Writing to Learn Math: Writing helps students stretch their thinking and make sense of new ideas.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Writing Prompt

Writing helps students stretch their thinking and make sense of new ideas.

When students wrestle their thoughts into shape and create explanations, they do the same sort of work that mathematicians do every day. It’s difficult for children (or anyone) to capture a thought and cage it in words. But it’s great practice for life.

Students may supplement their writing with illustrations. Sketch drawings can be a wonderful aid to mathematical thinking.

For the poetry prompts, students should aim for evocative descriptions, vivid verbs, and playful words. If your child can’t think of where to start a poem, try brainstorming a list of sensory details.

You may reuse writing prompts as often as you like. Change the question, if you wish — but even when the prompt remains the same, the students have changed since the last time they wrote about it. Today is a new day, so they are seeing with fresh eyes and thinking different thoughts.

Journaling Prompt 277: Words Help Us Think

Make a math “word wall” on your journaling page. Write all the math vocabulary words you know.

Decorate them with frames, or write them crossword-style or in different directions. Or make it fancy any way you like.

Optional: Include definitions or pictures explaining the hardest-to-remember words.

It’s a great topic because the calculations are relatively easy, mostly multiplication and division. But there’s a lot of thinking that goes into figuring out just what calculations we need to do.

Counting is not only for little kids! It’s part of discrete mathematics, which means the study of things that are not continuous—things that come in discrete chunks—and it’s important in game design, computer programing, and cryptography.

This post is a section of my Prealgebra & Geometry Games book where my daughter and I explore the fundamental counting principle, the key to solving many counting and probability puzzles.

The Science of Counting

We spent a bit of time counting basic numbers and counting items in sets. Then we moved to the more interesting type of mathematical counting. Combinatorics is the art of counting possibilities: all the choices we can make, the things we may do, the ways we might combine stuff.

For example, imagine you are choosing your avatar for a new online game. You need to pick an outfit, and you have the following options:

• Leggings: brown or black, tucked into high leather boots.
• Top: green, purple, or blue.
• Cloak or no cloak.
• Head piece: hat, scarf, eye patch, or none.

Focus on the first two choices, the leggings and top. Imagine these choices as paths through a mental landscape, branching off each other to form a tree diagram. Your first option creates two paths: brown leggings, or black leggings. Each of these paths leads to a new question: Which top? At that point, you must choose between three branching paths.

The total number of branches in a tree diagram is the product of the choices at each branch point. In this case, 2 × 3 = 6 possible combinations.

The Fundamental Counting Principle

As we add more choices, the paths continue to branch out. Our tree diagram grows unwieldy, and we desperately need a new counting tool.

The fundamental counting principle comes to our rescue. We can count the possibilities of a situation by multiplying our options.

If you have A options for your first choice, B options for your second choice, C options for your third choice, and so on, then you can find the total number of potential outcomes by multiplying:

A × B × C ×…

To choose our game avatar, we have to make four choices—leggings, top, cloak (or not), and head piece—with two, three, two, and four options respectively. So we calculate the total number of possible outfits:

2 × 3 × 2 × 4 = 48

For More Information

We spent a whole year in middle school playing with the science of counting. Here are two of my favorite resources:

• Keone Hon’s lesson on counting and probability
• Jason Batterson’s puzzle sets

Batterson later published his puzzles in the book Competition Math for Middle School (affiliate link), which includes solutions.

This game builds understanding of decimals, and also of the inverse relationship between multiplication and division.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Decimal Product Game

Math Concepts: multiplying decimals, decimal place value.

Players: two players or two teams.

Equipment: printed game board, colored markers or a set of matching tokens for each player, two paperclips or other small tokens to mark the factors.

Set-Up

The free 44-page PDF Multiplication & Fraction Printables file features two decks of mathematical model playing cards, plus hundred charts and all the game boards for the Math You Can Play: Multiplication & Fractions book.

How to Play

The first player places a paperclip on any one of the factors at the bottom of the board. The second player places the other paperclip on a factor — the same or different — and then marks the product of those two numbers by coloring the square or placing a token.

On each succeeding turn, a player moves just one paperclip to a new number and then marks the product of those two factors. If both players agree that all possible moves have already been colored in, the player whose turn it is may make a fresh start by moving both paperclips.

Whichever player marks four (or more) squares in a row — horizontally, vertically, or diagonally — wins the game. The squares must touch each other at edges or corners, with no gaps. If neither player can make four in a row, then the player who has the most sets of three in a row wins.

History

This variation on The Product Game comes from John Golden’s blog. He writes about his conversation with fifth-grade students:

“We discussed how to get hundredths. I think they knew on some level it was tenths times tenths, but had a bit of the ‘we haven’t been taught this yet’ syndrome. I used the analogy of a dime being a tenth of a dollar, so what’s a tenth of a dime?

“‘A penny!’

“And what part of a dollar is a penny? How many does it take to make one dollar? It is terrific that they are used to seeing one cent written as .01.”

Writing to Learn Math: At its heart, geometry is all about seeing connections and relationships.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Geometry Prompt

At its heart, geometry is all about seeing connections and relationships. How can students break shapes apart, put them together, move them around the page, turn them, or distort them? Which properties change, and which stay the same?

Every activity has the potential to spawn hundreds of variations. Alter something in the prompt to make a fresh investigation. Tweak the size, shape, or other properties of interest. What new things can your children see in the math? What questions can they ask?

For older students, use algebra to put some teeth in the relationships they see. Give the points names. Identify the line segments. Can your students write any equations about them? Which distances are equal to other distances, or areas equal to other areas? How can they know for sure? When they add new points, lines, or circles to the diagram, what new connections do they find?

Journaling Prompt 217: Pentagram

Draw a pentagon (five-sided shape). Draw diagonal lines inside the pentagon so that each corner is connected with a line to every other corner.

The diagonals make a shape called a pentagram.

What do you notice? What patterns do you see?

Where might you add new lines to the shape?

How many questions can you ask about this shape? Can you figure out the answers to any of your questions?

Hooray!

If you’d like to know more about the book, check out my interview with Cindy Rollins on The New Mason Jar podcast:

• S11E144: Charlotte Mason’s Living Math with Denise Gaskins

…and here’s the book description:

Apply the Principles of Education to Help Children Develop Mathematical Reasoning

“The Principality of Mathematics is a mountainous land, but the air is very fine and health-giving. People who seek their work or play in this principality find themselves braced by effort and satisfied with truth.”

—Charlotte Mason

Are you a parent or teacher? Here’s how to:

• Feed your children’s minds with big ideas.
• Give them confidence they can learn and make sense of things.
• Develop intuition and creative thinking about math.

Explore the world of mathematics with your children. Notice connections. Wonder about relationships. Delight in understanding.

Author Denise Gaskins shows you how to introduce your child to the living ideas of mathematics. Because when the ideas of math come to life for our children, their minds find joy in seeing numbers and shapes connect to each other and exploring these relations.

From early learning to algebra, Charlotte Mason’s Living Math offers a practical, activity-filled exploration of what it means to learn math as a family.

Chapters include:

• Practice Your Principles: Discover the fundamental principles of math education, the essentials of understanding who children are and how they learn.
• How Shall We Teach?: Get practical tips for guiding your kids, drawing out their curiosity with puzzles, and helping them master math topics.
• When Students Get Stuck: Help children find their own way around any barrier with a classic problem-solving approach they can use throughout their lives.
• Supplement Your Lessons: Find out how books, games, and math art contribute to a living education.
• And much more.

You’ll love how Charlotte Mason’s Living Math transforms your children’s experience of math, awakening their imagination about numbers, shapes, and patterns. Because the Principality of Mathematics is a wonderful place to explore with your kids!

So grab your copy of Charlotte Mason’s Living Math today, and launch your family’s adventure of learning.

But if you’re impatient—or if you want the exclusive, deluxe, full-color paperback or hardcover edition—you can order your copy now on my Playful Math Store.

This section comes from the chapter called “How Shall We Teach?” which also includes tips about textbooks, a little-known truth about the math facts, and what to do if you never understood math yourself.

Moving Toward Mastery

With any mathematical idea, our children need time to build knowledge. As we recognize their progress, we can shape our plans to better fit the next step in their development.

Here are four stages in growth from first contact to confidence:

• Exploration: become aware of the new idea, gathering experience by trial and error, playing with the concept, discovering how much it can do, searching out its limits.

• Translation: describe the results of their exploration, learning to express the relations they discover in some type of mathematical language (words, pictures, or symbolic notation).

• Consolidation: develop confidence in moving between experience and symbolic expressions, organizing their discoveries, asking new questions, and possibly launching their own related investigations.

• Mastery: use their knowledge fluently in a variety of situations, making it a foundation to support new concepts.

We celebrate when our children can translate a mathematical situation into symbols and calculate answers, even while recognizing they have not yet reached full understanding. As learning to identify a new bird or plant doesn’t confer expertise in biology, so there is much more to math than writing symbols.

“Our own school experiences can make it hard for us to teach without being tempted to ‘help them master’ a concept that they may or may not be ready to master,” according to living math expert Julie Brennan. “What we never learned in school was the concept of playing around with math, allowing ideas to ‘percolate,’ so to speak, before mastery occurs. And that process may take time.”

So how do we know when our children understand the math they’ve studied? When they can use it flexibly and creatively, as Mason explains:

This game builds calculations skills and helps develop fluency.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Bowling

Math Concepts: addition, subtraction, multistep calculation, column addition (for final score).

Players: any number.

Equipment: one deck of math cards, game board, ten tokens for covering numbers, pencil or pen, and (optional) scratch paper for adding up the scores.

Set-Up

You can print the bowling game board and score sheet from the Number Game Printables Pack, or draw your own game board and use the printable score sheets from the Apollo’s Templates website.

The 23-page printable (pdf) Number Game Printables Pack file includes hundred charts, graph paper, and game boards from the first two Math You Can Play books.

How to Play

Place the deck face down where all players can reach. When it’s your turn, roll your bowling ball by flipping up three cards. On your game board, cover the pins you knock down:

• The numbers on your cards.
• Any numbers you can make by adding or subtracting two cards.
• Any numbers you can make by adding and subtracting with all three of the cards. Each card may be used only once in each calculation.

For example, if you roll a two, four, and ace, you can cover those numbers and also:

• 3 = 4 − 1
• 7 = 1 + 2 + 4
• What else can you find?

If you cover all the numbers, you rolled a strike, and that’s the end of your turn. If you don’t cover all the numbers, discard those cards and turn up three more for your next ball. Can you cover the remaining numbers now? If so, that’s a spare.

Your score for each turn is how many numbers you cover, with special rules for strikes and spares as explained below.

After you roll two balls, the next player takes a turn. Keep going until each player has ten turns. If the draw pile runs out of cards, reshuffle the discards to replenish it.

How to Use the Score Sheet

The rules for keeping score in bowling seem confusing until you’ve played a few games. Each larger square box on the score sheet is called a frame and represents your score for one turn. The two (or three) smaller square boxes are for keeping track of each time you roll your ball.

One complete game of bowling consists of ten turns, with a maximum score of 300 points. You might think that with ten numbers to cover in each turn, the most you could score would be 100 points, but strikes and spares get bonuses.

For each turn, write down how many numbers you covered with your first ball in the first small box, or if you got a strike, mark an X there. If you threw a second ball, mark down how many more numbers you covered in the second small box, or if you got a spare, write a slash (/). If you did not get a strike or spare, add the numbers you wrote down and put that total score at the bottom of the frame.

A spare earns one bonus ball, and a strike earns two, which means you can’t finish scoring your frame yet. On your next turn, the balls you throw will count not only for their own frame but also as bonus points on your strike or spare. After you’ve thrown your next ball (or two balls, for a strike), add ten for the strike or spare plus the value of the bonus(es) and write in that total at the bottom of the unfinished frame.

Notice that the bonus balls get counted twice, once as part of the strike or spare and once for their own frame. And if you throw three strikes in a row, the third one will be counted three times:

• As the second bonus throw in the first strike frame.
• As the first bonus throw on the second strike.
• And finally as part of its own frame, along with its own bonus balls.

When you get to the last frame, you will see three small boxes. That’s because any players lucky enough to throw a strike or a spare still get their bonus balls. If you don’t get a strike or spare, ignore that third little box.

Finally, add up all the points from all ten frames to make your total score for the game.

Writing to Learn Math: What did the author mean? Put the thought in your own words. Do you agree or disagree? Why?

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Quotation Prompt

Let students choose how they want to react to the quotation. Or offer one of the following questions:

• What did the author mean? Put the thought in your own words.
• Do you agree or disagree? If you agree, can you think of someone who would disagree? Why?
• Is this quote a general principle, or only for specific situations? Describe a time when it might apply, or when it might not.
• Tell a time in your life when you lived up to the quotation — or when you wish you had.
• How does the quote relate to math, science, history, or another subject?

Short exercises are great writing practice. But occasionally you’ll want to assign deeper essay topics, such as:

• Look up the author’s name online. Who are/were they, and why do people care what they said?
• Quotes are often misattributed. Did the author really say this?
• What have others said about the same topic? Search out a variety of quotes related to this one. How are they similar? How are they different?
• Does thinking about the quotation make you want to change anything, in yourself or in the world? How could you put that idea into action?

Quotation from W.S. Anglin

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.

—W.S. Anglin

Whenever I need a diversion from the pressing day-to-day busyness of life, I like to crack open one of his books. And get humbled. He stumps me as often as not, but I enjoy the challenge.

Today, I’ve got two of his somewhat-easier puzzles from Amusements in Mathematics, dealing with the probability of obsolete coins tossed or pulled from a bag.

Okay, so the penny isn’t obsolete quite yet, even though the US government quit making them. But the “penny” in these puzzles is the old British pence, which went out of date seemingly ages ago, along with all the other non-decimal coins that made old math puzzles so confusing.

Probability puzzles are great for middle school students because the arithmetic itself is simple. The challenge is in figuring out exactly what calculation to do.

Have fun playing math with your kids!

Things to Know

If you haven’t played with probability before, here are a few things you need to know…

How to Count

In the study of probability, we are not counting existing things, but possible things. We can’t know for sure which result will happen, but we can count the results that might happen.

Sometimes, the situation is simple enough that you can make a list or chart of all the possibilities. When that gets too complicated, the fundamental counting principle comes to your rescue.

But you won’t need the fundamental counting principle for these puzzles, so I’ll explain more about it next week.

Probability

We calculate probability as a ratio, a fraction that compares the results we are interested in to all the possible things that might happen:

Probability = Winners/Possibilities

So you need to count two things. First, count all the possible results of your situation. Then, count the results that “win” whatever challenge you are studying.

Odds

The odds of winning your challenge situation are not the same as the probability of winning. Odds are much more confusing because the word is used so many ways.

Odds for, or odds on, is the ratio of the winning possibilities to the losers.

Odds on = Winners/Losers

Another common usage is odds against, or the ratio of losing possibilities to winners.

Odds against = Losers/Winners

Expected Value

In a game involving money or points, the expected value is the average amount you would win per turn in an infinite number of random plays.

We calculate expected value by finding the value of every possible outcome multiplied by the probability that result will occur, and then adding all those amounts together.

In Dudeney’s puzzle, the expected value is the amount you should be willing to pay to play the game. In reality, the man offering you the game will charge more than that, because the one inviolable rule of gambling is that the house always wins. Over the long run, it’s impossible to earn back the amount you pay to play.

Dudeney’s Two Questions in Probability

Dudeney writes…

There is perhaps no class of puzzle over which people so frequently blunder as that which involves what is called the theory of probabilities. I will give two simple examples of the sort of puzzle I mean. They are really quite easy, and yet many persons are tripped up by them.

A friend recently produced five pennies and said to me:

“In throwing these five pennies at the same time, what are the chances that at least four of the coins will turn up either all heads or all tails?”

His own solution was quite wrong, but the correct answer ought not to be hard to discover.

Another person got a wrong answer to the following little puzzle which I heard him propound:

“A man placed three sovereigns and one shilling in a bag. How much should be paid for the permission to draw one coin from it?”

It is, of course, understood that you are as likely to draw any one of the four coins as another.

My Comments

The sovereigns were each worth 20 shillings, each of which was worth 12 pence.

I question Dudeney’s last statement about the coins. Don’t mints normally try to make coins in different sizes and with smooth or ridged edges, so you can identify what’s in your pocket by feel?

Anyway, I prefer to think in terms of dollars, so you can imagine that the bag contains four bills: three $20’s and a single dollar. Perhaps folded into rings or shirts.

Not Quite Solutions

I don’t want to ruin your fun by giving away the answers. If you found an answer, but you’re not sure whether it’s right, you can check for yourself at the Internet Archive (answer 30).

But I will warn you of the most common errors.

For the penny problem, were you careful to consider all the possibilities? There are only six possible combinations of heads and tails, but most of those combinations can be made more than one way. So there are actually more than thirty ways the coins might land.

For the money-in-a-bag challenge, did you remember to take the shilling (or dollar bill) into account, as well as the more valuable coins? According to Dudeney, most people forget that every draw results in a win of some amount, even if it feels like a loss because of the small value.

This game pushes students to consider area and perimeter beyond the basic shapes.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Area Battle

Math Concepts: area, square units, perimeter, regular and irregular polygons.

Players: two to four.

Equipment: homemade game cards, pencils or markers.

The FREE 68-page printable (pdf) Prealgebra & Geometry Printables file includes Area Battle card templates, along with all the game boards for the Math You Can Play: Prealgebra & Geometry book.

Set-Up

Students cut square grid paper into 5 × 5 squares. Each student creates a personal deck of eleven cards, as follows:

• Regular or irregular polygons, one with each shaded-in area: 1, 2, 3, …, 8, and 9 square units. The area must be easy to count, so build each polygon from a combination of squares and half-squares.
• Two additional polygon cards with areas of the player’s choice.

Claim your deck: Mark the back of each card with your insignia (initials, emoticon, math symbol, etc. — your choice). Decorate as desired, so long as it doesn’t bleed through the paper and obscure your polygons.

How to Play

Stack your card deck face down in front of you. Draw the top two cards for your hand.

On your turn, call “high” or “low” for whether the greatest or least area wins the trick. Then each player chooses a card from their hand and holds it face down. When all players are ready, turn the cards up, and the winner captures them all. Each player keeps a pile of prisoner cards.

In case of a tie, those players reveal their other card, and the longest perimeter takes the trick. If it’s still a tie, leave the cards on the table for the winner of the next trick.

Players draw one or two new cards to replenish their hands before the next turn.

The game ends when at least one player runs out of cards. Whoever has captured the most prisoners wins the game. Give the prisoner cards back to their owner, unless you are playing the Spoils of War variation.

Variations

In between games, players can make new cards. But they must always play with a deck that fits the rules: nine cards with the areas 1–9, plus two extras.

Perimeter Battle: On each turn, the low or high perimeter takes the trick. In case of a tie, the greatest area wins the second fight.

Spoils of War: If you win the game, you may claim one card from your prisoner pile to replace the matching card from your deck. Erase the identification and add your own symbol on the back. You may offer your old card to the losing player as a consolation prize, or keep it as a backup card.

History

When John Golden created and tested this game, the students gave it “universal thumbs up as to whether they would recommend it to other teachers. They were requesting time to play in class later, which is surely a good sign.”

Writing to Learn Math: Measurement is our way of connecting numbers to the things we find in the world, in daily life.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Measurement & Data Prompt

Measurement is our way of connecting numbers to the things we find in the world, in daily life. Those numbers become data that students can examine, compare, and reason about.

Some measurements are clear and easy to determine, such as the length of a stick or the weight of a bunch of bananas. But other measurements are fuzzy and open to debate. For example, how can anyone measure the value of an idea or the intelligence of a puppy?

My Measurement & Data prompts give students a chance to collect and examine a variety of measurements and to practice different ways of representing data with charts or graphs.

Older students may want to examine how data shape the way people understand their society. Two websites to explore: slowrevealgraphs.com and nytimes.com/column/whats-going-on-in-this-graph.

Journaling Prompt 241: Number Race

Write the numbers 1–12 down the side of your page, each on its own line. Outline a row of four boxes next to each number (or more boxes for a longer race).

Roll two dice, and shade in a box for the sum.

The number that fills its boxes first wins the race.

Try another race. Does the same number always win? Which number can never win, and why? What other questions can you ask?

Do you ever take your kids’ math tests? It helps me remember what it is like to be a student. I push myself to work quickly, trying to finish in about 1/3 the allotted time, to mimic the pressure students feel. And whenever I do this, I find myself prone to the same stupid mistakes that students make.

Even teachers are human.

In this case, it was a multi-step word problem, a barrage of information to stumble through. In the middle of it all sat this statement:

“…and there were 3/4 as many dragons as gryphons…”

My eyes saw the words, but my mind heard it this way:

“…and 3/4 of them were dragons…”

What do you think — did I get the answer right? Of course not! Every little word in a math problem is important, and misreading even the smallest word can lead a student astray. My mental glitch encompassed several words, and my final tally of mythological creatures was correspondingly screwy.

But here is the more important question: Can you explain the difference between these two statements?

If Johnny Can’t Read, Then He Can’t Do Math

Few people realize that a math test is also a test of reading comprehension. In some cases the reading level required far exceeds the level of math.

Before they can solve a word problem, children must be able to understand what is written. They must hold that information in their working memory as they think about how to find a solution. They need to comprehend what they read, paraphrase it—concentrating on the relevant facts—and translate that information into a mathematical expression. Frequently they have to read between the lines and understand something that is implied, not explicitly stated.

That quote is from an article at Teen Literacy Tips blog. Does a literature teacher have anything useful to say about solving math problems? Well, the fact that word problems are also called story problems should clue us in to a significant connection.

When students struggle with word problems, more often than not it is a language issue that confuses them.

Consider, for example, the maddening confusion that may be caused by these two simple statements:

• Eight divided in half is four.
• Eight divided by one-half is sixteen.

These sentences seem to contradict each other, yet both are true. Can you explain the difference? If your students keep a math journal, you might use this question as a writing prompt.

(I’ve included one possible answer at the bottom of this post.)

My Problem with Dragons and Gryphons

In my word problem, it turned out there were fifty-six fantasy creatures in all. I got that part of the answer, but then I needed to find the number of gryphons.

This is how I did it…

“… and 3/4 of them were dragons …”

Imagine all fifty-six creatures in a zoo, split evenly into four pens. Three of the pens hold various breeds of dragon, and the fourth pen has the gryphons.

4 pens = 56 creatures

1 pen = 56 ÷ 4 = 14 gryphons

But that was not at all what the problem said. If I had been paying better attention to what I read, this is how I should have solved the problem…

“… and there were 3/4 as many dragons as gryphons …”

This zoo is wiser than the first one, because it splits the troublesome creatures into smaller groups, so there’s less chance of a fight that might endanger visitors. The gryphons are split into four pens, so each pen holds one-fourth of the gryphons. The dragons have three pens all to themselves, which makes three-fourths as many as the gryphons. Thus, we have a total of seven fenced-in areas.

7 pens = 56 creatures

1 pen = 56 ÷ 7 = 8 creatures per pen

4 pens = 4 × 8 = 32 gryphons

Math Journal Prompt

To make the reading issue more difficult, consider this: All the statements in the following list are equivalent. Compare each statement to the second diagram above, the correct one. Can you see each relationship?

• There are 25% fewer dragons than gryphons.
• For every four gryphons, there are three dragons.
• The ratio of dragons to gryphons is 3:4.
• Four out of every seven creatures is a gryphon.
• There are one-third more gryphons than dragons.
• There are one-fourth fewer dragons than gryphons.
• If you tag a creature at random from the group, the probability of choosing a dragon is three-sevenths.

Can you think of any other ways to say it? This would be another good math journal writing prompt.

The more we can work with our children on reading, paraphrasing, and translating word problems into mathematical expressions, the better prepared they will be to face the challenges they will meet in future math classes and in the textbook of life.

One Possible Answer to Dividing in/by Half

When you divide a number in half, you split it into two equal parts.

But when you divide a number by 1/2, you are finding how many halves it takes to make that number — that is, you are cutting it into half-size pieces and counting how many there are.

And in that case, because each whole thing is two halves, there will be twice as many pieces as the number you started with.

This game challenges players to make logical deductions about number permutations.

Many parents remember struggling to learn math. We hope to provide a better experience for our children. And one of the best ways for children to enjoy learning is through hands-on play.

So what are you waiting for? Let’s play some math!

Codebreaker

Math Concepts: permutations, logical deduction.

Players: two or more.

Equipment: paper and pencil, or whiteboard and marker.

How to Play

One player (the codemaker) thinks of a three-digit number between 000 and 999, writes this number on a piece of paper, and folds the paper to hide it. The other players (codebreakers) draw a two-column chart labeled Guesses and Clues. Players can each keep separate written records, or all may share a single chart on a whiteboard.

The codebreakers take turns guessing a three-digit number, which they record in the guess column. The codemaker responds with the appropriate clues:

• If one of the digits is the right number in the right place, say “There’s one number correct,” and draw a smiley face in the clues column. Do not tell which digit won the smiley.

• If one of the digits is a correct number but in the wrong place, say “There’s one number right, but it’s not where it belongs,” and draw an empty circle in the clues column.

• If more than one digit matches the secret code, tell the total number of correct and almost-right digits—and draw that many smiley faces and plain circles—but never reveal which digit matches which clue.

With two players, a round of play is two numbers, so each player has a chance to be the codemaker. Whoever breaks the code with the fewest guesses wins. For group play, whoever guesses the number gets to be codemaker for the next round.

Variations

For beginners, don’t allow repeating digits, and make it easier by using only the numbers 1–5. A secret number like 142 would be fine, but not 223. Advanced players can use four- to six-digit numbers and allow repetition.

Or try playing without paper, keeping track of guesses and clues in your head. This is a mind-bending challenge, but it can be done.

History

In the wee morning hours on a cross-country road trip, I made up the paperless version of Codebreaker to keep the driver awake. In my mind, it was a modification of the commercial board game Mastermind, but according to Wikipedia, similar guessing games with numbers or words date back at least a century.

Writing to Learn Math: You can spark creative thought by removing any need to worry about spelling or punctuation rules.

Do you want your children to develop the ability to reason creatively and figure out things on their own?

Help kids practice slowing down and taking the time to fully comprehend a math topic or problem-solving situation with these classic tools of learning: Notice. Wonder. Create.

Notice: Look carefully at the details of the numbers, shapes, or patterns you see. What are their attributes? How do they relate to each other? Also notice the details of your own mathematical thinking. How do you respond to a tough problem? Which responses are most helpful? Where did you get confused, or what makes you feel discouraged?

Wonder: Ask the journalist’s questions: who, what, where, when, why, and how? Who might need to know about this topic? Where might we see it in the real world? When would things happen this way? What other way might they happen? Why? What if we changed the situation? How might we change it? What would happen then? How might we figure it out?

Create: Create a description, summary, or explanation of what you learned. Make your own related math puzzle, problem, art, poetry, story, game, etc. Or create something totally unrelated, whatever idea may have sparked in your mind.

Math journaling may seem to focus on this third tool, creation. But even with artistic design prompts, we need the first two tools because they lay a solid groundwork to support the child’s imagination.

How To Use a Freewrite Prompt

You can spark creative thought by removing any need to worry about spelling or punctuation rules. During a freewriting session, students should write fast and raw until they reach the end of the page.

If students can’t think of what to write, they might consider how the journalist’s 5W1H questions apply to their prompt: who, what, where, when, why, how? Or they can reword their previous sentence, or look for a way to add extra details. It’s always valuable to rethink and revise our writing to make it better express the ideas in our heads.

If all else fails, students can keep writing anything that comes into their minds, even if it seems to have nothing to do with math. They may be surprised to find mathematical ideas pop up in the most unexpected places.

Freewriting prompts may be reused, especially if you change one or two words to make them new. Most of the questions are general enough to spawn entire books, so there will always be ideas the student didn’t have time to think about before.

Or let students propose their own topics. Make a long list of prompt sentences and cut the paper into strips, then crumple the strips into a jar. When it’s time to write, they can pull out a prompt and let the pencil run with it.

If children have trouble filling a whole page, let them write to a timer instead. Set it for five minutes or however long fits their energy level.

Journaling Prompt 282: Success

One thing I’m proud of in math is…

[Based on a problem I made up for my co-op students, once upon a time…]

Conversion factors are special fractions that contain problem-solving information. Why are they called conversion factors?

• “Conversion” means change, and conversion factors help you change the numbers and units in your problem.

• “Factors” are things you multiply with. So to use a conversion factor, you will multiply it by something.

For instance, if I am driving an average of 60 mph on the highway, I can use that rate as a conversion factor. I may use the fraction:

Or I may flip it over to make:

It all depends on what problem I want to solve.

After driving two hours, how far have I gone?

But if I am planning to go 240 more miles, how much longer will it take?

Any rate can be used as a conversion factor. You can recognize them by their form: this per that. Miles per hour, dollars per gallon, cm per meter, and many, many more.

Of course, you will need to use the rate that is relevant to the problem you are trying to solve. If I were trying to figure out how far a tank of gas would take me, it wouldn’t be any help to know that an M1A1 Abrams tank would get about 1/3 mile per gallon. I won’t be driving one of those.

Using Conversion Factors Is Multiplying by One

If I am driving a steady 60 mph on the highway, then driving for one hour is exactly the same as driving 60 miles, and:

This may be easier to see if you think of kitchen measurements. Two cups of sour cream are exactly the same as one pint of sour cream, so:

If I want to find out how many cups are in 3 pints of sour cream, I can multiply by the conversion factor:

Multiplying by one does not change the original number. In the same way, multiplying by a conversion factor does not change the original amount of stuff. It only changes the units that you measure the stuff in. When I multiplied 3 pints times the conversion factor, I did not change how much sour cream I had, only the way I was measuring it.

Conversion Factors Can Always Be Flipped Over

If there are:

Then there must also be:

If I draw house plans at a scale of:

Then I know that:

If there are:

Then there are:

Or if a single-engine airplane is burning fuel at 8 gallons/hour, then the pilot has only 1/8 hour left to fly for every gallon left in his tank.

This is true for all conversion factors, and it is an important part of what makes them so useful in solving problems. You can choose whichever form of the conversion factor seems most helpful in the problem at hand.

How can you know which form will help you solve the problem? Look at the units you have, and think about the units you need to end up with. In the sour cream measurement above, I started with pints and I wanted to end up with cups. That meant I needed a conversion factor with cups on top (so I would end up with that unit) and pints on bottom (to cancel out).

Just as numbers cancel out when the same number is on the top and bottom of a fraction (2/2 = 2 ÷ 2 = 1), so do units cancel out if you have the same unit in the numerator and denominator. So pints/pints= 1.

Important Tip 1: Some teachers don’t like to use the word “cancel” because students forget that it’s a shortcut for saying “divide by each other to equal one.” So the forgetful kids cross out numbers or units and put a zero in their place. But you would never do that, right?

You Can String Conversion Factors Together

String several conversion factors together to solve more complicated problems. How many cups of milk are there in a gallon jug?

As you write out your string of factors, you will want to draw a line through each unit as it cancels out, and then whatever is left will be the units of your answer.

Important Tip 2: Notice that only the units cancel — not the numbers. Even after I canceled out the quarts, the 4 was still part of my calculation.

Let’s Try One More

The true power of conversion factors is their ability to change one piece of information into something that at first glance seems to be unrelated to the number with which you started.

Suppose I drove for 45 minutes at 55 mph in a pickup truck that gets 18 miles to the gallon, and I wanted to know how much gas I used.

To find out, I start with a plain number that I know (in this case, the 45 minutes) and use conversion factors to cancel out units until I get the units I want for my answer (gallons of gas). How can I change minutes into gallons?

I need a string of conversion factors:

How Old Are You, Anyway?

If you want to find your exact age in nanoseconds, you need to know the exact moment at which you were born. But for a rough estimate, just knowing your birthday will do. First, find out how many days you have lived:

Once you know how many days you have lived, you can use conversion factors to find out how many nanoseconds that would be.

You know how many hours are in a day, minutes in an hour, and seconds in a minute.

And just in case you weren’t quite sure:

Have fun playing around with conversion factors. You will be surprised how many problems these mathematical wonders can solve.