Teachers could create individual lessons on A-C. For example, for A, students would graph a radical function using defined domain values in a table and transform the function to interpret data. For teachers to promote relational understanding, a lesson such as using the parent function, in this case f(x)=√x, to investigate, describe, and predict the effect of changes on the graphs. As a teacher presenting this lesson, I would create a Desmos lesson where students investigate, using sliders, the transformations of the function. To shake things up a bit, I would create stations where they would investigate the different transformations. Let me break it down a little farther because I know that sounds a little confusing. There could be a station for vertical shifts, where students investigate, using slides, how the vertical shift effects the graph of the function. In that station they would be given 5 minutes to explore the sliders and make predictions on more examples of vertical shifts. They would also graph, by hand, a couple examples of the square root function. Each example would gradually be more advanced. The students would be in groups of 2-3, so if students begins to struggle, they could actively consult with their peers.

Another station would be students investigating the horizontal shift of the square root function. The same process would hold from the vertical shift. Another station could be investigating the domain of the function. There is so much one could do with a lesson such as this so the few stations I have mentioned don’t quite cover everything that could be done. I would assess students by giving a handout combining all the stations into a few examples that promote productive struggle. “Such instruction embraces a view of students’ struggles as opportunities for delving more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions” (National Council of Teachers of Mathematics [NTCM], 2014, pp. 48). A lesson like this could be used for A-C of this standard.

During this lesson, students would be focusing on defining functions, graphing simple functions by hand, and graphing more complex functions using technology. Students would be accomplishing all these objectives. The more traditional way of presenting this problem would be to present a few examples of the function and assign 10-15 problems for students to work through. This way of teaching does not attend to the goal of relational understanding for students. The goal for teachers is for students to gain as much relational understanding as they can. This saves from spending valuable time re-teaching.

This Alabama Course of Study Standard could be easily adapted for students. The standard states for simple cases, students should be graphing functions by hand. Teachers could adapt lessons by using graphing technology such as Desmos or a graphing calculator for students who need a visual. Although the students aren’t physically graphing the functions by hand, the student thinking is still there. Students still have the opportunity get all the understanding they would from graphing by hand with the extra features technology such as Desmos has to offer. This again, allows students to gain relational understanding of graphing functions.

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