World Maths Day 2013: Register Now

It’s time to register for World Maths Day, which will take place on March 6, 2013. Last year, more than five million students from all around the world combined to correctly answer nearly 500 million math problems.

Would you like to help break the record this year? Register now so you can practice in advance!

About World Maths Day

  • Play with students from schools all around the world. Individuals and homeschoolers are welcome, too.
  • The competition is designed for ages 4-18 and all ability levels. Teachers, parents and media can also register and play.
  • It’s simple to register and participate. Start practicing as soon as you register.
  • And best of all, it’s absolutely free.

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Multiplication Matching Cards

PUFM 1.5 Multiplication, Part 2

Poster by Maria Droujkova of NaturalMath.com. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

Multiplication is taught and explained using three models. Again, it is important for understanding that students see all three models early and often, and learn to use them when solving word problems.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

I hope you are playing the Tell Me a (Math) Story game often, making up word problems for your children and encouraging them to make up some for you. As you play, don’t fall into a rut: Keep the multiplication models from our lesson in mind and use them all. For even greater variety, use the Multiplication Models at NaturalMath.com (or buy the poster) to create your word problems.

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Build Mathematical Skills by Delaying Arithmetic, Part 4

To my fellow homeschoolers,

While Benezet originally sought to build his students’ reasoning powers by delaying formal arithmetic until seventh grade, pressure from “the deeply rooted prejudices of the educated portion of our citizens” forced a compromise. Students began to learn the traditional methods of arithmetic in sixth grade, but still the teachers focused as much as possible on mental math and the development of thinking strategies.

Notice how waiting until the children were developmentally ready made the work more efficient. Benezet’s students studied arithmetic for only 20-30 minutes per day. In a similar modern-day experiment, Daniel Greenberg of Sudbury School discovered the same thing: Students who are ready to learn can master arithmetic quickly!

Grade VI

[20 to 25 minutes a day]

At this grade formal work in arithmetic begins. Strayer-Upton Arithmetic, book III, is used as a basis.

The processes of addition, subtraction, multiplication, and division are taught.

Care is taken to avoid purely mechanical drill. Children are made to understand the reason for the processes which they use. This is especially true in the case of subtraction.

Problems involving long numbers which would confuse them are avoided. Accuracy is insisted upon from the outset at the expense of speed or the covering of ground, and where possible the processes are mental rather than written.

Before starting on a problem in any one of these four fundamental processes, the children are asked to estimate or guess about what the answer will be and they check their final result by this preliminary figure. The teacher is careful not to let the teaching of arithmetic degenerate into mechanical manipulation without thought.

Fractions and mixed numbers are taught in this grade. Again care is taken not to confuse the thought of the children by giving them problems which are too involved and complicated.

Multiplication tables and tables of denominate numbers, hitherto learned, are reviewed.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

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Build Mathematical Skills by Delaying Arithmetic, Part 3

To my fellow homeschoolers,

How can our children learn mathematics if we delay teaching formal arithmetic rules? Ask your librarian to help you find some of the wonderful living books about math. Math picture books are great for elementary students. Check your library for the Time-Life “I Love Math” books or the “Young Math Book” series. You’ll be amazed at the advanced topics your children can understand!

Benezet’s students explored their world through measurement, estimation, and mental math. Check out my PUFM Series for mental math thinking strategies that build your child’s understanding of number patterns and relationships.

Grade IV

Still there is no formal instruction in arithmetic.

By means of foot rules and yard sticks, the children are taught the meaning of inch, foot, and yard. They are given much practise in estimating the lengths of various objects in inches, feet, or yards. Each member of the class, for example, is asked to set down on paper his estimate of the height of a certain child, or the width of a window, or the length of the room, and then these estimates are checked by actual measurement.

The children are taught to read the thermometer and are given the significance of 32 degrees, 98.6 degrees, and 212 degrees.

They are introduced to the terms “square inch,” “square foot,” and “square yard” as units of surface measure.

With toy money [or real coins, if available] they are given some practise in making change, in denominations of 5′s only.

All of this work is done mentally. Any problem in making change which cannot be solved without putting figures on paper or on the blackboard is too difficult and is deferred until the children are older.

Toward the end of the year the children will have done a great deal of work in estimating areas, distances, etc., and in checking their estimates by subsequent measuring. The terms “half mile,” “quarter mile,” and “mile” are taught and the children are given an idea of how far these different distances are by actual comparisons or distances measured by automobile speedometer.

The table of time, involving seconds, minutes, and days, is taught before the end of the year. Relation of pounds and ounces is also taught.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

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Cool Fibonacci Conversion Trick

photo by Muffet via flickr

Maria explains how to use the Fibonacci Numbers to convert distance measurements between miles and kilometers:

P.S.: Congratulations to Maria for her Math Mammoth program being featured in the latest edition of Cathy Duffy’s 100 Top Picks for Homeschool Curriculum! And Home School Buyer’s Co-op has a sale on Cathy Duffy’s book through the end of July.


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Build Mathematical Skills by Delaying Arithmetic, Part 2

To my fellow homeschoolers,

Most young children are not developmentally ready to master abstract, pencil-and-paper rules for manipulating numbers. But they are eager to learn about and explore the world of ideas. Numbers, patterns, and shapes are part of life all around us. As parent-teachers, we have many ways to feed our children’s voracious mental appetites without resorting to workbooks.

To delay formal arithmetic does not mean that we avoid mathematical topics — only that we delay math fact drill and the memorization of procedures. Notice the wide variety of mathematics Benezet’s children explored through books and through their own life experiences:

Grade I

There is no formal instruction in arithmetic. In connection with the use of readers, and as the need for it arises, the children are taught to recognize and read numbers up to 100. This instruction is not concentrated into any particular period or time but comes in incidentally in connection with assignments of the reading lesson or with reference to certain pages of the text.

Meanwhile, the children are given a basic idea of comparison and estimate thru [sic] the understanding of such contrasting words as: more, less; many. few; higher, lower; taller, shorter; earlier, later; narrower, wider; smaller, larger; etc.

As soon as it is practicable the children are taught to keep count of the date upon the calendar. Holidays and birthdays, both of members of the class and their friends and relatives, are noted.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

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PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

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PUFM 1.3 Addition

Photo by Luis Argerich via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

The basic idea of addition is that we are combining similar things. Once again, we meet the counting models from lesson 1.1: sets, measurement, and the numberline. As homeschooling parents, we need to keep our eyes open for a chance to use all of these models — to point them out in the “real world” or to weave them into oral story problems — so our children gain a well-rounded understanding of math.

Addition arises in the set model when we combine two sets, and in the measurement model when we combine objects and measure their total length, weight, etc.

One can also model addition as “steps on the number line”. In this number line model the two summands play different roles: the first specifies our starting point and the second specifies how many steps to take.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

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What to Do with a Hundred Chart #27

[Photo by geishaboy500.]

It began with a humble list of 7 things to do with a hundred chart in one of my out-of-print books about teaching home school math. Over the years I added a few new ideas, and online friends contributed still more, so the list grew to its current length of 26. Recently, thanks to several fans at pinterest, it has become the most popular post on my blog:

Now I am working several hours a day revising my old math books, in preparation for publishing new, much-expanded editions. And as I typed in all the new things to do with a hundred chart, I thought of one more to add to the list:

(27) How many numbers are there from 11 to 25? Are you sure? What does it mean to count from one number to another? When you count, do you include the first number, or the last one, or both, or neither? Talk about inclusive and exclusive counting, and then make up counting puzzles for each other.

Share Your Ideas

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.


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Free Video Lecture: How to Memorize Numbers

Would you like to improve your memory and make it easier to do mental math? The Great Courses is offering a free lecture by mathemagician Arthur Benjamin. But act quickly: This offer expires next Friday, October 14th.

Credit card numbers. Phone numbers. Driver’s license numbers. Social Security numbers. Birthdays. These are just a few of the many numbers you deal with every day — numbers that often need to be memorized for quick and easy recall.

And it turns out that there are fun and amazingly effective ways to memorize large numbers and long strings of digits — ways that can not only make everyday life easier but that can be used to perform larger calculations that were previously beyond your mental capacity.

Watch this free video lecture to discover just how easy—and fun—memorizing numbers can be!

— Secrets of Mental Math: Free Video Lecture on How to Memorize Numbers

How to Homeschool High School

With four out of my five students now graduated, I’ve experimented with a lot of high school options. By far my favorite resource — for the kids and for my own continuing education — has been The Great Courses (formerly The Teaching Company). They offer a wide variety of courses, such as:

One warning about The Great Courses: Unless you are truly desperate for a course, never pay full price. All of their courses go on sale at least once a year, and the discounts are well worth the wait. Many libraries carry these courses, however, so if you find one you like but don’t want to buy, you may be able to get it through inter-library loan.


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Lex times 11, by Dan DeChiaro, via flickr

How to Conquer the Times Table, Part 5

Photo of Lex times 11, by Dan DeChiaro, via flickr.

We are finishing up an experiment in mental math, using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.

Take your time to fix each of these patterns in mind. Ask questions of your student, and let her quiz you, too. Discuss a variety of ways to find each answer. Use the card game Once Through the Deck (explained in part 3)as a quick method to test your memory. When you feel comfortable with each number pattern, when you are able to apply it to most of the numbers you and your child can think of, then mark off that row and column on your times table chart.

So far, we have studied the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order). Then we memorized the doubles and mastered the facts built on them. And then last time we worked on the square numbers and their next-door neighbors.

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photo by Karen via flickr

How to Conquer the Times Table, Part 4

Photo of Miss Karen (and computer) times 3, by Karen, via flickr.

If you remember, we are in the middle of an experiment in mental math. We are using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible. So far, we have studied the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order). Then we memorized the doubles and mastered the facts built on them.

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Javier times 4, by Javier Ignacio Acuña Ditzel via flickr

How to Conquer the Times Table, Part 3

Photo of Javier times 4, by Javier Ignacio Acuña Ditzel, via flickr.

If you remember, we are in the middle of an experiment in mental math. We are using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible. Talk through these patterns with your student. Work many, many, many oral math problems together. Discuss the different ways you can find each answer, and notice how the number patterns connect to each other.

So far, we have mastered the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order).

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Eeva times 6, by Eric Horst via flickr

How to Conquer the Times Table, Part 2

Photo of Eeva times 6, by Eric Horst, via flickr.

The question is common on parenting forums:

My daughter is in 4th grade. She has been studying multiplication in school for nearly a year, but she still stumbles over the facts and counts on her fingers. How can I help her?

Many people resort to flashcards and worksheets in such situations, and computer games that flash the math facts are quite popular with parents. I recommend a different approach: Challenge your student to a joint experiment in mental math. Over the next two months, without flashcards or memory drill, how many math facts can the two of you learn together?

We will use the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.

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Memorizing the Math Facts

The most effective and powerful way I’ve found to commit math facts to memory is to try to understand why they’re true in as many ways as possible. It’s a very slow process, but the fact becomes permanently lodged, and I usually learn a lot of surrounding information as well that helps me use it more effectively.

Actually, a close friend of mine describes this same experience: he couldn’t learn his times tables in elementary school and used to think he was dumb. Meanwhile, he was forced to rely on actually thinking about number relationships and properties of operations in order to do his schoolwork. (E.g. I can’t remember 9×5, but I know 8×5 is half of 8×10, which is 80, so 8×5 must be 40, and 9×5 is one more 5, so 45. This is how he got through school.) Later, he figured out that all this hard work had actually given him a leg up because he understood numbers better than other folks. He majored in math in college and is now a cancer researcher who deals with a lot of statistics.

Ben Blum-Smith
Comment on Math Mama’s post What must be memorized?

The entire discussion (article and comments) is well worth reading:

You may also enjoy:


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ace of spades card deck

Game: Target Number (or 24)

[Photo by stevendepolo.]

Math concepts: addition, subtraction, multiplication, division, powers and roots, factorial, mental math, multi-step thinking
Number of players: any number
Equipment: deck of math cards, pencils and scratch paper, timer (optional)

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math problems for girls by woodleywonderworks

Mental Math: Addition

[Photo by woodleywonderworks.]

The question came from a homeschool forum, though I’ve reworded it to avoid plagiarism:

My student is just starting first grade, but I’ve been looking ahead and wondering: How will we do big addition problems without using pencil and paper? I think it must have something to do with number bonds. For instance, how would you solve a problem like 27 + 35 mentally?

The purpose of number bonds is that students will be comfortable taking numbers apart and putting them back together in their heads. As they learn to work with numbers this way, students grow in understanding — some call it “number sense” — and develop a confidence about math that I often find lacking in children who simply follow the steps of an algorithm.

["Algorithm" means a set of instructions for doing something, like a recipe. In this case, it means the standard, pencil and paper method for adding numbers: Write one number above the other, then start by adding the ones column and work towards the higher place values, carrying or "renaming" as needed.]

For the calculation you mention, I can think of three ways to take the numbers apart and put them back together. You can choose whichever method you like, or perhaps you might come up with another one yourself…

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photo by geishaboy500 via flickr

20+ Things to Do with a Hundred Chart

[Photo by geishaboy500.]

Are you looking for creative ways to help your children study math? Even without a workbook or teacher’s manual, your kids can learn a lot about numbers. Just spend an afternoon playing around with a hundred chart (also called a hundred board or hundred grid).

Here are a few ideas to get you started…

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jaycoxfilm-if-every-picture-tells-a-story

Math Game: What Number Am I?

Photo by jaycoxfilm.

Math concepts: mental calculations, math vocabulary, and anything else you want to include
Number of players: any number, but I think it works best with two players who alternate asking questions
Equipment: imagination and, if necessary, scratch paper

Many years ago, I read a magazine article by mathematical music critic Edward Rothstein, wherein he described a game he invented for his daughter:

  • “What number am I? If you add me to myself, you get four.”

Rather than explaining the rules of the game, let me tell you a story…

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mcgee-205

Diagnosis: Math Workbook Syndrome

Photo by otisarchives3.

I discovered a case of MWS (Math Workbook Syndrome) one afternoon, as I was playing Multiplication War with a pair of 4th grade boys. They did fine with the small numbers and knew many of the math facts by heart, but they consistently tried to count out the times-9 problems on their fingers. Most of the time, they lost track of what they were counting and gave wildly wrong answers.

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function-machine

The Function Machine Game

Math concepts: odd numbers, even numbers, greater-than/less-than, rounding off, addition, subtraction, multiplication, division, fractions, negative numbers, prime numbers, square numbers, problem solving, mental math
Number of players: two or more
Equipment: pencil (or pen) and paper for every player

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Math homework photo by MC Quinn.

How to Teach Math to a Struggling Student

photo by MC Quinn via flickr (CC BY 2.0)

Paraphrased from a homeschool math discussion forum:

Help! My daughter struggles with arithmetic. I guess she is like me: just not a math person. She is an outstanding reader. When we do word problems, she usually has no trouble. She’s a whiz at strategy games and beats her dad at chess every time. But numbers — yikes! When we play Yahtzee, she gets lost trying to add up her score. The simple basics of adding and subtracting confuse her.

Since I find math difficult myself, it’s hard for me to know what she needs. What’s missing to make it click for her? She used to think math was fun and tested well above grade level, but I listened to some well-meaning advice and totally changed the way we were schooling. I switched from using workbooks and games to using Saxon math, and she got extremely frustrated. Now she hates math.

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Caution children at play

Math Games by Kids

Photo by Mike Licht, NotionsCapital.com.

The cold came back and knocked me flat, but there are compensations. The downtime gave me a chance to browse my overflowing bookmarks folder, and I found something to add to my resource page. Princess Kitten and I enjoyed exploring these games and quizzes from Ambleweb.

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Euclid game

Euclid’s Game on a Hundred Chart

Math concepts: subtraction within 100, number patterns, mental math
Number of players: 2 or 3
Equipment: printed hundred chart (also called a hundred board), and highlighter or translucent disks to mark numbers — or use this online hundred chart

Set Up

Place the hundred chart and highlighter where all players can reach them.

How to Play

  • Allow the youngest player choice of moving first or second; in future games, allow the loser of the last game to choose.
  • The first player chooses a number from 1 to 100 and marks that square on the hundred chart.
  • The second player chooses and marks any other number.
  • On each succeeding turn, the player subtracts any two marked numbers to find and mark a difference that has not yet been taken.
  • Play alternates until no more numbers can be marked.

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Dirty numbers

Game: Hundred Chart Nim

Photo by Håkan Dahlström via flickr.

Math concepts: addition and subtraction within 100, logical strategy
Number of players: 2 or 3
Equipment: printed hundred chart (also called a hundred board) and beans, pennies, or other tokens with which to mark numbers — or use this online hundred chart

Set Up

Place the hundred chart and a small pile of tokens where both players can reach them.

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Percents: Key Concepts and Connections

[Rescued from my old blog.]

Paraphrased from a homeschool math discussion forum:

“I am really struggling with percents right now, and feel I am in way over my head!”

Percents are one of the math monsters, the toughest topics of elementary and junior high school arithmetic. Here are a few tips to help you understand and teach percents.

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negative number tree

Negative Numbers for Young Students

[Rescued from my old blog.]

Would you like to introduce your students to negative numbers before they study them in pre-algebra? With a whimsical number line, negative numbers are easy for children to understand.

Get a sheet of poster board, and paint a tree with roots — or a boat on the ocean, with water and fish below and bright sky above. Use big brushes and thick poster paint, so you are not tempted to put in too much detail. A thick, permanent marker works well to draw in your number line, with zero at ground (or sea) level and the negative numbers down below.

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Kids Do the Craziest Things

[Rescued from my old blog.]

My youngest daughter wanted to do Singapore math today. Miquon Red is her main math text this quarter, but we add a bit of Singapore Primary Math 1B whenever she’s in the mood. We turned to the lesson on subtracting with numbers in the 30-somethings. The first problem was pretty easy for her:

30 – 7 = []

I reminded her that she already knows 10 – 7. She agreed, “10 take away 7 is 3.” Then her eyes lit up. “So it’s 23! Because there are two tens left.”

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