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After teaching a lesson on vertical translations, you begin a section on horizontal translations. You want to select an example to introduce the concept of horizontal translations. Which of the following would serve your purpose least well?
Starting function: \(f(x)=x^2\)
Starting function: \(f(x)=2^x\)
Starting function: \(f(x)=2x\)
Starting function: \(f(x)=\cos x\)
(e) All of the examples are equally good
Task of Teaching
Choosing an example to introduce a concept
The shifted graph in option (a) is obtained by horizontally translating the original parabola. This shifted graph cannot be obtained as a vertical translation, as any vertical translation would move the vertex away from the horizontal axis, where the original parabola’s vertex is located. Therefore, option (a) is not a correct response to the item, because it clearly exemplifies horizontal translation.
The shifted graph in option (b) is obtained by horizontally translating the original exponential curve. This shifted graph cannot be obtained as a vertical translation: if original graph is translated up, then the image moves further and further away from the given shifted graph. If the original graph is translated down, then the image will intersect the horizontal axis, while the given shifted graph does not. Therefore, option (b) is not a correct response to the item, because it clearly exemplifies the horizontal translation.
The shifted graph in option (c) can be obtained by both horizontal and vertical translations. For example, shifting the original graph up by 5 units will produce the same image as shifting the original graph left by 2.5 units. More generally, shifting the original graph vertically by n units is equivalent to shifting it by –n/2 units. Therefore, option (c) is the correct response to the item, because it does not clearly exemplify horizontal translation.
The shifted graph in option (d) can be obtained with multiple possible horizontal translations or by reflection. Therefore, option (d) is better than option (c) to introduce the idea of horizontal translations, because it cannot be obtained with a vertical translation, though this option is also not as good an example as options (a) or (b).
Option (e) is incorrect because some options are worse than others for the purpose of introducing the concept of horizontal translation.
Mathematical task of teaching:
Choosing an example to address a student claim
In this case, the teacher must introduce a new topic, horizontal translations, after students have some experience with vertical translations. The teacher must choose an example that (1) attends to the mathematics of the new lesson; (2) makes that mathematics explicit and accessible to the entire class; and (3) challenges the students to distinguish the new concept from the old one.
What is the SCK assessed?
Interpretation: mathematical work is to identify that the best example illustrates a horizontal translation of any graph that can’t be obtained by a vertical translation of this graph (SCK), and the teacher needs to know that some transformations can be obtained through multiple possible translations (CCK)
Choosing an example: mathematical work is to
- Imagine how each answer choice could be transformed using alternate translations (e.g., what would it look like to translate the parabola horizontally)
- Notice which transformations can be obtained with different translations
- Notice which transformations just use translation and which also require scaling
- Awareness that translations aren’t the only kind of transformations