In at least 250 words, please respond to the following:
- What are some benefits to having students graph linear relations/functions?
- How are these benefits different than those that can be achieved just by looking at a linear equation or a t-chart of values?
- Which of the graphing methods do you think is most beneficial for your students and why?
Peer Response 1
Linear relations/functions are rather abstract concepts for students. We may refer to them as linear functions without explaining what the term linear function means especially if students are not graphing the linear relations/functions. There are several benefits to having students graph linear relations/ functions. The first benefit is providing a representation of the linear function. This representation helps students gain a better conceptual understanding of linear functions. A second benefit is providing a second method of solving and/or checking solutions. When students complete their work using more than one representation, not only are they checking their work, they are building flexibility and fluidity with their thinking. This flexibility allows for deeper problem solving and critical thinking as students are not trapped using a specific procedure.
Looking at a linear equation or T-chart does not develop conceptual understanding. These benefits differ from just looking at the linear equation or t-chart because they provide a clear understanding of the relationship between the values. Actually seeing the y-intercept and the relationship between the x and y values on a graph helps clarify the scenario in a word problem.
The graphing method most beneficial to my students will vary based on my studentsâ€ needs. I think the t-chart is most helpful for the majority of my elementary students. Students learn to graph ordered pairs and work with patterns in elementary school. The t-chart builds on this prior knowledge for graphing a linear equation. For my more advanced students, the slope-intercept and/or intercept methods may provide extension through more challenging material.
Peer Response 2
Linear functions are typically the first algebraic functions that students learn how to graph because they have a familiar visual shape and they can be easily connected to â€œreal-worldâ€ contexts. Having students graph linear functions is important because it gives students a foundation in using coordinate planes and connecting abstract equations to concrete ordered pairs, which is essential when it comes to graphing other families of functions that have more complex and subtle characteristics. Visualizing linear functions is also important to the math that students do in calculus, where they need to be able to draw and analyze tangent lines to continuous curves.
Different methods of graphing (or different methods of any mathematical solution method) typically exist because the â€œbestâ€ method is dependent on the particular function you are dealing with. If, for example, one or both of the intercepts does not fit on the grid you need to graph the function on, then it may be better to create a t-table and find some points that will be suitable to graph. This is especially true if you have an equation in point-slope form. On the other hand, if the intercepts can be found from the equation by inspection with relative ease (typically the case with equations in slope-intercept form), then the intercept method is usually the fastest in finding the graph of the line. Or you could also combine the two methods by graphing the y-intercept, and then using the slope of the line to generate a few more points that are close to the y-intercept.
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