Category Archives: Middle Elementary

Noticing Fractions in a Sidewalk

fraction-circle

My daughters didn’t want to admit to knowing me, when I stopped to take a picture of the sidewalk along a back street during our trip to Jeju. But aren’t those some wonderful fractions?

What do you see? What do you wonder?

Here is one of the relationships I noticed in the outer ring:

\frac{4 \frac {2}{2}}{20} = \frac {1}{4}

sidewalk

And this one’s a little trickier:

\frac{1 \frac {1}{2}}{12} = \frac {1}{8}

Can you find it in the picture?

Each square of the sidewalk is made from four smaller tiles, about 25 cm square, cut from lava rock. Some of the sidewalk tiles are cut from mostly-smooth rock, some bubbly, and some half-n-half.

I wonder how far we could go before we had to repeat a circle pattern?

Join the Summer Math Photo Challenge

How to Play: Take a walk around your neighborhood to see what sort of math you can find. Post a photo to Twitter or Instagram, with hashtags to mark it as a Summer Math Challenge photo. Describe what you’re seeing. Re-tweet and share others’ photos if you like. Encourage your friends to play!


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Socks Are Like Pants, Cats Are Like Dogs

Support This New Book from Natural Math

Socks Are Like Pants, Cats Are Like Dogs by Malke Rosenfeld and Gordon Hamilton is filled with a diverse collection of math games, puzzles, and activities exploring the mathematics of choosing, identifying and sorting. The activities are easy to start and require little preparation.

The publisher’s crowdfunding goal is $4,000. The book is almost ready to go to press, and I can hardly wait to see it!


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Review Game: Once Through the Deck

[Feature photo above by Shannon (shikiro famu) via Flicker (CC BY 2.0).]

Math Concepts: basic facts of addition, multiplication.
Players: one.
Equipment: one deck of math cards (poker- or bridge-style playing cards with the face cards and jokers removed).

The best way to practice the math facts is through the give-and-take of conversation, orally quizzing each other and talking about how you might figure the answers out. But occasionally your child may want a simple, solitaire method for review.

How to Play

Shuffle the deck and place it face down on the table in front of you. Flip the cards face up, one at a time.

For each card, say out loud the sum (or product) of that number plus (or times) the number you want to practice. Don’t say the whole equation, just the answer.

Go through the deck as fast as you can. But don’t try to go so fast that you have to guess! If you are not sure of the answer, stop and figure it out.

History

Brian at The Math Mojo Chronicles demonstrates the game in this video, which my daughter so thoroughly enjoyed that she immediately ran to find a deck of cards and practiced her times-4 facts. It’s funny, sometimes, what will catch a child’s interest.


Multiplication-Games

This game first appeared as part of my Times Table Series and will be included in the upcoming Math You Can Play multiplication book, planned for late 2015.


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Math Games with Factors, Multiples, and Prime Numbers

Students can explore prime and non-prime numbers with two free favorite classroom games: The Factor Game (pdf lesson download) or Tax Collector. For $15-20 you can buy a downloadable file of the beautiful, colorful, mathematical board game Prime Climb. Or try the following game by retired Canadian math professor Jerry Ameis:

Factor Finding Game

FactorFindingGame

Math Concepts: multiples, factors, composite numbers, and primes.
Players: only two.
Equipment: pair of 6-sided dice, 10 squares each of two different colors construction paper, and the game board (click the image to print it, or copy by hand).

On your turn, roll the dice and make a 2-digit number. Use one of your colored squares to mark a position on the game board. You can only mark one square per turn.

  • If your 2-digit number is prime, cover a PRIME square.
  • If any of the numbers showing are factors of your 2-digit number, cover one of them.
  • BUT if there’s no square available that matches your number, you lose your turn.

The first player to get three squares in a row (horizontal, vertical, or diagonal) wins. Or for a harder challenge, try for four in a row.


Feature photo at top of post by Jimmie via flickr (CC BY 2.0). This game was featured in the Math Teachers At Play (MTaP) math education blog carnival: MTaP #79. Hat tip: Jimmie Lanley.


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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.

Continue reading The Math Student’s Manifesto

Teaching the Standard Algorithms

[Feature photo above by Samuel Mann, Analytical Engine photo below by Roͬͬ͠͠͡͠͠͠͠͠͠͠͠sͬͬ͠͠͠͠͠͠͠͠͠aͬͬ͠͠͠͠͠͠͠ Menkman, both (CC BY 2.0) via Flickr.]

Babbage's Analytical Engine

An algorithm is a set of steps to follow that produce a certain result. Follow the rules carefully, and you will automatically get the correct answer. No thinking required — even a machine can do it.

This photo shows one section of the first true computer, Charles Babbage’s Analytical Engine. Using a clever arrangement of gears, levers, and switches, the machine could crank out the answer to almost any arithmetic problem. Rather, it would have been able to do so, if Babbage had ever finished building the monster.

One of the biggest arguments surrounding the Common Core State Standards in math is when and how to teach the standard algorithms. But this argument is not new. It goes back at least to the late 19th century.

Here is a passage from a book that helped shape my teaching style, way back when I began homeschooling in the 1980s…

Ruth Beechick on Teaching Abstract Notation

Understanding this item is the key to choosing your strategy for the early years of arithmetic teaching. The question is: Should you teach abstract notation as early as the child can learn it, or should you use the time, instead, to teach in greater depth in the mental image mode?

Beechick-EasyStartArithmetic

Abstract notation includes writing out a column of numbers to add, and writing one number under another before subtracting it. The digits and signs used are symbols. The position of the numbers is an arbitrary decision of society. They are conventions that adult, abstract thinkers use as a kind of shorthand to speed up our thinking.

When we teach these to children, we must realize that we simply are introducing them to our abstract tools. We are not suddenly turning children into abstract thinkers. And the danger of starting too early and pushing this kind of work is that we will spend an inordinate amount of time with it. We will be teaching the importance of making straight columns, writing numbers in certain places, and other trivial matters. By calling them trivial, we don’t mean that they are unnecessary. But they are small matters compared to real arithmetic thinking.

If you stay with meaningful mental arithmetic longer, you will find that your child, if she is average, can do problems much more advanced than the level listed for her grade. You will find that she likes arithmetic more. And when she does get to abstractions, she will understand them better. She will not need two or three years of work in primary grades to learn how to write out something like a subtraction problem with two-digit numbers. She can learn that in a few moments of time, if you just wait.

— Ruth Beechick
An Easy Start in Arithmetic (Grades K-3)
(emphasis mine)


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2015 Mathematics Game

[Feature photo above by Scott Lewis and title background (right) by Carol VanHook, both (CC BY 2.0) via Flickr.]

2015YearGame

Did you know that playing games is one of the Top 10 Ways To Improve Your Brain Fitness? So slip into your workout clothes and pump up those mental muscles with the Annual Mathematics Year Game Extravaganza!

For many years mathematicians, scientists, engineers and others interested in math have played “year games” via e-mail. We don’t always know whether it’s possible to write all the numbers from 1 to 100 using only the digits in the current year, but it’s fun to see how many you can find.

Math Forum Year Game Site

Rules of the Game

Use the digits in the year 2015 to write mathematical expressions for the counting numbers 1 through 100. The goal is adjustable: Young children can start with looking for 1-10, middle grades with 1-25.

  • You must use all four digits. You may not use any other numbers.
  • Solutions that keep the year digits in 2-0-1-5 order are preferred, but not required.
  • You may use +, -, x, ÷, sqrt (square root), ^ (raise to a power), ! (factorial), and parentheses, brackets, or other grouping symbols.
  • You may use a decimal point to create numbers such as .2, .02, etc., but you cannot write 0.02 because we only have one zero in this year’s number.
  • You may create multi-digit numbers such as 10 or 201 or .01, but we prefer solutions that avoid them.

My Special Variations on the Rules

  • You MAY use the overhead-bar (vinculum), dots, or brackets to mark a repeating decimal. But students and teachers beware: you can’t submit answers with repeating decimals to Math Forum.
  • You MAY NOT use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. Math Forum allows these, but I’ve decided I prefer my arithmetic straight.

Click here to continue reading.