I’ve finally gotten a chance to check out the recently released SBAC item preview. I am impressed with the rigor of the problems that they have created. The tasks make it clear that students will no longer be successful simply by memorizing procedures. Rather, they will need to have a deep understanding of concepts.
Below is a list of the of sample problems for the different grade levels.
So far, I’ve only looked at the Problem Solving tasks. For some of the problems, students are asked to choose the correct answers. Yes, answers is plural. There may be one correct answer, but there may be more. Some have students choose from a list of answer options and move them to correctly fill in the blank. After each of these types of problems, the participant can check to see if the answer is correct and receive a score.
Other tasks are free response items and are not scored, but a rubric is provided. In this post, I will look at two related HS problem solving tasks: Circle 1 and Circle 2.
My approach to coordinate geometry has always been to have students apply the Pythagorean Theorem when finding the distance between two points and when deriving the equation of a circle with a given center and a given radius. Students who understand that the Pythagorean Theorem is the foundation for finding a distance between two points and who understand that the distance between two points is the foundation for finding the equation of a circle (set of all points that are a given distance from the center) can easily approach this problem. Although the problem description suggests that trig can be used, I did not approach it in this way.
In my approach, the essence of the problem is creating congruent right triangles, in which the hypotenuse is the radius of each circle. There is not even a need to find the length of the hypotenuse in order to answer the question. In fact, applying the distance formula would, in my opinion, cloud the issue. Once the vertical and horizontal legs of the initial right triangle are known, the student can simply copy those vertical and horizontal distances starting at the point on the initial circle. The ending point will be the center of the second circle. However a student needs to understand that there is a unique right triangle that connects the center of the circle to every point on the circle. This requires a conceptual understanding of two important ideas: (1) the definition of a circle as the set of all points (x,y) that are equidistant from a given center and (2) the distance between two points is the hypotenuse of the right triangle whose legs are vertical and horizontal line segments on the coordinate plane.
Each problem has an explanation as well as a rubric. Below is what is provided for the Circle 1 problem. Note that the student only gets 1 out of 2 points for the correct answer. Also, the student can get an incorrect answer due to a slight error, but if the reasoning shows that they knew what they were doing, they will score 1 point. The reasoning is given as much weight as the correct answer! Hallelujah!
Circle 2 uses the same original circle, but throws in the fact that the area of the second circle is 1/4 the area of the original circle. For this problem, the student needs to understand that a radius that is 1/2 the radius of the original circle will be needed in order to result in an area that is 1/4 the area of the original circle.
The Practice Standards that students needs to use to solve this problem include:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments. (explaining their approach)
5. Use appropriate tools strategically. (using the coordinate plane as a tool)
6. Attend to precision.
7. Look for and make use of structure. (use the fact that the structure of circles being made up of an infinite number of right triangles.)
Both of the problems are the type of conceptually-based problem solving that I espouse, so I am excited that the SBAC folks are are the same wavelength. If this is what will be expected of students, it is clear that many high school teachers will have to change their approach from a procedural-based approach to a conceptual approach. I’m glad to finally have some rich examples to show teachers in the SBAC states. I’m anxious to peruse the other types of tasks. I’ll save that for another post.