Chapter 11 5-6

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.

5-6 Overview

The wide 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. Traditionally, tremendous amounts of time were spent helping children to develop proficiency and accuracy with pencil-and-paper procedures. Now, the societal reality is that adults needing to perform calculations quickly and accurately have electronic tools that are both more accurate and more efficient than those 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.

Much research in the past decade has looked at students' understanding of operations with large whole numbers and with decimals and fractions. A valuable summary of the findings is presented in Number Concepts and Operations in the Middle Grades, published by Lawrence Erlbaum and Associates and available from NCTM. Part of the preface describes what has been the focus of the research:

In the primary grades, children experience number as whole numbers and operate on them by adding and subtracting. ..... Despite the fact that the unit throughout this period is a single whole entity (the whole number "1," a rather simple looking concept), acquiring a mature conceptualization of unit is a protracted and cognitively demanding process.
In the middle grades, the operations change from addition and subtraction to multiplication and division. And the numbers change, from whole numbers to rational numbers. Underneath all of the surface level changes is a fundamental change with far-reaching ramifications: a change in the nature of the unit.

Both multiplication and division demand different understandings of whole numbers than those necessary for addition and subtraction. And, obviously, working with fractions and decimals requires a restructuring of a child's notion of unit. The topics that should comprise the numerical operations focus of the fifth and sixth grade mathematics program, and thus raise all of these challenges for students, are:

multi-digit whole number multiplication and division

decimal multiplication and division

fraction operations

integer operations

STANDARD 11: NUMERICAL OPERATIONS

5-6 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-4 expectations, experiences in grades 5-6 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.

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 blocks (in the standard form of a 2-digit multiplication problem) in each of which will be written a digit. 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 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.

I. 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 Bars to solve problems like: What would be the fairest way to divide four cakes among five people? 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 finish off 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 (7/8 = 1/2 + 1/4 + 1/8). They then try to find Egyptian fractions for 2/3, 2/5, and 4/5.
Students begin an exploration of integer operations using the rises and falls of a hot air balloon as a model of movement along a number line; the higher the balloon gets, the larger the number on the number line opposite it. The model explores the effect on the balloon's position of additions and subtractions of bags of sand.

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

Students take a class session to explore the exponent key, the x^2 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 make up a keypress sequence involving it the result of which they can predict before they key it in.
Students work through The Powers of the Knight lesson that is described in the vignette on page 48 of the New Jersey Mathematics Standards. It introduces a classic problem of geometric growth which engages them as they encounter notions of exponential notation.
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 with areas of 1, 4, 9, and 16, they then are asked to find a square whose area is 2.

K. develop, analyze, 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 Expectation H 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 pencil-and-paper procedures for completing those operations.
Students develop rules for integer operations by using Postman Stories, as described in Bob Davis' Explorations 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).

L. develop, analyze, 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 Christmas 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 he or she have fit in the room or just reach in the window? How much did they weigh?

M. develop, analyze, and explain arithmetic sequences.

Students solve this problem: How many rectangles are in a string of 6 squares stuck together in a line? By logically counting first the 1 by 1 squares, then the 2 by 1 rectangles, then the 3 by 1 rectangles, and so on, they discover an interesting sequence that they can then extend to answer the question.
Students explore the increase in the number of diagonals of a polygon as the number of sides increases. They attempt to find a formula to figure out how many diagonals an n-sided polygon would have.

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 Rolling Numbers. They use four white dice and one red one to generate for 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 = 1, the following expression works: (2+5)/(3+4)=1. Questions about order of operations and about appropriate use of parentheses frequently arise.