## Math Game: Place Value Fish

Math Concepts: addition, subtraction, place value to six or seven digits.
Players: two or more.
Equipment: pencil and paper.

### Set-Up

Each player needs a sheet of blank or lined paper, and a pencil.

At the top of your page, write a 6-digit number. All the digits must be different, and none of them can be zero.

### How to Play

On your turn, you go fishing for points. Ask one other player, “Give me your _____’s.” The blank is for the single-digit number of your choice.

The other player answers, “You get _____.” This blank is for the value of that digit in the other player’s number.

For example, suppose you asked for 5’s. If the other player has a 5 in the tens place of his number, you get 50 points. But if 5 was in the ten-thousands place, you would get 50,000. And if there is no 5 at all, you get zero.

You add those points to your number. The other player subtracts the points from his number.

Then it’s the next player’s turn to go fishing.

### Notice These Rules

Your number may change with each turn (except when you get zero). Always use your most recent number to add or subtract the fishing points.

If you have more than one of the digit asked for (like the player on the left above, who has two 7’s), you may choose which one to give away. That is, you can give the other player 70 points and not even mention the 7,000.

### Endgame

Keep taking turns until every player gets five chances to fish for points. After five rounds, whoever has the highest score wins the game.

UNLESS the winner made an arithmetic error.

Be sure to check each other’s math, because any player who makes a mistake automatically loses the game.

### Share the Fun

If you try this math game with your kids, I’d love to hear how it goes. Please drop a comment below.

And tell us about your favorite math game, so we can all play that, too. 😀

CREDITS: This game comes from Michael Schiro’s book Mega-Fun Math Games: 70 Quick-and-Easy Games to Build Math Skills. Feature photo (top) by Ruben Ortega via Unsplash.

## A Puzzle for Palindromes

If you haven’t seen the meme going around, this is a palindrome week because the dates (written American style and with the year shortened to ’19) are the same when reversed.

Here’s a math puzzle for palindrome week — or any time you want to play with math:

• Print a 100 chart.
• Choose a color code.
• Play!

What do you think: Will all numbers eventually turn into palindromes?

You can find all sorts of hundred charts on my Free Math Printable Files page.

Find out more about the Palindromic Number Conjecture in Mark Chubb’s article An Unsolved Problem your Students Should Attempt.

Or play with Manan Shah’s advanced palindromic number questions.

## Math Game: Six Hundred

Today I’m working on the next book in my Math You Can Play series, culling the games that don’t fit. Six Hundred is a fine game, but I can’t figure out how it landed in the prealgebra manuscript…

Math Concepts: addition, multiplication, parity (odd or even).
Players: any number.
Equipment: six regular 6-sided dice (my math club kids love this set), free printable score sheet, pen or pencil.

### Set-Up

A full game consists of eighteen rounds of play. Players may share the dice and score sheet, taking turns around the table. But for a large group you may want to have extras, so that two or more people can be rolling their dice at the same time.

### How to Play

On your turn, roll all six dice up to three times. After each roll, you may set aside one or more dice to keep for scoring, if you wish. Once a die has been set aside, you may not change your mind and roll it again.

After the third roll, choose an unused category on your score sheet. Count the dice according to the rules for that section, and write down your score. If your dice do not fit anywhere, then you must take a zero in the category of your choice.

When all players have filled their score sheet and recorded any appropriate bonuses (or penalties), whoever has the highest score wins.

### Scoring

Dice are scored in eighteen categories, in four sections, as follows. The maximum possible score is 600 points.

##### Numbers

Record the sum of only the dice showing that number. For example, if you rolled 1, 1, 3, 4, 4, 4, you could score 2 in the Ones category. Or you could score 12 in the Fours category, or zero in the Fives.

Bonus: If the combined Numbers score is 80 or more, add 35 points to your total.

##### Rungs (1–4)

Score the total of all six dice. Like a ladder, the score in each rung must be greater than the one before it. Rung 1 gets the lowest number, and Rung 4 the highest.

You may fill in the rungs in any order. But if you write 18 in Rung 2, then the score in Rung 1 must be 17 or less, and the score in Rung 3 must be at least 19.

Penalty: If the Rung scores don’t fit the ascending value rule, this category is worth zero.

##### Clusters

Score the total of all six dice, if they fit the rules for that category.

• Four of a Kind: at least four dice show the same number.
• Five of a Kind: at least five dice show the same number.
• Odds: all six dice show odd numbers.
• Evens: all six dice show even numbers.
##### Patterns

Score the amount shown for each pattern.

• Series: 30 points you roll 1, 2, 3, 4, 5, 6.
• Pairs: 30 points if you roll three pairs of matching numbers. Four dice showing the same number may be counted as two pairs.
• Triplets: 30 points if you roll two sets of three dice with the same numbers, such as three 2s and three 5s.
• Sextet: 36 points when all six dice show the same number.
##### Game Bonus

If you score at least one point in all eighteen categories, or if the only zero you take is for the sextet, then award yourself an additional 36 points.

### History

Players around the world have played poker-style dice games for ages. I grew up with Yahtzee, but you may know the game by Yatzy, Yacht, Generala, or another name.

Reiner Knizia included this mathematical version in his book Dice Games Properly Explained. And I found it online at Michael Ayers’s Stick Insect blog.

John Golden posted a simpler “Mathzee” game played with five dice on his Math Hombre blog — and while you’re there, be sure to check out his amazing Math Games page.

CREDITS: Feature photo (top) by rekre89 via Flickr (CC BY 2.0).

## Math Game: Number Train

Math Concepts: number symbols, numerical order, thinking ahead.
Players: two or more.
Equipment: one math deck of playing cards (remove face cards and jokers), or a double deck for more than four players; additional cards to use as train cars.

### Set-Up

Give each player four to six miscellaneous cards (such as the face cards and jokers you removed from the card deck) to serve as the cars of their number trains.

Lay these cards face down in a horizontal row, as shown. Shuffle the math card deck and spread it on the table as a fishing pond.

### How to Play

On your turn, draw one card and play it face up on one of your train cars. The numbers on your train must increase from left to right, but they do not need to be in consecutive order. If you do not have an appropriate blank place for your card, you have two choices:

• Mix the new card back into the fishing pond.

• Use the new number to replace one of your other cards, and then discard the old one.

The first player to complete a train of numbers that increases from left to right wins the game.

### Variations

House Rule: Decide how strict you will be about the “increases from left to right” rule and repeated numbers. Does “1, 3, 3, 7, 8” count as a valid number train? Or will the player have to keep trying for a card to replace one of the threes?

For older players: You can adapt Number Train to play with more advanced students:

This post is an excerpt from my book Counting & Number Bonds: Math Games for Early Learners, available now at your favorite online book dealer.

One of my favorite stores, Rainbow Resource Center, is offering several of my books at a great discount.

Check them out!

## Playing Complex Fractions with Your Kids

This week, I’m working on graphics for my upcoming book 70+ Things to Do with a Hundred Chart. I had fun with this complex fraction image.

It looks a bit cluttered. Possible tweak: Remove the brackets and instead use a thicker dividing line to show the thirds.

While I’m thinking about that, would you like a sneak peek at an activity from the book?

You don’t need a set of worksheets or lesson plans to learn math. All you need is an inquiring mind and something interesting to think about.

Play. Discuss. Notice. Wonder.

Enjoy.

Here’s how you can play complex fractions with your kids…

Print a few blank 120 charts and turn them sideways, so each chart has ten rows with twelve squares in each row.

Cut out the rows to make fraction strips with twelve squares on each strip.

Color a different set of squares on each strip. On some strips, arrange the colored squares all together at one end. On other strips, mix them around.

If we count each strip as one whole thing, what fraction of its squares are colored?

Match the strips that represent the same fraction.

On some of the strips, there will be more than one way to name the fraction. For example, if six squares are colored, we can call that 6/12 or 2/4 or 1/2 of the strip. These alternate names are easiest to see when the colored squares are all at one end of the strip, because you can fold the strip to show the halves or fourths.

How many different fraction names can you find for each set of colored squares?

### Look for Complex Fractions

We could also call the strip with six colored squares “1 1/2 thirds” of the whole strip. Can you show by folding why that name makes sense?

Or we could call the strip with five colored squares “2 1/2 sixths.”

When we have a fraction within a fraction like this, we call it a complex fraction, because it is more complicated than a common (or simple) fraction.

Another way to say it: Complex fractions have other fractions inside them.

A complex fraction is like a puzzle, challenging us to find its secret identity — the common fraction that names the same amount of stuff.

For example, how much is 3 1/3 fourths? One fourth would be three of the twelve squares on a fraction strip. So three fourths would be three sets of those three squares, or nine squares. Then we need to add one-third of the final fourth, which is one of the remaining three squares. So 3 1/3 fourths must be ten squares in all.

3 1/3 fourths = 10/12 = 5/6

How many complex fractions can you find in your set of fraction strips?

### Challenge Puzzles

Can you figure out how much a one-and-a-halfth would be?

That is one piece, of such a size that it takes one and one-half pieces to make a complete fraction strip.

A one-and-a-halfth is a very useful fraction and was a favorite of the ancient Egyptian scribes, who used it to solve all sorts of practical math problems.

How about a one-and-a-thirdth? How many of those pieces make a whole strip? What common fraction names the same amount of stuff?

Or how much would a two-thirdth be? In that case, it only takes two-thirds of a piece to make a complete strip. So the whole piece must be greater than one. A two-thirdth’s secret identity is a mixed number. Can you unmask it?

Make up some challenge fraction mysteries of your own.

### Update…

I’m still working on the graphics for my hundred chart book. Here’s the latest version of the complex fraction strips.

I like this one much better.

What do you think?

CREDITS: The slogan “Make Math Your Own” comes from Maria Droujkova, founder and director of the Natural Math website. Maria likes to say: “Make math your own, to make your own math!”

70+ Things to Do with a Hundred Chart is now available from Tabletop Academy Press.

## Math Journals for Elementary and Middle School

This fall, my homeschool co-op math class will play with math journaling.

But my earlier dot-grid notebooks were designed for adults. Too thick, too many pages. And the half-cm dot grid made lines too narrow for young writers.

So I created a new series of paperback dot-grid journals for my elementary and middle school students.

I hope you enjoy them, too!

## Math Journaling Prompts

So, what can your kids do with a math journal?

Here are a few ideas:

I’m sure we’ll use several of these activities in my homeschool co-op math class this fall.

### Noticing and Wondering

Learning math requires more than mastering number facts and memorizing rules. At its heart, math is a way of thinking.

So more than anything else, we need to teach our kids to think mathematically — to make sense of math problems and persevere in figuring them out.

Whenever your children need to learn a new idea in math, or whenever they get stuck on a tough homework problem, that’s a good time to step back and make sense of the math.

Kids can write their noticings and wonderings in the math journal. Or you can act as the scribe, writing down (without comment) everything child says.

For more tips on teaching students to brainstorm about math, check out these online resources from The Math Forum:

Problem-solving is a habit of mind that you and your children can learn and grow in. Help your kids practice slowing down and taking the time to fully understand a problem situation.

### Puzzles Are Math Experiments

Almost anything your child notices or wonders can lead to a math experiment.

For example, one day my daughter played an online math game…

A math journal can be like a science lab book. Not the pre-digested, fill-in-the-blank lab books that some curricula provide. But the real lab books that scientists write to keep track of their data, and what they’ve tried so far, and what went wrong, and what finally worked.

Here are a few open-ended math experiments you might try:

##### Explore Shapes
• Pick out a 3×3 set of dots. How many different shapes can you make by connecting those dots? Which shapes have symmetry? Which ones do you like the best?
• What if you make shapes on isometric grid paper? How many different ways can you connect those dots?
• Limit your investigation to a specific type of shape. How many different triangles can you make on a 3×3 set of dots? How many different quadrilaterals? What if you used a bigger set of dots?
##### Explore Angles

• On your grid paper, let one dot “hold hands” with two others. How many different angles can you make? Can you figure out their degree without measuring?
• Are there any angles you can’t make on your dot grid? If your paper extended forever, would there be any angles you couldn’t make?
• Does it make a difference whether you try the angle experiments on square or isometric grid paper?
##### Explore Squares
• How many different squares can you draw on your grid paper? (Don’t forget the squares that sit on a slant!) How can you be sure that they are perfectly square?
• Number the rows and columns of dots. Can you find a pattern in the corner positions for your squares? If someone drew a secret square, what’s the minimum information you would need to duplicate it?
• Does it make a difference whether you try the square experiments on square or isometric grid paper?

### Or Try Some Math Doodles

Create math art. Check out my math doodling collection on Pinterest and my Dot Grid Doodling blog post. Can you draw an impossible shape?

## How Would YOU Use a Math Journal?

I’d love to hear your favorite math explorations or journaling tips!

P.S.: Do you have a blog? If you’d like to feature a math journal review and giveaway, I’ll provide the prize. Send a message through my contact form or leave a comment below, and we’ll work out the details.

## FAQ: Struggling with Arithmetic

My son can’t stand long division or fractions. We had a lesson on geometry, and he enjoyed that — especially the 3-D shapes. If we can just get past the basics, then we’ll have time for the things he finds interesting. But one workbook page takes so long, and I’m sick of the drama. Should we keep pushing through?

Those upper-elementary arithmetic topics are important. Foundational concepts. Your son needs to master them.

Eventually.

But the daily slog through page after page of workbook arithmetic can wear anyone down.

Many children find it easier to focus on math when it’s built into a game.

Take a look at Colleen King’s Math Playground website. Or try one of the ideas on John Golden’s Math Hombre Games blog page.

Or sometimes a story helps, like my Cookie Factory Guide to Long Division.

### Math Textbook Tips

Games are great for practicing math your child has already learned. But for introducing new concepts, you’ll probably want to follow your textbook.

Still, even with textbook math, there are ways to make the journey less tedious:

• Most children do not need to do every problem on a workbook page, or every page in a section. There is a lot of extra review built into any math program.

• You don’t have to finish a section before you work whatever comes after it. Use sticky bookmarks to keep track of your position in two or three chapters at a time. Do a little bit of the mundane arithmetic practice, and then balance that with some of the more interesting topics your son enjoys.

• As much as possible, do math out loud with a whiteboard for scratch work. Somehow, working with colorful markers makes arithmetic more bearable.

• Set a timer for math, and make the time short enough that he feels the end is in sight. I suggest no more than thirty minutes a day for now. And whenever the timer rings, stop immediately — even if you are in the middle of a problem.

### The Timer Can Be a Life-Saver

Doing math in short sessions helped us avoid the emotional melt-downs my daughter used to have.

Thinking is hard work, and if I asked for too much, she would crash.

Because I sat with her and worked together every problem, I knew what she understood and when we could skip a problem. Or sometimes even jump several pages. Which meant that, even with short lessons, we still got through our book on time.

### Arithmetic Is Like Vegetables

But as I said before, textbooks include a whole lot of repetition.

Too much repetition deadens the brain.

So we also took long breaks from our textbook program. Entire school-year-long breaks, just playing with math. Letting “enrichment” activities be our whole curriculum.

As healthy as vegetables are, you would never limit your son to eating just lima beans and corn.

Similarly, be sure to feed him a varied math diet.

For example, you can follow his interest in geometry beyond the standard school topics.

Explore tessellations, Escher art, and impossible shapes such as the Penrose triangle.

Building Lego scenes is a practical application of 3-D geometry. He might even want to try stop motion animation.

Talk about how math works in real life. Ponder the choices on John Stevens’s “Would You Rather?” blog or try some of the challenges at Andrew Stadel’s Estimation 180 website. Many of these require three-dimensional reasoning.

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.

CREDITS: Frustrated Child photo by by Pixabay on Pexels.com. Penrose Lego by Erik Johansson via Flickr (CC BY 2.0). Homework Hands photo by Tamarcus Brown on Unsplash.

## FAQ: Forgetting What They Learned

“As we go through each lesson, it seems like my daughter has a good handle on the concepts, but when we get to the test she forgets everything. When I ask her about it, she shrugs and says, ‘I don’t know.’ What do you do when your child completely loses what she has learned?”

Forgetting is the human brain’s natural defense mechanism. It keeps us from being overwhelmed by the abundance of sensory data that bombards us each moment of every day.

Our children’s minds will never work like a computer that can store a program and recall it flawlessly months later.

Sometimes, for my children, a gentle reminder is enough to drag the forgotten concept back out of the dust-bunnies of memory.

Other times, I find that they answer “I don’t know” out of habit, because it’s easier than thinking about the question. And because they’d prefer to be doing something else.

And still other times, I find out they didn’t understand the topic as well as I thought they did when we went through it before.

No matter how we adults try to explain the concepts, some kids want to be answer-getters. They don’t want to do the hard work of thinking a concept through until it makes a connection in their minds. They want to memorize a few steps and crank through the lesson to get it over with.

In all these cases, what helps me the most is conversation.

My children and I always talk about our math. I ask questions like “What do you think? What do you remember? Can you explain the question to me? What are they asking for?”

And, whether the child’s answer is right or wrong, I practice my poker face. Trying not to give anything away, I ask, “How did you figure it out? Can you think of a way to confirm your answer?”

### Talking Math with Your Kids

If you have preschool and elementary-age kids, read Christopher Danielson’s inspiring book and blog:

For middle school and older students, check out Fawn Nguyen’s wonderful collection of Math Talks. Be sure to read the “Teachers” page for tips and talking points:

“You don’t need special skills to do this. If you can read with your kids, then you can talk math with them. You can support and encourage their developing mathematical minds.”

— Christopher Danielson

### Playful Ways to Learn or Review Math

Games are a great way to practice math. Check out these (free!) math games for all ages:

And if you have elementary-age children, here are a few grade-level tips to help them learn (and remember) math concepts:

CREDITS: Girl in field photo by SOURCE Hydration Systems and Sandals technology via Flickr. (CC BY 2.0) Nigerian classroom photo by Doug Linstedt and young girl studying by pan xiaozhen on Unsplash.

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.

I’ve been working on my next Playful Math Singles book, based on the popular Things to Do with a Hundred Chart post.

My hundred chart list began many years ago as seven ideas for playing with numbers. Over the years, it grew to its current 30+ activities.

Now, in preparing the new book, my list has become a monster. I’ve collected almost 70 ways to play with numbers, shapes, and logic from preschool to middle school. Just yesterday I added activities for fraction and decimal multiplication, and also tips for naming complex fractions. Wow!

Gonna have to edit that cover file…

In the “Advanced Patterns” chapter, I have a section on math debates. The point of a math debate isn’t that one answer is “right” while the other is “wrong.” You can choose either side of the question — the important thing is how well you support your argument.

Here’s activity #69 in the current book draft.

### Have a Math Debate: Adding Fractions

When you add fractions, you face a problem that most people never consider. Namely, you have to decide exactly what you are talking about.

For instance, what is one-tenth plus one-tenth?

Well, you might say that:

$\frac{1}{10}$  of one hundred chart
+ $\frac{1}{10}$  of the same chart
= $\frac{2}{10}$  of that hundred chart

But, you might also say that:

$\frac{1}{10}$  of one chart
+ $\frac{1}{10}$  of another chart
= $\frac{2}{20}$  of the pair of charts

That is, you started off counting on two independent charts. But when you put them together, you ended up with a double chart. Two hundred squares in all. Which made each row in the final set worth $\frac{1}{20}$  of the whole pair of charts.

So what happens if you see this question on a math test:

$\frac{1}{10}$  + $\frac{1}{10}$  = ?

If you write the answer “$\frac{2}{20}$”, you know the teacher will mark it wrong.

Is that fair? Why, or why not?

CREDITS: Feature photo (above) by Thor/geishaboy500 via Flickr (CC BY 2.0). “One is one … or is it?” video by Christopher Danielson via TED-Ed. This math debate was suggested by Marilyn Burns’s blog post Can 1/3 + 1/3 = 2/6? It seemed so!