Chapter 11 9-12

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 11: NUMERICAL OPERATIONS

All students will develop their understanding of numerical operations through experiences which enable them to construct, explain, select, and apply various methods of computation including mental math, estimation, and the use of calculators, with a reduced role for pencil-and-paper techniques.

9-12 Overview

In ninth through twelfth grades, 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 our students will live and work, strong skills in numerical operations have perhaps even more importance than they once did. By giving our 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, we prepare them 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 only new topics to be introduced in this standard for these grade levels are work with factorials and matrices as useful tools to be used 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 pencil-and-paper. Students need to continue to develop the alternatives to pencil-and-paper as they learn more operations on other types of numbers, 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

STANDARD 11: NUMERICAL OPERATIONS

9-12 Expectations and Activities

The expectations for these grade levels appear below in boldface type. Each expectation is followed by activities which illustrate how the expectation can be addressed in the classroom.

Building upon K-8 expectations, experiences in grades 9-12 will be such that all students:

O. select and use an appropriate method for computing from among mental math, estimation, pencil-and-paper, 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.

P. 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 though of as (20+3)(10+4)) and (2x+3)(x+4).
Students devise their own procedures and "rules" for operations on variables with exponents by performing trials of equivalent computations on whole numbers.

Q. develop, analyze, apply, and explain methods for solving problems involving factorials, exponents, and matrices.

Students work through the Breaking the Mold unit that is described in the vignette on page 63 of the New Jersey Mathematics Standards. 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 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? How many different 6-place New Jersey license plates are possible? How many different phone numbers can be given out in one area code?