In this task (see figure 17) students were required to write an equation of the path of a ball in a miniature golf and determine whether they make a hole in one or not. This task was adopted from Larson (2004) Algebra II class textbook. They were given the ordered pairs for the starting point, the banking position, and the hole. The task required their application of absolute value function understanding.
Following is a summary and a description of their responses to the task 3 questions. Inter- pretation of the responses and their analysis using RBC+C model and Ernest (2006) semiotic systems theory will follow.
Table 12. Participantsβ Responses to Task 3 Application of Absolute-value Functions Participant Question Response Multiple
Representa- tions Referred Pseudo- Conceptual Understanding Indicator/s Semiotic System Component
Ron Make the shot? πππ VD, AL, G, N Surface Association S(object) (R semantic ) Justification y = β12
7|π₯ β 6| + 8
Subs ordered pairs (6,8), (2.5, 2)& (9.5, 2) VD, AL, G S(object) M(mathematical con- tent) Alternative op- tion Graphing
absolute value function
VD, AL, G Surface Association R(semantic) S(process) M(informal theory) Tyron Make the shot? πππ VD, AL, N, G Surface Association S(object)
(R semantic ) Justification y =β12
7|π₯ β 6| + 8
Sub ordered pairs (6,8), (2.5, 2)& (9.5, 2) VD, AL, G S(object) (R semantic ) Alternative op- tion Graphing
absolute value function
VD, AL, G R(semantic)
S(process) M(informal theory) Stacey Make the
shot?
πππ VD, AL, G Surface Association S(object) (R semantic ) Justification y =β12
7|π₯ β 6| + 8
Sub ordered pairs (6,8), (2.5, 2)& (9.5, 2)
VD, AL, G Surface Association M(informal theory) S(object)
(R semantic ) Alternative op-
tion
Graphing
absolute value function
VD, AL, G Surface Association M(informal theory) S(object)
(R semantic ) Yolanda Make the shot? π¦ππ VD, AL, N, G Surface Association R(semantic)
S(process) Justification y =β12
7|π₯ β 6| + 8
Sub ordered pairs (6,8), (2.5, 2)& (9.5, 2)
VD, AL, N, G Surface Association M(informal theory) S (Process) M(mathematical con- tent) R(semantic) Alternative option Graphing
absolute value function
VD, AL, N, G Surface Association M(informal theory) S (Process)
M(mathematical con- tent) R(semantic)
Interpretation of Task 3 Responses.
All four students started off responding to the question by recognizing that the task in- volves an absolute value function but more importantly identify that the vertex of the absolute value function as (6, 8). Ron, Stacey, and Yolanda started off not sure that they would make the shot. Stacey explicitly stated that she was not sure is she will make it βI am not sure if I will make it but I will try I think I might make the shotβ line 1. All four students correctly recognized
ordered pair (6, 8) as the vertex of the absolute value function. They then build-with a strategy and substituted the coordinates of the vertex (6, 8) and the point (2.5, 2) and determined the ab-
solute value function of the path of the ball as y =β12
7 |π₯ β 6| + 8. To determine if they made
the shot or not, Ron, Stacey, and Yolanda substituted the values of the ordered pair(9.5, 2) and ended up with a true statement 2= 2 . New understanding of the absolute value function was
constructed when they used the true statement to infer that the ordered pair (9.5, 2) which repre- sents the hole was on the path of the ball hence they would make the shot. Tyron adopted a dif- ferent position to determine if he made the shot or not. He recognized that the axis of symmetry split the function into two equal parts. Tyron build-with a strategy when he drew the axis of sym- metry π₯ = 6 recognized and determined that from the y-coordinate, both points
(2.5, 2) πππ (9.5, 2) had the same co-ordinate. Also he determined that the absolute value func- tion |π₯ β 6| signified that points on both sides of axis of symmetry that were at the same hori- zontal distance from the x-coordinate x = 6 had same y-coordinates. Since the ordered pairs (2.5, 2) πππ (9.5, 2) were both 3.5 units horizontally from axis of symmetry on both sides of the axis it implied that the hole was on the path of the absolute value function on the opposite side across from starting point and hence he would be able to make the shot.
Analysis of Task 3 Responses.
All four students recognized immediately what the goal of the task was. Semiotic system component that students gravitated to was the transformation of rules. Ron, Stacey, and Yolanda correctly identified the procedures to apply by using algebraic manipulation of the absolute value function to determine whether they made the shot or not. They processed the absolute value function in the path of the ball and determined it to be y =β12
7 |π₯ β 6| + 8. They then used or-
dered pair of the hole (9.5, 2) in the absolute value function and determined if they made the shot of not. True statement 2 = 2 implied that they made the shot. Use of multiple representa- tions i.e., graph and algebraic symbol triggered and supported a symbol sense where the vertex (6, 8) and the axis of symmetry were correctly identified. Tyronβs demonstrates a deeper concep- tual understanding of the underlying meaning structure of the signs in the absolute value func- tion. He makes a logical link between the different elements of the function and uses them to re- spond to the task question. He makes a connection between the axis of symmetry and the abso- lute sign |π₯ β 6| and infers that the initial golf ball position (2.5, 2) and the hole are horizontally at equal distance of 3.5 from axis of symmetry π₯ = 6 on both sides. In absolute value function, βexcept for the vertex for every y-coordinate there are two x-coordinatesβ. Tyron then infers that he will make the shot because these two ordered pairs (2.5, 2) and (9.5, 2) are same distance horizontally from axis of symmetry. Use of multiple representations allowed for the students to re-image their conceptual understanding of the absolute value functions, develop a reflective ap- proach to understanding each sign in the function and make a logical external link between the parameters of the functions and the problem content in this task. Representational versatility i.e. ability to work seamlessly between representations was demonstrated when Tyron made the con- nections between the significance of the axis of symmetry and the two ordered pairs.