I am motivated by my recent experiences tutoring math students to write the following missive on discipline associated with tackling new problem types. I don’t feel like it is something we should be using directly as tutors, but I often find myself going through this 5-step process by example with the current problem type, so that the student, and I, both know at the end that if s/he simply practices the method we have agreed on, s/he’ll be able to accurately and quickly do the next such problem that arises.
HOW TO TACKLE NEW MATHEMATICS PROBLEM TYPES
- Understand the definitions surrounding these types of problems. Then understand why it might be important to know how to do this. It might take some reading and/or asking to know this, but knowing why is important because it keeps you motivated. Be able to say why it’s important to you. You might have a low level why and a high level why – see the example below for what this means.
- Work through a few problems from the book.
- Use these examples to craft a solution method that works for you. The book presents a method. A teacher, professor or tutor will probably have a related method, but with some differences. Any valid method can be used as long as you can show your work with it, since teachers usually ask you to show your work. The key is that the method be something you can remember and repeat accurately and quickly.
- Memorize things you need to memorize to help you with your accuracy and speed.
- Practice what you need to practice, until you can do it quickly and accurately.
Example: let the problem type be finding the slope of a line from a pictured graph of that line. Let’s go through what each of these 5 steps above might look like.
- Slope.
- Definition of slope: the angle, inclination, steepness, or gradient of a straight line. Defined to be positive if it rises as one moves to the right, and negative if it descends as one moves to the right.
- Low level why: if you know the slope, there are a bunch of other questions about the line that can be answered quickly, such as where does this line cross the x-axis or cross the y-axis? Slope is necessary to at least two common methods of characterizing a particular line, the gradient-intercept form and the point slope form.
- High level why or practical reasons for this concept: a common use of slope is in geography, when describing the steepness or gradient of the surface of the ground. Many geographic information services (GIS) analyze digital elevation data and derive slope and aspect data sets. Slope is an important landscape metric that is used in characterizing and modeling things like landforms, surface runoff, habitats, soils, wildfire risk.
- Work through a few problems of the type allow you to begin to understand a method that will work for you. Use the book, use a tutor, review a teacher or professor lecture … whatever it takes.
- Here is an example method for finding slope from a graph that works for many students. Notice that this method is a lot more difficult to describe than it is to execute. That’s often what we’re looking for – something procedural, that you developed yourself, that you can therefore remember and makes it easy for you to do a problem of this type quickly whenever you’re called upon to do so. It doesn’t have to be different than how you were shown to do these problems. But it can be, if that works for you and it is correct.
- Find any two points on the line, using the graph to do so. It can pay to be a little clever, if only to make the arithmetic you’re about to do a little easier. That cleverness is simply to attempt to find one or both points so that they are characterized by x and y values both of which are integers. But that’s not always possible so don’t work too hard at that.
- Write the two points down, each as an ordered pair with the x value first and y value second, using the standard ordered pair syntax. When you do this, write the second point down exactly underneath the first point. For example:
(41, -6)
(-7, 10)
- Now put a big “minus” sign in front of the second point as a reminder of what you are about to do, example
(41, -6)
— (-7, 10)
- Now subtract the two x values and then the two y values, and write the answer as another ordered pair. Be careful with the minus signs, it’s easy to get mixed up. Example
(41, -6)
— (-7, 10)
– – – – – –
(48, -16)
- What you’ve just done is compute the “rise” or delta-y, and “run” or delta-x, i.e. (dX, dY). The slope is defined as dY/dX so the last step is easy: divide the second number by the first, and simplify the resulting fraction as necessary. Example: the slope m is dY / dX =b -16 / 48 = -1/3.
- Double check!! if the slope you calculated is positive, does the graphed line rise as it goes to the right along the x-axis? If negative, does it descend? Does it look right? That is, if it is a fraction between -1 and 1, is the line in the graph going up or down slowly? If it is greater than 1 or less than 1 does the line in the graph go up or down fairly quickly?
- Summary: find two good points, write them one on top and the second directly underneath. “Subtract” the 2nd ordered pair from the first, being careful with the minus signs!, to achieve a new ordered pair (run, rise). Divide rise by run, then simplify. That’s your slope m.
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- 4. Memorize some things to make it easier for you to do and check your work quickly:
- Slope, often denoted m, can be defined as rise, or change in the Y axis, divided by (or “over”) run, or change in the X axis. Again, slope is “rise over run”.
- A line with slope 1 goes up at a 45 degree angle. A line with slope -1 goes down at a 45 degree angle.
- A line with slope 0 is parallel to the X axis, and perpendicular to the Y axis.
- A line with an “infinite” slope, that is any finite number (the rise) divided by zero (the run), is parallel to the Y axis.
- If you use ordered pairs, you can “subtract” the second point from the first and get a new ordered pair (run, rise).
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- 5. Practice until it’s second nature to do this. There is no substitute for practice.
Does this look like a lot of work? It IS a lot of work. Well, it is at first. Then you become accustomed to learning how to solve new problem types, and it goes more quickly. This is an aspect of learning how to learn, and having some discipline to do things like this will serve you extremely well in life, that is, for much more than just mathematics. Definitions and why, look through or work some examples, craft a correct method that works for you, memorize things that will help you execute the method quickly and accurately, and practice until it’s second nature. That’s it. Just do it.