Graph of smiley face

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Figure 7-36. The graph of smiley face.

They were told that the x axis which was coloured as red was the domain. They were then

asked whether it was a function or not. They were then asked to explain the reasons for

their answers. Table 7–7 summarizes all students’ responses and explanations:

Function or not Explanation

Aysel Not a function Colloquial definition Ali Function/changed

to not a function

Colloquial definition wrongly used/ignoring elements left in the domain/when mentioned 1 in the domain changed his mind

Arif Function/changed to not a function

Used colloquial definition when reminded of −1 on x

axis.

Ahmet Not sure Vertical line test/drawing of set-correspondence diagrams Demet Not sure Focused on x axis under the areas of three pieces of the

graph/no further explanation

Deniz Not a

function/changed to function

The numbers on y axis is not the same as the numbers on x axis

Belma Function Exemplar based response (the graph is like a parabola) Cem Not a function The shape is different

Belgin Not sure The shape is diferent

Table 7-7. A summary of students’ responses to the graph of smiley face.

Aysel used the colloquial definition to consider the graph as a function:

‘I think this is not a function…like in the other function. There should not be elements left in the domain, but 1 does not take any value in y. I think it’s not (a

function)’ (Aysel).

She seemed to change her response to consider the graph as a function. Applying vertical

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‘There are elements left in the domain even I apply verticals…this is a graph of a function. Here for instance, I drew one through, 2 takes only one value, it intersects once, but at the end 1 also should have taken a value. I think it’s not a function…but can there be elements that make function undefined?...no no it’s not (a function)’ (Aysel).

Ali first considered it as a function since he thought of the domain as the line segments

under the graph. He explained by drawing a set diagram as shown in Figure 7–37 below:

Figure 7-37. Ali’s written explanations for the graph of smiley face.

‘This is a function…because …2 in y can take the same value, two different values, two elements of the domain’ (Ali).

When his attention was drawn to corresponding value for 1, he changed his mind and did

not consider it as a function.

Arif first considered it as a function from R to R. He assigned a few numbers on the x axis

with the numbers on y axis as shown in Figure 7–38 below:

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He was then asked to find the corresponding value for −1 on axis. He said that he could

not find it. When he was asked whether this affected it being a function, he said that ‘there

can’t be elements left in the domain, there can be in the range. It’s not a function’.

Ahmet considered it as a function by applying the vertical line test to one piece of the

graph on the right, as shown in Figure 7–39 below:

Figure 7-39. Ahmet’s written explanations for the graph of smiley face.

He then drew set correspondence pictures leaving one element in the domain unassigned

(See Figure 7–39 above):

‘This (b) is in the domain but it does not go to anywhere. (Focusing on the graph) For instance, 0 is in the domain but it does not go to anywhere in y…there are no

corresponding values for −1 and 1’ (Ahmet).

When asked for 0, he said that its corresponding value is −1.5. Referring to the second set

diagram, he said that f(a)=x but f(b) is empty. He then focused on 1 on the x axis, but

he could not decide whether or not the fact that 1 does not have any corresponding value

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Demet could not decide about the graph. She focused on the numbers on the x-axis, −2

and −3, 2 and 3 (under the areas of two pieces of the graphs) and −1 and −2 (on the y

axis) as shown in Figure 7–40 below:

Figure 7-40. Demet’s written explanations for the graph of smiley face.

She could not decide whether it is a function or not.

Deniz did not consider it as a function because the y axis is labelled between −2 and 2

while the x axis is labelled between −3 and 3. When he was told that he could put −3 and

3 on the y axis, he considered it as a function (See Figure 7–41 below).

Figure 7-41. Deniz’s written explanations for the graph of smiley face.

Belma considered it as a function since the graph is similar to a parabola. She tried to join

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Figure 7-42. Belma’s written explanations for the graph of smiley face.

‘These are like parabolas, however I can’t say anything more…1 (is assigned) with 1, −1 with −1…function if we join them they are like parabolas, increasing and

decreasing, sine and cosine…’ (Belma).

Cem did not consider it as a function since he is seeing such a thing like this for the first

time.

Belgin was not sure about the graph. She said the following:

‘I don’t wanna do this (question), the shapes are very different’ (Belgin).