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 the expanded paperback edition of Let’s Play Math: How Families Can Learn Math Together and Enjoy It. Coming in early 2016 to your favorite online bookstore…

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Math Teachers at Play #94 via mathematicsandcoding

94

Check out the new math education carnival at Tom Bennison’s blog. Games, puzzles, teaching tips, and all sorts of mathy fun:

If you enjoy this carnival, why not send in a blog post of your own for next month? We love posts on playful ways to explore and learn math from preschool discoveries through high school calculus.

Entries accepted at any time!


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Calling All Math Teacher Bloggers and Homeschoolers: Carnival Time!

by Bob Jagendorf via flickr[Image by Bob Jagendorf (CC BY-NC 2.0) via Flickr.]

The monthly Math Teachers at Play (MTaP) math education blog carnival is almost here. If you’ve written a blog post about math, we’d love to have you join us! Each of us can help others learn, so in a sense we are all teachers.

Posts must be relevant to students or teachers of school-level mathematics (that is, anything from preschool up to first-year calculus). Old posts are welcome, as long as they haven’t been published in past editions of this carnival.

Have you noticed a new math blogger on your block that you’d like to introduce to the rest of us? Feel free to submit another blogger’s post in addition to your own. Beginning bloggers are often shy about sharing, but like all of us, they love finding new readers.

Don’t procrastinate: The deadline for entries is this Friday, January 22. The carnival will be posted next week at mathematicsandcoding.

Would You Like to Host the Carnival?

Thank you so much to the volunteer bloggers who have stepped up to carry this MTaP math education blog carnival through the years! I would never be able to keep the carnival going on my own.

If you’d like to join in the fun, we have plenty of openings for 2016. Read the instructions on our Math Teachers at Play page. Then leave a comment or email me to let me know which month you’d like to take.

Explore the Other Math Carnivals

While you’re waiting for next week’s Math Teachers at Play carnival, you may enjoy:


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Hotel Infinity: Part Five

Hotel Infinity1Tova Brown concludes her exploration of the Hilbert’s Hotel Paradox with a look at the cardinality of the real numbers.

You run a hotel with an infinite number of rooms. You pride yourself on accommodating everyone, even guests arriving in infinitely large groups — but some infinities are more infinite than others, as it turns out.

Tova Brown
Hotel Infinity: Part Five

Check out Tova Brown’s growing collection of videos that explore advanced math concepts through story-telling.


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Hotel Infinity: Part Four

Hotel Infinity1Tova Brown dives deeper into Hilbert’s Hotel Paradox, considering the difference between rational numbers and reals.

You run an infinitely large hotel, and are happy to realize that you can accommodate an infinite number of infinite groups of guests.

However, a delicate diplomatic situation arises when a portal to another universe opens, introducing a different kind of guest, in a different kind of group.

Can you make room for them all?

Tova Brown
Hotel Infinity: Part Four

Click here to read Part Five…


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Hotel Infinity: Part Three

Hotel Infinity1Tova Brown continues to examine Hilbert’s Hotel Paradox, pondering infinite sets of infinite sets.

As the proprietor of an infinitely large hotel, you pride yourself on welcoming everyone, even when the rooms are full. Your hotel becomes very popular among infinite sports teams, as a result.

Recruitment season presents a challenge, however, when many infinite teams arrive at once. How many infinite teams can stay in a single infinite hotel?

Tova Brown
Hotel Infinity: Part Three

Click here to read Part Four…


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Hotel Infinity: Part Two

Hotel Infinity1Tova Brown explores the second part of Hilbert’s Hotel Paradox. What’s infinity plus infinity?

Running an infinite hotel has its perks. Even when the rooms are full you can always find space for new guests, so you proudly welcome everyone who appears at your door.

When two guests arrive at once, you make room. When ten guests arrive, you accommodate them easily. When a crowd of hundreds appears, you welcome them all.

Is there no limit to your hospitality?

Tova Brown
Hotel Infinity: Part Two

Click here to read Part Three…


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