# Reblog: Solving Complex Story Problems

[Dragon photo above by monkeywingand treasure chest by Tom Praison via flickr.]

Over the years, some of my favorite blog posts have been the Word Problems from Literature, where I make up a story problem set in the world of one of our family’s favorite books and then show how to solve it with bar model diagrams. The following was my first bar diagram post, and I spent an inordinate amount of time trying to decide whether “one fourth was” or “one fourth were.” I’m still not sure I chose right.

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

Cimorene spent an afternoon cleaning and organizing the dragon’s treasure. One fourth of the items she sorted was jewelry. 60% of the remainder were potions, and the rest were magic swords. If there were 48 magic swords, how many pieces of treasure did she sort in all?

[Problem set in the world of Patricia Wrede's Enchanted Forest Chronicles. Modified from a story problem in Singapore Primary Math 6B. Think about how you would solve it before reading further.]

How can we teach our students to solve complex, multi-step story problems? Depending on how one counts, the above problem would take four or five steps to solve, and it is relatively easy for a Singapore math word problem. One might approach it with algebra, writing an equation like:

$x - \left[\frac{1}{4}x + 0.6\left(\frac{3}{4} \right)x \right] = 48$

… or something of that sort. But this problem is for students who have not learned algebra yet. Instead, Singapore math teaches students to draw pictures (called bar models or math models or bar diagrams) that make the solution appear almost like magic. It is a trick well worth learning, no matter what math program you use …

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# Reblog: Putting Bill Gates in Proportion

[Feature photo above by Baluart.net.]

Seven years ago, one of my math club students was preparing for a speech contest. His mother emailed me to check some figures, which led to a couple of blog posts on solving proportion problems.

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

## Putting Bill Gates in Proportion

A friend gave me permission to turn our email discussion into an article…

Can you help us figure out how to figure out this problem? I think we have all the information we need, but I’m not sure:

The average household income in the United States is $60,000/year. And a man’s annual income is$56 billion. Is there a way to figure out what this man’s value of $1mil is, compared to the person who earns$60,000/year? In other words, I would like to say — $1,000,000 to us is like 10 cents to Bill Gates. ### Let the Reader Beware When I looked up Bill Gates at Wikipedia, I found out that$56 billion is his net worth, not his income. His salary is $966,667. Even assuming he has significant investment income, as he surely does, that is still a difference of several orders of magnitude. But I didn’t research the details before answering my email — and besides, it is a lot more fun to play with the really big numbers. Therefore, the following discussion will assume my friend’s data are accurate… [Click here to go read Putting Bill Gates in Proportion.] ## Bill Gates Proportions II Another look at the Bill Gates proportion… Even though I couldn’t find any data on his real income, I did discover that the median American family’s net worth was$93,100 in 2004 (most of that is home equity) and that the figure has gone up a bit since then. This gives me another chance to play around with proportions.

So I wrote a sample problem for my Advanced Math Monsters workshop at the APACHE homeschool conference:

The median American family has a net worth of about $100 thousand. Bill Gates has a net worth of$56 billion. If Average Jane Homeschooler spends $100 in the vendor hall, what would be the equivalent expense for Gates? # 10 Questions to Ask About a Math Problem [Photo by CourtneyCarmody via flickr.] It’s important to teach our children to ask questions, about math and about life. As I wrote in my series about homeschooling with math anxiety, “School textbooks only ask questions for which they know the answer. When homeschoolers learn to think like mathematicians, we will ask a different type of question.” So I was delighted to see this new post from Bon Crowder: Ten Questions to Ask About a Math Problem. Click the link and read the whole thing! Why a list of questions about math problems? Before creating them, I decided the questions should do the following: • Allow the student to dig in deeper to the math problem, and the math behind the problem. • Help the student to think about the problem in ways they wouldn’t normally. • Let the student get creative in thinking about the problem. And of course doing these things regularly will train them to continue to do this with all math problems through their lives. — Bon Crowder Ten Questions to Ask About a Math Problem Get all our new math tips and games: Subscribe in a reader, or get updates by Email. # Problem-Solving Poll: What’s Your Answer? [Photo by Alex E. Proimos via flickr.] Patrick Vennebush, author of Math Jokes 4 Mathy Folks (the book and the blog) wants to know how you and your children would answer a tricky math problem. I have often heard that, “Good teachers borrow, great teachers steal.” So today, I am stealing one of Marilyn Burns’s most famous problems. She takes this problem to the streets, and various adults give lots of different answers. When I’ve used it in workshops, even among a mathy crowd, I get lots of different answers, too. What’s your answer? “A man buys a truck for$600, then sells it for $700. Later, he decides to buy it back again and pays$800 dollars. However…”

Go to Patrick’s blog to read the whole problem and submit your answer. Let everybody in the family try it!

Update: Patrick posted the solution and percentages correct for students of various ages.

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# Rate × Time = Distance Problems

I love how Richard Rusczyk explains math problems. It’s a new school year, and that means it’s time for new MathCounts Mini videos. Woohoo!

# Tell Me a (Math) Story

feature photo above by Keoni Cabral via flickr (CC BY 2.0)

My favorite playful math lessons rely on adult/child conversation — a proven method for increasing a child’s reasoning skills. What better way could there be to do math than snuggled up on a couch with your little one, or side by side at the sink while your middle-school student helps you wash the dishes, or passing the time on a car ride into town?

As soon as your little ones can count past five, start giving them simple, oral story problems to solve: “If you have a cookie and I give you two more cookies, how many cookies will you have then?”

The fastest way to a child’s mind is through the taste buds. Children can easily visualize their favorite foods, so we use mainly edible stories at first. Then we expand our range, adding stories about other familiar things: toys, pets, trains.

# How to Translate Word Problems

Hooray! The MathCounts Mini videos are back. September’s edition is all about translating word problems into algebra:

Do your students like making videos? This year, MathCounts is challenging students in grades 6-8 to create a math problem video. Check out the details:

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# Babymath: Story Problem Challenge III

<a href="http://www.flickr.com/photos/goetter/2352128932/"Photo by Raphael Goetter via Flickr

Alex and Leon enjoyed their baby sister, but they were amazed at how much work taking care of a baby could be. One particularly colicky night, everyone in the family took turns holding the baby, rocking the baby, patting her back, and walking her around before she finally succumbed to sleep.

Then Alex collapsed on the couch, and Leon sank into the recliner. They teased each other with these story problems.

# Probability Issue: Hints and Answers

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original posts. If you’re stuck, read the hints. Then go back and try again. Figure them out for yourself — and then check the answers just to prove that you got them right.

This post offers hints and answers to puzzles from these blog posts:

# Hobbit Math: Elementary Problem Solving 5th Grade

[Photo by OliBac. Visit OliBac's photostream for more.]

The elementary grades 1-4 laid the foundations, the basics of arithmetic: addition, subtraction, multiplication, division, and fractions. In grade 5, students are expected to master most aspects of fraction math and begin working with the rest of the Math Monsters: decimals, ratios, and percents (all of which are specialized fractions).

Word problems grow ever more complex as well, and learning to explain (justify) multi-step solutions becomes a first step toward writing proofs.

This installment of my elementary problem solving series is based on the Singapore Primary Mathematics, Level 5A. For your reading pleasure, I have translated the problems into the world of J.R.R. Tolkien’s classic, The Hobbit.

[Note: No decimals or percents here. Those are in 5B, which will need an article of its own. But first I need to pick a book. I'm thinking maybe Naya Nuki...]

## Printable Worksheet

In case you’d like to try your hand at the problems before reading my solutions, I’ve put together a printable worksheet:

# Best Articles about Solving Word Problems

[Photo by scui3asteveo.]

I’m still working on that Best of Blog page. [It's done! :D] Here are the 20 best Let’s Play Math! blog articles about solving word problems (also known as story problems)…

## Solving Word Problems

More recent posts tagged ‘Word problems’

## Bar Diagrams Help Students Think

More recent posts tagged ‘Bar diagrams’

## Other Post in the Best of Blog Series

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# Math Teachers: Be Less Helpful

I especially like the “4 layers of a story problem” concept, which helped me understand how Dan converts textbook problems into What Can You Do With This? challenges.

# Narnia Math: Elementary Problem Solving 4th Grade

[Photo by armigeress.]

In 4th grade, math problems take a large step up on the difficulty scale. Students are more mature and can read and follow more complex stories. Multi-step word problems become the new norm, and proportional relationships (like “three times as many”) show up frequently. As the year progresses, fractions grow to be a dominant theme.

As a math teacher, one of my top goals is that my students learn to solve word problems. Arithmetic is (relatively) easy, but many children struggle in applying it to “real world” situations.

In previous posts, I introduced the problem-solving tools of word algebra and bar diagrams, either of which can help students organize the information in a word problem and translate it into a mathematical calculation. The earlier posts in this series are:

In this installment, I will continue to demonstrate the problem-solving tool of bar diagrams through a series of ten 4th grade problems based on the Singapore Primary Math series, level 4A. For your reading pleasure, I have translated the problems into the universe of a family-favorite story by C. S. Lewis, The Lion, the Witch and the Wardrobe.

## Update

I’ve put the word problems from my elementary problem solving series into printable worksheets:

# Algebra: A Problem in Translation

[Photo by *Irish.]

In my post Elementary Problem Solving: The Tools, I introduced word algebra as a way to help students think their way through a story problem. In the next two posts, I showed how the tool worked with simple word problems.

Now, before I move on to focus exclusively on bar diagrams, I would like to show how word algebra can help a student solve a typical first-year algebra puzzle.

A homeschooling friend who avoided algebra in high school, trying to help her son cope with a subject she never understood, posted: “Help! Our answer is different from the book’s.” Here is the homework problem:

Josh earned $72 less than his sister who earned$93 more than her mom. If they earned a total of \$504, how much did Josh earn?

# Elementary Problem Solving: Review

[Bill Watterson identifies the trouble with math problems, through the eyes of Calvin and Hobbes.]

It’s time to revive and (hopefully!) finish my long-neglected series on solving word problems in elementary mathematics. I’ve been having fun making up the problems, so now I just have to write the posts. Coming up soon:

Since it has been more than two years since the last entry, however, I wanted to take a few minutes to recap our progress so far and to refer new readers back to the original posts:

# How to Solve Math Problems

[Photo by Aaron Escobar. This post is a revision and update of How to Solve Math Problems from October, 2007.]

What can you do when you are stumped by a math problem? Not just any old homework exercise, but one of those tricky word problems that can so easily confuse anyone?

The difference between an “exercise” and a “problem” will vary from one person to another, even within a single class. Even so, this easy to remember, 4-step approach can help students at any grade level. In my math classes, I give each child a copy to keep handy:

[Note: Page 1 is the best for quick reference, especially with elementary to middle school children. Page 2 lists the steps in more detail, for the teacher or for older students.]

# Christmas in July Math Problem

[Photo by Reenie-Just Reenie.]

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

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

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

How many of each color light bulb did Ethel have?

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

# Word Problems in Russia and America

Andrei Toom calls this an “extended version” of a talk he gave a few years ago at the Swedish Mathematical Society. At 159 pages [2010 updated version is 98 pages], I would call it a book. Whatever you call it, it’s a must-read for math teachers:

Main Thesis: Word problems are very valuable in teaching mathematics not only to master mathematics, but also for general development. Especially valuable are word problems solved with minimal scolarship, without algebra, even sometimes without arithmetics, just by plain common sense. The more naive and ingenuous is solution, the more it provides the child’s contact with abstract reality and independence from authority, the more independent and creative thinker the child becomes.

# Solving Complex Story Problems II

[Oops! I found one more post from my old blog. It apparently slipped off the back of my metaphorical desk and has been sitting with the dust bunnies.]

Here is a math problem in honor of one of our family’s favorite movies

Han Solo was doing some needed maintenance on the Millennium Falcon. He spent 3/5 of his money upgrading the hyperspace motivator. He spent 3/4 of the remainder to install a new blaster cannon. If he spent 450 credits altogether, how much money did he have left?

[Modified from a word problem in Singapore Primary Math 5B. Stop and think about how you would solve it before reading further.]

# How to Solve Math Problems

## Update

For the 2009 school year, I revised these handouts into a one-page reference that I could slip into the back of each student’s homemade white board. For details, see:

## That’s a Tough One!

What can you do when you are stumped? Too many students sit and stare at the page, waiting for inspiration to strike — and when the solution doesn’t crack their heads open and step out, fully formed, they complain: “Math is too hard!”

So this year I have given my Math Club students a couple of mini-posters to put up on the wall above their desk, or wherever they do their math homework. The first gives four questions to ask yourself as you think through a math problem, and the second is a list of problem-solving strategies.

# Ben Franklin Math: Elementary Problem Solving 3rd Grade

The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to solve math problems? I must help them develop the ability to translate “real world” situations into mathematical language.

In two previous posts, I introduced the problem-solving tools algebra and bar diagrams. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

## Working Math Problems with Poor Richard

This time I will demonstrate these problem-solving tools in action with a series of 3rd-grade problems based on the Singapore Primary Math series, level 3A. For your reading pleasure, I have translated the problems into the universe of a well-written biography of Ben Franklin, Poor Richard by James Daugherty.

[Photo by Betsssssy.]

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?

# Penguin Math: Elementary Problem Solving 2nd Grade

The ability to solve word problems ranks high on any math teacher’s list of goals. How can I teach my students to reason their way through math problems? I must help my students develop the ability to translate “real world” situations into mathematical language.

In a previous post, I analyzed two problem-solving tools we can teach our students: algebra and bar diagrams. These tools help our students organize the information in a word problem and translate it into a mathematical calculation.

Now I want to demonstrate these problem-solving tools in action with a series of 2nd grade problems, based on the Singapore Primary Math series, level 2A. For your reading pleasure, I have translated the problems into the universe of one of our family’s favorite read-aloud books, Mr. Popper’s Penguins.

# Elementary Problem Solving: The Tools

[This article begins a series rescued from my old blog. Moving has been a long process, but I'm finally unpacking the last cardboard box! To read the entire series, click here: elementary problem solving series.]

Most young students solve story problems by the flash of insight method: When they read the problem, they know almost instinctively how to solve it. This is fine for problems like:

There are 7 children. 2 of them are girls. How many boys are there?

As problems get more difficult, however, that flash of insight becomes less reliable, so we find our students staring blankly at their paper or out the window. They complain, “I don’t know what to do. It’s too hard!”

We need to give our students a tool that will help them when insight fails.

# Are You Smarter than a 3rd-6th Grader?

Here are a few challenging word problems from Singapore:

I did fine on the 3rd-grade problems, but I stumbled a bit on the 4/5th-grade “How much sugar…” problem. The toy cars were tricky, but manageable. I misread the problem with the chocolate and sweets at first — I think of chocolates as a sub-category of sweets, but in this problem they are totally different. (Perhaps “sweets” are what I would call “hard candy”?) Finally, I had to resort to algebra for the last two Grade 6 questions.

How many can you solve?

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# Puzzle: Random Blocks

In the first section of George Lenchner’s Creative Problem Solving in School Mathematics, right after his obligatory obeisance to George Polya (see the third quote here), Lechner poses this problem. If you have seen it before, be patient — his point was much more than simply counting blocks.

A wooden cube that measures 3 cm along each edge is painted red. The painted cube is then cut into 1-cm cubes as shown above. How many of the 1-cm cubes do not have red paint on any face?

And then he challenges us as teachers:

Do you have any ideas for extending the problem?
If so, then jot them down.

This is strategically placed at the end of a right-hand page, and I was able to resist turning to read on. I came up with a list of 15 other questions that could have been asked — some of which will be used in future Alexandria Jones stories. Lechner wrote only seven elementary-level problems, and yet his list had at least two questions that I had not considered. How many can you come up with?

# Solving Complex Story Problems

[Dragon photo above by monkeywingand treasure chest by Tom Praison via flickr.]

Let’s play around with a middle-school/junior high word problem:

Cimorene spent an afternoon cleaning and organizing the dragon’s treasure. One fourth of the items she sorted was jewelry. 60% of the remainder were potions, and the rest were magic swords. If there were 48 magic swords, how many pieces of treasure did she sort in all?

[Problem set in the world of Patricia Wrede's Enchanted Forest Chronicles. Modified from a story problem in Singapore Primary Math 6B. Think about how you would solve it before reading further.]

How can we teach our students to solve complex, multi-step story problems? Depending on how one counts, the above problem would take four or five steps to solve, and it is relatively easy for a Singapore math word problem. One might approach it with algebra, writing an equation like:

$x - \left[\frac{1}{4}x + 0.6\left(\frac{3}{4} \right)x \right] = 48$

…or something of that sort. But this problem is for students who have not learned algebra yet. Instead, Singapore math teaches students to draw pictures (called bar models or math models or bar diagrams) that make the solution appear almost like magic. It is a trick well worth learning, no matter what math program you use.

# How Can We Teach Problem Solving?

We continue to plan our co-op courses for next fall. Some of the classes I had hoped for will not happen, and my children are going to have to make some tough choices between the remaining topics. Unfortunately, they have not yet mastered the ability to be in two classrooms at once.

I have three math courses to plan, and I think I will focus as much as I can on teaching math through problems, even at the elementary level. These are once-a-week enrichment classes for homeschooled students, so I assume they have a “normal” math program at home. I want to introduce a few topics they might not otherwise see, to deepen their understanding of the topics they have studied, and to give them a taste of that “Aha!” feeling that comes from conquering a challenging problem. Has anybody done something like this, and can you recommend some good resources?