I need help with a Mathematics question. All explanations and answers will be used to help me learn.
We have explored several representations of linear functions – growth patterns in squares, t-charts, graphs, and equations. In at least 250 words, please respond to the following:
- Discuss the benefits and difficulties of using multiple representations when teaching linear functions or relationships.
- What do each of these add to our understanding of linear relationships?
- What are the difficulties associated with each of these representations?
Peer Response 1
In my opinion, learning math effectively is all about building connections between concepts. Using multiple representations for learning about linear functions (and functions in general) is a great way to teach students how to build those connections and gives them tools for how to engage with much more advanced functions that they encounter in future math classes, functions that may be simple to conceptualize in an equation but difficult to conceptualize graphically (such as the Dirichlet function), or vice versa (such as complicated piecewise functions). From a practical teacher standpoint, one difficulty in emphasizing multiple representations is that it becomes more difficult to assess students’ understanding of each representation in depth.
Each of the representations is a way of organizing the information in linear functions. A t-chart allows quick side-by-side comparison between two variables. A graph shows the same comparison but in visual form. An equation is a statement about how the function behaves after an arbitrary number of â€œstepsâ€ in the pattern and can also contain structural information about how the pattern grows and the starting value of the pattern.
One difficulty in using multiple representations is that when going from a squares pattern to a graph or equation, you have to deal with the distinction between discrete and continuous variables. That is, you have an equation or graph that can be defined for all real numbers that models a pattern that is only defined for the natural numbers. On the other hand, drawing this distinction leads to a discussion on the important topic of domain and range.
Peer Response 2
There are many benefits to using multiple representations when teaching linear functions and relationships. It is important to for students to see the connections between different representations and understand they mean (or represent) the same thing. For example, a chart or table, graph, diagram, or equation can all be used to represent the same relationship. Many times, students see these representations in isolation which can make it difficult for them to understand they all represent the same relationship. To avoid this misunderstanding, it is important for students to see various relationships expressed in different forms starting in the primary grades. If students are exposed to this early they will be better prepared when they begin finding slope, using slope-intercept form, and writing equations.
Multiple representations allow us to understand that a linear function increases or decreases at a constant rate. Charts and graphs provide us with a visual of the relationship, while equations allow us to find an infinite number of values for the relationship.
I found the growth patterns in squares to be the most confusing initially. I really had to pay attention to where it started and what changed each time. After watching the lecture, I felt very confident with how I completed the week 2 assignment. When I began to respond to the forum question in week 2 I realized there was no step/size zero so I did not have a starting point. I had to decompose the figures to find the pattern which I think students may struggle with a bit. I have seen students become confused with t-charts when the x values are not sequential, like problem 1 on this week’s homework. It causes the rate of change to appear different which can be very difficult for students to comprehend.