewelry store, it should go to one of the best location. Then you can rebuild the pet store somewhere else.
   
   
6. You won't need the fire station or police station in the early levels.
   
   
7. When a store needs repaired, if you can upgrade it (if you have enough cash), it gets repaired for the upgrading fee, so that becomes more cost-effective than simply repairing it.
   
   
8. If a fire comes and you don't have a fire station, you can sell that particular store before it burns down and you lose it.
   

Hope this helps!

British Money and Fractions & Decimals 3 - New Math Mammoth books

I have two new books available for the Math Mammoth Blue Series:

1) British Money

Math Mammoth British Money is a worktext that covers money-related topics usually encountered during years 2-4. The book contains both textbook explanations and exercises, and is designed to be very easy to teach from, requiring very little teacher preparation (you do need to find practise coins before the lessons).

Please read more, and see free sample pages.


2) Fractions & Decimals 3

Math Mammoth Fractions & Decimals 3 continues the study of fraction and decimal topics, on the 6th grade level. This book assumes the student already has studied fractions and decimals in the past, for example using Math Mammoth Fractions 2 and Math Mammoth Decimals 2. The goal of the book is to go through all of the fraction and decimal arithmetic, using up to six decimal digits and larger denominators in fractions than what is commonly encountered in 4th and 5th grade materials.

Read more, and see free sample pages.

Carnival time again

Please check out Math Teachers at Play #25!

I especially liked the fun math game provided by Pumpkin Patch blog, and the fun(ish) algebra worksheets from The Exponential Curve blog. There's even a murder mystery included...

Coupon code for free shipping at Lulu

If you'd like to purchase a printed version of any of Math Mammoth books at Lulu, use the coupon code FREEMAIL305 to get $3.99 off the shipping charge (that is the shipping charge for one single book within the US). Only applicable for US addresses. Offer expires May 1, 2010.

Teaching long multiplication a.k.a multiplying in columns)

The two videos below show how you could teach multi-digit multiplication, or the multiplication algorithm, or multiplying in columns to students.

Teaching multiplication algorithm

Multiplication algorithm with a 2-digit multiplier

I approach this in steps. First, to teach students to multiply 4 × 87 or 5 × 928 (one factor is single-digit):

1) Teach students to multiply single-digit numbers by whole tens and hundreds.
2) Teach them the partial products algorithm;
3) Use the above as a stepping stone and teach the usual multiplication algorithm.

Then we can go on to the two-digit multiplier:

4) Teach the partial products again.
5) Teach the regular form of the algorithm.


Let's look at these steps in more detail.

Step 1. This means teaching students to multiply 5 × 80 or 7 × 400 or 3 × 40 or 9 × 900 (mentally!). The shortcut is to multiply without the zero or zeros, then tag the zero or zeros to the result.

But, where does it come from? For example, 5 × 80 is the same as 5 × 8 × 10. We first multiply 5 × 8 and then multiply that result by 10.

Step 2 is the partial products algorithm. Here, we write the numbers under each other, just like in the regular multiplication algorithm. But the multiplying is done in two (or three) parts: first the ones, then the tens, and then the hundreds (if any).



Step 3. After this, you would teach the usual multiplication algorithm. Point out to students how the two multiplications and the addition (the carry) are done at the same time, instead of as separate calculations.

This step needs practiced quite a bit before moving on so that students become confident in the carrying process.

Step 4a has to do with learning to multiply 50 × 46 or 70 × 352 or 600 × 529 in columns, using the regular algorithm. It is like multiplying 5 × 46 and tagging a zero, or multiplying 6 × 529 and tagging two zeros, but we place those extra zeros in the result first, before multiplying.

    3                 1 5
    4 6               5 2 9
x   5 0           x   6 0 0
--------        ------------
2 3 0 0         3 1 7 4 0 0

The bolded and underlined zeros are placed there before multiplying. The second video makes this maybe even clearer.

Step 4b has to do with multiplications by a 2-digit multiplier, such as 45 × 89. Here, we'd first multiply 40 × 89, then 5 × 89 (using the regular algorithm), and then add the two results. So this means, first of all, three separate calculations.

Step 5: We show students the regular form of the multiplying in columns with a two-digit multiplier, and point out how those three separate calculations are now compactly written under each other.

That's it in a nutshell! Please watch the videos to make this even more clear.

More new worksheet generators

At HomeschoolMath.net, you can now make free worksheets for place value & scientific notation (such as write a number in expanded form or in scientific notation), and also for proportions, including simple proportion word problems.

The links are:
Place value & scientific notation worksheets
Proportion worksheets

Cute cat videos website with Weebly

Warning: this is totally off-topic of math.

Lately I've made an experiment of sorts, and that is I've checked what it is like to create a website with Weebly. Weebly is a service that lets you build a website with a drag-and-drop interface.

My testing website is called Cute Cat Videos. I've collected a few videos of cats doing tricks, silly cats, cats and kittens with other animals, and such.



So far, so good. You can create a simple, small website with Weebly, without technical knowledge. It could be good for anyone (even kids) who want to build a small website without learning html, css, or any of that stuff.

I liked the selection of templates. Drag and drop seems to work fine. You can add custom html. You can add videos or other media, or Google Adsense.

Of course it has its limitations - don't expect too much. For example, in the two-column layout, I can't edit the html to change the background color. Or edit the html for the images so they wouldn't have borders. Or customize the title tags for the inside pages. But, for small needs it could work well.

Currclick spring sale

Currclick is having a sprint sale from March 23 till April 5. Math Mammoth books are on sale for 20% off, in case you are interested!

Carnival time...

We have great carnival this time: Math Teachers at Play #24. Check it out!

Some of my picks are an easy card game for practicing addition/subtraction, some fun worksheets for addition and subtraction, and consideration about how many balloons you'd need to lift up a house (think the movie Up)

I also really liked the idea of giving students the answer, and asking them to explain why and how it is gotten, as is explained in Why is this answer right?

But there are many more and Denise has worked hard to create this beautiful carnival so check it out!

Review of Soma cube and Tangram from Monkey Pod Games

The Soma cube is a dissection puzzle. It has seven pieces that have to be assembled into a 3x3x3 cube. So, that is the main challenge, but you can also arrange the pieces into all kinds of other interesting 3-D shapes.


The seven pieces of the Soma cube

I received a wooden Soma cube from Monkey Pod games. The cube comes in a neat, good-looking wooden box that can be showcased in your bookcase as a decorative item. But, it's certainly not meant for decoration only! It is quite a fun challenge! The Soma cube from Monkey Pod games comes with a little booklet that shows you the solution so you can get the pieces back to the box, but it also shows you several dozen other 3-d shapes to make with the pieces.

I enjoyed trying to put together some of those. And the kids enjoyed just playing with the pieces, trying to come up with something interesting, and of course they did! It was great fun for them.

I let a neighbor family borrow the cube, and the results were similar: adults enjoyed the challenges, and kids enjoyed the fun of building something. One adult visitor spent over an hour trying to recreate several different shapes given in the booklet. It's really easy to get totally absorbed in doing these!

The other item I received from Monkey Pod games was a tangram. Again, the pieces and the box were of quality good-looking wood (made from monkeypod wood, in fact!).

You probably know what a tangram is - it is a puzzle of seven flat pieces that can be arranged into lots of different 2-dimensional shapes.

The little booklet that came with the tangram showed dozens and dozens of shapes to try to make with the tangram pieces. My kids immediately took to it, and had fun trying to make the shapes, especially the animal ones. I did some too. It is certainly fun, but not necessarily evident what pieces to use at first.

Overall, the games (or puzzles) from Monkey Pod games can provide hours and hours of educational fun that stretches your brain. They come with little instructional guides, which you can also download from the website.

Also, these puzzles are great gifts since they look so good (not plastic or cheapy looking). They are also great items to showcase at parties. Don't be surprised if your guests get totally immersed in them!

Monkey Pod games has dozens of different kinds of wooden puzzles and other items to choose from.

Take a look at for example the disentanglement puzzles and the interlocking wooden puzzles at Monkey Pods website. Two examples are below.

It's like being in a candy store! And the prices are not bad either, a lot of the puzzles are around $10. Amazon sells them as well.

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Disclaimer: I was given the Soma cube and the wooden Tangram by Monkey Pod Games. Other than that, I have received no other compensation. The opinions are mine. Some of the links above are Amazon affiliate links so if you click to Amazon through them and then buy something, I will earn a few cents.