Getting Started 
Misconception/Error The student is unable to draw a diagram that reflects the parameters of the problem. 
Examples of Student Work at this Level The student draws two rectangles without a shared side or a rectangle along with one of its diagonals. The student may also be unable to find three sets of possible dimensions.

Questions Eliciting Thinking Can you explain, in your own words, how the duplex will look?
What shape will the building have?
How many units are there within the building? How do the floor areas of the two units compare? 
Instructional Implications Guide the student to sketch a diagram of the duplex, showing both units and the shared wall. Remind the student that each unit is to have an area of 1200 square feet. Ask the student to provide possible sets of dimensions for the units and to check that the product of each set is 1200. Remind the student that l represents the length of one residential unit and w represents its width. Ask the student to label each of the cinder block walls using these variables. Guide the student to write an expression in terms of l and w that represents the length of wall needed for the entire duplex [e.g., ]. Then, ask the student to represent the area of each of the residential units of the duplex in terms of l and w (i.e., 1200 = lw). Assist the student in using the area equation to represent the width, w, in terms of the length, l (i.e., w = ). Then guide the student to use this expression to write the function that describes the length of cinderblocks needed for all walls in terms of l.
Ask the student to use graphing technology to graph the function. Have the student review the meaning of both the independent and dependent variables and then describe how the graph can be used to determine the dimensions that minimize the number of cinder blocks required for the walls. 
Moving Forward 
Misconception/Error The student is unable to correctly write a function that models the indicated length. 
Examples of Student Work at this Level The student sketches a rectangular diagram divided into two congruent parts to represent the duplex and provides three possible sets of dimensions. However, when attempting to write the indicated function, the student is unable to represent one of the dimensions in terms of the other. The student:
 Writes a function in three variables [e.g., ] that may or may not be correct.
 Writes the formulas for the area and perimeter of a rectangle but does not apply them to represent the width in terms of the length.

Questions Eliciting Thinking What do you know about the areas of the floors? What is the product of the length and the width? Can you write the width in terms of the length?
How many lengths and how many widths comprise the outside walls and common walls? 
Instructional Implications Ask the student to label each of the cinder block walls using the given variables. Guide the student to write an expression in terms of l and w that represents the total length of the walls. Ask the student to represent the area of each of the residential units of the duplex in terms of l and w (i.e., 1200 = lw). Assist the student in using the area equation to represent the width, w, in terms of the length, l (i.e., ). Then guide the student to use this expression to write the function that describes the length of cinderblocks needed for all walls in terms of l.
Ask the student to use graphing technology to graph the function. Have the student review the meaning of both the independent and dependent variables and then describe how the graph can be used to determine the dimensions that minimize the number of cinder blocks required for the walls. 
Almost There 
Misconception/Error The student is unable to describe how to use the graph of the function to find the minimum length of cinder blocks required for the walls. 
Examples of Student Work at this Level The student sketches a rectangular diagram divided into two congruent parts to represent the duplex and provides three possible sets of dimensions. The student represents one of the dimensions of a duplex unit in terms of the other and writes a function that represents the length of the walls in terms of one of the variables. The student may:
 Graph the function but is unable to use the function to find the length (or width) that minimizes the length of wall.
 Give the dimensions that minimize the length of wall but is unable to justify them.

Questions Eliciting Thinking What do the variables in your function represent?
Which variable is graphed on which axis?
Can you interpret the coordinates of one of the points on the graph? 
Instructional Implications Review the meaning of the variables in the function and ask the student to interpret the coordinates of a specific point on the graph. Then guide the student to observe that meaningful values of the variables (in the context of the problem) must be positive. Ask the student to locate the part of the graph that includes points with positive coordinates. Then ask the student to consider which point (on the portion of the graph in the first quadrant) has the smallest value of the independent variable. Finally, guide the student to observe that the xcoordinate of the lowest point on the graph in the first quadrant describes a dimension (either the length or width, depending on the function) associated with the minimum length of wall. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student sketches a rectangular diagram divided into two congruent parts to represent the duplex and provides three possible sets of dimensions (e.g., 20 x 60, 30 x 40, 50 x 24) of each residential unit. The student represents the width of a unit in terms of its length, l, and writes a function that represents the length of the walls needed for the entire duplex in terms the variables [e.g., ]. The student says the lcoordinate of the lowest point on the graph in the first quadrant describes the length associated with the minimum length of wall.
Note: The dimensions of the rectangle that minimize the length are l = 40 feet and w = 30 feet.

Questions Eliciting Thinking Does it matter in which quadrant you look for the lowest point?
Which coordinate of the lowest point is the minimum wall length?
What shape rectangle do you suppose maximizes area for a given perimeter? 
Instructional Implications Ask the student to use the graph to estimate the dimensions that minimize the length of wall. Then ask the student to determine the total length of the walls for the optimal solution. Finally ask the student to increase the optimal length by five feet and then compute the width and the total length for the revised length and width. 