Visualizing Algebraic Transformations.

I just taught algebraic transformations in class and I did it a bit differently than I've usually done.  Usually, I just emphasize the equations and if you see this here, then the graph does this.  I changed things up this time because I've attended several webinars that encourage providing a visual element to go with the equations.

All along, I've encouraged students to graph the parent equation on Desmos and the new equation so they can see where the new graph is.  This made it easier for them to count the units left or right, up or down.  It was fun watching them involved in tracking how the new one moved and changed.

When we started the section that had the graph starting at a location other than at 0,0, and asking student to write a new equation for the transformed graph.  I asked students to attempt to write the new equation to match the transformation and then graph it to see if it matched the required movement.  It was interesting to watch students look at the new graph, change various numbers until it worked.  I admit, I gave them the official information but let them do more exploration rather than making them follow all the official rules.

After they had a chance to explore things, I showed them how to use the information given to rewrite the equations using the official rules. Many of my students "saw" the relationship between the equations and the movements.  Even the students who got the hang of using the equations with the new movement, graphed both equations to verify they were correct.  

In the process, I also discovered that about half the students didn't pay attention to the numbering on the axis lines.  Several ended up with an incorrect equation because they didn't see that the graph was labeled by two's rather than one's.  I suggested they take time to read the markings before they began looking for the new equation.

I noticed that students had a bit of difficulty with a reflection over the y - axis because for most of the functions they used, it didn't change anything.  They also had difficulty with horizontal and vertical shrink and stretch because they almost look the same and the difference is based on nuances.  Honestly, I teach all of this but it's hard to discuss when they will ever have to identify the shrinks and stretches. 

I think the next time I teach this, I'll incorporate more activities found on Desmos to give students a better grasp of transformations.  I think most people are comfortable just graphing things using some sort of graphing calculator and don't think they need to know any of this but I think they need to be aware of things.  This way they can look at the equation they graphed to see if it looks correct.

With all these new graphing calculator, it is not as necessary to know how to read an equation and graph it from the information in an equation. Let me know what you think, I'd love to hear.  Have a great day.