Discovering Slope of a Line

A Free Algebra Exploration to Deepen Understanding & Increase Retention - Slope Formula

As math teachers we know the frustration of having kids who seemed to understand something during the lessons but then suddenly forget the formula on the test, or get things backwards. 
 
For slope, it tends to be kids putting it upside-down or messing up the negatives.  They may also be lacking the basic concepts of greater slopes being “steeper” and are then unable to make sense of their errors or catch simple mistakes. 
 
There tends to be a lack of retention in a lot of basic concepts like this.  It can be aggravating as a teacher to grade work that should be relatively “simple” like finding slope between two points, and see so many little errors.
 
Well, I have some great news.  If you see this with your class, you may just be missing the simple element of inquiry!  Inquiry provides students with the opportunity to discover concepts themselves, which makes the learning more meaningful and powerful. 
 
Teaching through guided explorations allows kids to develop a formula themselves, leading to a deeper understanding and stronger retention.
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​Inquiry - Based Learning
 
Inquiry (or discovery) learning has so many awesome benefits.  This method is sure to help your students learn AND retain new concepts.  When students learn through inquiry they are asking the questions; they are discovering the answers!
         
          When students learn through investigation,
 

• Understanding of the deeper concept behind a mathematical principle is increased.

 

• Students are able to reproduce the rules, formulas, and procedures for themselves without memorizing because they understand where the rules came from.

 

• Students gain confidence in math class.  They also become more self-motivated.

 

• Retention of the lesson material increases.

 

• Students get a boost in their problem-solving skills.  They gain resilience to stick with a challenge and learn to think it through for themselves, and develop strong critical thinking skills.

 

• Students become accountable for their own learning.

 

Introducing Slope with Inquiry
 
The next time you introduce slope, you definitely want to try out this inquiry activity.
 
1.  First, download this “Discovering Slope” sheet (within the Free Guide to Teaching Slope)

2.  Group your students into pairs and give each pair one sheet.  Partner work is great for inquiry activities, so students can talk through their discoveries and make progress together.  Their discussions help solidify concepts, while helping them get unstuck. 

On the sheet there are 5 sets of staircases & ramps that the students will need to put in order of “ steepness”.
 
3.  Tell them to read the directions and complete the page.  The next part is tricky- sit back, relax, and try not to help!  
This will probably go against your instincts as a teacher, but you need to stay strong!  Your students learn through their struggle.
 
Try to avoid giving hints.  Your students need this practice too; it will help them build up persistence.
 
Don’t let them give up.  They will end up making great observations once they jump in and embrace the uncertainty!
 
4.  Walk around and listen to the conversations that are happening.  Take note of any interesting observations that you may want students to share in the discussion afterward.   
They’ll discover that they need to use both height and length… but how?  Some groups may be stumped for a few minutes.  There may be frustration they have to work through.  Then, they will realize they need a kind of a ratio between the two.
 
They may try it upside-down at first (run / rise), but then eventually they should get to the formula for rise / run themselves! 
 
They’ll work to count the grid spaces, and later can understand the subtraction model to develop the formula for when they are given just coordinates and no graph.
 
They will now always remember and understand much more deeply that slope is "RISE OVER RUN" because they figured it out for themselves.  This is the goal of the guided discovery / inquiry process!! :)
 
5.  Give encouragement to groups that are truly stuck and frustrated, but try not to give hints about where to begin.  Train your class to try something that makes sense, and see where they get.   
That is the purpose of the discovery process.  Each time you let them struggle on their own, they build those skills and will do better next time.
 
6.  Gather back together as a class for a follow-up discussion.  In the discussion, clear up any misconceptions that popped up. Allow students to share their approaches and discoveries.  
Try to direct the conversation to noticing that, once simplified, two of the slopes had the same number.  Or did any of your students disagree that they were equivalent slopes because the directions were different?
 
Take all observations.  This will help later when you teach the differences between positive and negative slopes.
 
 
 
Once this activity is done, you’ll be ready to lead your class into a lecture with guided notes on the Algebra notation, the definition of slope and practice problems.  Click here for my Slope Doodle Notes!

I hope these materials make this your most successful year of slope lessons yet!  Enjoy the journey, and have a great year of Algebra! :)

Feel free to share other ideas for exploring slope in the comments!  I bet doing a life-size measuring layout with actual step-stools would be incredible if you have the time and can set it up!

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