Appendix A: Examples of Activities Given to Students in the Experimental Section

Appendix A: Examples of Activities Given to Students in the Experimental Section

Activity 1. Given a plane with mx = 1 and my = 2, find the slope in the direction of the vector <2, 3>. We will refer to this as m<2,3>.

Solution:

Place a plane consistent with these 3D slopes and identify the right triangle whose rise and run are associated with this slope.

Obtain the rise for.

Obtain the run for .

Using the Pythagorean Theorem and the above right triangle, .

Calculate the slope.

Activity 2. Find the slope of the segment from (1, 1, 0) to (4, 5, 6).
Solution:

Find the run:

The run is between points(1, 1, 0) and (4, 5, 0) in the xy-plane.

Using the Pythagorean Theorem, it can be determined that the run is 5.

Find the rise:

The rise is the distance from (4, 5, 0) to (4, 5, 6), which is 6.

Find the slope:

The slope can be visualized by placing the right triangle with vertices (1, 1, 0), (4, 5, 0), and (4, 5, 6) along with its associated rise and run in three-dimensional space.

The slope is .

Appendix B: Interview Tasks and Results

Task 1 [Parts 1 and 2 - geometric to numerical register, Part 3 – geometric to algebraic register]:

Part 1. The points (1,1,2) and (3,1,6)areshown on the kit. Can you find the slope of the line that goes from (1,1,2) to (3,1,6)?

Part 2. If students successfully completePart 1, the points (1,1,1) and (4,5,6) will be placed on the kit. Can you find the slope of the line that goes from (1,1,1) to (4,5,6)?

Part 3. If students successfully complete Part 2, twopoints are placed on the kit, and it is indicated that we do not know the scale. The first point is (x0,y0,z0) and the second point is (x1,y1,z1). What is the slope between the two points?

Task 1 / Control SectionResults / Experimental SectionResults
Part 1 / 5 Unsuccessful (AACCC)
1 Successful (A) / 1 Unsuccessful (C)
5 Successful (AAACC)
Part 2 / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Part 3 / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)

Table B1. Results for students from the experimental and control sectionsonTask 1.

Task 2 [Parts 1 and 2 - geometric to numerical register]:

A plane is placed in front of the students (using the kit) that has slope 1 in the x direction and 2 in the y direction. (If the students did not understand “x” and “y” directions, the interviewer then tried again with alternate language such as “East”, “the direction of the vector <1,0>”, etc. finally drawing the direction on the xy-plane.)

Part 1. What is the slope if we walk in the direction of the vector <1,0> on this plane?

Part 2. What is the slope if we walk in the direction of the vector <1,1> on this plane?

Task 2 / Control SectionResults / Experimental SectionResults
Part 1 / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Part 2 / 6 Unsuccessful (AAACCC) / 4 Unsuccessful (ACCC)
2 Successful (AA)

Table B2. Results for students from the experimental and control sectionsonTask 2.

Task 3 [Parts 1 and 2 - algebraic to numerical register]:

The students are given a plane withthe formula z=2x + 3y+1.

Part 1. What is the slope if we walk in the y direction on this plane?

Part 2. What is the slope if we walk in the direction of the vector <1,1> on this plane?

Note: If the students did not understand vector directions, the interviewer then tried again with alternate language such as “East”, “the x direction”, etc. finally drawing the direction on the xy-plane.

Task 3 / Control SectionResults / Experimental SectionResults
Part 1 / 4 Unsuccessful (ACCC)
2 Successful (AA) / 1 Unsuccessful (C)
5 Successful (AAACC)
Part 2 / 6 Unsuccessful (AAACCC) / 4 Unsuccessful (ACCC)
2 Successful (AA)

Table B3. Results for students from the experimental and control sectionsonTask 3.

Task 4 [verbal to algebraic register]:

If we walk 2 meters eastward on a plane we rise 10 meters and if we walk 3 meters northward we rise 12 meters. If the plane passes through the origin (0,0,0), what is the formula for the plane?

Task 4 / Control SectionResults / Experimental SectionResults
Obtained Slopes / 6 Unsuccessful (AAACCC) / 1 Unsuccessful (C)
5 Successful (AAAC)
Obtained Formula / 6 Unsuccessful (AAACCC) / 3 Unsuccessful (CCC)
3 Successful (AAA)

Table B4. Results for students from the experimental and control sectionsonTask 4.

Task 5 [geometric to algebraic register]:

A plane with mx = 2, my = 1 and intercept (0,0,1) is placed on the kit. Find the formula for this function.

Task 5 / Control SectionsResults / Experimental SectionsResults
Obtained surface with mx / 6 Unsuccessful (AAACCC) / 4 Unsuccessful (ACCC)
2 Successful (AA)
Obtained surface with my / 6 Unsuccessful (AAACCC) / 4 Unsuccessful (ACCC)
2 Successful (AA)
Obtained formula. / 6 Unsuccessful (AAACCC) / 4 Unsuccessful (ACCC)
2 Successful (AA)

Table B5. Results for students from the experimental and control sectionsonTask 5.

Task 6 [Part 1 – treatment in geometric register; Part 2 - geometric to numerical register]:

If f is represented by the surface (shown in kit) before you,

Part 1.

Find a point where fxis (i) equal to zero, (ii) greater than zero and (iii) less than zero.
Find a point where D<..71,..71>fis (i) equal to zero, (ii) greater than zero and (iii) less than zero.

Part 2.

Label the point (1,1),f(1,1)) on the kit.

Approximate fx(1,1).
Approximate D<u1,u2>f(1,1) where <u1,u2> goes in the direction <1,1>.

Note if the studentsplaceda tangent line in the right direction and attemptedto get rise/run, the intervieweraccepted practically anything remotely possible.

Task 6 / Control SectionResults / Experimental SectionResults
Part 1A / 3 Unsuccessful (ACC)
3 Successful (AAC) / 1 Unsuccessful (C)
5 Successful (AAACC)
Part 1B / 4 Unsuccessful (ACCC)
2 Successful (AA) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Part 2A / 6 Unsuccessful (AAACCC) / 3 Unsuccessful (ACC)
3 Successful (AAC)
Part 2B / 6 Unsuccessful (AAACCC) / 3 Unsuccessful (ACC)
3 Successful (AAC)

Table B6. Results for students from the experimental and control sectionsonTask 6

y
x / 1 / 3 / 5
2 / 1 / 5 / 1
5 / 4 / 3 / 2
8 / 5 / 3 / 5

Task 7 [Part 1 – treatment in numerical register; Part 2 - numerical to algebraic register]:

If the function f is represented by the above table,

Part 1. Find the best approximation of fx(5,1)

Part 2. If g is the tangent plane of f at the point (2,1), find a formula for g.

Task 7 / Control SectionResults / Experimental SectionResults
Part 1 / 5 Unsuccessful (AACCC)
1 Successful (A) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Part 2 / 6 Unsuccessful (AAACCC) / 3 Unsuccessful (CCC)
3 Successful (AAA)

Table B7. Results for students from the experimental and control sectionsonTask 7

Task 8 [Part 1 - algebraic to numerical register; Part 2 - algebraic to geometric register]:

The function f is represented by the formula f(x,y)=x2y3, and we are interested in fx(2,1)

Part 1. Find the value of fx(2,1).

Part 2. With the axes above, draw the cross section needed to visualize fx(2,1) in 2-D. Label the axes and the formula of this cross section. Draw the line the slope of which is associated with fx(2,1).

Task 8 / Control SectionResults / Experimental SectionResults
Part 1 / 1 Unsuccessful (C)
5 Successful (AAACC) / 1 Unsuccessful (C)
5 Successful (AAACC)
Part 2 / 5 Unsuccessful (AAACC)
1 Successful (C) / 5 Unsuccessful (AAACC)
1 Successful (A)

Table B8. Results for students from the experimental and control sectionsonTask 8

Task 9 [algebraic to numerical register]:

The function f is represented by the formula f(x,y)=x2 + y2, and we are interested in approximating fx(1,0). Fill in the following table as we use successively closer approximations for the indicated derivative (a limit). After you fill this table out, try to guess fx(1,0).

Two Points used / (1,0) and (__,__) / (1,0) and (__,__) / (1,0) and (__,__)
Run / 1 / 0.1 / 0.01
Rise
Approximation of fy(1,0)

Table B9. Table to be filled out by students for Task 9.

Task 9 / Control SectionResults / Experimental SectionResults
Identified Points / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Identified Rise / 6 Unsuccessful (AAACCC) / 3 Unsuccessful (CCC)
3 Successful (AAA)
Approximated Derivative / 6 Unsuccessful (AAACCC) / 3 Unsuccessful (CCC)
3 Successful (AAA)

Table B10. Results for students from the experimental and control sectionsonTask 9.

Task 11 [numerical to geometric register]:

Given the linear function f is a plane, show using the kit, a plane consistent with fx(1,1) = 2 and fy(1,1) = -2.

Task 10 / Control SectionResults / Experimental SectionResults
Obtained surface with fx(1,1) = 2 / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Obtained surface with fy(1,1) = -2 / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)
Obtained surface consistent with both / 6 Unsuccessful (AAACCC) / 2 Unsuccessful (CC)
4 Successful (AAAC)

Table B11. Results for students from the experimental and control sectionsonTask 10.