Experiences will be such that all students in grades K-2:

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

• Students use unifix cube towers of two colors to show all the ways to make "7" (for example, 3+4, 2+5, 0+7, and so on). This activity focuses more on developing a sense of "sevenness" than on addition concepts, but good strong number concepts make the standard operations much easier to understand.
• Kindergartners and first graders use workmats depicting various settings in which activity takes place to make up and act out story problems. On a mat showing a vacant playground, for instance, students place counters to show 3 kids on the swings and 2 more in the sandbox. How many kids are there in all? How many more are on the swings than in the sandbox? What are all of the possibilities for how many are boys and how many are girls?
• Students work through the Sharing a Snack lesson that is described in the vignette on page 44 of the New Jersey Mathematics Standards. It challenges students to find a way to fairly share a large number of cookies among the members of the class, promoting discussion of early division, fraction, and probability ideas.

• Students play "One More Than" Dominoes by changing the regular rules such that a domino can be placed next to another only if it has dots showing "one more than" the other. Dominoes of any number can be played next to others that show 6 (or 9 in a set of double nines). "One Less Than" Dominoes is also popular.
• Students work through the Elevens Alive lesson that is described in the vignette on page 39 of the New Jersey Mathematics Standards. It asks them to consider the parts of eleven and the natural, random, occurrence of different addend pairs when tossing eleven two-colored counters.
• Second graders regularly use the Doubles and Near Doubles, the Make Ten, and the Counting On and Counting Back strategies for addition and subtraction. 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.
• Students play games like Addition War to practice their basic facts. Each of two children has half of a deck of playing cards with the face cards removed. They each turn up a card and the person who wins the trick is the first to say the sum (or difference) of the two numbers showing.

• Second graders use popsicle sticks bundled as tens and ones to try to find a solution to the first two-digit addition problem they have formally seen: Our class has 27 children and Mrs. Johnson's class has 26. How many cupcakes will we need for our joint party? Solution strategies are shared and discussed with diversity and originality praised. Other problems, some requiring regrouping and others not, are similarly solved using the student-developed strategies.
• Students use calculators to help with the computation involved in a first-grade class project: to see how many books can be read by the students in the class in one month. Every Monday morning, student reports contribute to a weekly total which is then added to the monthly total.
• Students look forward to the hundredth day of school on which there will be a big celebration. On each day preceding it, the students use a variety of procedures to determine how many days are left before day 100.

• Kindergartners explore part/whole relations with Pattern Blocks by seeing which blocks can be replicated with other blocks. For example: Can you make a shape that is the same as the yellow hexagon with 2 blocks of some other color? with 3 blocks of some other color? with 6 blocks of some other color? and so on.
• Students use paper folding to begin to identify and name common fractions. If you fold this rectangular piece of paper in half and then again and then again, how many equal parts are there when you open it up? Similarly folded papers, each representing a different unit fraction allow for early comparison activities.

• The daily Calendar Routine provides the students with plenty of opportunities to do computation. Questions like these arise almost every day: There are 27 children in our class. 24 are here today. How many are absent? Of the 24, 14 are buying lunch. How many brought their lunch? It's now 9:12. How long until we go to gym at 10:30? The students always choose a computation method with which they feel comfortable. They are frequently asked why they chose the method they chose and whether an exact answer to the question was important.
• Students regularly have Human versus Calculator Races. Given a list of addition and subtraction basic facts, one student uses mental math strategies and another uses a calculator. They quickly come to realize that the human has the advantage.
• Students regularly answer multiple choice questions like these with their best guesses of the most reasonable answer: A regular school bus can hold: 20 people, 60 people, 120 people? The classroom is: 5 feet high, 7 feet high, 10 feet high?

• Students regularly practice a whole variety of oral counting skills, both forward and backward, by various steps. For instance: Count by ones - start at 1, at 6, at 12, from 16 to 23; Count by tens - start at 10, at 30, at 110, at 43, at 67, from 54 to 84, and so on.
• Students are used to estimating sums and differences both before doing either pencil-and-paper computation or calculator computation and after so doing to confirm the reasonableness of their answers.
• Students are given a set of index cards on each of which is printed a two-digit addition pair (23+45, 54+76, 12+87, and so on). As quickly as they can they sort the set into three piles: More than 100, Less than 100, and Equal to 100.

• Students explore three-addend problems like 4 + 5 + 6 =, looking first to see if adding the numbers in different orders produces different results and, later, looking for pairs of compatible addends (like 4 and 6) to make the addition easier.
• Students make up humorous stories about adding and subtracting zero. I had 27 cookies. My mean brother took away zero. How many did I have left?
• Second graders, exploring multiplication arrays, make a 4 x 5 array of counters on a piece of construction paper and label it: 4 rows, 5 in each = 20. Then they rotate the array 90 degrees and label the new array, 5 rows, 4 in each = 20. Discussions follow which lead to intuitive understandings of commutativity.

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:

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?

• 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.

• 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.

• 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.

• 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?

• 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?

• 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?

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.

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.

• 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.

• 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.

• 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).

• 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?

• 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.

• 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.

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:

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.

• 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.

• 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!

• 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.

• 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.

• 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.

• 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

The major work in this area, then, that will take place in the high school grades, is continued opportunity for real-world applications of operations, wise choices of appropriate computational strategies, and integration of the numerical operations with other components of the mathematics curriculum.

The only new topics to be introduced in this standard for these grade levels are work with factorials and matrices as useful tools to be used in problem solving situations.

Estimation, mental math, and technology use should fully mature in the high school years 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 learn more operations on other types of numbers, but the work here is almost exclusively 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 ninth through twelfth grade mathematics program are:

Building upon K-8 expectations, experiences in grades 9-12 will be such that all students:

O. 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 frequently use all of these computational strategies in their ongoing mathematics work. Inclinations to over-use the calculator, in situations where other strategies would be more appropriate, are overcome with five minute "contests," speed drills, and warm-up exercises that keep the other skills sharp and point out their superiority in given situations.
• Numerical problems in class are almost always worked out in "rough" form before any precise calculation takes place so that everyone understands the ballpark in which the computed answer should lie and which answers would be considered unreasonable.

• Students work on the painted cube problem to enhance their skill in writing algebraic expressions: A 3-inch cube is painted red. It is then cut into 1-inch cubes. How many of them have 3 red faces? 2 red faces? 1-red face? No red faces? Repeat the problem using an original 4-inch cube, then a five-inch cube, then an n-inch cube.
• Students develop a procedure for binomial multiplication as an extension of their work with 2-digit whole number multiplication arrays. Using Algebra Tiles, they uncover the parallels between 23 x 14 (which can be though of as (20+3)(10+4)) and (2x+3)(x+4).
• Students devise their own procedures and "rules" for operations on variables with exponents by performing trials of equivalent computations on whole numbers.

• Students work through the Breaking the Mold unit that is described in the vignette on page 63 of the New Jersey Mathematics Standards. It uses a science experiment with growing mold to involve students in discussions and explorations of exponential growth.
• Students use their graphing calculators to find a curve that best fits the data from population growth in the state of New Jersey over the past 200 years.
• Students discover the need for a factorial notation and later incorporate it into their problem solving strategies when solving simple combinatorics problems like: How many different five card poker hands are there? How many different 6-place New Jersey license plates are possible? How many different phone numbers can be given out in one area code?