## KenKen Classroom Puzzles Start Next Week

KenKen arithmetic puzzles build mental math skills, logical reasoning, persistence, and mathematical confidence. Puzzle sets are sent via email every Friday during the school year — absolutely free of charge.

What a great way to prepare your kids for success in math!

### How to Play

For easy printing, right-click to open the image above in a new tab.

Place the numbers from 1 to 6 into each row and column. None of the numbers may repeat in any row or column. Within the black “cages,” the numbers must add, subtract, multiply, or divide to give the answer shown.

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## FAQ: Trouble Finding the Right Math Program

“I can’t find a home school math program my son likes. We’ve tried Singapore Math, Right Start, Saxon, and Math Mammoth. We subscribed to a month of IXL Math to keep him in practice, but he hates that, too. I know I shouldn’t have changed so many times, but this was our first year of homeschooling, and I was trying to please him. But I’m running out of things to try. Do you think Life of Fred might work?”

You’ve tried all those math programs in one year? Many people recommend that new homeschoolers take a few months off to “detox” from the classroom setting, to relax and enjoy the freedom of making their own choices. But your son might want a few months to detox from his homeschool experience.

I suggest you set aside all those books and focus on games and informal math. Try to avoid schoolish lessons until your son starts to enjoy learning for its own sake. The Internet offers an abundance of creative math ideas.

• For example, download the Wuzzit Trouble or DragonBox apps to play with, but don’t make it a homework assignment.
• Or let him choose one of the activities at Gordon Hamilton’s Math Pickle website and explore it for a day or a week or as long as it remains interesting.
• Browse through the Primary Level 1 or Level 2 puzzles and games at the Nrich Mathematics website for more ideas.

Look for more playful math on my blog’s resource pages:

### Explore Big Concepts: Infinity

Math that captures a child’s imagination can make the more tedious work seem bearable. For instance, in the 1920s, mathematician David Hilbert created a story about an imaginary grand hotel with an infinite number of rooms.

### Explore Big Concepts: Fractals

Take a mental trip to infinity by playing with fractals. Cynthia Lanius’s online Fractals Unit for Elementary and Middle School Students offers a child-friendly starting point.

Fractals are self-similar, which means that subsections of the object look like smaller versions of the whole thing.

Most children enjoy exploring the concept of infinity with hands-on fractal patterns, such as this Sierpinski triangle made of tortilla chips. Talk about what you notice and wonder: How does the triangle grow? How many chips will we need for the next stage?

### The Daily Four

If you worry that your son needs to keep practicing traditional arithmetic during his break, try making him a series of Daily Four pages:

• Fold a sheet of plain paper in half both ways, making four quarter sections.
• Write one math problem in each part. Choose them from any of your math books.
• Make sure each problem is different — one addition, one fractions, one multiplication, or whatever — and that none of them are hard enough to cause frustration.
• Don’t worry about an answer sheet. Show him how to use a calculator to check his work.

You can make up a whole week’s worth of these problem sheets at once, with a balanced mix of problems for each day. Your son won’t feel overwhelmed, but you’ll know he’s reviewing his number skills.

Or download some of the Corbettmaths 5-a-Day practice sheets for him. Some problems may seem too easy while others require concepts he hasn’t studied yet. Easy review won’t hurt anything, but do let him skip the problems that feel too hard.

Credits: “Rock Surfer Boy” by Ken Bosma and “Boy” by Isengardt via Flickr. (CC BY 2.0) Hotel Infinity video by Tova Brown.

This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

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## Review: Math & Magic in Wonderland

Are you looking for a fun book to read over the summer? I just finished Lilac Mohr’s delightful Math & Magic in Wonderland, and I loved it.

Highly recommended, for kids or adults!

### About the Book

A Jubjub bird disguised as a lark,
Borogroves concealing a snark,
When you’re in Tulgey Wood, you must
Be careful whom it is you trust…

With the discovery of Mrs. Magpie’s Manual of Magic for Mathematical Minds, Lulu and Elizabeth embark on an exciting journey to a realm inspired by Lewis Carroll’s poetry. The twins must use ingenuity and sagacity to solve classic logic puzzles that promise to uncover the book’s secrets and earn them The Vorpal Blade. In this interactive novel, the reader is invited to play along with the two heroines on their grand mathematical adventure.

Do you have the smarts to help Lulu and Elizabeth outwit the frumious Bandersnatch?

It’s time to enter Wonderland and find out!

–from the back cover of Math & Magic in Wonderland by Lilac Mohr

### What I Liked

Puns, poetry, and plenty of puzzles. Tangrams, tessellations, truth-tellers and liars. History tidbits and many classics of recreational mathematics.

The sisters Lulu and Elizabeth seem real — though perhaps more widely read than most of us. They are different from each other. They make mistakes and have disagreements. But they never deteriorate into the cliché of sibling rivalry that passes for characterization in too many children’s books.

In each chapter, the girls must solve a language, math, or logic puzzle to proceed along their journey. Then a “Play Along” section offers related puzzles for the reader to try.

No matter how challenging the topic, the book never talks down to the reader.

### What I Didn’t Like

… Um … Honestly, I can’t think of anything.

Since it’s traditional to criticize the editing of self-published books, I will say this: There was at least one place where the wording seemed a bit awkward. I would have phrased the sentence differently. But don’t ask me to identify the page — I was too caught up in the story to bother jotting down such a quibble. And I tried flipping through the book as I wrote this post, but I can’t find it again.

Unless you hate logic puzzles and despise Lewis Carroll’s poetry.

But for everyone else, this book is truly a gem. If you like The Cat in Numberland or The Man Who Counted, then I’m sure you’ll enjoy Math & Magic in Wonderland.

Disclaimer: Like almost all book links on my blog, the links in this post take you to Amazon.com, where you can read descriptions and reviews. I make a few cent’s worth of affiliate commission if you make a purchase — but nowhere near enough to influence my opinion about the book.

### And Now for the Giveaway

Lilac offered a paperback copy of Math & Magic in Wonderland for one lucky reader of Let’s Play Math blog.

The giveaway is done. Congratulations, Keshua!

But the comments section below remains open, and I’d still love to hear your answers:

• Tell us about your favorite language, math, or logic puzzle book! Or share a book you’ve been wanting to read.

## FAQ: He Won’t Stop Finger-Counting

“My oldest son has somehow developed the horrid habit of counting on his fingers. We worked on the math facts all summer. He knows the answers in simple form, such as 9 + 4, but if it’s in a bigger problem like 249 + 54, he counts up to add or counts down to subtract, all using fingers. My younger children have no problem with mental math, but he can’t seem to get it. Are there any tips or tricks to stop this?”

Counting on fingers is not a horrid habit, it is a crutch. Please think for a moment about the purpose of crutches. The blasted things are an uncomfortable nuisance, but there are times when you can’t get anywhere without them. And if you need them, it does you no good for a friend to insist you should crawl along on your own.

That is how your son feels right now about his fingers. He is struggling with something his younger siblings find easy, and he can tell that you are frustrated. His confidence is broken, in a cast, and needs time for healing. So he falls back on what he knows he can do, counting up the answer.

Think positive: this means he still believes that math ought to make sense — that to understand what he is doing is more important than to guess at an answer. You want him to value sense-making, because otherwise he will try to memorize his way through middle school and high school math. That is the road to disaster.

“Schools spend a lot of time working with young children to get these facts memorized, but many children aren’t ready for that task yet. They’ll count on their fingers, and may be reprimanded for it.
“What happens when a person becomes embarrassed about counting on their fingers? If they still want to think, they’ll hide it. That’s the better option. The worse option that way too many students choose? They start guessing. When math becomes too incomprehensible, or not living up to someone else’s expectations becomes too painful, many students give up on math, and then they just guess.
“We count on our fingers as part of a thinking process. Perhaps the thing I want to figure can be memorized. But if I haven’t memorized it yet myself, the most efficient way to figure it will likely involve fingers.

—Sue VanHattum
Philosophy

### The Problem of Transfer

What you describe is called the problem of transfer, and it is one of the huge, unsolved problems of education.

We can train someone to do a simple, limited task such as answering flash cards. But how do we get that knowledge to sink in, to become part of the mind, so they can use it in all sorts of different situations?

No one has figured that out.

There is no easy solution. It requires patience, and providing a variety of experiences, and patience, and pointing out connections, and asking the student to think of connections, and lots more patience.

### Some Things to Try

It might help to do fewer math problems in a day, so you can take time to work more deeply on each one. Talk together about the different ways you might solve it. Make it a challenge: “Can we think of three different ways to do it?”

In math, there is never just one way to get a solution. Thinking about alternatives will help your son develop that transfer of skills.

Or pick up some workbooks that target mental math methods. The Mental Math workbook series by Jack Hope and Barbara and Robert Reys will help him master the techniques your younger kids learned without effort. It may still take him longer to do a calculation than what you are used to with the other children, but these books will give him a boost in recognizing the types of mental tools he can use.

Here are a few of my previous blog posts that include mental math tips:

Or perhaps encourage him to keep using his fingers, but to switch to a more efficient system, such as Chisenbop. According to math education expert Jo Boaler, research shows that finger-counting supports mathematical understanding.

### Mental Math: A Battle Worth Fighting

Jumping into mental math is hard for an older child who wasn’t taught that way. I believe it’s a battle worth fighting, because those mental math techniques build understanding of the fundamental properties of numbers.

But the main goal is for him to recognize his options and build flexibility, not to do each calculation as fast as possible.

And be sure he no longer needs those crutches before you try to take them away.

Photo credits: “Stryde Walking To School on his New Crutches” by Jim Larrison and “Silhouette of a boy” by TimOve via Flickr. (CC BY 2.0)

This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, as are many of the articles in my Let’s Play Math FAQ series.

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## Multiplication Is Not Repeated Addition: Update

[Photo “Micah and Multiplication” by notnef via Flickr (CC-BY 2.0).]

Some Internet topics are evergreen. I noticed that my old Multiplication Is Not Repeated Addition post has been getting new traffic lately, so I read through the article again. And realized that, even after all those words, I still had more to say.

So I added the following update to clarify what seemed to me the most important point.

I’d love to hear your thoughts! The comment section is open down below . . .

## Language Does Matter

Addition: addend + addend = sum. The addends are interchangeable. This is represented by the fact that they have the same name.

Multiplication: multiplier × multiplicand = product. The multiplier and multiplicand have different names, even though many of us have trouble remembering which is which.

• multiplier = “how many or how much”
• multiplicand = the size of the “unit” or “group”

Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

## A New Type of Number

In multiplication, we introduce a totally new type of number: the multiplicand. A strange, new concept sits at the heart of multiplication, something students have never seen before.

The multiplicand is a this-per-that ratio.

A ratio is a not a counting number, but something new, much more abstract than anything the students have seen up to this point.

A ratio is a relationship number.

In addition and subtraction, numbers count how much stuff you have. If you get more stuff, the numbers get bigger. If you lose some of the stuff, the numbers get smaller. Numbers measure the amount of cookies, horses, dollars, gasoline, or whatever.

The multiplicand doesn’t count the number of dollars or measure the volume of gasoline. It tells the relationship between them, the dollars per gallon, which stays the same whether you buy a lot or a little.

By telling our students that “multiplication is repeated addition,” we dismiss the importance of the multiplicand. But until our students wrestle with and come to understand the concept of ratio, they can never understand multiplication.

## For Further Investigation

If you’re interested in digging deeper into how children learn addition and multiplication, I highly recommend Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

To learn about modeling multiplication problems with bar diagrams, check out the Mad Scientist’s Ray Gun model of multiplication:

And here is an example of the multiplication bar diagram in action:

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## New Hundred Chart Game: Odd-Even-Prime Race

[Photo by geishaboy500 (CC BY 2.0).]

Counting all the fractional variations, my massive blog post 30+ Things to Do with a Hundred Chart now offers nearly forty ideas for playing around with numbers from preschool to prealgebra.

Here is the newest entry, a variation on #10, the “Race to 100” game:

(11.5) Play “Odd-‌Even-‌Prime Race.″ Roll two dice. If your token is starting on an odd number, move that many spaces forward. From an even number (except 2), move backward — but never lower than the first square. If you are starting on a prime number (including 2), you may choose to either add or multiply the dice and move that many spaces forward. The first person to reach or pass 100 wins the game.
[Hat tip: Ali Adams in a comment on another post.]

And here’s a question for your students:

• If you’re sitting on a prime number, wouldn’t you always want to multiply the dice to move farther up the board? Doesn’t multiplying always make the number bigger?

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## Noticing Fractions in a Sidewalk

My daughters didn’t want to admit to knowing me, when I stopped to take a picture of the sidewalk along a back street during our trip to Jeju. But aren’t those some wonderful fractions?

What do you see? What do you wonder?

Here is one of the relationships I noticed in the outer ring:

$\frac{4 \frac {2}{2}}{20} = \frac {1}{4}$

And this one’s a little trickier:

$\frac{1 \frac {1}{2}}{12} = \frac {1}{8}$

Can you find it in the picture?

Each square of the sidewalk is made from four smaller tiles, about 25 cm square, cut from lava rock. Some of the sidewalk tiles are cut from mostly-smooth rock, some bubbly, and some half-n-half.

I wonder how far we could go before we had to repeat a circle pattern?