In third and fourth grade, students continue to develop their number sense by using manipulatives and technology. The key components of number sense, as identified in the K-12 Overview, include an awareness of numbers and their uses in the world around us, a good sense of place value concepts, approximation, estimation, and magnitude, the concept of numeration, and an understanding of comparisons and the equivalence of different representations and forms of numbers.

Third and fourth graders are refining their understanding of whole numbers and are just beginning to develop understanding of numbers like decimals and fractions that require substantially different ways of thinking about numbers. An excellent activity that can be used to impress upon these students the omnipresence of numbers around them is the keeping of a journal reflecting all of the uses of numbers that they can find in magazines and books. In fourth grade, they focus particularly on uses of fractions and decimals. They may include decimal prices in advertisements, fraction-off sales, and decimal or fraction measurements. Discussions of the uses found and the meanings of the numbers involved can provide interesting insights.

Their numeration work in earlier grades, having focused on models of number, has enabled students to use relatively sophisticated models like play money or chip trading to represent whole numbers up to three digits. The regular and consistent use of concrete models is essential for the continuing development of their understanding of numeration. They should use base-ten models not only to extend their experiences with whole numbers to four places, and then symbolically beyond that, but also to create meaning for decimals.

In addition, models are essential for the initial explorations of the meaning of fractions. Fraction Circles or Fraction Bars help children establish rudimentary meaning for fractions, but have the drawback of using the same size unit for all of the pieces. Cuisenaire Rods or paper folding can be used to accomplish the same goals without this drawback.

Children at these grade levels also continue their learning about equivalence. They should be engaged in activities using concrete models to generate equivalent forms of many different kinds of numbers. For multi-digit numbers, equivalences such as: 367 = 3 hundreds, 6 tens and 7 ones = 3 hundreds, 5 tens and 17 ones = 2 hundreds, 14 tens and 27 ones are useful in promoting a confident feeling about place value and will help in understanding multi-digit computation. Early explorations of equivalent fractions (1/2 = 2/4) and equivalent decimals (3 tenths = 30 hundredths) can accompany the exploration of the basic equivalences between fractions and decimals (1/2 = 0.5).

Estimation should be a routine part not only of mathematics lessons, but of the entire school day. Children should be regularly engaged in estimating both quantities and the results of operations. They should respond to questions that arise naturally during the course of the day like: About how many kids do you think there are in the auditorium? About how many paper cranes will each student have to fold if the class needs to make 200 altogether? and How many floor tiles do you think are on the floor? After several children have had chances to make estimates about numbers like these, they should defend their estimates by giving some rationale for thinking they are close to the actual number. These discussions can be invaluable in helping them develop number sense.

Technology plays an important role in number sense at these grade levels. Calculators can be wonderful exploration tools when examining new numbers. Students will themselves raise questions about decimals when someone divides 30 by 60 inadvertently instead of 60 by 30 and wonder what the 0.5 in the display means. Computers provide software that creates environments in which students manipulate base-ten models on-screen and explore initial fraction and decimal concepts.

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

whole number meanings through many digits

place value and number base

initial meanings for fractions and decimals

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 3 and 4.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

1. Use real-life experiences, physical materials, and technology to construct meanings for whole numbers, commonly used fractions, and decimals.

• Students are comfortable using a full array of base-ten models including money, base-ten blocks, and chip trading materials to represent both whole numbers and decimals.

• Students use computer software that provides easy pictorial representation of large whole numbers, decimals, and fractions like the MECC Math Tools, the Silver Burdett Math Workshop, and the Wasatch Math Construction Tools.

• Students use geoboards to model common fractions. For example, they search for multiple ways to show 1/4 on the geoboard.

• Students use Cuisenaire Rods to model fractions, frequently switching the rod or length used as the whole to avoid the misconception that, for instance, the yellow rod is always one-half.

2. Develop an understanding of place value concepts and numeration in relationship to counting and grouping.

• Pairs of students play Race to Five Hundred and its opposite, Race to Zero, with base ten blocks. In the first game, each student, in turn, rolls a red die and a green die and makes a two digit number from the faces showing (using the red die as the tens digit). He or she then takes that many tens and ones from the bank. Whenever there are ten tens or ten ones in a player's collection, the player must trade for a larger block. The first player to collect 5 hundreds is the winner. In Race to Zero, the players start with 5 hundreds and give back blocks according to their dice rolls.

• Students use a die to generate random digits from 1 to 6. After each roll, they decide where to place the digit in a 4-digit whole number. The goal is to produce as large a number as possible. If a ten-sided die or spinner with ten equal sectors is available, students should use it to generate random digits from 0 to 9 and repeat the activity.

• Students work in groups to decide what the next base-ten block after the thousands block would look like.

• Students read and listen to children's literature that is related to a numeration theme like Millions of Cats and The 500 Hats of Bartholomew Cubbins.

3. See patterns in number sequences, and use pattern-based thinking to understand extensions of the number system.

• Students use the constant function feature of their calculators to program a skip count. They press + 12 === to watch the calculator display count by twelves, trying to anticipate what number will come next and making predictions to each other. Any number can replace the 12 to change the difficulty level of the activity.

• Students also use their calculators to play Guess My Rule games. One student secretly programs the calculator by typing something like x 2 =. Thereafter, every time a number is pressed followed by an equals sign, the original number will be multiplied by two. A second student must guess the rule that was programmed. Rules like + 3 =, ÷ 4 =, and - 2 = also work.

• Students create and solve arrow puzzles on a hundred number chart. By naming a number and then giving directions for movement on the chart, instructions are given to arrive at some other number. For example: 72, down, down, right, right, up leaves the student at the number 84. After examples of different patterns are demonstrated on the chart, students point out patterns and try to solve puzzles mentally.

• Students make a table to reflect how many handshakes there would be if everyone shook hands in groups of different sizes. For example, for 2 people, 1 handshake; for 3 people, 3 handshakes; for 4 people, 6 handshakes. As they extend the table for larger groups, the students look for a pattern in the emerging numbers.

• Students search for patterns in the addition of even and odd numbers by using unifix cubes to represent the numbers and trying to arrange the sum into two stacks of equal height. (This will work for even numbers, but not for odd ones.)

4. Develop a sense of the magnitudes of whole numbers, commonly used fractions, and decimals.

• Students imagine collecting 10,000 of something. They discuss what objects would be reasonable to collect (such as bottle caps, pennies for charity, or pebbles from the beach), how much space this collection might take up, and how much it would weigh?

• Students cut and paste sheets of base-ten graph paper to make models of the different powers of ten: 1, 10, 100, 1000, 10,000.

• Students locate numbers as points on a number line strung across the room, continuing to attach labels as they learn more about numbers. Paper clips or tape are used to fasten equivalent forms of a number to the same point.

• Students estimate and investigate how long a million seconds is using calculators.

• Students write a Logo or BASIC computer program which will count to 100, printing the numbers to the screen as it runs, and timing how long it takes. Then they predict how long it would take the program to count to one thousand, to one hundred thousand, and to one million. They make the required changes to the program and check their predictions.

5. Understand the various uses of numbers including counting, measuring, labeling, andindicating location.

• Students keep a 24-hour diary recording all of the ways they use or see others use numbers. They pool all of these uses of numbers and classify them into categories that they design.

• As part of a geography unit, students make a map of a fantasy island, using a Cartesian coordinate system to help describe the location of various places on the island. They use other numbers to describe geographical properties of the sites: elevation, longitude and latitude, population, and the like.

• The students brainstorm ways to describe their math book in terms of numbers: its width, the number of pages, the publication date, a student-generated "quality rating" of the book, the area of the cover, and so on.

6. Count and perform simple computations with money.

• Students establish a school store and make transactions on a regular basis, with different students assigned as clerk each day.

• Students read Dollars and Cents for Harriet and then decide how they would spend five dollars.

• Students practice making change with coins by counting up to the amount given. For example, if the bill is $1.73, and $2.00 is the amount given, the students would count up to $2.00 by starting with two pennies and saying, "$1.74" and "$1.75"; then they add one quarter to bring the total to $2.00. They would then count this change to find its value of $0.27.

• Students play Treasure Math Storm on the computer or use IBM's Exploring Measurement, Time and Money.

7. Use models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.

• When modeling 3- and 4-digit numbers with a base-ten model like base-ten blocks or place value chips, the students are frequently asked questions like: Show all the ways you can make 327. Children thus begin to see that 3 hundreds, 2 tens, and 7 ones; 2 hundreds, 12 tens, and 7 ones; 2 hundreds, 11 tens, and 17 ones; and 32 tens and 7 ones all represent the same number. Students are assessed by asking them to show 327 in two different ways.

• Students use shadings on ten-by-ten grids to represent fractions and decimals that are equivalent. For example, the representation for 0.4 is the same as that for 4/10.

• Students develop their own questions, the answers to which are equivalent to some target number. For example, if the target number is 24, students may ask the following questions: What is 20 + 4? What is 2 x 12? What is 2 x 2 x 2 x 3? How much is 2 dozen? How many is 3 less than the number of children in our class? or How much would something cost if you paid a quarter and got back a penny in change?

• Students use geoboards, pattern blocks, Cuisenaire Rods, paper folding, and tangrams to explore common fractions. They may be challenged to model 3/4, for instance, with all of the different models.

• Students use money to represent decimals. For example, 8 dimes = $0.80 = .8. They also represent fractional parts of a dollar as a decimal (a quarter = 1/4 = 25 cents = .25).

• Students use graham crackers, candy bars, pizzas, and other food to illustrate fractions.

• Students work through the Sharing Cookies lesson that is described in the First Four Standards of this Framework. They realize that 8 is not readily divisible by 5 and try to find ways to solve that sharing problem using real cookies.

• Students play Bowl a Fact by rolling three dice and using the numbers shown to make number sentences whose answers equal numbers from 1 to 10. For each different answer, they knock down the bowling pin labeled with that number. For example, if they roll 2, 5, and 3, they can make these number sentences: 2 + 5 + 3 = 10, 5 + 3 - 2 = 6, 5 x 2 - 3 = 7, 5 - 3 + 2 = 4, and 3 x 2 - 5 = 1, and therefore knock down the 10, 6, 7, 4 and 1 pins. If they cannot knock down all ten pins on the first roll, they roll the dice again and try to get the remaining pins. The students are assessed by giving all of them the same outcomes of two rolls of the three dice to play the game.

8. Compare and order whole numbers, commonly used fractions, and decimals.

• Students use base-ten materials such as, blocks, sticks or money to make models of pairs of 3- or 4-digit numbers like 405 and 450 and compare them to see which is larger. Responses and reasons can be written in a journal.

• Students play Guess the Point. A long number line with endpoints of 130 and 470, for example, is drawn on the board with the intermediary points labeled as multiples of ten above the line. The labels are then covered by a long piece of butcher paper that can be lifted to reveal them. A student places a finger somewhere on the line and others must estimate the numerical label of the point chosen. The paper is then lifted to check the accuracy of their responses.

• Pairs of students play Hi-Lo with whole numbers and decimals. One student thinks of a number within a given range such as 1 to 1000. The other student tries to guess the number, receiving feedback after each guess as to whether the guess was too high or too low, and keeping a written record of the guesses and the feedback. The goal is to find the number using as few guesses as possible.

• When using Cuisenaire Rods, students choose a base rod to represent one whole, and then determine the values of all of the rest of the rods. They then use the rods to model the comparison of the relative sizes of two fractions with different denominators.

9. Explore real-life settings which give rise to negative numbers.

• Students record daily low Celsius temperatures throughout the winter and draw a line graph of those temperatures. Students discuss changes in the graph and the meaning of the line dipping below the zero degree line.

• Students examine a videotape of a section of a football game and record the results of a series of plays as a series of integers - gains as positive integers and losses as negative integers (for example: -3, +5, +9, first down; -5, -4, +6, punt). They use their record to determine the total yardage gained during the drive.

• Students use an almanac to find the altitudes of selected cities around the country, and discuss what it means for a city to be below sea level.

References

Gàg, Wanda. Millions of Cats. New York: Coward, McCann, & Geoghegan, 1928.

Maestro, Betsy and Maestro, Giulio. Dollars and Cents for Harriet. New York: Crown Publishers, Inc., 1988.

National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, VA, 1989.

Seuss, Dr. The 500 Hats of Bartholomew Cubbins. New York: Vanguard, 1938.

Software

Exploring Measurement, Time, and Money. IBM.

Math Construction Tools. Wasatch.

Math Tools. Minnesota Educational Computing Consortium (MECC).

Math Workshop. Silver Burdett.

Treasure Math Storm. The Learning Company.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/.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.