Understanding Math: Algebraic Multiplication

Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a Difference?; Understanding Math, Part 4: Area of a Rectangle; and Understanding Math, Part 5: Multiplying Fractions.

Understanding-AlgebraWe’ve examined how our vision of mathematical success shapes our children’s learning. Do we think math is primarily a tool for solving problems? Or do we see math as a web of interrelated concepts?

Instrumental understanding views math as a tool. Relational understanding views math as an interconnected system of ideas. Our worldview influences the way we present math topics to our kids. And our children’s worldview determines what they remember.

In the past two posts, we looked at different ways to understand and teach rectangular area and fraction multiplication. But how about algebra? Many children (and adults) believe “math with letters” is a jumble of abstract nonsense, with too many formulas and rules that have to be memorized if you want to pass a test.

Which of the following sounds the most like your experience of school math? And which type of math are your children learning?

Instrumental Understanding: FOIL

Every mathematical procedure we learn is an instrument or tool for solving a certain kind of problem. To understand math means to know which tool we are supposed to use for each type of problem and how to use that tool — how to categorize the problem, remember the formula, plug in the numbers, and do the calculation.

When you need to multiply algebra expressions, remember to FOIL: multiply the First terms in each parenthesis, and then the Outer, Inner, and Last pairs, and finally add all those answers together.

The FOIL method for multiplying two binomials.
The FOIL method for multiplying two binomials.

Relational Understanding: The Area Model

Each mathematical concept is part of a web of interrelated ideas. To understand mathematics means to see at least some of this web and to use the connections we see to make sense of new ideas.

The concept of rectangular area has helped us understand fractions. Let’s extend it even farther. In the connected system of mathematics, almost any type of multiplication can be imagined as a rectangular area. We don’t even have to know the size of our rectangle. It could be anything, such as subdividing a plot of land or designing a section of crisscrossed colors on plaid fabric.

We can imagine a rectangle with each side made up of two unknown lengths. One side has some length a attached to another length b. The other side is x units long, with an extra amount y stuck to its end.

We don’t know which side is the “length” and which is the “width” because we don’t know which numbers the letters represent. But multiplication works in any order, so it doesn’t matter which side is longer. Using the rectangle model of multiplication, we can see that this whole shape represents the area \left ( a+b \right )\left ( x+y \right ) .

An algebraic rectangle: each side is composed of two unknown lengths joined together.
An algebraic rectangle: each side is composed of two unknown lengths joined together.

But since the sides are measured in pieces, we can also imagine cutting up the big rectangle. The large, original rectangle covers the same amount of area as the four smaller rectangular pieces added together, and thus we can show that \left ( a+b \right )\left ( x+y \right )=ax+ay+bx+by .

Four algebraic rectangles: the whole thing is equal to the sum of its parts.
Four algebraic rectangles: the whole thing is equal to the sum of its parts.

With the FOIL formula mentioned earlier, our students may get a correct answer quickly, but it’s a dead end. FOIL doesn’t connect to any other math concepts, not even other forms of algebraic multiplication. But the rectangular area model will help our kids multiply more complicated algebraic expressions such as \left ( a+b+c \right )\left ( w+x+y+z \right ) .

The rectangle model of multiplication helps students keep track of all the pieces in a complex algebraic calculation.
The rectangle model of multiplication helps students keep track of all the pieces in a complex algebraic calculation.

Not only that, but the rectangle model gives students a tool for making sense of later topics such as polynomial division. And it is fundamental to understanding integral calculus.

In calculus, students use the rectangle model of multiplication to find irregular areas. The narrower the rectangles, the more accurate the calculation, so we imagine shrinking the widths until they are infinitely thin.
In calculus, students use the rectangle model of multiplication to find irregular areas. The narrower the rectangles, the more accurate the calculation, so we imagine shrinking the widths until they are infinitely thin.

To be continued. Next up, Understanding Math Part 7: The Conclusion…


CREDITS: “Math Workshop Portland” photo (top) by US Department of Education via Flicker (CC BY 2.0). LPM-ebook-300This is the sixth post in my Understanding Math series, adapted from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, available at your favorite online book dealer.

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2016 Mathematics Game

PostcardNewYearsResolutionSoapBubbles1909

[Feature photo above from the public domain, and title background (below) by frankieleon (CC BY 2.0) via Flickr.]

2016-math-game

Have you made a New Year’s resolution to spend more time with your family this year, and to get more exercise? Problem-solvers of all ages can pump up their (mental) muscles with the Annual Mathematics Year Game Extravaganza. Please join us!

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 2016 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 2-0-1-6 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 use a double factorial, n!! = the product of all integers from 1 to n that have the same parity (odd or even) as n. I’m including these because Math Forum allows them, but I personally try to avoid the beasts. I feel much more creative when I can wrangle a solution without invoking them.

Click here to continue reading.

November Math Calendars

High school math teacher Chris Rime has done it again. Check out his November 2015 printable math calendars for Algebra 1 (in English or Spanish), Algebra 2, and Geometry students. Enjoy!

algebra-2-november-2015-preview

Things to Do with a Math Calendar

At home:
Post the calendar on your refrigerator. Use each math puzzle as a daily review “mini-quiz” for your children (or yourself).

In the classroom:
Post today’s calculation on the board as a warm-up puzzle. Encourage your students to make up “Today is…” puzzles of their own.

As a puzzle:
Cut the calendar squares apart and trim off the dates. Then challenge your students to arrange them in ascending (or descending) order.

Make up problems to fill a new calendar for next month.
And if you do, please share!


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Math Teachers at Play #76

StudentTeam

76[Feature photo (above) by U.S. Army RDECOM. Photo (right) by Stephan Mosel. (CC BY 2.0)]

On your mark… Get set… Go play some math!

Welcome to the 76th edition of the Math Teachers At Play math education blog carnival — a smorgasbord of links to bloggers all around the internet who have great ideas for learning, teaching, and playing around with math from preschool to pre-college.

By tradition, we start the carnival with a puzzle in honor of our 76th edition. But if you would like to jump straight to our featured blog posts, click here to see the Table of Contents.

Continue reading Math Teachers at Play #76

The Linear Inequality Adventures of Ohio Jones

Ohio Jones 2

Ohio Jones 1

Last week, Kitten and I reached her textbook’s chapter on graphing linear equations, and a minor mistake with negative numbers threw her into an “I can’t do it!” funk. It’s not easy teaching a perfectionist kid.

Usually her mood improves if we switch to a slightly more advanced topic, and luckily I had saved these worksheets on my desktop, waiting for just such an opportunity. Today’s lesson:

  • Some fun(ish) worksheets
    “For tomorrow, students will be graphing systems of inequalities, so I decided to create a little Ohio Jones adventure (Indiana’s lesser known brother)…”

I offered to give her a hint, but she wanted to try it totally on her own. It took her about 40 minutes to work through the first few rooms of the Lost Templo de los Dulces and explain her solutions to me. I’m sure she’ll speed up with experience.

So far, she’s enjoying it much more than the textbook lesson. It’s fascinating to me how the mere hint of fantasy adventure can change graphing equations from boring to cool. Thanks, Dan!


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Math Teachers at Play #70

Reimann-hexagon

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.

Math Playtime With Blocks

Feature video by Stuart Jeckel via youtube.

DO Try This at Home

And ask questions!


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