Each day we will solve and discuss typical AP calculus questions. Not primarily for calculus sake, but to show how to adapt AP questions for use in the middle and early high school math classes. To do so, we cut the noncalculus portions of the questions out and give them to our algebra and geometry classes. For example, if the AP question asks students to find the area of a region enclosed by two functions and a vertical line, in the adapted problem, 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. Examining the middle and early high school versions of the AP Calculus questions, one sees easily how minor changes in the content and vocabulary can help better prepare students for success.

The fundamental concepts of a first course in calculus are Limits, Functions, Their Graphs, Derivatives, and Integrals. Without a grasp of these concepts any attempt to participate seriously in a scientific discussion of the world around us is hopeless. Exercise: Why?

When should we introduce the vocabulary necessary to talk about Functions and their Graphs, Limits of Sequences, Series, and Functions Derivatives (Slopes), and Definite Integrals (Averages, Areas)? The first time the word “function” is seriously covered in an activity in the Comprehensive Curriculum is in 7 th grade (Unit 3, Activity 14). In 7 th grade (G7U2A7), an “input-output table” is discussed without using the word “function” at all. Throughout the early grades the word “function” is avoided by using the more familiar word “rule.” Sequences are called patterns, series do not exist, there are no limits, lines have no slopes (derivatives), and averages and areas can not be called definite integrals. Why?

Grade 8, Unit 6 Activity 6: How Much Do I Get? Pose the following to the students and have them explore which is the better salary option. You have been hired to do yard work for the summer, and you will be paid every day for 15 days. But first you have to choose your salary option as (1) get paid $10 the 1st day, $11 the 2nd day, $12 the 3rd day, etc., or (2) get paid $.01 the first day, $.02 the 2nd day, $.04 the 3rd day, $.08 the 4th day, etc. Have students create a table or chart of their values and a rule that explains the relationship in the chart. Have students decide which type of sequence each of the salary options illustrates and generate the 15 terms in each sequence. Next, have students determine the amount they would get paid at the end of the 15 days to determine the best salary option. Have the students graph each of these salary options on the same graph in different colors and make observations about the relationship of the two options. Lead the class in a discussion about how the information in the chart, graph or rule all relates to the situation.

Good problem are accessible throughout the grade levels: from elementary school to dissertation level mathematics. That is, good problems have a vertical component. A good problem is suitable for a mathematical analysis that combines different strands of mathematics, especially algebra and geometry. That is, good problems have a horizontal component. A good problem strengthens the mathematical habits of mind and compels the student to think on a deeper level. It is open-ended and generalizes in a way that mathematicians would regard as sophisticated, mature, interesting, and pleasing mathematically. That is, good problems have an infinite component.

 Look at extreme cases  Go beyond a particular answer to a particular problem  Extend problems beyond specific numbers and situations  Look for geometric and algebraic approaches to problems  Look for different problems with identical mathematical analyses  Be curious  Pay attention to the “structure” of the relationships being derived  Be historical  Look for hidden beauty

Grade 8, Unit 6, Activity 9: Playing Around with Fibonacci! 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 988, … Lead the class in a short discussion about the Fibonacci sequence of numbers. … Have them select any three consecutive terms and find the product of the first and third numbers, square the second number and look for a pattern. Have students write conjectures on paper and check these conjectures to see if they hold true for other sequences. Once the students have had time to write the conjectures, have them create a second Fibonacci-like sequence starting with 4 and determine whether the pattern holds true. Give students time to write a conjecture and test their conjecture on other sequences. Challenge the students to begin another Fibonacci-like sequence with another single digit and test the earlier conjectures so that they can determine a conjecture that holds true for any Fibonacci-like sequence of numbers.

In our experience, students have a difficult time dealing with the algebraic aspects of calculus. The difficulties are often so overwhelming, that teaching calculus past chapter two becomes impossible. Without sufficient practice and exposure to a more challenging algebra curriculum, our college bound students have little chance of being successful in the ever growing number of undergraduate programs and careers that demand such skills.