Word Problems from Literature: Playful Math Singles

A Playful Math Single

Word Problems from Literature

Help Students Master Problem Solving in Elementary to Middle School Math

Denise Gaskins

Second Edition

Copyright © 2023 Denise Gaskins

All rights reserved.

Ebook Version 2.0

The Playful Math Singles from Tabletop Academy Press are short, topical books featuring clear explanations and ready-to-play activities.

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Readers Love Denise’s Playful Math Books

Reading one of Gaskins’ books is like going to a really great teacher workshop — part philosophy, part practical ideas, and all excellent. She just oozes expertise and enthusiasm.

—Amy at Hope Is the Word blog

It revolutionized our homeschool this year.

—Caitlin Fitzpatrick Curley, My-Little-Poppies.com

I’m glad my son’s introduction to math will be in the form of games and not a dry workbook.

—Nicki, online reader review

With this approach I can teach my kids to think like mathematicians without worrying about leaving gaps. I can’t wait to take my children by the hand and head off to explore the wonderful world of maths.

—Lucinda Leo, NavigatingByJoy.com

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Contents

Title Page

Preface to the Second Edition

Word Problems as Mental Manipulatives

Lay the Foundation: One-Step Problems

Build Modeling Skills: Multistep Problems

Master the Technique: From Multiplication to Fractions

Extend Your Skills: Measurement and Decimals

Reap the Reward: Ratios and More Fractions

Conquer Monster Topics: Percents, Rates, and Proportional Reasoning

Move Toward Algebra: Challenge Problems

Write Your Own Word Problems from Literature

Conclusion: Making Sense of Word Problems

Appendixes

About the Author

Online Resources

Answers to the Student Workbook

Special Thanks

Copyright and Credits

Preface to the Second Edition

As a math coach, I love showing adults and children how to look at math with fresh eyes, to explore the adventure of learning math as mental play, which is the essence of creative problem-solving. Mathematics is not just rules and rote memory. Math itself is a game, playing with ideas.

I’ve written several books to help families play math together. But at heart, I’ve always been a fiction fan — especially fantasy fiction. And this book, Word Problems from Literature, lets me bring that love of story to the surface.

This is one of my all-time favorite books, and I've had so much fun with this new edition: adding stories, writing make-your-own-problem prompts, sneaking extra teaching tips into the worked-out solutions, creating an almost-magical guide to helping kids reason their way through math problems.

To provide support when your children get stuck, I added my favorite problem-solving tip, the Four Questions that push students to apply their own common sense, emphasizing the importance of reasoning about math. And to give you a boost when you get stuck, I beefed up the explanations of the advanced puzzles, adding several tips on understanding and teaching fraction division and other monster calculations.

There's a new section in the student workbook on What If I’m Not Good at Math? to help children develop a problem-solving mindset. Plus, I wrote more than 30 new word problem prompts to get your students writing their own creative math stories.

Most of all, I’ve tried as much as possible to encourage both adults and students to work at making sense of the math, seeing how the numbers relate, avoiding the crutch of standard textbook rules so you can experience the joy of figuring things out for yourself.

Those of us who made it through school math by memorizing and following rules eventually paid the price. We came to the point where our minds could hold no more, where the rules we had learned all jumbled together, where we felt lost in the dark as the rock began to crumble beneath our metaphorical feet while the wizard rammed his staff into the ground and cried, You shall not pass!

Some people reach the point of mental exhaustion with upper-elementary fractions or middle school ratios and percents, while others make it into algebra or geometry before they crash. A rare few have a good enough mental filing system that they can proceed as far as calculus before it all falls apart.

Estimates vary, but anywhere from half to three-fourths of adults suffer from some level of math anxiety due to their school experience.

Can we spare our children this fate?

We must strive to teach math in a way that makes sense, where children don't just memorize the rules we tell them but see clearly how math concepts connect, drawing their own conclusions, building their understanding into a solid foundation for future learning.

And we must give our students the tools to build on this foundation, problem-solving tools that will help them face and conquer any new math monster that comes their way.

This book will help you do that.

Have fun playing math with your kids!

—Denise Gaskins,

rural Illinois, July 2022

Storying —

encountering the world

and understanding it contextually

by shaping ideas,

facts,

experience itself

into stories —

is one of the most fundamental means

of making meaning.

As such, it is an activity

that pervades all learning.

— Gordon Wells

~1~

Word Problems as Mental Manipulatives

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

—Paul Halmos

Archimedes tried to find the distance around a circle and almost discovered calculus. Pierre de Fermat predicted the result of a gambling game and laid the foundations of probability. Leonhard Euler went for an afternoon walk over the bridges of Konigsberg and invented topology. Georg Cantor created a way to count infinity and opened a whole new world of modern math.

Through the centuries, mathematics has grown as mathematicians struggled with and solved challenging puzzles.

Problems are the raw material of math, the ore we dig, grind up, and melt, refining it to produce ideas. Our understanding of math grows as we play with problems, puzzle them out, and look for connections to other situations. The threads that connect these problems become the web of ideas we call mathematics. Each puzzle we solve adds a new thread to the web, or strengthens one that already exists, or both.

If we want our children to learn real math, we need to offer them plenty of problems to solve. A child may work through several pages of number calculations by rote, following memorized steps, but a good problem demands more thought.

The story in a word problem puts flesh on the abstract bones of arithmetic, encouraging children to ponder what it means for one thing to be bigger than another, or smaller, or faster, or slower, or made up of several parts.

Math professor Herb Gross says: We teachers so often hear students summarize a course by saying, ‘I could do everything except the word problems.’ Sadly, in the textbook of life, there are only word problems.

Our children will meet numbers in many guises throughout their lives. Few of these will be as straight-forward as a textbook word problem, but real-life problems and school math stories will always have certain things in common. Quantities will be related to each other in a given proportion. Situations will be complex, and solutions may require many steps.

Story problems give students a chance to grapple with these complexities in a controlled environment, where we can increase the difficulty in stages over several years.

The Purpose of Word Problems

Word problems feed a student’s imagination. Like other puzzles, word problems are often artificial, but that needn’t diminish our pleasure in solving them.

In working a puzzle, we expect to find difficulties and setbacks. That’s part of the game. Similarly, when students approach word problems as puzzles, they become less concerned with rushing to an answer and more interested in figuring out the relationships within the story.

Playing with puzzles strengthens mathematical understanding, according to retired professor Andre Toom. Originally from Russia, Toom taught math in several colleges and universities around the world.

Here’s how he explains the purpose of story problems:

"Word problems are very valuable — not only to master mathematics, but also for general development.

"Especially valuable are word problems solved with minimal scholarship, without algebra, even sometimes without arithmetic, just by plain common sense. The more naive and ingenuous is the solution, the more it provides the child contact with abstract reality and independence from authority, the more independent and creative thinker the child becomes.

"When we teach children to solve problems in school, we do not expect them to meet exactly and literally the same problems in later life. Mathematical education would be next to useless if its only use were literal. We want much more, we want to teach children to solve problems in general.

"In this respect traditional word problems are especially valuable, because to solve a word problem, you have to understand what is said there.

"This function of word problems is very poorly understood. The main educative value of word problems is that they serve as mental manipulatives, paving children’s road to abstract thinking.

"For example, coins, nuts, and buttons are clearly distinct and countable, and for this reason are convenient to represent relations between whole numbers. The youngest children need some real, tangible tokens, while older ones can imagine them, which is a further step of intellectual development. That is why coin problems are so appropriate in elementary school.

"Pumps and other mechanical appliances are easy to imagine working at a constant rate. Problems involving rate and speed should be common already in middle school. Trains, cars, and ships are so widely used in textbooks not because all students are expected to go into the transportation business, but for another, much more sound reason: These objects are easy to imagine moving at constant speeds.

There is an important similarity between children’s play and mathematics: In both cases, creative imagination is essential.

—Andre Toom

The Trouble with Word Problems

How can we teach our students to reason their way through math problems? We must help them develop the ability to translate real-world situations into mathematical language.

Most young children solve story problems by the flash-of-insight method, hearing the problem and knowing by instinct how to solve it. This is fine for simple problems like Four kittens played with a yarn ball. Two more kittens came to join the fun. Then how many kittens were playing with the yarn ball?

When problems grow more difficult, however, that flash of insight becomes less reliable. We find our children fidgeting with their paper, staring out the window, complaining, I don’t know what to do. It’s too hard.

They need a tool that will work when insight