These work samples were drawn from two different classrooms although the task given to the students was the same:

Design a dart board that has four regions with the following features:

 score value probability % 100 points 10% 50 points 20% 25 points 30% 10 points 40%

The dart board may be any shape (circle, square, rectangle, triangle, etc.) and must have an area from 1,000 square centimeters to 3,000 square centimeters. Assume the probability is proportional to the area of the region. Make a scale drawing with dimensions and explain your solution in words.

The task calls for the student to set up a total area that satisfies given constraints. Then the student must partition this area correctly into regions of sizes proportional to the given percentages. The scale drawing requires understanding of appropriate measurement and proportional reasoning. A firm grasp of area measurement is needed for a successful solution.

Probability, while mentioned, is not actually called for by the task. The assumption that equates the probability of hitting a region with the area of the region presumes that darts would always land on the board and that players’ aim at the target would be ineffective.

Nevertheless, the task lends itself to a wide variety of solutions. Some approaches are quite involved and complex. Other satisfactory solutions might be equally insightful, yet less complicated.

 Circumstances of performance These samples of student work were produced under the following conditions: - alone in a group in class - as homework with teacher feedback with peer feedback timed opportunity for revision

Sample 1
 These work samples illustrate standard-setting performances for the following parts of the standards: e Number and Operation Concepts: Interpret percent as part of 100. f Number and Operation Concepts: Reason proportionally. a Geometry and Measurement Concepts: Be familiar with assorted two- and three-dimensional objects. d Geometry and Measurement Concepts: Determine and understand length, area, and volume. b Problem Solving and Mathematical Reasoning: Implementation. a Mathematical Skills and Tools: Compute accurately with arithmetic operations on rational numbers. b Mathematical Communication: Organize work, explain a solution orally and in writing, and use other techniques to make meaning clear to the audience.

What the work shows
e Number and Operation Concepts: The student interprets percent as part of 100…
Both students showed the relationship of their diagrams to 100 points and represented that relationship as percent.

 f Number and Operation Concepts: : The student uses ratios and rates to express “part–to–part” and “whole–to–whole” relationships, and reasons proportionally to solve problems. The student in Sample 1 used different colors (reproduced here in grayscale) to identify accurately the areas corresponding to the percents of the whole as specified by the task. The student in Sample 2 appropriately applied the formula for area of a parallelogram. The student calculated the area, taking care to recognize the area of the dart board as being the total of the ten regions. a Geometry and Measurement Concepts: The student is familiar with two- and three-dimensional objects including…rhombi…. The student in Sample 2 accurately calculated the area of the rhombi and showed how the rhombi tessellate to create the larger board. Sample 2

d Geometry and Measurement Concepts: The student determines and understands length, area, including perimeter and surface area; uses units, square units.
The student in Sample 1 used square units accurately (cm²) to divide a large square into one hundred congruent smaller squares.
The student in Sample 2 exhibited command of the concepts of area and percent throughout the work.
The dart board sketch is drawn to scale. This diagram provides strong evidence of proportional reasoning, part of . The student appropriately used the centimeter as the unit of measure.

 Sample 2 b Problem Solving and Mathematical Reasoning: Implementation. The student makes the basic choices involved in planning and carrying out a solution; that is, the student: • invokes problem solving strategies, such as illustrating with sense–making sketches to clarify situations; • determines, where helpful, how to break a problem into a simpler one. In Sample 1, the student simplified the problem by representing the whole as a square with one hundred congruent parts. In Sample 2, the student broke the task into smaller, more manageable pieces.

a Mathematical Skills and Tools: The student computes accurately with arithmetic operations on rational numbers.
In Sample 2, the student accurately multiplied rational numbers.

b Mathematical Communication: The student organizes work, explains facets of a solution, labels drawings, and uses other techniques to make meaning clear to the audience.
The student in Sample 1 provided a legend to explain clearly the value of each square, and designated colors to represent the point values of each region. In addition, the student described how the number of squares colored for each region corresponds to the percent represented by that region: “I took 40% and colored 40 boxes.”
Here, as elsewhere in Sample 2, the prose makes clear the means by which the student built on previous steps to determine proportional areas and percents. Both work samples contain minor grammatical errors, but the mathematical solutions are correct.