## 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 …

### Update: My New Book

You can help prevent math anxiety by giving your children the mental tools they need to conquer the toughest story problems.

Read Cimorene’s story and many more in Word Problems from Literature: An Introduction to Bar Model Diagrams—now available at all your favorite online bookstores!

And there’s a paperback Student Workbook, too.

## 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 ## 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.

## 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 young 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.

## 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.

UPDATE: Problems have been genericized to avoid copyright issues.