An Old Friend: "X Marks the Spot"

There is a way to teach factoring quadratics that seems to be catching on at my school. I'd like to take credit for it, but I've been doing it this way for a while and I honestly can't remember if I came up with this myself or not. If you're reading this and I stole it from you, I am in your debt.

First, you get the students accustomed to multiplying binomials using the box method, and factoring basic quadratics by using the box method backwards. I've made notes (download) that walk you through both processes.

Then you teach the students this trick, which I call "x marks the spot" for no particular reason: I give the students a series of large X marks, each with a number above it and a number below it. The students have to write numbers on either side of the X that add to make the top value, and that multiply to make the bottom value. For example:

In this case, you want two numbers that will add to make 10, and multiply to make 16.

Yes indeed, they are 8 and 2. So you would write 8 on the left, and 2 on the right, or the other way around. Students love these (here's a page of them).

Once you've got everybody comfortable with X marks the spot and with using the box, you put these two pieces together, and you get a technique that will directly factor any quadratic, or determine that it's not factorable, if that's the case.

Let's say you want to factor  6x² – 11x – 10.   First you make an X :

 The number on the top of the X is the value of "b" in the quadratic expression; the number on the bottom of the X is the value of "ac" in the quadratic expression (6 * -10 = -60).

We find the other two numbers as before.

-15 and 4 add to make -11, and multiply to make -60.

Now we bring in the box.

As always, the first and last term of the quadratic are loaded into the first and last cells in the box.

Now, the numbers we got in the X  (the -15 and the 4) become the coefficients on the linear parts.

The box is full. We can now start anywhere- with any row, or any column- and extract the greatest common factor. We write this outside the box.

The rest of it falls into place.

We have it now:  6x² – 11x – 10  =  (2x - 5)(3x + 2) .

A fellow teacher recently created some terrific materials for this lesson: a graphic organizer and a scaffolded practice packet.

It's not the most meaningful way to factor quadratics, but it works every time. The main reason I like to teach it this way is that even though it's a long process, it's easy to break the learning process up into bite-sized pieces. Finding the greatest common factor of two terms is a skill that can be learned on a different day than factoring simple quadratics with the box, and X marks the spot is just a fun activity on its own. Once everyone is comfortable with all the separate pieces, it's not such a leap to just put them all together.  

The Big Graph (How "Roaming" Began)

"Roaming" is a simple and adaptable format I came up with for having the students work on problem sets together. We do this just after learning a new concept, or sometimes as a test review activity.

It goes like this: let's say the students are learning to solve linear equations in one variable. Each student gets a worksheet with 10 boxes on it, and in each box is an expression, such as "3x" or "2 - 5x" or "x - 10x" or "5". For variety, I might make three or four versions of this worksheet, but it's not necessary for everyone to have a unique worksheet.

Now the students are required to get up and start networking with each other. (Some groaning.) Each student finds someone to work with. Let's say Dave is working with Lisa. Dave chooses an expression on his page to use, and so does Lisa. Dave writes Lisa's expression next to his own, and puts an equal sign between them; Lisa does the same with Dave's expression on her own paper. That way, Dave has the same equation on his paper that Lisa has on hers, only the sides of the equation are swapped. Dave and Lisa work together to solve the equation, making sure they check that they got the same answer:

Dave's paper: Dave's expression equals Lisa's expression.

Lisa's paper: Lisa's expression equals Dave's expression.

Or it might occur that they construct an equation that is a false statement, or possibly even an identity.

Then Dave and Lisa split off and find new people to work with. In this way, each person in the class works with 10 other people in the class.

That, anyway, is the theory. In practice, the biggest problem I've had with it is that certain students are likely to circumvent the requirement to actually work with each person as they roam around. For example, instead of working through the problem with Lisa, Dave might be inclined to ask her for an expression, write it down, and then immediately abandon Lisa and move on to Josh, either solving the equations quickly as he roams or just laying them all out before he even begins to solve them. Sometimes I've seen students try to stay seated and call out to their classmates around the room to get the necessary information!

So, of course, it doesn't run by itself- I have to float around and make sure it's really happening as I intended. But the benefits are pretty clear. If I'm a student who doesn't know how to solve a linear equation, it's going to give me some confidence to see one of my peers doing it (more than simply watching the teacher would); this activity simply gives me that benefit over and over again. It's also a kind of community building activity, because they can actually be face to face with many different people in the class that they might otherwise never talk to. And, not least of all, this activity (along with other formats, which I will share soon) gives me a way to avoid having everyone working solo on problem sets at their desks, something I don't mind doing occasionally but would never do routinely.

Roaming is quite adaptable- I've used it for many different concepts and purposes. For example, if we're practicing slope, each student could be given a single x-y pair, and then two students could find the slope of the line connecting their two points. The same student can use the same point over and over again and get a different result with every new partner.

The concept of roaming first occurred to me when I had a classroom with a nice 8-inch square tile floor. All math classrooms need to have a square tile floor. It is a low-tech way to make graphing into a kinesthetic experience. I used that floor to teach graphing in all kinds of ways, either by placing laminated dots or taping down twine.

Eventually, I realized that if each student were responsible for a single line graphed on the floor, the lines would intersect multiple times, creating a problem set; any two lines formed a system of linear equations, and if the students were to solve each system algebraically, they could check their work by locating the actual intersection point on the floor to see if it matched.

I gave them instructions something like this:

Directions

1.  Graph the line given to you above with your piece of twine and tape.

2.  Using another piece of tape, label your line with your name. (Do this on the edge somewhere, where it won’t get in the way.)

3.  As other classmates graph their lines, find someone whose line crosses yours. The two of you can then put your equations together as a system and find an exact solution (with algebra) for the point of intersection.

For each system, be sure to consider:

         Is it better to solve this system by elimination or by substitution?

4.  Record your work on each system, and write each person’s name by the system they were a part of.

5. Once you find a solution, use a sticky note, placed under the twine, to label the point of intersection.

By the end of the class, we've got a whole lot of lines on the floor and a whole lot of points marked, and it looks pretty great.

You'll notice that several of the lines are parallel. That's because we don't want to have too many intersections or it gets out of hand. It's best to map out exactly what you want the finished picture to look like ahead of time. That way, you can have the picture in front of you as the students lay down the twine, and if something doesn't look right you can easily identify the error.

This activity has only crashed and burned once. What that class really required was some careful coaching by example before we even got started. They didn't understand what to do and were too resistant to simply learn by doing.

Every other time, however, this has been a success. Some classes understand what I'm after right away and jump right into it; others need lots of explanation and prodding, but once things get rolling it's a smooth ride. And it's not unusual for a student to say something like, "Hey, can we do stuff like this every day?"

Absolute Value Statements (and: Could You Generalize That?)

There was a question once on the PSSA (Pennsylvania state test) when I taught in Philadelphia that presented a range of acceptable values for the width of a nail head produced by a factory. There was an ideal (target) width that the nails were supposed to have, and then there was a certain amount of error that was allowed. The students were supposed to recognize that the interval of acceptable values could be expressed by an absolute value inequality, and then write that inequality.

It struck me that this was a great way to teach absolute value statements in the first place. With this, we could show that such statements actually have some utility in the world. Here are two (long) lessons that resulted. 

The Nail Factory (download): This is a four-part lesson that approaches the PSSA question's premise from different directions.

In the first part, the students measure nails (or at least virtual nails, on paper) and decide which ones satisfy a certain inequality and which don't.

In the second part, the students draw their own nails to show examples of what would be acceptable and what would be rejected. This is the kind of lesson that would require short discussions along the way; if you have a document camera, you could show student work at this point. The savvy student would draw the good nails all about the same size, and draw the rejected nails by exaggerating short and long lengths.

In the third part, the inequality changes, but is not revealed. Students measure examples of accepted and rejected nails, and then select from four possible choices for what the new inequality might be. You may want to ask if any of the students have an idea for making this process more efficient- and you may notice some students measuring the shortest and longest nails out of the "accepted" pile to match up with the possible statements. We're getting closer to the essential idea here- that the value we're subtracting from x in the statement is actually the center of the interval of acceptable lengths.

In the fourth part, the inequality changes again. The students measure the nails once again, but this time, they must write the statement themselves, and are not given choices. Hopefully by this point, there have been enough discussions surrounding this idea that this task will be within their reach. Of course, students may produce slightly different versions of the inequality. In fact, there's plenty of room for controversy at more than one point in this lesson; some of the nails are very close to the edge of being acceptable, and students may disagree over whether a certain nail should be accepted or not.

Shot Putters (download): This lesson (officially titled "Exploring Absolute Value") gets at the same essential idea that an absolute value statement has a solution set with a center and a radius, but it also asks us to use that knowledge to write a general expression. I've had great discussions with students about the mystery guest "Rufus," and how we could write an expression for his shot put "score" in terms of the a, b, and c that appear in his statement. I realize that this is a highly manufactured scenario. Perhaps inspiration will strike one day and I'll think of a more authentic way to teach this. In the mean time, I like this lesson anyway, because there just aren't enough opportunities for the students to use logic to build variable expressions- at least, not enough non-routine examples. Most of them are exactly the same and therefore pose no challenge (e.g. "Farrah makes t-shirts, she has a one-time expense of $___, and spends $___ making each one..." etc.).

What Is a Logarithm?

In this activity (download), students build an intuitive understanding of what a logarithmic expression is. They also "discover," or empirically observe, how the basic laws of logarithms work. There's a lot of filling in the blanks here. Like many of my activities, the goal here is to start a conversation, so that the students can contribute just as much to the introduction of a new topic as the teacher.

Where Should My $1500 Go?

Topics: Exponential Functions, Compound Interest, and e.

This lesson (download) takes a student through the process of deciding at which bank to deposit $1500 in a savings account. The banks and their terms for interest are entirely fanciful, but the main purpose of the lesson is to guide the student to this understanding: that compounding interest at shorter and shorter intervals always increases the benefit, but by an amount that becomes less and less; in other words, the benefit approaches a limit as "n" goes to infinity. The extension activity (download) shows the student that for a principal of 1 and an interest of 100%, the limit of the formula for compound interest (as "n" goes to infinity) is equal to e.