# Natural Math Multiplication Course

This April, the creative people at Moebius Noodles are inviting parents, teachers, playgroup hosts, and math circle leaders to join an open online course about multiplication. My preschool-2nd grade homeschool math group is eager to start!

Each week there will be five activities to help kids learn multiplication by exploring patterns and structure, with adaptations for ages 2-12.

The course starts April 6 and runs for four weeks.

## Preliminary Syllabus

Week 1: Introduction.
What is multiplication? Hidden dangers and precursors of math difficulties. From open play to patterns: make your own math. 60 ways to stay creative in math. Our mathematical worries and dreams.

Week 2: Inspired by calculus.
Tree fractals. Substitution fractals. Multiplication towers. Doubling and halving games. Zoom and powers of the Universe.

Week 3: Inspired by algebra.
Factorization diagrams. Mirror books and snowflakes. Combination and chimeras. Spirolaterals and Waldorf stars: drafting by the numbers. MathLexicon.

Week 4: Times tables.
Coloring the monster table. Scavenger hunt: multiplication models and intrinsic facts. Cuisenaire, Montessori, and other arrays. The hidden and exotic patterns. Healthy memorizing.

Sounds like lots of fun!

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# Reblog: A Mathematical Trauma

Feature photo (above) by Jimmie via flickr.

My 8-year-old daughter’s first encounter with improper fractions was a bit more intense than she knew how to handle.

I hope you enjoy this “Throw-back Thursday” blast from the Let’s Play Math! blog archives:

Photo (right) by Old Shoe Woman via Flickr.

Nearing the end of Miquon Blue today, my youngest daughter encountered fractions greater than one. She collapsed on the floor of my bedroom in tears.

The worksheet started innocently enough:

$\frac{1}{2} \times 8=\left[ \quad \right]$

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# Multiplication Models Card Game

[Poster by Maria Droujkova of NaturalMath.com. This game was originally published as part of the Homeschooling with a Profound Understanding of Fundamental Mathematics Series.]

Homeschooling parents know that one of the biggest challenges for any middle-elementary math student is to master the multiplication facts. It can seem like an unending task to memorize so many facts and be able to pull them out of mental storage in any order on demand.

Too often, we are tempted to stress the rote aspect of such memory work, which makes our children lose their focus on what multiplication really means. Before practicing the times table facts, make sure your student gets plenty of practice recognizing and using the common models for multiplication.

To help your children see what multiplication looks like in real life, explore the multitude of Multiplication Models collected at the Natural Math website. Or try some of the hands-on activities in the Family Multiplication Study.

You may want to pick up this poster and use it for ideas as you play the Tell Me a (Math) Story game. Word problems are important for children learning any new topic in math, because they give children a mental “hook” on which to hang the abstract number concepts.

And for extra practice, you can play my free card game…

# Things To Do with a Hundred Chart #29

Here’s a new entry for my 20+ Things to Do with a Hundred Chart post:

(29) Blank 100 Grid Number Investigations: Challenge your students to deduce the secret behind each pattern of shaded squares. Then have them make up pattern puzzles of their own.
[Created by Stuart Kay. Free registration required to download pdf printable.]

## 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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# WOW – Multiplication! An Open Online Course for Parents and Teachers

Once again, the authors of Moebius Noodles are teaming up to offer a free course for families, math clubs, playgroups, and others who want to explore adventurous mathematics with kids of any age.

This short course will last only two weeks, and the topic is multiplication:

Both of my homeschool math circles (one with preschool-1st grade, and one with teens) thoroughly enjoyed the month-long problem solving course this summer, and we expect the new one to be just as much fun. Will you join us?

As with most of the Moebius Noodles courses, Maria and Yelena have adapted the activities for all ages from toddlers to adults. Where young ones go on a scavenger hunt for pretty snowflakes and cool truck wheels, older kids build bridges from multiplication to symmetry, spatial transformations, and proportions.

Visit the registration page to sign up no later than September 8. The main course activities will happen September 9th through 22nd. Expect to spend about two hours a week.

# Multiplying Negative Numbers with Rectangles

I love using rectangles as a model for multiplication. In this video, Mike & son offer a pithy demonstration of WHY a negative number times a negative number has to come out positive:

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# What Do You Notice? What Do You Wonder?

If you want your children to understand and enjoy math, you need to let them play around with beautiful things and encourage them to ask questions.

Here is a simple yet beautiful thing I stumbled across online today, which your children may enjoy:

It reminds me of string art designs, but the app makes it easy to vary the pattern and see what happens.

• What do your students notice about the patterns?
• What questions can they ask?

I liked the way the app uses “minutes” as the unit that describes the star you want the program to draw. That makes it easier (for me, at least) to notice and understand the patterns, since minutes are a more familiar and intuitive unit than degrees, let alone radians.

# 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.

# Trouble with Times Tables

[feature photo above by dsb nola via flickr.]

Food for thought:

Imagine that you wanted your children to learn the names of all their cousins, aunts and uncles. But you never actually let them meet or play with them. You just showed them pictures of them, and told them to memorize their names.

Each day you’d have them recite the names, over and over again. You’d say, “OK, this is a picture of your great-aunt Beatrice. Her husband was your great-uncle Earnie. They had three children, your uncles Harpo, Zeppo, and Gummo. Harpo married your aunt Leonie … yadda, yadda, yadda.

— Brian Foley
Times Tables – The Worst Way to Teach Multiplication

On the other hand, if you want your children to develop relationships with the numbers, to learn the math facts naturally, then be sure to tell lots of math stories. And when you are ready to focus on multiplication, be sure to study the patterns and relationships within the times tables.

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# PUFM 1.5 Multiplication, Part 1

Photo by Song_sing 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.

My apologies to those of you who dislike conflict. This week’s topic inevitably draws us into a simmering Internet controversy. Thinking my way through such disputes helps me to grow as a teacher, to re-think on a deeper level things I thought I understood. This is why I loved Liping Ma’s book when I first read it, and it’s why I thoroughly enjoyed Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

Multiplication of whole numbers is defined as repeated addition.

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

Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not… Adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.

— Keith Devlin
It Ain’t No Repeated Addition

# Multiplication Challenge

Can you explain why the multiplication method in the following video works? How about your upper-elementary or middle school students — can they explain it to you?

Pause the video at 4:30, before he gives the interpretation himself. After you have decided how you would explain it, hit “play” and listen to his explanation.

# 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.

# 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.

# 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).

# 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.

# How to Conquer the Times Table, Part 1

Photo of Evil Erin times 5 (and dog times 2), via flickr.

Perhaps the biggest challenge for any middle-elementary math student is to master the multiplication facts. It can seem like an unending task to memorize so many facts and be able to pull them out of mental storage in any order on demand. Too often, the rote aspect of such memory work overwhelms students, eclipsing their view of the principles behind the math. Yet rote memory is not enough: A student may be able to recite the times tables perfectly and still be reduced to counting on fingers in the middle of a long division problem.

We will use the world’s oldest interactive game — conversation — to learn the multiplication facts one bite at a time. But first, let’s take some time to think about what multiplication really means.

# Game: Times Tac Toe

Photo by jma.work via flickr.

I’ve been working on a book of math games for homeschoolers and other teachers. I hope to get it published later this year, but the editing drags on. Would anybody like to draw the illustrations?

Meanwhile, here’s a 2-player game your students may enjoy…

By Denise Gaskins Posted in Games

# Quotable: Times Tables Are Not Math

Important note: times tables are not math. Math doesn’t need to be made fun; it already is fun. Memorizing your times tables is a rote activity, it requires a fair bit of repetition for most, and it may need to be made fun. Just saying.

It’s a great game! Do click over to Dan’s blog and check it out:

And while we’re on the topic of times tables, Maria posted an article, too:

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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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# 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)

# Prime Numbers Are like Monkeys

[Photo by mape_s.]

I’m afraid that Math Club may have fallen victim to the economy, which is worse in our town than in the nation in general. Homeschooling families have tight budgets even in the best of times, and now they seem to be cutting back all non-essentials. I assumed that last semester’s students would return, but I should have asked for an RSVP.

Still, Kitten and I had a fun time together. We played four rounds of Tens Concentration, since I had spread out cards on the tables in the library meeting room before we realized that no one was coming. Had to pick up the cards one way or another, so we figured we might as well enjoy them! She won the first two rounds, which put her in a good mood for our lesson.

I had written “Prime numbers are like monkeys!” on the whiteboard, and Kitten asked me what that meant. That was all the encouragement I needed to launch into my planned lesson, despite the frustrating dearth of students. The idea is taken from Danica McKellar’s book Math Doesn’t Suck.

# Math Facts: 5 Minutes a Day

Y of x reminded me about one of my old favorite websites for math fact practice with a purpose:

5-10 minutes of daily practice will cement the math facts in your student’s mind, while at the same time doing a good deed. For each correct answer, a Free Rice sponsor donates a very small amount of rice to feed hungry people worldwide through the UN World Food Program.

Even very small amounts of rice add up. Since Free Rice started in 2007, its sponsors have bought more than 63 billion grains of rice, just by paying for one right answer click at a time.

You and your students can practice other topics as well:

# Cute Math Facts for Visual Thinkers

[Photo by angela7dreams.]

A forum friend posted about her daughter’s adventure in learning the math facts:

She loves stories and drawing, so I came up with the “Math Friends” book. She made a little book, and we talked about different numbers that are buddies.

# Contig Game: Master Your Math Facts

[Photo by Photo Mojo.]

Yahtzee and other board games provide a modicum of math fact practice. But for intensive, thought-provoking math drill, I can’t think of any game that would beat Contig.

Math concepts: addition, subtraction, multiplication, division, order of operations, mental math
Number of players: 2 – 4
Equipment: Contig game board, three 6-sided dice, pencil and scratch paper for keeping score, and bingo chips or wide-tip markers to mark game squares

# Hat Tip: Maria D.

[Photo by SouthbankSteve.]

One of my favorite places to lurk on the Internet is the Living Math Forum, and I especially enjoy reading the posts by MariaD. She led me (in a roundabout way) to the educational resources I posted yesterday. So, in keeping with my good intentions, I am adding one more listing to my blogroll page:

# 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…

# Christmas in July Math Problem

[Photo by Reenie-Just Reenie.]

In honor of my Google searchers, to demonstrate the power of bar diagrams to model ratio problems, and just because math is fun…

Eccentric Aunt Ethel leaves her Christmas tree up year ’round, but she changes the decorations for each passing season. This July, Ethel wanted a patriotic theme of flowers, ribbons, and colored lights.

When she stretched out her three light strings (100 lights each) to check the bulbs, she discovered that several were broken or burned-out. Of the lights that still worked, the ratio of red bulbs to white ones was 7:3. She had half as many good blue bulbs as red ones. But overall, she had to throw away one out of every 10 bulbs.

How many of each color light bulb did Ethel have?

Before reading further, pull out some scratch paper. How would you solve this problem? How would you teach it to a middle school student?

# What’s Wrong with “Repeated Addition”?

[Photo by Alejandra Mavroski.]

Myrtle called it The article that launched a thousand posts…, and counting comments on this and several other blogs, that may not be too much of an exaggeration. Yet the discussion feels incomplete — I have not been able to put into words all that I want to say. Thus, at the risk of once again revealing my mathematical ignorance, I am going to try another response to Keith Devlin’s multiplication articles.

Let me state up front that I speak as a teacher, not as a mathematician. I am not qualified, nor do I intend, to argue about the implications of Peano’s Axioms. My experience lies primarily in teaching K-10, from elementary arithmetic through basic algebra and geometry. I remember only snippets of my college math classes, back in the days when we worried more about nuclear winter than global warming.

I will start with a few things we can all agree on…