In the ninth through twelfth grades, the themes described in the K-12 Overview - estimation, mental computation, and appropriate calculator and computer use-become the focus of this standard. What is different about this standard at this level when compared to the traditional curriculum is its mere presence. In the traditional academic mathematics curriculum, work on numerical operations was basically finished by eighth grade and focus then shifted exclusively to the more abstract work in algebra and geometry. But, in the highly technological and data-driven world in which today's students will live and work, strong skills in numerical operations have perhaps even more importance than they once did. By giving older students a variety of approaches and strategies for the computation that they encounter in everyday life, approaches with which they can confidently approach numerical problems, they will be better prepared for their future.

The major work in this area, then, that will take place in the high school grades, is continued opportunity for real-world applications of operations, wise choices of appropriate computational strategies, and integration of the numerical operations with other components of the mathematics curriculum.

The new topics to be introduced in this standard for these grade levels involve factorials, matrices, operations with polynomials, and operations with irrational numbers as useful tools in problem solving situations.

Estimation, mental math, and technology use should fully mature in the high school years as students use these strategies in much the same way that they will as adults. If earlier instruction in these skills has been successful, students will be able to make appropriate choices about which computational strategies to use in given situations and will feel confident in using any of these in addition to paper-and-pencil techniques. Students need to continue to develop alternatives to paper-and-pencil as they learn about operations with matrices and other types of number, but the work here is almost exclusively on the continuing use of all of the strategies in rich, real-world, problem solving settings.

The topics that should comprise the numerical operations focus of the ninth through twelfth grade mathematics program are:

operations on real numbers

translation of arithmetic skills to algebraic operations

operations with factorials, exponents, and matrices

The cumulative progress indicators for grade 12 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 9, 10, 11 and 12.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

6^*. Select and use appropriate computational methods from mental math, estimation, paper-and-pencil, and calculator methods, and check the reasonableness of results.

• Students frequently use all of these computational strategies in their ongoing mathematics work. Inclinations to over-use the calculator, in situations where other strategies would be more appropriate, are overcome with five minute "contests," speed drills, and warm-up exercises that keep the other skills sharp and point out their superiority in given situations.

• Numerical problems in class are almost always worked out in "rough" form before any precise calculation takes place so that everyone understands the "ballpark" in which the computed answer should lie and which answers would be considered unreasonable.

• Students use estimation in their work with irrational numbers, approximating the results of operations such as sqrt(15) + sqrt(17) or sqrt(32) sqrt(8), and developing general rules.

• Students discuss the advantages and disadvantages of using graphing calculators or computers to perform computations with matrices.

• Students demonstrate their ability to select and use appropriate computational methods by generating examples of situations in which they would choose to use a calculator, to estimate, or to use mental math.

• Students solve given computational problems using an assigned strategy and discuss the advantages and disadvantages of using that particular strategy with that particular problem.

13. Extend their understanding and use of operations to real numbers and algebraic procedures.

• Students work on the painted cube problem to enhance their skill in writing algebraic expressions: A 3-inch cube is painted red. It is then cut into 1-inch cubes. How many of them have 3 red faces? 2 red faces? 1-red face? No red faces? Repeat the problem using an original 4-inch cube, then a five-inch cube, then an n-inch cube.

• Students develop a procedure for binomial multiplication as an extension of their work with 2-digit whole number multiplication arrays. Using algebra tiles, they uncover the parallels between 23 x 14 (which can be thought of as (20+3)(10+4)) and (2x+3)(x+4).

• Students work through the Ice Cones lesson that is described in the First Four Standards ofthis Framework. They discover that in order to graph the equation to determine the maximum volume of the cones, they need to use algebraic procedures to solve for h in terms of r.

• Students devise their own procedures and "rules" for operations on variables with exponents by performing trials of equivalent computations on whole numbers.

• Students use algebra tiles to develop procedures for adding and subtracting polynomials.

• Students use compasses and straightedges to construct a Golden Rectangle and find the ratio of the length to the width (1 + sqrt(5))/2.

• Students consider the ratios of successive terms of the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...), where each term after the first two is the sum of the two preceding terms, finding that the ratios get closer and closer to the Golden Ratio (1 + û5)/2.

14. Develop, apply, and explain methods for solving problems involving factorials, exponents, and matrices.

• Students work through the Breaking the Mold lesson that is described in the Introduction to this Framework. It uses a science experiment with growing mold to involve students in discussions and explorations of exponential growth.

• Students use their graphing calculators to find a curve that best fits the data from the population growth in the state of New Jersey over the past 200 years.

• Students discover the need for a factorial notation and later incorporate it into their problem solving strategies when solving simple combinatorics problems like: How many different five card poker hands are there? In how many different orders can four students make their class presentations? In how many different orders can six packages be delivered by the letter carrier?

• Students compare 2¹⁰¹, (2⁵⁰) ², and 3x2¹⁰⁰ to decide which is largest. They explain their reasoning.

• Students read "John Jones's Dollar" by Harry Keeler and discuss how it demonstrates exponential growth. They check the computations in the story, determining their accuracy.

References

Keeler, Harry Stephen. "John Jones's Dollar," in Clifton Fadiman, Ed. Fantasia Mathematica. New York: Simon and Schuster, 1958.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.