13895144613_d79092bea2_z

The Math Student’s Manifesto

[Feature photo above by Texas A&M University (CC BY 2.0) via Flickr.]

Note to Readers: Please help me improve this list! Add your suggestions or additions in the comment section below…

What does it mean to think like a mathematician? From the very beginning of my education, I can do these things to some degree. And I am always learning to do them better.

(1) I can make sense of problems, and I never give up.

  • I always think about what a math problem means. I consider how the numbers are related, and I imagine what the answer might look like.
  • I remember similar problems I’ve done before. Or I make up similar problems with smaller numbers or simpler shapes, to see how they work.
  • I often use a drawing or sketch to help me think about a problem. Sometimes I even build a physical model of the situation.
  • I like to compare my approach to the problem with other people and hear how they did it differently.

(2) I can work with numbers and symbols.

  • I know how numbers relate to each other.
  • I’m flexible with mental math. I understand arithmetic properties and can use them to make calculations easier.
  • I’m not intimidated by algebra symbols.
  • I don’t rely on memorized rules unless I know why they make sense.

(3) I value logical reasoning.

  • I can recognize assumptions and definitions of math terms.
  • I argue logically, giving reasons for my statements and justifying my conclusion.
  • I listen to and understand other people’s explanations.
  • I ask questions to clarify things I don’t understand.

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Two Ways to Do Math

Two-Ways-to-Do-Math

Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Raoul Bott, via The MacTutor History of Mathematics archive. Background photo courtesy of Swedish National Heritage Board (CC BY 2.0) via Flickr.


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dividing sausages

Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

Fractions: 1/5 = 1/10 = 1/80 = 1?

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

  • How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

What Do You Notice?

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Reimann-hexagon

Math Teachers at Play #70

800px-Brauchtum_gesteck_70_1[Feature photo above by David Reimann via Bridges 2013 Gallery. Number 70 (right) from Wikimedia Commons (CC-BY-SA-3.0-2.5-2.0-1.0).]

Do you enjoy math? I hope so! If not, browsing this post just may change your mind.

Welcome to the 70th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of 42+ links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college. Let the mathematical fun begin!

By tradition, we start the carnival with a puzzle in honor of our 70th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Click here to continue reading.

One Right Way

One-Right-Way

Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Herb Gross, via Math as a Second Language. Background photo courtesy of kristos_b (CC BY 2.0) via Flickr.


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A Good Problem Requires Dreaming Time

A-good-problem

Wednesday Wisdom features a quote to inspire my fellow homeschoolers and math education peeps. Today’s quote is from Howard Eves, An Introduction to the History of Mathematics. Background photo courtesy of Brenda Clarke (CC BY 2.0) via flickr.


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Carnival Parade in Aachen 2007

Math Teachers at Play #66

[Feature photo above by Franz & P via flickr. Route 66 sign by Sam Howzit via flickr. (CC BY 2.0)]
Route 66 Sign

Welcome to the Math Teachers At Play blog carnival — which is not just for math teachers! If you like to learn new things and play around with ideas, you are sure to find something of interest.

By tradition, we start the carnival with a couple of puzzles in honor of our 66th edition.

Let the mathematical fun begin!

Puzzle 1

how crazy 66

Our first puzzle is based on one of my favorite playsheets from the Miquon Math workbook series. Fill each shape with an expression that equals the target number. Can you make some cool, creative math?

Click the image to download the pdf playsheet set: one page has the target number 66, and a second page is blank so you can set your own target number.

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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!

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How to Think like a Mathematician

Would you like to learn how to think like a mathematician? Stanford professor (and NPR “Math Guy”) Keith Devlin is teaching a free online course through Coursera. It starts in just a few weeks. I’ve signed up. Will you join us?

The prerequisite is to be taking or have finished high school math. If (like me) you took it so long ago that you can’t quite remember, don’t worry: The focus of the course is not on long-forgotten mathematical procedures, but on “learning to think in a certain (very powerful) way.”

Mathematical thinking is not the same as doing mathematics — at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself.

The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box — a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

— Keith Devlin
Introduction to Mathematical Thinking

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Olympic Logic

I love logic puzzles! Nrich Maths offers four fun Olympics Logic puzzles. And be sure to check out the rest of their Nrich Olympics Math as well.

Medals Count

Given the following clues, can you work out the number of gold, silver and bronze medals that France, Italy and Japan got in this international sports competition?

  • Japan has 1 more gold medal, but 3 fewer silver medals, than Italy.
  • France has the most bronze medals (18), but fewest gold medals (7).
  • Each country has at least 6 medals of each type.
  • Italy has 27 medals in total.
  • Italy has 2 more bronze medals than gold medals.
  • The three countries have 38 bronze medals in total.
  • France has twice as many silver medals as Italy has gold medals.

Go to Nrich Maths and try all four puzzles!


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Princess in the Dungeon Game

Yet more fun from Rosie at Education Unboxed. I found these while looking for videos to use in my PUFM Subtraction post. Rosie says:

This is seriously embarrassing and I debated whether to put this video online or not because this is NOT my normal personality, but my girls made up this game and will play it for over an hour and ask for it repeatedly… so I figured someone out there might be able to use it with their kids, too.

If you know me, please don’t ever ask me to do this in public. I will refuse.

Princess in the Dungeon, Part 1 – Fractions with Cuisenaire Rods

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More Than One Way to Solve It, Again

photo by Annie Pilon via flickr

We continue with our counting lessons — and once again, Kitten proves that she doesn’t think the same way I do. In fact, her solution is so elegant that I think she could have a future as a mathematician. After all, every aspiring novelist needs a day job, right?

If only I could get her to give up the idea that she hates math…

Permutations with Complications

How many of the possible distinct arrangements of 1-6 have 1 to the left of 2?

Competition Math for Middle School, by J. Batterson

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by Eirik Newth via flickr

More Than One Way to Solve It

More-Than-One-Way

Photo by Eirik Newth via flickr.

In a lazy, I-don’t-want-to-do-school mood, Princess Kitten was ready to stop after three math problems. We had gotten two of them correct, but the last one was counting the ways to paint a cube in black and white, and we forgot to count the solid-color options.

For my perfectionist daughter, one mistake was excuse enough to quit. She leaned her head against me as we sat together on the couch and said, “We’re done. Done, done, done.” If she could, she would have started purring — one of the most manipulative noises known to humankind. I’m a soft touch. Who can work on math when there’s a kitten to cuddle?

by tanjila ahmed via flickr

Still, I managed to squeeze in one more puzzle. I picked up my whiteboard marker and started writing:

DONE
DOEN
DNOE
DENO
DNEO
ONED
ODNE

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photo by George Parrilla via flickr

The (Mathematical) Trouble with Pizza

Photo by George Parrilla via flickr.

Kitten complained that some math programs keep repeating the same kind of problems over and over, with bigger numbers: “They don’t get any harder, they just get longer. It’s boring!”

So we pulled out the Counting lessons in Competition Math for Middle School. [Highly recommended book!] Kitten doesn’t like to compete, but she enjoys learning new ideas, and Batterson’s book gives her plenty of those, well organized and clearly explained.

Today’s topic was the Fundamental Counting Principle. It was review, easy-peasy. The problems were too simple, until…

Pizzas at Mario’s come in three sizes, and you have your choice of 10 toppings to add to the pizza. You may order a pizza with any number of toppings (up to 10), including zero. How many choices of pizza are there at Mario’s?

[The book said 9 toppings, but I was skimming/paraphrasing aloud and misread.]

  • Can you figure out the answer?

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Quotable: What to Do When You’re Stuck

When a kid is feeling bad about being stuck with a problem, or just very anxious, I sometimes ask him to make as many mistakes as he can, and as outrageous as he can. Laughter happens (which is valuable by itself, and not only for the mood — deep breathing brings oxygen to the brain). Then the kid starts making mistakes. In the process, features of the problem become much clearer, and in many cases a way to a solution presents itself.

Maria Droujkova
Natural Math discussion of math club activities

Does It Work?

While I was collecting entries for the Math Teachers at Play #35 blog carnival, I ran across this post by Dave Lanovaz:


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dragon and moon

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:

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Quotable: Problem Solving

I do my best to make my students think, but they still try to become good little algorithm followers.

David Cox
The 2 Product Property


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dwarfswolves

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:

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Burnt Algebra

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?

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Calvin and Hobbes, math atheist

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:

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Math, Math, Math, math, mathh,,,,maaah,,,,by Aaron Escobar

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.]

HowToSolveIt2

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.]

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writing

Writing to Learn Math II

[Photo by Andy Hay.]

In addition to all the funny Google searches, I get plenty of normal inquiries about math topics. People come here looking for help with fractions, word problems, and math club activities — no surprise, those — but I would never have predicted the popularity of the search topic “writing in math class.”

Last year, I compiled a variety of math journal resources, but I’ve found many more since then, especially for older (high school and college) students. So if you’re looking for new ways to get your math students writing…

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function-machine

The Function Machine Game

Math concepts: odd numbers, even numbers, greater-than/less-than, rounding off, addition, subtraction, multiplication, division, fractions, negative numbers, prime numbers, square numbers, problem solving, mental math
Number of players: two or more
Equipment: pencil (or pen) and paper for every player

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ProblemSolvingToolBox

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.

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Junto books diagram

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.

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Bar diagram for Janie and Bill

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.

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Trouble with Percents

Can your students solve this problem?

There are 20% more girls than boys in the senior class.
What percent of the seniors are girls?

This is from a discussion of the semantics of percent problems and why students have trouble with them, going on over at MathNotations. (Follow-up post here.) Our pre-algebra class just finished a chapter on percents, so I thought Chickenfoot might have a chance at this one. Nope! He leapt without thought to the conclusion that 60% of the class must be girls. After I explained the significance of the word “than”, he solved the follow-up problem just fine.


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Have more fun on Let’s Play Math! blog: