Is sloppy notation for solving linear systems reducing understanding?

Whenever I prepare to teach a concept in my class I typically flip through my previous years’ stuff (notes, handouts, blog posts) to see how I approached things before. I also look in the textbook, especially to see the notation that is used.

I’m about to start teaching my MPM2D class how to solve linear systems. I’m going to start with the “Substitution” method, so I’ve been browsing the archives.

Solving linear systems is challenging, in my experience. Students tend to lack a good understanding of why we can substitute expressions for variables, and then they assume it’s simply arcane magic they need to memorize (albeit temporarily).

Today it occurred to me that our notation might be causing some of the problems, and improving clarity might improve understanding.

The Graphical Interpretation

When we are solving a linear system of two equations, we are looking for the point(s) of intersection between the lines represented by the equations. Graphically, we want to find the point at which the lines cross each other. Students generally are fine with this idea. They get the picture (haha).

linearSystem.png

The trouble starts with algebra (doesn’t it always?).

The Algebraic Interpretation

We have two linear equations, and we are looking for the point(s) which satisfy both equation. That is, we want to find an x -value for which both lines have the same y -value.

The equations are statements about how the x – and y -coordinates of points on the lines are related. For example, here are two lines:

y=2x+5

y=\frac{1}{2}x-7

These lines have a single intersection point, at (-8,-11) . So -8 is the x -value for which both lines have the same y -value, -11.

The Substitution Method

To find this, my textbook, my past self, and my colleagues would all employ the Substitution Method, which says that if you can isolate a variable in one equation, you can substitute the corresponding expression in for that variable in the other equation to find the intersection point. This works because we are looking for the point for which both equations are true simultaneously.

In the example above, the first equation tells us that y=2x+5 . If we apply that restriction to the second equation, by replacing y with 2x+5 , we get

2x+5 = \frac{1}{2}x-7

This is the substitution for which the method is named. We now have an equation with one unknown, x . Solving this equation tells us that at the intersection point we have x=-8 . We can now substitute x=-8 back into either original equation and solve to get y=-11 .

My Concern

Maybe it doesn’t really matter, but I’m concerned with how we often write out the algebra without explanatory words and with imprecise notation. I don’t mean during a lesson; in that first, iconic example in class we write out words like crazy, trying to make all the magic connections apparent. I mean during later solutions – practice exercises, one-on-one help, and student work.

I think we should be saying this instead:

Suppose there is a point of intersection (x_1,y_1) . Then

y_1=2x_1+5 and y_1=\frac{1}{2}x_1-7

2x_1+5 =\frac{1}{2}x_1-7

\frac{3}{2}x_1=-12

x_1=-8

y_1=2x_1+5

y_1=2(-8)+5

y_1=-11

So the point of intersection is (-8,-11) .

See the tiny, subtle difference? We’re picking a specific point and calling it (x_1,y_1) .

The reason I’m wondering if this might be better is that students often ask me why they can substitute 2x+5 in for y in the other equation. And their concern is reasonable, because 2x+5 isn’t always equal to \frac{1}{2}x-7 ; that’s only true at the intersection point.

We write out the sentence “When the lines intersection we have 2x+5 = \frac{1}{2}x-7 ” and go from there during our lesson or on our notes. That’s what my textbook does. But we rarely require this “extra” writing, and instead let a page of symbols replace clear communication and thorough thinking.

Should I try it?

Do you think I should use (x_1,y_1) and see what happens? Does it even matter? I’m not just being pedantic*; I’m hoping that better communication will lead to better understanding.

Let me know if you’ve done this before, if you think I should try it, or if I’m out to lunch on this one.

*I’m also being pedantic, I’m sure. 

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Solving quadratics – what’s most important?

I’m currently teaching MPM2D, Grade 10 Academic math, and I’ve been having conversations about what’s most important for students to master in the quadratics part of the course. We’ve learned about the different forms of quadratic equations (standard, factored, vertex) and how transformations apply to give us vertex form, and we’ve started factoring.

The expectations I’m looking at right now are summarized here:

  • expand and simplify second-degree polynomial expressions, using a variety of tools and strategies
  • factor polynomial expressions involving common factors, trinomials, and differences of squares using a variety of tools and strategies
  • determine,through investigation,and describe the connection between the factors of a quadratic expression and the x -intercepts (i.e.,the zeros) of the graph of the corresponding quadratic relation, expressed in the form y=a(x-r)(x-s)
  • interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x -intercepts of the corresponding relations
  • express y = ax^2 + bx + c in the form y=a(x-h)^2+k  by completing the square in situations involving no fractions, using a variety of tools
  • sketch or graph a quadratic relation whose equation is given in the form y = ax^2 + bx + c , using a variety of methods
  • explore the algebraic development of the quadratic formula
  • solve quadratic equations that have real roots, using a variety of methods

Of all that stuff, what’s most important? There is time pressure in this course, and I’m trying to focus on what’s truly essential moving forward. I also want to make sure that my evaluation reflects the importance of each understanding and skill.

Have you taught this course (or the sequels, MCR3U and MCF3M)? What do you think?

Which kind of calculator promotes good algebraic thinking?

I teach high school math. Students bring scientific calculators to class, or they sometimes have to borrow one from me. I have two types available: immediate execution calculators and formula calculators. I’ve been wondering lately whether one type of calculator is better for learning algebra than the other.

Here’s how they work (see Wikipedia for a longer explanation: https://en.wikipedia.org/wiki/Calculator_input_methods).

Immediate Execution

The TI-36X Solar 2004 version immediate execution calculator.

These calculators work by performing calculations along the way as you type in values and operations. For example, you can evaluate the expression

\sin(3 \times 45)

by typing 3, multiply, 45, then the sine key. As you press operations and operands the calculator will evaluate what it can according to the rules of order of operations, or BEDMAS. For binary operands (those taking two values to produce a result, like multiplication), you put the values in order. For unary operations (those taking just one value, like squaring or taking a sine), the value must be present on the calculator screen when you press the operator key. These calculators usually have a bracketing feature to allow the user to work through complex expressions without using memory storage.

Formula

The Sharp EL-510R formula calculator.

These calculators work by waiting until the user has typed in a complete expression to evaluate, then evaluating the entire expression. The order of button-pushing is pretty much as the symbols are written in the expression, making them easier to use for a lot of folks. Once a value is calculated, it’s stored in an “answer” variable in case it’s needed for the next evaluation.

Algebraic Expressions and BEDMAS

When we write out algebraic expressions, we have a number of conventions to follow. The most important convention is order of operations, which people usually learn to remember with the mnemonic BEDMAS or PEDMAS:

  • Brackets (Parentheses)
  • Exponents
  • Division and Multiplication
  • Addition and Subtraction

When evaluating (simplifying) an expression, you first simplify the smaller expressions inside brackets. Then you evaluate exponents, then division and multiplication in the order they appear, and finally addition and subtraction in the order they appear. It’s useful to think of brackets as isolating sub-expressions, which then follow the same rules. It’s also useful to think of this order as the “strength” of the operation: multiplication is a stronger operation than addition, so it holds its operands more tightly together, and it gets evaluated first.

When a student is learning order of operations, it often feels like a set of arcane rules. There is no reason, from the student perspective, that it has to be this way. In fact, it didn’t really need to be this way, but the convention was established and now it’s important to abide by it (if you want to be understood, that is).

How a calculator helps (and hinders) learning arithmetic

People often lament that today’s youth can’t perform basic arithmetic in their head. It’s unfortunately true; I often see students reach for their calculator to evaluate 35 \div 5 or even 4 \times 6 . These are facts which prior generations had drilled relentlessly and now have available as “instant” knowledge. Younger people typically haven’t spent enough time practising these computations to develop facility with them. This is partly because the calculator is so readily available.

(Aside for parents: If you have kids, please do make them practise their age-appropriate facts. It’ll help them in the same way practising reading makes things easier)

This will draw a lot of heat, I’m sure, but I think calculators do have a strong place in even K-6 learning. They let students explore quickly without the burden of computation getting in the way of non-computational learning. It’s the same effect that web-based, dynamic geometry software can have on learning relationships between figures, lines, etc. (if you’re looking for awesome dynamic geometry software, try GeoGebra – free and wonderful).

But calculators are a hindrance when students are learning to compute fluently. They allow a student to bypass some of the thinking part of the exercise. Don’t let students (or your kids) use a calculator when they don’t have to. Only use them when students need the speed for the task they’re completing.

How a calculator helps (and hinders) learning algebra (?)

Here’s the part I don’t know about, but I’m speculating about.

I think immediate execution calculators require students to understand the algebraic expressions we write, where formula calculators bypass the thinking part of evaluating expressions.

As with arithmetic, if practising evaluating expressions is not part of the learning, and might be getting in the way of the goals for learning, then either type of calculator is fine.

But as students are developing their understanding of algebra and the order of operations, the immediate execution calculator displays the results of operations as they are evaluated, while the formula calculator obscures the evaluations in favour of a single result.

When a student types 5, add, 6, square, equals into an immediate execution calculator, they see the value 36 as soon as they press the square button. There is a reminder that the square operation is immediate. Similarly when a student wants to evaluate \sin(30+45) they must type 30, add, 45, equals*, then sine, emphasizing that the bracketed portion has to be evaluated first (i.e. before the sine function is applied).

*A student can use brackets, which is equivalent to pressing equals before sine. Also, I hope anyone using the sine function knows that 30+45 is 75 and doesn’t need a calculator’s help for the addition.

Is there research?

I perused the InterTubes to find research into this question, but either it’s not out there or I’m not skilled enough to find it.

I want to know whether one calculator is better than the other for a student who is learning to evaluate expressions.

Has no one looked into this? Help?

Summative Task for Quadratics – #MCF3M

My Grade 11 e-Learning math class is completing a unit on quadratic equations. I have a few things happening for their summative assessment, but the part I find most interesting is the following “experiment”. It’s heavily based on the Leaky Tower task from TIPS4RM at EduGAINS.ca. I’m going to test it out tonight with my kids before I finalize the evaluation criteria and post the task. If you have feedback, I’d love to hear it. I’ll be adding photos to help explain the setup.

Leaking Bottle – Summative Task – Part 1

You’ll be completing a short experiment and writing a report to go with it. You can get help from a classmate, family member, etc. while running the experiment, but just as an extra set of hands. No one should be helping you with the math part.

Preparation

Gather the supplies you’ll need:

  • a clear, disposable, empty, plastic bottle
  • a ruler
  • a watch, phone, or other time-keeping device OR a video-recording device.

—photo here—

Carefully poke a hole in the bottle about 3cm from the bottom. Seriously, be careful here. You might try using something sharp, like a pin or a nail, to start the hole, then widen it with a pencil. You want the final hole to have a diameter of 3-7mm. Don’t worry about being super-precise.

—photo here—

Hold a ruler next to your bottle, or tape a ruler to your bottle if you need both of your hands free. You want to be able to measure the water level, so put the “zero” end of the ruler at the bottom.

—photo here—

Cover the hole and fill the bottle with water. If your bottle has a tapered top (like the one pictured here), only fill it up in the cylindrical section (i.e. before it starts to narrow). You can cover the hole with your finger, or you might try a piece of tape (if you use tape, fold the end on itself so it’s easier to remove).

—photo here—

Data Collection

If you’re recording video (easier, I think), start recording. If you’re just using a watch or other timing device, wait for a “good” time, like a whole minute, for a starting point.

Uncover the hole, letting the water in the bottle flow out into a sink or another container. Don’t make a mess; nobody wants a mess.

—photo here—

If you’re using a watch, use the ruler to record the water level every 5 or 10 seconds or so. Pick an easy time to keep track of. Record measurements until the flow of water stops.

If you’re recording a video, let the water finish flowing out, then stop the video. Play the video back, noting the height of the water every 5 or 10 seconds or so.

Analysis

You now have a table of values: time (independent variable) and height measurements (dependent variable). If you didn’t get good data (you lost track of time, the video didn’t work, etc.), perform the experiment again. It doesn’t take long.

  1. Using Desmos, create a scatter plot for your measurements.
  2. Find an equation to fit the data as best you can.
  3. Identify the key points on the graph.
  4. How should the equation you found be restricted? i.e. what should the domain and range be?
  5. Write the equation you found in Standard Form and Vertex Form.

Leaking Bottle – Summative Task – Part 2

One small change

Repeat the above experiment, but this time put another hole about 7-10cm above the first one. Uncover them at the same time, so water will flow out of both holes.

—photo here—

Your analysis will be a little more complex, because you won’t have a single, nice equation that can accurately model the data.

  1. Using Desmos, create a scatter plot for your measurements.
  2. Find an equation (or equations!) to fit the data as best you can.
  3. Identify the key points on the graph.
  4. How should the equation(s) you found be restricted? i.e. what should the domain(s) and range(s) be?
  5. Write the equation(s) you found in Standard Form and Vertex Form.

OrdOp – the math card game using Order of Operations

I played a game with my math students today. It’s called OrdOp, and we used it to practice our mental computation skills. We used standard playing cards, but the “real” version uses cards numbered 0 to 25.

If you want to try it, here are the rules (and printable cards).

OrdOp – standard playing card rules

OrdOp – custom cards and rules

OrdOp – how to play (video)

Physical phenomena for quadratic relations

I’m working on a quadratics unit for my MCF3M (online) and MBF3C (F2F) classes. The Ms need to be able to do a few more things, but both groups have to be able to model quadratic “stuff” using an equation.

I’ll be using desmos.com pretty heavily, and I got some great ideas from Heather Theijsmeijer (@HTheijsmeijer).

I’m trying to find some examples of physical phenomena that I can have students in either class play with to practise/demonstrate modelling. Here are my ideas so far:

Throwing or Bouncing a Ball

This is the first thing I thought of. A ball follows a nice parabolic path in the air if it’s moving horizontally.

My plan is to have students use a phone or camera to record a video or a rapid burst of images, overlay a set of axes, and fit a curve to the path. My iPhone can record at 120fps, which is great. I also found a handy post at Stack Overflow that explains how to extract images from a video, so that might be helpful too.

Pouring Water from a Hose

Set your hose at an angle, turn on the water, and snap a picture. Parabola. Beauty. Maybe put a piece of grid paper behind it, or just import it into Desmos.

Rolling a Ball Up An Incline

This one’s messy, but I think it might work.

Dip a marble in some ink or paint. Set a piece of grid chart paper on an incline (say, a piece of plywood) and roll the ball on an angle up the paper. When it crests and rolls back down, it should have left parabolic paint. On graph paper.

Other ideas?

I’m open to suggestions. I have stuff like photos of suspensions bridges, etc., but I really want something students can generate on their own.

Key Learnings from our e-Learning Collaborative Inquiry

I learned a lot over the past couple of days. My board brought in Donna Fry (@fryed) and Tim Robinson (@timrobinsonj) to guide our group’s learning, and several folks from the board’s central program team were helping out as well. Check out #elADSB for a bunch of new Twitterers, too.

Here are some prompts that we were asked to respond to, and here are my (rather brief) thoughts.

1. ​​What questions, wonderings do you have ​with respect to our problem of practice?

How do we encourage collaboration that isn’t false collaboration? How do we ensure that the collaboration is meaningful and valuable? I don’t want to force collaboration among students when it doesn’t make sense.

What does a rich task look like in a course which has a lot of technical, procedural learning? Is it enough that there is inquiry in the task, or does it need to be “authentic” and relevant? I’m concerned about fauxthenticity: forcing an unrealistic application out of a concept just so that we can say it’s “real-world”.

2. What key learnings have you had over yesterday and today?

I need to provide more structure for self-monitoring and self-reflection for my students, and then ensure that they follow through on that self-assessment. They’re still kids, and they need a firmer hand with organization and checking on their learning, or they may neglect important stuff (or even delude themselves into thinking they understand concepts that they don’t).

I need to consider including more face time in my course – maybe having “Math Chat With Mr. G” or something on certain afternoons… Even if some students can’t get me live, I shouldn’t prevent it for those who can.

3. Moving forward, as a result of your learning, how do you envision your courses/practice changing?

I need to be more careful about connecting students with each other instead of encouraging them to rely on me.

I need to set up a fast, reliable, easy-for-me-to-check-in-on self-monitoring system for my students.

I will set up Adobe Connect or Google Hangout time periodically.

I’m going to work on figuring out when requiring collaboration makes sense, when students should be working independently, and when they just have a choice (most often?).

I’m going to make some plans and then ask someone else to give me suggestions about them before I implement them. :)