As indicated in the K-12 Overview, the widespread availability of computing and calculating technology has given us the opportunity to significantly reconceive the role of computation and numerical operations in our fifth and sixth grade mathematics programs. Some skills are less important while others, such as 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 in grades 5 and 6, therefore, 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. At these grade levels, students are consolidating their understanding of whole number operations (especially multiplication and division) and beginning to develop computational skills with fractions and decimals. A sample unit on fractions for the sixth-grade level can be found in Chapter 17 of this Framework.

Much research in the past decade has focused on students' understandings of operations with large whole numbers and work with fractions and decimals. Each of these areas requires students to restructure their simple conceptions of number that were adequate for understanding whole number addition and subtraction.

Multiplication requires students to think about different meanings for the two factors. The first factor in a multiplication problem is a "multiplier." It tells how many groups one has of a size specified by the second factor. Thus, students need different understandings of the roles of the two numbers in the operation of multiplication than their earlier understandings of addition, in which both addends meant the same thing.

A similar restructuring is necessary for dealing appropriately with operations involving fractions and decimals. This restructuring revolves around the role of the "unit" in these numbers. In earlier grades, students thought about 5 or 498 as numbers that represented that many things. The understood unit, one, is the number which was used to count a group of objects. With fractions and decimals, though, the unit, still one and still understood, is a harder concept to deal with because its essential use is to help define the fraction or decimal rather than as a counter. When we speak of 5 poker chips or 35 students, our message is reasonably clear to elementary students. But when we speak of 2/3 of the class or 0.45 of the price of the sweater, the meaning is significantly less clear and we must be much more explicit about the role being played by the unit.

The topics that should comprise the numerical operations focus of the fifth and sixth grade mathematics program are:

multi-digit whole number multiplication and division
decimal multiplication and division
fraction operations
integer operations

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 5 and 6.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 5-6 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.

• Fifth and sixth grade students have calculators available to them at all times, but frequently engage in competitions to see whether it is faster to do a given set of computations with the calculators or with the mental math techniques they've learned.

• Fifth graders make rectangular arrays with base-ten blocks to try to figure out how to predict how many square foot tiles they will need to tile a 17' by 23' kitchen floor.

• Students are challenged to answer this question and then discuss the appropriate use of estimation when an exact answer is almost certain to be wrong: The Florida's Best Orange Grove has 15 rows of 21 orange trees. Last year's yield was an average of 208.3 oranges per tree. How many oranges might they expect to grow this year? What factors might affect that number?

• Students play multiplication max out. Each student has a 2 x 2 array of blanks (in the standard form of a 2-digit multiplication problem) into each of which a digit will be written. One student rolls a die and everyone must write the number showing into one of the blanks. Once a number is written, it cannot be moved. Another roll-another number written, and so on. The object is to be the player with the largest product when all four digits have been written. If a player has the largest possible product that can be made from the four digits rolled, there is a bonus for maxing out.

8. Extend their understanding and use of arithmetic operations to fractions, decimals, integers, and rational numbers.

• Students work in groups to explore fraction multiplication and division. They use fraction circles and fraction strips to solve problems like: How can you divide four cakes among five people evenly? They solve the problems and then write in their math journals about the methods they used and the reasons they believe their answers to be good ones.

• Students complete their study of fraction addition and subtraction by reading about Egyptian fractions. The Egyptians wrote every fraction as a unit fraction or the sum of a series of unit fractions with different denominators, for example, 7/8 = 1/2 + 1/4 + 1/8. They try to find Egyptian fractions for 2/3 (1/2 + 1/6); 2/5 (1/3 + 1/15); and 4/5 (1/2 + 1/5 + 1/10).

• Students demonstrate their understanding of division of fractions on a test by drawing a picture to show that "1 1/2 ÷ 1/2" means: How many halves are there in 1 1/2?

• Students use two-color chips to explore addition of integers. They each take ten chips and toss them ten times. Each time, the students record the number of yellow chips as a positive number (points earned) and the number of red chips as a negative number (points lost). For each toss, the student writes a number sentence, such as 6 + -4 = 2 for 6 points earned and 4 lost. Students may also keep a running total of points overall.

• Students read and discuss If You Made a Million by David Schwartz, relating money to decimals.

• Students read and discuss "Beasts of Burden" in The Man Who Counted: A Collection of Mathematical Adventures by Malba Tahan. In this story, three brothers must divide their Father's 35 camels so that one gets 1/2 of the camels, another 1/3, and the last 1/9. The narrator and a wise mathematician help them solve the problem by adding their camel to the 35, making 36. One brother then gets 18, another 12, and the third receives 4 making a total of 34. The narrator and mathematician take back their camel as well as the one left over.

• Students demonstrate their understanding of addition and subtraction of unlike fractions on a test by finding the errors made by a fictitious student and explaining to that student what he/she did wrong.

• Students read Shel Silverstein's poem A Giraffe and a Half and make up stories about mixed fractions.

• Students discuss whether 2/3 x 5/4 is more or less than 2/3. They explain their reasoning.

• Students listen to John Ciardi's poem "Little Bits" and discuss the concept that a whole can be described by an infinite number of equivalent names, such as 1/2 + 1/4 + 1/8 + 1/8 or 3/3.

• Students solve missing link problems, like the one below, in which they must find number(s) and/or symbols that will make a true sentence:

  ______ + 3.1 - ______ - 5.4 = 8.7

9. Extend their understanding of basic arithmetic operations on whole numbers to include powers and roots.

• Students explore the exponent key, the x² key, and the square root key on their calculators. The groups are challenged to define the function of each key, to tell how each works, and to create a keypress sequence using these keys, the result of which they predict before they key it in.

• Students work through The Powers of the Knight lesson that is described in the Introduction to this Framework. It introduces a classic problem of geometric growth which engages them as they encounter notions of exponential notation.

• Students work through the Pizza Possibities lesson that is described in the First FourStandards of this Framework. In it, students discover that the number of pizzas possible doubles every time another choice of topping is added. They work through the Two-Toned Towers lesson that is also described in the First Four Standards and note the similarities in the problems and in their solutions.

• Students join the midpoints of the sides of a 2 x 2 square on a geoboard to form a smaller square. They determine the area of the smaller square and explore the lengths of its four sides.

10. Develop, apply, and explain procedures for computation and estimation with whole numbers, fractions, decimals, integers, and rational numbers.

• Students working in groups develop a method to estimate the products of two-digit whole numbers and decimals by using the kinds of base-ten block arrays described in Indicator 6 above. Usually just focusing on the "flats" results in a reasonable estimate.

• Students follow up a good deal of experience with concrete models of fraction operations using materials such as fraction bars or fraction squares by developing and defending their own paper-and-pencil procedures for completing those operations.

• Students develop rules for integer operations by using postman stories, as described in Robert Davis' Discovery in Mathematics. The teacher plays the role of a postman who delivers mail to the students. Sometimes the mail delivered contains money (positive integers) and sometimes bills (negative integers). Sometimes they are delivered to the students (addition) and sometimes picked up from them (subtraction).

• Students model subtraction with two-color chips by adding pairs of red and yellow chips. (First, they must agree that an equal number of red and yellow chips has a value of 0.) For example, to show 4 - (-2), they lay out 4 red chips, add 2 pairs of red and yellow chips (whose value is 0), and then take away 2 yellow chips. They note that 4 - (-2) and 4 + 2 give the same answer and try to explain why this is so.

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

• Students develop an estimate of pi by carefully measuring the diameter and circumference of a variety of circular objects (cans, bicycle tires, clocks, wooden blocks). They list the measures in a table and discuss observations and possible relationships. After the estimate is made, pi is used to solve a variety of real-world circle problems.

• Students use holiday circulars advertising big sales on games and toys to comparison shop for specific items between different stores. Is the new Nintendo game, Action Galore, cheaper at Sears where it is 20% off their regular price of $49.95 or at Macy's where it's specially priced at $41.97?

• One morning, as the students arrive at school, they see a giant handprint left on the blackboard overnight. They measure it and find it to be almost exactly one meter long. How big was the person who left the print? Could she have fit in the room to make the print, or did she have to reach in through the window? How could you decide how much she weighs?

• Students read Jim and the Giant Beanstalk by Raymond Briggs. Jim helps the aging giantby measuring his head and getting giant eyeglasses, false teeth and a wig. The students use the measures given in the book to find the size of the giant's hand and then his height.

• Students develop a sampling strategy and use proportions to determine the population of Bean City (NCTM Addenda Booklet, Grade 6), whose inhabitants consist of three types of beans.

• Students discuss different ways of finding "easy" percents, such as 50% of 30 or 15% of 25. They then generate percent exercises that can be solved mentally and share them with their classmates.

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 " 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 rolling numbers. They use four white dice and one red one to generate four working numbers and one target number. They must combine all of the working numbers using any operations they know to formulate an expression that equals the target number. For example, for 2, 3, 4, 5 with target number 1, the following expression works: (2+5)/(3+4)=1. Questions about order of operations and about appropriate use of parentheses frequently arise.

References

Briggs, Raymond. Jim and the Giant Beanstalk. Coward-McCann, Inc., 1970.

Ciardi, John. "Little Bits," in You Read to Me, I'll Read to You. New York: Lippincott, 1962.

Davis, Robert. Discovery in Mathematics. New Rochelle, NY: Cuisenaire, 1980.

Hiebert, J., and Behr, M. (Eds.) Number Concepts and Operations in the Middle Grades. Reston, VA: National Council of Teachers of Mathematics, 1988.

National Council of Teachers of Mathematics. Addenda Booklet, Grade 6. Reston, VA, 1992.

Schwartz, David. If You Made a Million. New York: Lothrop, Lee, & Shepard Books, 1989.

Silverstein, Shel. A Giraffe and a Half. New York: Harper and Row, 1964.

Tahan, Malba. "Beasts of Burden," in The Man Who Counted: A Collection of Mathematical Adventures. W. W. Norton, 1993.

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.