Table of Contents > Cluster 3 > Open-ended Items > Sample 2

HSPA Math Macro 3e: Statistical distributions in real-world situations

Sample Open-ended Item #2

Sample Item

The picture below shows stages 1 and 2 of a geometric progression that follows this rule: In a triangle, a line is drawn from the middle of each leg to the middle of the hypotenuse. The legs of the new triangles are 1/2 the length of the previous triangle's legs. Stage 1 and Stage 2 of an right triangle are shown below: Find the area of the triangle in Stage 1. Draw stage 3 and Stage 4 Find the total area of the all shaded triangles in Stage 3 and Stage 4. Will the total area of the shaded triangles in any stage ever exceed the area of the triangle in Stage 1? Explain your answer.

Sample Responses

Click on a link below to view individual responses and scorer comments to this sample test item:

"0" Response | "1" Response | "2" Response | "3" Response | Top

"0" Response

This response scored a "0" because the answers are incorrect and they show no understanding of the application of appropriate formulas. 

"0" Response | "1" Response | "2" Response | "3" Response | Top

"1" Response

This response scored a "1" because it does show an understand of area and the appropriate formula to use when calculating the area of a triangle, but the application of "fractal" rules and iteration was incorrect, and no work was shown past the first part of the question.

"0" Response | "1" Response | "2" Response | "3" Response | Top

"2" Response

This response scored a "2"  because the steps are correct for stage 3, but there is no stage 4 shown. The explanation for the second part of the question shows unclear logic.

"0" Response | "1" Response | "2" Response | "3" Response | Top

"3" Response

This response scored a "3" because it shows a complete understanding of the rules of fractals and shows correct calculations for area. The sample also clearly communicates the reason that the fractal will not ever exceed the original area.  

"0" Response | "1" Response | "2" Response | "3" Response | Top

NJPEP: Virtual Academy
Designed by the faculty at FDU's School of Education in partnership with the consultants of IDE Corp.