The alphabet provides plenty of practice in regards to slope. Printed letters can be created using lines – for example, an ‘A’ is just a positive, negative and zero slope line. Or ‘N’ is composed of two undefined slope lines and a negative slope line. A ‘C’ on the other hand would simply be made of a “non-linear” line. This is a great activity for students who need extra practice with the basics: positive, negative, undefined and zero slopes.
Below are the links for two handouts:
This handout is for students to simply identify the lines of letters as positive, negative, zero, undefined or non-linear.
This handout is where letters are described in slope and students decipher the message. This one could be done better, a la funsheets style, but I haven’t figured out how to attach documents to this blog that can actually open.
If there’s one thing I’ve figured out about teaching the basics of slope, it’s that there’s not one single method that will reach every single student. (This is true of any topic). However, it is still possible to reach every student since different methods work for different students. Here’s a few slope memory tricks that I’ve used when remediating students, if they just don’t get it after being shown the traditional ways:
Mr. Slope Guy
This was actually the favorite method of my below-level high school students. On every assessment relating to linear equations, the first thing most students did was sketch this on the top page as a guide. This isn’t my creation, but I can’t remember where and when I came across this to give the proper credit.
Since we write from left to write, people inherently will write the word “slope” from left to write, and this gives students a visual. Without moving the paper around, write the word “slope” on the line and if you find yourself writing upwards, it’s positive. Writing down is negative. Straight across is zero. And since there’s not really a place to write the word “slope” on a vertical line (without moving the paper), that’s undefined.
This only works for distinguishing positive from negative slopes, but simply tracing the line with a fingertip from left to right lets students physically feel the direction of the line as to whether it’s going up or down. I prefer that students write the word “slope” as mentioned above since writing is inherently left to right and tracing is not, but some students prefer this method.
One test asked “what is the slope of a horizontal line,” and a student told me that she couldn’t decide whether to write zero or undefined until she remembered that I had told them horiZontal has “z” for zero. Whatever works…
I recently made a slope worksheet where I drew figures on a coordinate plane, and students had to state the slope of each of the sides of the figures. Then it occurred to me that this would make a really great slope project.
Students could create their own line design on a coordinate plane, and label the slopes of the lines they used. It doesn’t sound neat when stated like that, but here’s an example of what a final product might look like. (I only wrote the slope for six segments, but you get the idea).
Stained glass and linear equations (or inequalities) are fairly common, but I think keeping it just as slope might be better. Students don’t have to have lines running all across the coordinate plane since they only have state the slope for smaller line segments.
To ensure students don’t just draw a few squares, students should be given a list of criteria. For example, direct students that their design must include 6 negative sloped lines, 6 positive, 4 zero slope and 4 undefined slope lines. Or 5 pairs of parallel and 5 pairs of perpendicular lines, or some similar variation. That way students have to use different sloped lines in their designs, and it also gives them a finite number of segments they have to write the slope for. This way, they’re not penalized if they produce more complex designs.
Here’s the Sample Project word document in case you want to use it as an example or if you want to modify it. And here’s a Student Template.
The other day, my students spent the day in the computer lab and explored the myriad of web sites on line that let them explore linear equations. I directed them to some fairly decent tutorials, and various online tools that allowed them to physically move lines to explore the effect it had on slope and equations. It was the first time I took them to the lab this year, and thought they would thoroughly enjoy this hands on opportunity. But they blandly went through the tutorials, and some even complained that they would prefer next time to stay in the class room, because it’s more interesting when I explain things to them.
However, the last 15-20 minutes of the day, I directed them to the website below. The student loved it. Many students who struggled with rate of change/slope and writing slope-intercept equations were proficient by the end of class. Unlike most other math “games,” the graphics on this are excellent and students actually enjoy the concept of the game, which is rather strange (which is why they liked it). There are roaches that traverse the coordinate plane at a particular line, for which the students must identify the slope or slope-intercept equation that will exterminate the roaches. They can even choose the method of exterminating the roaches (fumigator, shoe …).
Linear equations game
If you don’t have the time to do this during a class period, show the students the site and they will go there on their own time. Many of my students asked me to send them the link at their home address so they could play at home!
I’ve arrived at that one frustrating point in teaching Algebra – teaching linear equations. I don’t know why the concepts are so difficult for students to grasp. I’ve tried teaching it a myriad of ways, and have yet to discover the best method for teaching this section. It takes me nearly a month to teach all the things that go with slope, slope-intercept and standard form equations. And those are just the basic concepts of linear equations. (After the tortuous month is over, something kicks in and they finally get it and claim it’s SO easy).
And why it is difficult for students to remember that horizontal lines have a slope of zero and vertical lines have undefined slopes is beyond me.
I always start by trying to show them how to arrive at that conclusion using the slope formula. Since vertical lines invariably end up having zero in the denominator – which is THE big mathematical no-no – then their slope is undefined. They insist on calling it zero.
I resort to making them actually divide by zero on the calculator (after having arrived at some vertical line ratio using their slope formula), because it gives them “ERROR” since dividing by zero is a big mathematical no-no. And even though it says ERROR on their calculator, they still want to call it zero. The few who can actually remember that it’s undefined don’t want to call it undefined – they call it unfined, unrefined, or unidentified …
We analyze how linear equations with positive slopes increase or rise (from left to right) and how equations with negative slopes decrease or fall. Then we contrast that with a horizontal line, which neither rises or falls – and since it does “nothing” then it is “nothing” (0). But with a vertical line, you can’t tell if it is rising or falling; therefore since you can’t DEFINE which way the line is going, it is UNdefined.
Then I get desperate and resort to telling them that hOrizOntal has O’s in it, which looks like zeroes, because horizontal lines have a slope of zero. And that Vertical starts with a V, which strongly resembles the U in Undefined.
…that horiZontal has the letter Z in it, just like Zero …
…that Zero starts with a letter Z which actually has not one, but two horizontal lines ….