Chapter 11 3-4

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.

3-4 Overview

The wide availability of computing and calculating technology has given us the opportunity to reconceive the role of computation and numerical operations in our third and fourth grade mathematics programs. Traditionally, tremendous amounts of time were spent at these levels 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 in this realm, 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.

Third and fourth graders are primarily concerned with cementing their understandings of addition and subtraction and developing new meanings for multiplication and division. They should be in an environment where they can do so by modeling and otherwise representing a whole variety of real-world situations in which these operations are appropriately used. It is important that the variety of situations to which they are exposed include the full gamut of multiplication and division. There are several slightly different taxonomies of these types of problems, but minimally students at this level should be exposed to repeated addition and subtraction, array, area, and expansion problems. Students need to recognize and model each for each operation.

Basic facts in multiplication and division also continue to be very important. Students should be able to quickly and easily recall one-digit products and quotients. The most effective approach to enabling them to acquire this ability has been shown to be the focused and explicit use of basic fact strategies - conceptual techniques that make use of the childs understanding of the operations and number relationships to help recover the appropriate product or quotient. Doubles and Doubles and One More are useful strategies, but also useful are discussions and understandings regarding the regularity in the nines facts, the roles of one and zero in these operations, and the roles of commutativity and distributivity.

Students must still be able to perform two-digit multiplication and division with pencil and paper, but the widespread availability of calculators has made the particular procedure used to perform the calculations less important. It need no longer be the single fastest, most efficient algorithm chosen without respect to the degree to which children understand it. Rather, the teaching of this two-digit computation should take on more of a problem solving approach, a more conceptual, developmental approach. Students should first use the models of multi-digit number that they are most comfortable with (base ten blocks, money) to explore the new class of problems. Students who have never formally done two-digit multiplication might be asked to use their materials to help figure out how many pencils are packed in the case just received in the school office. There are 24 boxes with a dozen pencils in each box. Are there enough for every student in the school to have one? Other, similar, real-world problems would follow, some involving regrouping and others not.

After initial exploration, students share with each other all of the strategies theyve developed, the best ways theyve found for working with the tens and ones in the problem, and their own approaches to dealing with the place value issues involved. Most students can, with direction, take the results of those discussions and create their own pencil-and-paper procedures for multiplication and division. The discussions can, of course, include the traditional approaches but these ought not to be seen as the only right way to do these operations.

Estimation and mental math also become critically important in these grade levels as students are inclined to use calculators for more and more of their work. In order to use that technology effectively, third and fourth graders must be able to use estimation to know the range in which the answer to a given problem should lie before doing any calculation, to assess the reasonableness of the results of a computation, and to be satisfied with the results of an estimation when an exact answer is unnecessary. Mental mathematics skills, too, play a more important role in third and fourth grade. Simple two-digit addition and subtraction problems and those involving powers of ten should be performed mentally. Students should have enough confidence in their ability with these types of computations to do them mentally rather than looking for either a calculator or pencil and paper.

Technology should also be an important part of the environment in third and fourth grade classrooms. Calculators provide a valuable teaching tool when used to do student-programmed repeated addition or subtraction, to offer estimation and mental math practice with target games, and to explore operations and number types that the students have not formally encountered yet. They should also be used routinely to perform computation in problem solving situations that the students may not be able to perform otherwise. This use prevents the need to artificially contrive real-world problems so that the numbers come out friendly.

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

multiplication and division basic facts

multi-digit whole number addition and subtraction

two-digit whole number multiplication and division

decimal addition and subtraction

explorations with fraction operations

STANDARD 11: NUMERICAL OPERATIONS

3-4 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-2 expectations, experiences in grades 3-4 will be such that all students:

A. develop meaning for the four basic operations by modeling and discussing a variety of problems.

Students broaden their initial understanding of multiplication as repeated addition by dealing with situations where it is arrays, expansions, and combinations. These kinds of questions are not easily covered by the repeated addition meaning: How many stamps are on this 7 by 8 sheet? How big would this painting be if it was 3 times as big? How many outfits can you make with 2 pairs of pants and 3 shirts?
Students use counters to model both repeated subtraction and partition division meanings for division and write about the difference in their journals.
From the beginning of their work with division, children are asked to make sense out of remainders in problem situations. The answers to these three problems are different even though the division is the same: How many cars will we need to transport 19 people if each car holds 5? How many more packages of 5 ping-pong balls can be made if there are 19 balls left in the bin? How much does each of 5 children have to contribute to the cost of a $19 gift?

B. develop proficiency with basic facts through the use of a variety of strategies.

Students use Streets and Alleys as both a mental model of multiplication and a useful way to recover facts when needed. It simply involves drawing a series of horizontal lines to represent one factor and a series of vertical lines crossing them to represent the other. The number of intersections of the streets and alleys is the product!
Students use a double maker on a calculator for practice with doubles. They enter x 2 = on the calculator. Any number pressed then, followed by the equals sign, will show the numbers double. Students work together to try to say the double for some number before the calculator shows it.
Students regularly use the Doubles and Doubles and One More and the Use a Related Fact strategies for multiplication, but they also recover facts by knowledge of the role of zero and one in multiplication, of commutativity, and of the regular patterned behavior of nines. Practice sets of problems are structured so that strategy use is encouraged and the students are regularly asked to explain the procedures they are using.

C. construct, use, and explain procedures for performing whole number calculations in the various methods of computation.

Students work through the Product and Process lesson that is described in the vignette on page 41 of the New Jersey Mathematics Standards. It challenges students to use calculators and four of the five digits 1, 3, 5, 7, and 9 to discover the multiplication problem that gives the largest product.
Students explore lattice multiplication and try to figure out how it works.
Students use the skills they've developed with arrow diagrams (See Number Sense, Expectation C for Grades 3-4) to practice mental addition and subtraction of 2- and 3-digit numbers. To add 23 to 65, for instance, they start at 65 on their mental hundred number chart, go down twice and to the right three times.

D. use models to explore operations with fractions and decimals.

Students use Fraction Circles pieces (each unit fraction a different color) to begin to explore addition of fractions. Questions like: Which of these sums are greater than 1? and How do you know? are frequent.
Students use the base ten models that they are most familiar with for whole numbers and relabel the components with decimal values. Base ten blocks represent 1 whole, 1 tenth, 1 hundredth, and 1 thousandth. Coins, which had represented a whole number of cents, now represent hundredths of dollars.
Students operate a school store with school supplies available for sale. Other students, using play money, decide on purchases, pay for them, receive and check on the amount of change.

E. select and use an appropriate method for computing from among mental math, estimation, paper-and-pencil, and calculator methods and check the reasonableness of results.

Students play Addition Max Out. Each student has a 2 x 3 array of blocks (in standard 3-digit addition form) 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 sum when all six digits have been written. If a player has the largest possible sum that can be made from the six digits rolled, there is a bonus for Maxing Out.
Students discuss this problem from the NCTM Standards: The three fourth grade teachers decided to take their classes on a picnic. Mr. Clark spent $26.94 for refreshments. He then used his calculator to see how much the other two teachers should pay him so that all three could share the cost equally. He figured they each owed him $13.47. Is his answer reasonable?

F. use a variety of mental computation and estimation techniques.

Students frequently do warm-up drills that enhance their mental math skills. Problems like: 3000 x 7 =, 200 x 6 =, and 5000 x 5 + 5= are put on the board as individual children write the answers without doing any pencil-and-paper computation.
Students make appropriate choices from among front-end, rounding, and compatible numbers strategies in their estimation work depending on the real-world situation and the numbers involved.
Students use money and shopping situations to practice estimation and mental math skills. Is $20.00 enough to buy items priced at $12.97, $4.95, and 3.95? About how much would 4 cans of beans cost if each costs $0.79?

G. understand and use relationships among operations and properties of operations.

Students take 7x8 block rectangular grids printed on pieces of paper. They each cut along any one of the 7 block long segments to produce two new rectangles, for example, a 7x6 and a 7x2 rectangle. They then discuss all of the different rectangles pairs they produced and how they are all related to the original one.
Students write a letter to a second grader explaining why 2+5 equals 5+2.
Students explore modular, or clock, addition as an operation that behaves differently from the addition they know how to do. How is it different? How is it the same? How would modular subtraction and multiplication work?