Straight line graph

In the interview, all students were shown a straight line graph as shown in Figure 7–12

below:

Figure 7-12. Straight line graph in the interview.

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

reasons for their answers. Table 7–3 summarizes all students’ responses and their

explanations:

Function or not Explanation

Ali Function Exemplar based focus followed by colloquial definition/ use of set diagram

Aysel Function Exemplar based focus followed by colloquial definition Ahmet Function Vertical line test with reference to the colloquial definition

Belma Function Action on the graph (assigning numbers on _{numbers on} x to the

y)

Belgin Function Action on the graph (assigning numbers on _{numbers on} x to the

y)

Arif Function Action on the graph (confused with the domain and range _{/ assigning numbers on} x and y with each other)

Cem Function Visual hints

Demet Function No explanation

Deniz Could _decide not No explanation

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Eight out of nine students considered the straight line graph as a function. Two of them

(Ali and Aysel) used the colloquial definition to explain their answers. Both of them

referred to the exemplars of straight lines.

For instance, Ali said that it was f(x)=x. He was then asked to think of it as if he did not

know that it was the graph of f(x)= x. He then responded in terms of the colloquial

definition by drawing a set diagram picture as shown in Figure 7–13 below:

Figure 7-13. Ali’s written explanations for the straight line graph.

‘There shouldn’t be elements left in the domain…I said f = x but it’s not like this. I have to know the slope…every x value has an image in y, the definition’. (Ali)

Similarly, Aysel first referred to a cluster of exemplars, y=ax. She then continued to

respond in terms of the colloquial definition:

‘…function definition, it’s a special relation. Every element in the domain goes to only one element, there are not elements left in the domain. Everything in x, since it goes to infinity, all elements in x find their places in the function. Furthermore, one value in x does not go to more than one y. Therefore, it’s a function’. (Aysel)

Ahmet used the colloquial definition by applying the vertical line test as shown in Figure

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Figure 7-14. Ahmet’s written explanations for the straight line graph.

‘It’s a function because for every x value, there is a y value. I can’t see two y

values for x here. Do we draw lines parallel to x, or to y? One of them. If we draw verticals to y and if it intersects at one point…if it intersected at two points then it

wouldn’t be a function…if it intersected twice, then there would be two y values for

an x’. (Ahmet)

Three students (Belma, Belgin, Arif) assigned a few numbers on x axis to the numbers on

y axis. Belma considered it as a function and explained on the first graph in Figure 7–15:

Figure 7-15. Belma’s written explanations for the straight line graph.

‘It’s a function…because…from R to R, from the elements of the set of real

numbers to other elements. In other words every element is met with its element…when we give 1 for f(x), x is 1’.

She was then shown another straight line with a different slope (the second graph in Figure

7–15 above). She again assigned 1 to 1 and drew y= x rather than focusing on the given

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Belgin also considered the straight line as a function with a similar reason:

‘because every value…values in the domain are assigned to the values in the range…for instance, if we give 1 for x, then y is 1…there are no elements left in

the domain. This is a function’ (Belgin).

However, when she was given a straight line passing through the origin with a different

slope as shown in Figure 7–16 below, she did not consider it as a function. She plotted the

point (1,1) and (referring to 1 on the x axis) said that there is an element left in the

domain.

Figure 7-16. Belgin’s written explanations for the straight line graph.

Arif assigned 3 on the x axis to 1 in y axis. However he seemed to be confused with the

aspects of domain and range:

“This is a function. Because…if we draw parallel lines here, for every y, an image of

y in x, and if we draw on x, an image of x in y. Suppose y is 1, and x is 3…therefore it’s a function” (Arif).

Considering the straight line graph as a function, one student (Cem) focused on the visual

properties of the graph.

He realized that there weren’t numbers on the axes. Therefore, I have put numbers on the

axes. He then drew lines from negative numbers on the negative y axis to the graph as

shown in the Figure 7–.17 below. However, he did not assign these values to the numbers

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Figure 7-17. Cem’s written explanations for the straight line graph.

He was then given a straight line, which did not pass through the origin as shown in Figure

17 above. Although he considered this as a function, he could not explain the reason

correctly:

‘The lines coming vertically from here also come here…to −2’. (Cem)