Denise Gaskins' Let's Play Math

Understanding Math: Algebraic Multiplication

Posted on February 2, 2016February 8, 2024 by Denise Gaskins

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-Algebra We’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.

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.

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.

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.

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.

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, text added). This 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.

Math Teachers at Play #94 via mathematicsandcoding

Posted on February 1, 2016August 7, 2019 by Denise Gaskins

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

Maths Teachers at Play 94

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.

MTaP Submission Form

Entries accepted at any time!

Hotel Infinity: Part Five

Posted on January 15, 2016August 7, 2019 by Denise Gaskins

Hotel Infinity1 Tova 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.

Hotel Infinity: Part Four

Posted on January 14, 2016August 7, 2019 by Denise Gaskins

Hotel Infinity1 Tova 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…

Hotel Infinity: Part Three

Posted on January 13, 2016August 7, 2019 by Denise Gaskins

Hotel Infinity1 Tova 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…

Hotel Infinity: Part Two

Posted on January 12, 2016August 7, 2019 by Denise Gaskins

Hotel Infinity1 Tova 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…

Hotel Infinity: Part One

Posted on January 11, 2016August 7, 2019 by Denise Gaskins

Hotel Infinity1 Tova Brown’s introduction to Hilbert’s Hotel Paradox, a riddle about the nature of infinity…

Once upon a time, there was a hotel with an infinite number of rooms. You might be thinking this is impossible, and if so you’re right. A hotel like this could never exist in the real world.

But fortunately we’re not talking about the real world, we’re talking about math. And when we do math we can make up whatever rules we want, just to see what happens.

—Tova Brown
Hotel Infinity: Part One

Click here to read Part Two…

2016 Mathematics Game

Posted on January 1, 2016February 8, 2024 by Denise Gaskins

[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.

Math Teachers at Play #93 via Math Misery? Blog

Posted on December 23, 2015February 8, 2024 by Denise Gaskins

The December math education blog carnival is up for your browsing enjoyment:

Math Teachers At Play #93

The carnival is short this month, but full of treats. Enjoy!

“So here we are on December 23, 2015, the 93rd edition of Math Teachers At Play! As per tradition, what’s so fascinating about the number 93?

“First, it’s a prime number! No. Wait, that’s clearly false. So 93 is not a prime number. But that’s not very fascinating. Aha! But 93 is a semi-prime! since 93=31×3. Even more interesting is that 94 and 95 are semi-primes. So a question is, is there another triplet of positive integers that are also semi-primes? It’s a good question to ask your students! …”

Click here to go read the whole post at Math Misery? blog.

CREDITS: “celebrate trees” photo (top) by Lauren Manning via Flicker (CC BY 2.0).

Understanding Math: Multiplying Fractions

Posted on December 17, 2015February 8, 2024 by Denise Gaskins

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?; and Understanding Math, Part 4: Area of a Rectangle.

In this post, we consider the second of three math rules that most of us learned in middle school.

To multiply fractions, multiply the tops (numerators) to make the top of your answer, and multiply the bottoms (denominators) to make the bottom of your answer.

$fraction-rule$

Instrumental Understanding: Math as a Tool

$math-fractions$ Fractions confuse almost everybody. In fact, fractions probably cause more math phobia among children (and adults) than any other topic before algebra.

Children begin learning fractions by coloring or cutting up paper shapes, and their intuition is shaped by experiences with food like sandwiches or pizza. But before long, the abstraction of written calculations looms up to swallow intuitive understanding.

Upper elementary and middle school classrooms devote many hours to working with fractions, and still students flounder. In desperation, parents and teachers resort to nonsensical mnemonic rhymes that just might stick in a child’s mind long enough to pass the test.

The CrissCross Applesauce family is just one of the many fraction mnemonic tricks you can find online. For more information, check out NixTheTricks.com.

Relational Understanding: Math as a Connected System

Do you remember our exploration of the area of a rectangular tabletop?

Now let’s zoom in on our rectangle. Imagine magnifying our virtual grid to show a close-up of a single square unit, such as the pan of brownies on our table. And we can imagine subdividing this square into smaller, fractional pieces. In this way, we can see that five-eighths of a square unit looks something like a pan of brownies cut into strips, with a few strips missing:

One batch of brownies is one square unit, but part of the batch has been eaten. Now we have fractional brownies: five-eighths of the pan.

But what if we don’t even have that whole five-eighths of the pan? What if the kids came through the kitchen and snatched a few pieces, and now all we have is three-fourths of the five-eighths?

3/4 of 5/8: We can make a fraction of a fraction by cutting the other direction. We cut the strips into fourths, and the kids ate one part of each strip.

How much of the original pan of brownies do we have now? There are three rows with five pieces in each row, for a total of 3 × 5 = 15 pieces left — which is the numerator of our answer. And with pieces that size, it would take four rows with eight in each row (4 × 8 = 32) to fill the whole pan — which is our denominator, the number of pieces in the whole batch of brownies. So three-fourths of five-eighths is a small rectangle of single-serving pieces.

Compare the pieces we have left to the original batch. Each of the numbers in the fraction calculation has meaning. Can you find them all in the picture?

$fraction-rule$

Notice that there was nothing special about the fractions 3/4 and 5/8, except that the numbers were small enough for easy illustration. We could imagine a similar pan-of-brownies approach to any fraction multiplication problem, though the final pieces might turn out to be crumbs.

Of course, children will not draw brownie-pan pictures for every fraction multiplication problem the rest of their lives. But they need to spend plenty of time thinking about what it means to take a fraction of a fraction and how that meaning controls the numbers in their calculation. They need to ask questions and to put things in their own words and wrestle with the concept until it makes sense to them. Only then will their understanding be strong enough to support future learning.

Click here to continue reading: Understanding Math Part 6, Algebraic Multiplication…

CREDITS: “School Discussion” photo (top) by Flashy Soup Can via Flicker (CC BY 2.0, text added). This is the fifth 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.