Multiplication algorithm poll

Dave at Math Notations has an interesting poll up, till Feb. 29th. It asks about which multiplication algorithm should be taught in grades 3-5 for multidigit multiplication, and if mastery should be required.

If you're interested in that issue, go check it out and vote:

43x27? Which Algorithm should be taught? A MathNotations survey will end the debate!

(The part about ending the debate is mentioned tongue-in-cheek, I'm sure :)

Should you use a math dictionary?

When it comes to what is fashionable in homeschooling, these days lap books, notebooking, lapbooking, whatever term you use, are definitely "in". I haven't seen much anything for math, though. And no, I haven't made any lapbooks either! But I do have an idea for one:

Let your child/student make their own math dictionary! Just make a new page in it every time there is a new concept or term to study. The student can write the term, write an explanation, an example calculation, or draw a picture of it − or all of those.

Making such a math vocabulary book shouldn't take lots of time, because you don't usually encounter new words every single lesson in a typical math curriculum. The rest of the time the book can act as a reference or as a review medium.

There is one exception though, and that is GEOMETRY. In geometry, just about all the time you have new terms to learn. In fact, a big part of the geometry in elementary grades is simply learning the meaning of words such as parallel, perpendicular, trapezoid, vertical angles, symmetry, diameter, radius, circumference, and so on.

I know; I've written a geometry book. In it, I encourage kids to make such a geometry notebook for themselves.

There also exist very nice already written math dictionaries:

1) Online dictionaries.

The two mentioned below complement each other, and are free!

Visual Mathematics Dictionary - includes definitions of terms, often with pictures, and links to related terms, which I think is a very nice feature.

A Maths Dictionary For Kids 2007 - this one explains the term and often has some kind of interactive activitiy you can do right on the site, such as converting Celsius to Fahrenheit, or draw triangles, color or click things.

2) Printed dictionaries

Printed ones can make a nice reference for your home library.


Math Dictionary for Kids: The Essential Guide to Math Terms, Strategies, and Tables

This handbook includes illustrated, concise explanations of the most common terms used in general math classes. The 400 illustrated definitions and examples are categorized by subjects that include ... measurement; algebra; geometry; fractions and decimals; statistics and probability; and problem solving. Each term has a concise definition and an example or illustration.



Math Dictionary: The Easy, Simple, Fun Guide to Help Math Phobics Become Math Lovers

An A to Z list of some of the most common terms you'll hear in a math classroom.

The terms cover general arithmetic, geometry, algebra, graphing, probability, statistics, and much more. Advanced mathematical terms such as those used in calculus are not covered. Each definition is spelled out in plain terms, often with simple diagrams to illustrate, eliminating any confusion.


Visual Math Dictionary

This reference has over 600 definitions and additional resources including tables, rules and symbols. Math terms are explained simply and visually with ample examples in two colors and clear, concise wording. Visual descriptions of many important concepts abound, including rules for finding area and volume, data representation, financial math, rational numbers equivalents, factors, figurative numbers, the international system, transformations, prime and composite numbers and much more.

New Math Mammoth books: Multiplication 2, Division 2, Place Value 4

Some recent additions to Math Mammoth Blue Series books:

Math mammoth Multiplication 2

This book concentrates on multi-digit multiplication, first explaining what it is based on (multiplying in parts), then practicing the algorithm. also included: order of operations, multiplying with money, and lots of word problems.


See sample pages here: (PDF)

Contents & Introduction
Multiply by Whole Tens and Hundreds
Multiply in Parts
Multiplying in Columns, Standard Way
Error of Estimation
Order of Operations
Money and Change
Multiplying 3-digit by 2-digit

Math Mammoth Division 2

This book includes lessons on division, long division, the remainder, part problems, average, and problem solving.




See samples:
Contents and Introduction
Division Terms, Zero and One
Finding Parts with Division
Long Division 1
Long Division with 4-Digit Numbers
Average
Divisibility Rules

NOTE: Multiplication 2 and Division 2 now replace the earlier book called Multiplication Division 2.

Math Mammoth Place Value 4

In this book, the focus is on place value with
thousands, ten thousands, and hundred thousands. We also study numbers with millions a little. The book is most suitable for third or fourth grade.



See sample pages (PDF):
Contents
Thousands
Rounding
Estimating
A Little Bit of Millions

A reminder and games

Henry Cate is still looking for submissions for the Homeschooling Carnival logo contest. If you've been kicking around some ideas, please submit them soon.


Then, a new games website (for me, anyway): Exuberant games!

These are really simple, letting children practice the basic operations (addition and division facts) but in nice child-friendly settings.

Multiplication games are here

For example, my daughter enjoyed serving ice cream to the penguins who would turn sad with wrong answers (the Cone Crazy game).

Classic math mistakes

I found this link just today, via Mr. Barton's essential math freebies page.

It is a collection of classic math mistakes, presented as posters with silly titles.

You can post them on your wall and let students figure them out, because the poster doesn't actually explain the mistake; it just shows an example of it and the funny title.

Check it out:

Classic Math Mistakes - Gallery of Posters

Clock face problem

Divide the face of the clock into three parts with two lines so that the sum of the numbers in the three parts are equal.

Let's first understand the problem. We need to draw lines into the clock face. It doesn't say the lines need to end in the center or go through the center. The lines could seemingly go many various ways.

Again, to help a student who can't even get started, just tell them to draw some lines into the clock face. Here's one example:



Of course this is not a solution since it does not fulfill the condition that the sum of the numbers in the three parts needs to be equal. But it gets you started. You have some lines, so you can ask the student to add the numbers in the three parts and verify the sums are NOT equal.

How do we make the sums to be equal? Just random trying would take a long time.

The KEY to this problem is that since the sum in each of the three parts is equal, if you add those three sums, you will get the sum of all the numbers in the clock face. Thus, we can find out the partial sum by computing the total sum and dividing by 3.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78. 78 ÷ 3 = 26.

So the sum of the numbers in each part has to be 26. After this key point, solving this problem is fairly easy.

Now, we need to find numbers on the clock face that will add up to 26. Again, trial and error will probably produce the result fairly soon. But we can use some thinking too:

Let's consider first the largest number on the clock, 12. What other numbers should go with 12 to make 26?
We can add lots of its neigbors to it and try.

12 + 1 + 2 + 3 + 4 + 5 = 27 so just adding numbers from 1 on in order won't work.

That means we might need to couple 12 and 11.

12 + 11 = 23. Need three more. So... 12 + 11 + 1 + 2 will give 26. That's the one part that we can separate with one line.

Then continuing from the larger ones, 10 + 9 + 8 = 27. This is too much. Need another approach.

But let's draw lines across the clock face so that 10 can be joined with some smaller numbers. You will see soon that 10 + 9 + 3 + 4 = 26.



The last ones left are 5 + 6 + 7 + 8 = 26.

Now we can try hunt for another solution. Is there one?

Note that we can only draw two lines. You can't do this in very many different ways. You can draw two lines across the clock face, or perhaps draw one line across and another from the circumference to the first line, as in my first picture. In the one solution we've found, the two parts consisted of neighboring numbers and the one middle part did not. In either case, we need to use some neighboring numbers to make up the sums.

When you consider the numbers from 1 to 12 on the clock face, the only neighboring numbers that add up to 26 are the ones we used: 12 + 11 + 1 + 2 and 5 + 6 + 7 + 8. So we have to use at least one of those sets for the one part. If you try it out, you will quickly note no other solutions are possible than the one found above.

This may not be considered a totally rigorous argument by mathematical standards, but it is important for students to become convinced that there are no other solutions. Convincing others of the same is the first step towards mathematical PROOF.

Solving the plus sign problem

How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

This is a fifth grade problem taken from Word Problems for Kids by Canada's SchoolNet.

The first step, as always, is to understand the problem. The student needs to know what is an "addition sign" and a "digit". We're simply asked to put plus signs in between those numbers and add them up, and try to come up with 99.

Then, after we have a basic idea of what the problem is about, is the time to do something. You know, often the child may say, "I don't know how to start. I have no idea what to do!"

But in this case, as often happens, you'll get somewhere as soon as you'll do something. It's really simple: put some plus signs in there and just see what happens. Let's simply put the plus sign in between every digit:

9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45.

Great, we got something. We got 45 which is too small. Now the student should start thinking HOW to make the sum bigger?

Obviously, the only way to do that is to use ONE OR SOME two-digit numbers. We need to omit at least one of those plus signs!

So try something. For example:

98 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 136.
9 + 8 + 7 + 6 + 5 + 4 + 3 + 21 = 63.

Encourage the student to try a few other possibilities. He/she should notice that if you make the two-digit number using the large digits, the sum will be MORE than if you use the smaller digits for that.

(That is, of course, really simple.)

At this point the student might simply use "brute force" and write out all the possibilities with one two-digit number. That's alright; that will get him an answer!

However, there is also a quicker way, if we use some thinking. The target answer 99 is about half-way in between the two sums I have above (136 and 63), so I would try next to use the "middle digits" such as 4, 5, and 6 as the two-digit number:

9 + 8 + 7 + 6 + 54 + 3 + 2 + 1 = 90, which is too little.
9 + 8 + 7 + 65 + 4 + 3 + 2 + 1 = 99, which is the right answer!

Any other possibility with one two-digit number will either be less than 90 or more than 99, so that is the only solution using one two-digit number and seven plus signs.

But, good problem solvers will also consider checking if this is the ONLY solution. There is the possibility of using TWO two-digit numbers. Indeed, I quickly stumbled upon another solution that way:

9 + 8 + 7 + 6 + 5 + 43 + 21 = 99.

It's easy to note that this is the only possibility using six plus signs, since if you put the plus sign between some other digits, your sum will be MORE than this sum.

I hope this is helpful to some of you. I know the problem is quite easy. My intention is simply to point out how a typical problem solving process can go. Observing the "tricks of the trade" can help you to solve problems, and to teach others do the same.

Two problems for YOU to solve

I'd like for you to try solve these two 5th grade problems. They are from a very nice collection of word problems for kids by Canada's SchoolNet.

I'd like to use them to illustrate problem solving. But before I do that and give you the solutions, please try to solve them yourself. And don't post solutions here for now; I will have to reject such comments.

Solutions and discussion will follow in a few days.

THE FIRST PROBLEM IS:

* How many addition signs should be put between digits of the number 987654321 and where should we put them to get a total of 99?

THE SECOND IS:

* Divide the face of the clock into three parts with two lines so that the sum of the numbers in the three parts are equal.

Review of Math Mammoth Geometry 1

Sol Lederman from Wild About Math blog has posted a review of my Math Mammoth Geometry 1 book.

I appreciate the review very much and want to thank Sol for it.

Teaching problem solving

I have written a more complete presentation on the topic of teaching problem solving — including what not to do. I posted it on my HomeschoolMath.net website:

The do's and dont's of teaching problem solving in math

I'd like your input. For example if you can suggest other resources to be added to the end of that article, or some other ideas, leave a comment here on the blog or email me.

Math Mammoth LightBlue series books as downloads

Great news!

The LightBlue Series books are now available as downloads. These are designed to be a complete curriculum for a given grade. They follow the lesson format of my Blue Series books: explanations, then problems (worktext). In fact, the two series do share a lot of material.

The LightBlue is for you if you need a math curriculum for a given grade that covers all the topics needed. These are practically no-prep for the teacher, and in fact, kids can probably teach themselves the majority of the time (assuming they can read).

Please read more by following these links:

Math Mammoth Grade 1 Complete Worktext

Math Mammoth Grade 2 Complete Worktext

Math Mammoth Grade 3 Complete Worktext

I will have grade 4 ready sometime this spring, February or March I think.

Blogging in 2007 and in 2008

Welcome to 2008! Carnival of homeschooling, the 2-year anniversary edition is up at Why Homeschool. Henry notes that during 2007, over 300 bloggers contributed to the carnival. That is quite a bunch! I'm glad, for the hosts' sakes, that not all of them contribute any one time!

Also, they are hosting a contest for the Carnival of Homeschooling graphic.

Also, Sol at Wild About Math has invited math bloggers to find their most popular posts during 2007. Well, I'm kind of a latecomer on this because he wanted the comments before the end of 2007, but I thought maybe you'd enjoy knowing anyway.

Now, I used to have but a measly SiteMeter free version installed, so I frankly don't know which posts were visited the most. (I did install Google Analytics yesterday, instead, so from now on I will know!)

Based on the number of comments, these were among the most popular:

Multiplying in parts and the standard algorithm

So many percent more

Performing well below grade level

The ones I esteem the highest are the ones I have put into my newsletters during 2007. I always use content from my blog for my newsletter.

I've had posts about the four operations, drilling (or not?) the math facts, living math books, manipulatives, percent, integers, multiplying in parts, and balance problems. See the sidebar "tags" to find them, or visit the newsletter archives link above.

This coming year I simply wish to continue writing about topics that can help homeschooling parents in particular, and all math teachers in general.