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Adapting AP Mathematics Questions As a Pre-AP Strategy
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by Dixie Ross
Teacher
Pflugerville High School
Pflugerville, Texas
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| | My Vertical Team regularly engages in adapting AP free-response questions for use in the classes prior to AP Calculus or Statistics. I began this process years ago, when I realized that my AP Calculus students often erred on parts of AP questions that required no calculus at all. For example, they might need to write the equation of a line passing through two points, or set up an equation to show how the volume of water in a cone is proportional to the height. I cut the noncalculus portions of the questions out and gave them to my algebra and geometry classes. The students enjoyed seeing how their current learning would be used in advanced mathematics classes, and many became interested in taking AP Calculus, which was very new in our school.
Over the years, I started rewriting parts of the questions, replacing the calculus skills with the skills I was teaching in my other classes, always trying to maintain the stem and the style of the question. I also began collaborating on and sharing the questions with the other teachers in my department. By that time, we realized that we were doing "vertical teaming," that collaboration and the resulting collegiality were cornerstones for building subsequent efforts. Because they spent many hours in examining the questions and working alongside the AP teachers, the teachers of all courses became familiar with the concepts, skills, and habits necessary for success in AP.
When teachers in other districts are just beginning vertical teaming, I find it is unwise for them to try to adapt free-response questions right away. They are more effective when they start by examining questions that have already been adapted by others. I often am asked incredulously, "Do you really expect me to teach AP Calculus to my seventh graders??!!" No, of course not. I do expect the seventh-grade teachers, though, to be familiar with the AP course content and goals so that they know how their material pertains to subsequent courses. Once they examine the middle school versions of the AP Calculus questions, their fears are allayed. They see easily how minor changes in the content, vocabulary, assessment style, or pedagogy can help better prepare students for success.
The problem set we will use in our second Vertical Team meeting this year is included here. (See the related article, "A Year in the Life of an AP Math Vertical Team," below in "See also.")
 Adaptation of 2003 Calculus AB Form B question 5 for Middle Grades
Adaptation of 2003 Calculus AB Form B question 5 for Algebra 1
Adaptation of 2003 Calculus AB Form B question 5 for Geometry (1)
Adaptation of 2003 Calculus AB Form B question 5 for Algebra 2 / Precalculus
This meeting's problems are based on 2003 Calculus AB/BC Form B question 5. The focus topic for that meeting is analysis of functions, with a particular emphasis on the vocabulary used in describing function behavior. The questions all have the same graphical stem from the AP Exam. Teachers will work in small groups by grade level or subject area to consider the following:
- How does the adapted free-response question connect to the AP Calculus topic discussed?
- Does the question review, reinforce, or effectively introduce vocabulary, skills, or concepts from your curriculum?
- On which parts of the question might your students have difficulty, and how could you assist them?
- Where does this fit into your teaching timeline?
- How might students use technology in approaching this problem?
- What other questions could be asked using this stem?
Once teachers have several experiences in examining adapted AP questions, they are usually ready to adapt one themselves. Working in small groups, including teachers from multiple grade levels, they tackle one or two questions that seem particularly amenable. Each group should include an experienced AP teacher, who can explain some of the subtleties of the question. Often the form of the stem must be changed to make it accessible for students in lower grades; for instance, a stem that includes a trigonometric function can be changed to tabular or graphical form. Teachers begin the task in one meeting and then complete it at the next meeting so they have more time to contemplate and refine their product. Once it is somewhat finished, it is shared with others for suggestions or constructive criticism. After teachers use the questions in class, they make further improvements.
Here are some other adapted problems:
Adaptations of 2003 Calculus AB/BC Question 1
I used these two problems to introduce the Pre-AP concept to a group of mostly middle school teachers. I showed them the original AP question and commented on the test format and scoring process. The question asked students to find the area of the region enclosed by two functions and a vertical line. In the adapted problems, three lines now enclose the region, and students calculate the area of the resulting triangle. Although this is much simpler, the basic concept of finding the area of an enclosed region is still present, so that the connection to the AP question is maintained while students use current skills. The teachers worked on the two adapted problems in small groups, using the following questions to prompt discussion:
- How do the adapted questions relate to the original AP question?
- Could your students do these problems without assistance?
- How are these questions different from those in your textbook?
Adaptation of 2003 Calculus AB/BC question 1 for Middle Grades
Adaptation of 2003 Calculus AB/BC question 1 for Algebra 1
Adaptation of 2003 Statistics Form B Question 5
I used this problem with the same group, mostly middle school teachers, to increase awareness of AP Statistics, which is relatively new in our district. We followed the same process that we used for the AP Calculus Exam problems. The statistics problem allowed us to discuss randomness, independence of events, simulation, probability, the law of large numbers, and many other important statistical topics. The teachers were particularly interested in using this problem in their classes, because all of the emphasized math skills are important on our state assessment, which is used to determine student promotion.
Adaptation of 2003 Statistics Form B question 5 for Middle Grades
Another Adaptation of 2003 Calculus AB Form B Question 5
As we wrote the other adaptation of this problem, we saw the opportunity to ask questions about the area between the function and the x-axis, which ties in very nicely with the AP Calculus topic of the area of a bounded region. The other question, however, already had plenty of parts, and we didn't want to detract from our focus topic, the analysis of functions. We wrote this one as another example for teachers of how their curriculum is related to students' future studies.
Adaptation of 2003 Calculus AB Form B question 5 for Geometry (2)
Adapting AP problems is a lot like eating peanuts; it's hard to stop with just one. Teachers enjoy the creative challenge, and students benefit: both in reinforcing current learning and in preparing for the future.
Answers to all of these adaptations
Dixie Ross has over 17 years of teaching experience in all levels of mathematics from remedial to AP Calculus BC. She began as a mathematics teacher at Taylor High School in Taylor, Texas. She started the AP program there in 1989 and began working as a College Board consultant and workshop leader in 1994. She served on the Development Committees for the Math Vertical Teams Toolkit, the Setting the Cornerstones workshop, and the new Building Success in Mathematics workshop, and has also served as an Exam Reader. In 1999, she began teaching at Round Rock High School, where she is currently teaching AP Calculus (AB and BC) and pre-calculus.
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