How Multiple Representation Justifications Led to Generalized Understanding
As I mentioned in my last post, one of the main benefits of multiple representation justifications is they help students piece together specifics that lead to a generalization. Finding patterns and creating generalizations enables students to group and compress the content in their memory to make room for more content. Facilitating this process involves a clear understanding of the big picture. Then the trick is to pick out all of the little details that will help students put together the puzzle. This is my first year teaching Algebra 2, so I have struggled a bit with looking forward to next chapter’s big picture and preparing students for those specifics.
While going through these lessons with my students, I jotted down some of the questions either I or my students asked. Then I compiled all of the list of details that I hoped would help students discover some of the bigger pictures, and grouped the questions accordingly.
Some background knowledge before you read: My Algebra 2 students were investigating key features of various functions, such as parabolas and rational equations (y = 1/x²). Some key features they were asked to focus their justifications on were intercepts, domain/range, and asymptotes. These features are part of the big idea of investigating functions, which leads into investigating transformations of functions next chapter.
Questions: “Why don’t our graphs match?” “Do I need to change the window?” “Does the graph match the table?”
Content: One of the simplest connections student needed to make was that their graph should match their table. By justifying with multiple representations, students caught many of their errors themselves. One error students often found was that they weren’t plugging in their equations correctly (they missed parentheses or used the wrong symbol). They also started to connect the window feature in the calculator to the domain and range of their graph.
Questions: “What is the biggest/smallest number you can get?” “Does this range match the table?”
Content: Once students had a clear graph to look at they could make observations on the domain and range. Some students struggle to see the limitations while looking at the graph, so the table provided extra support. They were able to quickly determine if there were undefined values or if their function was increasing or decreasing to specific values.
Questions: “Are there any numbers you cannot plug into your equation?” “Are there any output values you did not see in the graph or table?” “What operation in the equation is limiting your domain or range?”
Content: Seeing the domain and range just from looking at the equation is a huge Algebra 2 concept that gets students ready for Pre-Calculus and Calculus. At this point students were somewhat familiar with limitations on basic operations (dividing by 0 is impossible or squaring any number will give you a positive number). Seeing a more complex equation and identifying how these operations limit the equation was a difficult jump. Many students relied on the graphs or tables to identify the domain and range and then figured out why by checking numbers in the equations.
Questions: “Where did you see that intercept on the graph or in the table?” “How do you use the equation to find the intercepts?”
Content: Most, if not all, of my students were able to use a graph and table to explain the x and y intercepts but many were certain what to do with the equation. Once I asked students to be more specific about where to look in the graph of table (look at where y = 0 for x-intercepts), then they were able to use the equation (plug 0 in for y and solve for x). This connection is key throughout Algebra 2, especially when solving quadratic equations, so I am glad students had time to work with this concept.
Questions: “Does an asymptote just mean there is no value there?” “Where in the graph can you see the asymptote?” “What is happening in the equation at values close to the asymptote?”
Content: Asymptotes show up very early in our Algebra 2 curriculum. At this point, the meaning of an asymptote is difficult for students to grasp, and the concept is mainly explained via the graph. I thought it would be fun to stretch students understanding and ask them to justify with the equation as well. This led to some awesome discussions that were essentially about limits, a Calculus topic! I didn’t use the term limits, but by having students investigate what happens to the function as x gets very large or very close to a particular value, they were able to realize what happens to the entire function. For example, we discussed how y = 1/(x²) is not defined at 0, but the fraction gets larger and larger as you plug in numbers close to 0. These conversations blew their minds! It was such a fun extension activity.
I realize that this post is full of questions and content that some of you will never use or may not care about, but let’s remember the big picture. By having students justify using multiple representations, they were able to make connections within the content that they had not considered before. I was impressed with how many big picture ideas my students ended up discussing even though I wasn’t entirely sure where the content was going in the following chapters. I think students walked out of these classes with a more workable definition of domain, range, and intercepts than than they came in with. Hopefully by writing about and discussing these big ideas, they will remember the content for future use in our class. Definitely, though, students had some great discussions with each other and were asking questions that led to a much deeper, complex understanding of the material.