This is best BEFORE teaching how to find asymptotes and graph rational functions.

Start with just a 5 minute introduction to what an asymptote means and what it looks like.  Then, the kids jump into Investigation 1.

The kids analyze the factored forms of the functions and figure out what is causing the vertical asymptotes and holes.

Then, the second investigation follows with horizontal and slant asymptotes.

For this one, I made a little reference "cheat sheet" for the students.  If they do not recognize what is happening on the calculator screen, they can take a peek at these samples.

For both parts of this activity, it works best if you have the students cut the cards apart.  This way, they can separate and differentiate them.  Then, they can look at each "collection" and start to hunt for the patterns and characteristics of each feature.

I like to have the kids work in pairs. 

Click here to download all the files for the "What Makes an Asymptote" Investigation lesson.

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Tricks for finding those PERFECT ideas that will fit YOUR particular classroom and content needs

• Know your search options:  Pinterest allows you to search by pins, pinners, or boards.  For example, if you are looking for bulletin board ideas for your sixth grade classroom, you can search "6th grade bulletin board" and then narrow your search using the buttons below.  You can then browse "boards" that are entire collections filled with 6th grade bulletin board ideas. 
  

• Start thinking of Pinterest as your new search engine.  For teaching ideas, it beats Google, in my opinion.  When I'm planning a lesson, I head straight to pinterest.  It's so quick and easy to use as a search engine, and gives me better results than anything else.  I can almost always find exactly the fresh idea I am looking for.

• Follow the right pinners, then start with your home feed each time you log in.  Pinterest is selecting pins just for you based on not only who you follow, but also pins that are similar to the topics that you have already shown interest in.  If you teach middle or high school math, I've collected links to some of my favorite math education boards and sorted them by category for you here.

• When you are looking up close at a pin you like, scroll down to keep searching.  Pinterest will offer you "related pins" and also show you more boards that contain that pin.  Use the related pins and boards to find more ideas that are exactly like what you are looking for.

• Here's a really cool way to find content that is a great fit for your own classroom:  Enter your favorite website or blog at the end of the pinterest URL in your web browser using this formula: https://www.pinterest.com/source/mathgiraffe.com but replace mathgiraffe.com with your own favorite teaching site if for some strange reason I'm not your favorite (very unlikely, I know).  If you choose a "source" blog that is exactly the type of content you enjoy in your classroom, then the pinners that you see listed will be teachers JUST LIKE YOU, who teach the same content in the same way!  Browse through the pins that came directly from your favorite site, and click on those pinners.  They will likely have boards packed with teaching ideas that you will love, since they share your interests.

How to make sure you don't lose that idea that you loved, and how to remember the inspiration it gave you

• Make your boards specific.  It may not seem like you need to do this at first, but as your boards fill up, it gets harder and harder to go back and find that great idea you had for wrapping up your unit on fractions.  I started my subject boards and a few topic boards, but one of my own Pinterest goals is to get even more specific, like "linear equations," so I can find what I am looking for quickly.

• Change the pin description!  It's so quick and easy to just click "pin," but then you have another person's caption on an image that you want to use later as a bookmark.  Include enough in your description to remind you what is actually on the site that the pin links to.  Also, write yourself a quick note so you remember what inspired you.  Try something like "This organization idea would be perfect for my assessment folders.  I'll add checklists to the inside labels."

• In addition to your subject and content boards, make some boards to save classroom management and decor ideas.  You can even save teacher outfits, ideas for parent night, and more!  Don't limit yourself, because eventually you will see an idea you want to save, and it won't quite fit your board topics.  You will never find it again if you save your "discipline tips" pin onto your "Science Lessons" board.

Tips for really utilizing all of the features and extras that are built-in on Pinterest

• Instead of instantly deleting email notifications from Pinterest, use them!! Save the emails that tell you who re-pinned your pins.  When you have time to browse for school ideas, click through the pinner names and board names on those emails.  Browse through those people's other teaching ideas.  If they liked what you pinned, then you will probably like what they pinned!

• If you see a pin that you like, but you are not sure if it is worthy of cluttering up your boards, click "like."  You will be able to find it later if you really want to by searching your own "likes."  You can also comment on pins.  If you are pinning straight from the original pin, feel free to ask questions or comment on the idea!  That pinner or blogger will get a notification and hopefully will answer you!

• If you pin something to the wrong board, you can "copy" or "move" it.  You can even do this with multiple pins all at once.  This is handy if you decide to make your boards more specific later on and want to re-organize.  It's a lot easier than re-re-pinning each one individually.

• Pinterest also allows you to set up "secret boards."  These are great for ideas that you want to save, but don't want students, parents, or friends seeing.

• If you have not already installed a "Pin It" button on your browser, be sure to take advantage of it!  You can do this directly through Pinterest here: https://about.pinterest.com/en/goodies  - The button sits on your browser toolbar, and when you find content on the web that you want to save, you just click it.  It will create an instant pin for you.  This is wonderful for bookmarking the websites that you already use or reference all the time for teaching.  You can now save them alongside all your other collections in Pinterest, so everything is in the same place.

• Collaborate with your teaching team using group boards.  You and your co-teachers can all share a board.  You can invite others so that you can all pin to the same board.  Share ideas as a team!
  

Inquiry-based learning teaches students how to find patterns, figure out properties, and discover new rules of mathematics.  They slowly learn how their brain actually progresses through the idea to get them from "lost" to "ohhhhh, I see what the pattern is."  After this becomes second nature, they will be more ready to attack future unknown situations.

The critical thinking skills that are required to ask questions that will lead to discovery are learned and acquired. They develop through practice, just like any other skill.  Students must practice the inquiry process, just like we as teachers must practice sitting back and letting them struggle.  It's not easy, but begins to feel more natural over time.

As students get more and more accustomed to the inquiry structure that you use, they will slowly strengthen their own skills and gain independence in problem solving. 

Here's an Algebra example: 
I always teach exponent rules with guided inquiry.  We work through the patterns and write out all the variables and cancel or expand until students develop product rule, power rule, and quotient rule for themselves.  Then, when we get to the negative exponents later in the unit, their skills with this process have grown to the point that they know how to come up with their own concrete examples.  They know how to look for patterns.  I have them develop their own sets of rules for negative exponents.

Work towards the skills that will lead your students to persistence in problem solving.

After discovering a mathematical property on their own, students will truly understand the concept behind the rule.  Instead of following a set procedure, they will understand WHY a rule works and HOW it was developed.

Let your students build a concept, not just follow a given process.

It's very rare that I give a formula as a part of notes.  I like to have students find a formula for themselves.  One example is for surface area of a cylinder.  I have a dissection lesson for classes to "dissect" a cylinder "specimen." 

They have to discover for themselves that the length of the rectangle is equal to the circumference of the circle, and then go from there.  Most middle school students across the world REALLY struggle with truly understanding this formula.  They have a hard time visualizing the base and why circumference plays a part in this formula.  However, after this hands-on lesson, the students really "get it" and never forget how the rectangular face wraps around.  You can read more about this lesson in my Cylinder Dissection blog post.

I do a similar structure for discovering circle theorems in High School Geometry.  Students draw their own chords and have to tell ME how arcs, tangents, central angles, and inscribed angles are related, instead of me telling THEM these theorems.

When students feel the pride that comes from discovering a theorem, property, or formula for themselves (just like a mathematician does!), they suddenly gain a new level of confidence in their own math abilities.

I have noticed students have a sudden willingness to try a new challenge or approach a different type of problem instead of giving up.  Kids believe in their ability to apply knowledge from one situation to another.

I feel like this has given my students huge advantages on standardized tests.  After I started incorporating more and more inquiry learning, the kids got more and more comfortable with being exposed to an unfamiliar problem type.

This is so important to note, and one of the benefits that I rarely see advertised by supporters of inquiry learning.  Here's the thing:

If a student develops a formula, rule, or property for himself, then he understands on a deeper level where it came from.  He won't have to memorize it at all. 

The student can reproduce the formula at any time because he "discovered" it.  This is especially true if it was done in a hands-on way.

During a test, or later in life, the development of the idea is what will help a student recall how a property works or how to re-create the formula.

Some tips as you start to incorporate more inquiry learning, and give your students these benefits -

1. Sit back.  Don't jump in to help.  Let students struggle through the first few inquiry lessons.  It's so hard to resist giving hints, but do your best to choose when to give assistance and who to offer it to.  They will get used to this process and it will really be best for them in the long run.  Help those who truly need it, but otherwise, let them develop their own skills.  Stick with it.  It will get easier!
2. Never (or very rarely) GIVE your students a rule or formula.  Get out of the habit of feeding information and properties to them.  For example, if you currently teach rules of divisibility, twist the lesson to have the kids find the rules and present them to you.
3. Require your students to write up their observations in complete sentences.  Follow up all inquiry lessons with a full written explanation.  Give your kids the benefits that come from working to explain the properties that were discovered in their own words.  Then, also have them translate into mathematical language to develop the formula as well.
4. Have students share both their questioning process and discoveries aloud.  They can work in pairs or small teams or they can present findings to the class.  Allow them to collaborate and share the strategies they used to figure out the property.  Encourage metacognition!
5. Plan carefully so that your students do have enough guidance to succeed.  Try a consistent structure like my SPORT method for structuring inquiry so that they know what to expect and can thrive in inquiry learning.
6. When possible, incorporate materials and manipulatives that make the lessons more hands-on.  This is a little more difficult in Algebra than it is in Geometry.  Check out some specific examples for making your Geometry lessons into hands-on discoveries here.