Chapter 11 7-8

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.

7-8 Overview

The wide availability of computing and calculating technology demands that we significantly reconceive the role of computation and numerical operations in our seventh and eighth grade mathematics programs. Traditionally, tremendous amounts of time were spent at these levels helping students to finish off their development of complex pencil-and-paper procedures for the four basic operations with whole numbers, fractions, and decimals. Now, the societal reality is that adults needing to perform those calculations quickly and accurately have electronic tools that are both more accurate and more efficient than the paper-and-paper procedures. At the same time, though, the new technology has presented us with a situation where some numerical operations, skills, and concepts are much more important than they used to be. Estimation, mental computation, and understanding the meanings of the standard arithmetic operations all play a more significant role than ever in the everyday life of a mathematically literate adult.

The major shift in the curriculum that will take place at these grade levels, therefore, is one away from drill and practice of pencil-and-paper symbolic procedures and toward real-world applications of operations, wise choices of appropriate computational strategies, and integration of the numerical operations with other components of the mathematics curriculum.

Seventh and eighth graders are relatively comfortable with the unit shift discussed in the Grades 5-6 Numerical Operations Overview. Operations on fractions and decimals, as well as whole numbers, should be relatively well developed by this point and the focus switches to a more holistic look at operations. Numerical Operations becomes less a specific object of study and more a process, a set of tools for problem setting. It is critical that teachers spend less time on focused work in this area so that the other areas of the Standards-based curriculum receive adequate attention.

One important set of related topics that do receive some significant attention here, though, is ratio, proportion, and percent. Seventh and eighth graders are cognitively ready for a serious study of these topics and to begin to incorporate proportional reasoning into their set of problem solving tools. Work in this area should start out informally, progressing to the student formulation of procedures that make proportions and percents the powerful tools they are.

Another two topics that receive greater attention here, even though they have been informally introduced earlier, are integer operations and powers and roots. Both of these types of operations further expand the students knowledge of the types of numbers we use and the ways in which we use them.

Estimation, mental math, and technology use begin to mature in seventh and eighth grades 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 continue to learn more operations on other types of numbers, but the work here is primarily 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 seventh and eighth grade mathematics program are:

rational number operations

integer operations

powers and roots

proportion and percent

STANDARD 11: NUMERICAL OPERATIONS

7-8 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-6 expectations, experiences in grades 7-8 will be such that all students:

H. 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 choose a stock from the New York Stock Exchange and estimate and then compute the net gain or loss each week for a $1,000 investment in the company.
Students use spreadsheets to "program" a set of regular, repeated, calculations. They might, for example, create a prototype on-line order blank for a school supply company that lists each of the ten items available, the individual price, a cell for each item in which to place the quantity ordered, the total computed price for each item, and the total price for the order.
Students regularly have Human versus Calculator Races. Given a list of specially selected computation exercises, one student uses mental math strategies and another uses a calculator. They quickly come to realize that the human has the advantage in many situations.

I. extend their understanding and use of arithmetic operations to fractions, decimals, integers, and rational numbers.

Given a decimal or a fractional value for a piece of a tangram puzzle, the students determine a value for each of the other pieces and a value for the whole puzzle.
Students use Fraction Squares to justify their understanding that the multiplication of two fractions less than one results in a product that is less than either.
Students regularly demonstrate their understanding of operations on these numbers by formulating their own reasonable word problems to accompany given number sentences such as 3/4 divided by 1/2 = 1 1/2.

J. extend their basic understanding of basic arithmetic operations on whole numbers to include powers and roots.

Students play Powers Max Out. Each student has a set of 5 blocks, in each of which will be written a digit. They are in the form VW^X + YZ. One student rolls a die and everyone must write the number showing in one of their blocks. They can never change it. Another roll - another number written, and so on. The object is to be the player with the largest valued expression when all five digits have been written. If a player has the largest possible value that can be made from the five digits rolled, there is a bonus for Maxing Out.
Students develop their own "rules" for operations on numbers raised to powers by rewriting the expressions without exponents. For example 7^2 x 7^4 = (7 x 7) x (7 x 7 x 7 x 7) = 7 x 7 x 7 x 7 x 7 x 7 = 7^6 . You just add them!

K. develop, analyze, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers.

Students use a videotape of a youngster walking forward and backward as a model for multiplication of integers. The "product" of running it forward (+) with the student walking forward (+) is walking forward (+). The other three combinations also work out correctly.
Students explore the possibility of using for decimal multiplication the array model with base-ten blocks that they have used for 2-digit whole number multiplication. How could the values of the blocks be changed to allow it to work? What new insights do we gain from the use of this model?
Students judge the reasonableness of the results of fraction addition and subtraction by "rounding off" the fractions involved to 0, 1/2, or 1.

L. develop, analyze, apply, and explain methods for solving problems involving proportions and percents.

Students use The Geometers Sketchpad software to draw a geometric figure on a computer screen, scale it larger or smaller, and then compare the lengths of the sides of the original with those of the scaled image. They also compare the areas of the two.
Students set up a part/whole proportion as one method of solving percent problems.
Students are comfortable using a variety of approaches to the solution of proportion problems including a unit-rate method, and a factor-of-change method in addition to the traditional cross multiplication method.

M. develop, analyze, and explain arithmetic sequences.

Students solve this problem: How many squares are on a checkerboard? By logically examining first the 1 by 1 squares, then the 2 by 2's, then the 3 by 3's, and so on, they discover an interesting sequence that they can then extend to answer the question.
Students work on the Good News problem from the NCTM Standards: Good News travels fast. Iris saved enough money from her paper route to buy a new bike. She immediately told two friends, who, ten minutes later, each repeated the news to two other friends. Ten more minutes later, these friends each told two others. If the news continues to spread like this, how many people will know about Iriss new bike after eighty minutes?
Students explore the sequence generated by the layers of a pyramid built with layers which are equilateral triangles of tennis balls. The top layer is 1 ball, the next is 3, the next is 6, and so on.

N. understand and apply the standard algebraic order of operations.

Students bring in calculators from home to examine their differences. Among other activities, they each key in " 6 + 2 x 4 = " and then compare their calculator displays. Some of the displays show 32 and others show 14. Why? Which is right? Are the other calculators broken?
Students play with the software How the West was Won, Two, Three, Four, which requires them to construct numerical expressions that use the standard order of operations.
Students use the digits 1, 2, 3, and 4 to find expressions for each of the numbers between 0 and 50. For example, 7 = (3x4)/2 + 1