Traditionally, tremendous amounts of time were spent at these grade levels helping students to finish their development of complex paper-and-pencil procedures for the four basic operations with whole numbers, fractions, and decimals. While some competency with paper-and-pencil computation is necessary, 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.

As indicated in the K-12 Overview, then, the major shift in the curriculum that will take place at these grade levels is one away from drill and practice of paper-and-pencil 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 this standard's Grades 5-6 Overview. Operations on fractions and decimals, as well as whole numbers, should be relatively well developed by this point, allowing the focus to shift 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 focused on numerical operations, per se, so that the other areas of the Standards-based curriculum receive adequate attention.

One important set of related topics that needs to receive some significant attention here, however, 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.

Two other 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 that are used and the ways in which they are used.

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 paper-and-pencil procedures. For example, students should evaluate simple problems involving fractions, such as what's two-thirds of 5 tablespoons? using mental math. Students need to continue to develop alternatives to paper-and-pencil as they learn more about 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		powers and roots
integer operations		proportion and percent

The cumulative progress indicators for grade 8 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 7 and 8.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 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 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 vs. calculator races. Given a list of specially selected computation exercises (e.g., 53 x 20, 40 x 10, 95 + 17 +5 ), 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.

8. 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 show why the multiplication of two fractions less than one results in a product that is less than either.

• Students demonstrate their understanding of operations on rational numbers by formulating their own reasonable word problems to accompany given number sentences such as 3/4 divided by 1/2 = 1 1/2.

9. Extend their 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 blanks, into 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 into one of their blanks. Once written, anumber can not be moved. 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, 7² x 7⁴ = (7 x 7) x (7 x 7 x 7 x 7) = 7 x 7 x 7 x 7 x 7 x 7 = 7⁶. You just add the exponents!

• Students read The King's Chessboard, The Rajah's Rice: A Mathematical Folktale from India, or A Grain of Rice. All of these stories involve a situation in which a quantity is doubled each day. Students use the story to discuss powers of 2 and to look for patterns in the sums of the powers of 2.

• Students use the relationship between the area of a square and the length of one of its sides to begin their study of roots. Starting with squares on a geoboard with areas of 1, 4, 9, and 16, they then are asked to find squares whose areas are 2, 5, and 13.

• Students work through the Rod Dogs lesson that is described in the First Four Standards of this Framework. They investigate how the surface area and volume of an object changes as it is enlarged by various scale factors.

10. Develop, 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 the tape forward (+) with the student walking forward (+) is walking forward (+). The "product" of running it backward (-) with the student walking forward (+) is walking backwards (-). The other two combinations also work out correctly.

• Students use base ten blocks laid out in an array to show decimal 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.

• Students explore the equivalence between fractions and repeated decimals by finding the decimal representations of various fractions and using the resulting patterns to find the fractional equivalents of some repeated decimals.

11. Develop, apply, and explain methods for solving problems involving proportions and percents.

• Students use The Geometer's 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 are comfortable using a variety of approaches to the solution of proportion problems. Example: If 8 pencils cost 40 cents, how much do 10 pencils cost? This problem can be solved by:

  unit-rate method	8 pencils for 40 cents means 5 cents per pencil or 10 x 5=50 cents for 10
  factor-of-change method	10 pencils is 10/8 of 8 pencils, so cost is (10/8)x40=50 cents
  cross multiplication method	8/40 = 10/x, 8x = 400, so x=50 cents.

• Students set up a part/whole proportion as one method of solving percent problems.

• Students spend $100 by selecting items from a catalog. They must compute sales tax and consider it in deciding what they will buy.

12. 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 "3 + 15 ÷ 3" and then compare their calculator displays. Some of the displays show 6 and others show 8. Why? Which is right? Are the other calculators broken? Students decide what key sequence would work for the calculators that do not use order of operations.

• Students play with the software How the West was One + Three x 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.

References

Barry, David. The Rajah's Rice: A Mathematical Folktale from India. San Francisco, CA: W. J. Freeman, 1995.

Birch, David. The King's Chessboard. Puffin Pied Piper Books, 1988.

Pittman, Helena Clare. A Grain of Rice. Bantam Skylark, 1986.

Software

Geometer's Sketchpad. Key Curriculum Press.

How the West Was One + Three x Four. Sunburst Communications.

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.