Fifth and sixth graders should have a good sense of whole numbers and their orders of magnitude and should be focusing mostly on developing number sense with decimals, fractions, and rational numbers, which require substantially different ways of thinking about numbers. They also should be exploring two relatively new topics: ratios and integers. The key components of number sense, as explained in the K-12 Overview, are an awareness of the uses of numbers in the world around us, a good sense of approximation, estimation, and magnitude, the concept of numeration, and an understanding of comparisons and the equivalence of different representations and forms of numbers.

Students at this age are capable of categorizing all of the ways in which numbers are used in our society. An excellent activity is to have them collect ways in which they see numbers being used during a twenty-four hour period. Their uses of numbers would probably include telephone numbers, addresses, ages, page numbers, clothing sizes, library book numbers, room numbers, and many others. Discussions of the similarities and differences among these uses should resolve themselves into some of the standard categorizations: counts, measures, labels, and indicators of location. The students' data can then be graphed according to these categories.

Their numeration work in earlier grades, having focused on models of whole numbers, has taken these students to the point where they are able 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. The focus now shifts to a real sense of the meanings of decimals and fractions and to providing models which adequately serve that purpose. Money continues to provide a superb setting for the learning of decimal concepts (at least up to two decimal places) because of the students' increasing familiarity with it, because of the vast array of real-world applications that it makes available, and because of its inherent motivational quality. Base-ten blocks are also useful as a slightly more abstract model. They have the added advantage of being able to represent any number with four digits in the place value system. A block model of 3274 with which the students are familiar can become a model of the decimal 3.274 if, instead of thinking of the smallest block as one unit, they think of the largest block as one unit.

In addition, models are essential for the continued exploration of fraction meaning and fraction operations. Fraction Circles or Fraction Bars help children establish rudimentary meaning for fractions, but have the drawback of using the same size unit for all the pieces. This is a fairly serious drawback leading to the misconception, for instance, that 1/3 is always less than 1/2 without regard to the units in which those fractions are expressed; students need to be aware, for example, that 1/3 of a large pizza is frequently larger than 1/2 of a small one. Cuisenaire Rods or paper folding can also be used to accomplish many of the same goals without the same drawback. A sample unit on fractions for the sixth-grade level can be found in Chapter 17 of this Framework.

Work with ratios, percents, and integers in grades 5 and 6 should be limited to informal exploration, with no use of more formal, symbolic procedures. Students should use models as they explore these topics. Theymight use 2 red tiles and 1 yellow tile to illustrate mixing paint in the ratio 2:1 and extend this pattern in order to make larger (and smaller) quantities. By using base ten blocks and 10 x 10 grids, they can visualize percent more easily. Two-color counters or the number line might be used to model positive and negative numbers (integers).

Students in grades 5 and 6 should begin to understand the ways in which different types of numbers are related. For example, they should understand that every whole number is a rational number, since it can be written as a fraction. Similarly, every decimal is a rational number. By the end of sixth grade, they should have had sufficient experiences with integers to realize that the integers consist of the whole numbers and their opposites (additive inverses).

Students at these grade levels continue their learning about equivalence, but there is a significant shift in what that means. As third and fourth graders, they have explored simple fractions and decimals, and their work with equivalence has focused primarily on the multiple ways to represent whole numbers (8 = 2 + 6 = 9 - 1, 23 = 2 tens + 3 ones = 1 ten + 13 ones, and so on). Now, as fifth- and sixth-graders, they should begin to focus on the representation of the same quantity with different types of numbers. Their work with equivalent fractions (1/2 = 2/4) and equivalent decimals (3 tenths = 30 hundredths), for example, should lead to exploration of the basic equivalences of fractions and decimals (1/2 = 0.5). They should be engaged in activities using concrete models to generate equivalent forms of many different kinds of numbers. They also begin to explore the role of ratios and percents in this mix. Ten-by-ten grid paper helps enormously with these activities, since all forms of a quantity can frequently be represented on it.

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 what fraction (percentage) of the kids in the playground do you think are wearing gloves? About one-third of our students stay for the after-school program in the afternoon; if there are 500 students in the school about how many of them stay? After children have had several 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 good number sense.

Technology plays an important role in number sense at these grade levels. Calculators can be wonderful exploration tools when examining new relationships. Many insights about the relationships between fractions and decimals, for instance, can be achieved by simply dividing the numerators of fractions by their denominators. Generalizations about what kinds of fractions produce what kinds of decimals start to flow very freely in such open-ended explorations. Computer software also creates environments in which students manipulate decimal models on-screen and explore fraction and decimal relationships.

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

fractions

decimals

equivalence

integers

ratio and percent

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:

10. Understand money notations, count and compute money, and recognize the decimal nature of United States currency.

• Students try to identify fake "bargains" or misleading information from store advertisements, and try to determine which of several options is the best. For example: Which of the following offers the cheapest price?
  

       XYZ Mega-Deal Tuna          ABC Tuna          Aunt Betty's Best Tuna
       10 oz can $2.80             3 oz can $0.87    4 oz can $1.04
  

• Students plan a fantasy driving trip to Walt Disney World for their family. They research and consider the variety of expenses to be incurred - lodging along the way and at the park, meals, souvenirs, gasoline, admission, and so on. A reasonable budget for the trip is the centerpiece of a report prepared by each student group. Their reports are assessed using a scoring rubric that includes mathematical correctness as well as creativity.

• Students use play money to model decimal numbers. They use decimal language and find fractional equivalents for each coin. For example: a dime = 0.1 = 1/10 = 1 tenth.

• Students first estimate and then use a calculator to find out how long it would take to spend one million dollars at a rate of one dollar per second.

11. Extend their understanding of the number system by constructing meanings for integers, rational numbers, percents, exponents, roots, absolute values, and numbers represented in scientific notation.

• Students each develop questions, the answers to all of which are equivalent to some target number. For example, if the target number is 24, students may ask the following questions: What is 8 x 3? What is (-25) - (-49)? What is 5² -1? What is 3 more than the sixth triangular number? What is 1 less than one-fourth of 100? or What is the smallest positive number with 8 factors?

• Students continue to refine their concepts of fractions using all available models to answer questions like: Is 1/4 always larger than 1/8? Is 1/4 of every pizza larger than 1/8 of every other pizza? Issues that point out the importance of defining the unit are special topics for discussion.

• Students use two-color counters to construct models of the set/subset meaning of fraction. You might ask: Given 3 red counters in a set of 12, what are the equivalent fractions thatrepresent the reds as a part of the set?

• Students also use two-color counters to model and begin to make sense of positive and negative integers. In this system, a positive 1 is represented by one color and a negative 1 by the other. Students determine the value of a pile of counters by pairing up counters, one of each color, setting aside all pairs, and counting the remaining counters.

• Students read Shel Silverstein's poem A Giraffe and a Half and discuss how to describe an amount that is more than one whole but less than two.

• Students read The Phantom Tollbooth and discuss the relationships between decimals and fractions in the book. For example, Milo meets half a child (actually, .58 of a child since the average family has 2.58 children).

• Students construct a time line to scale to show the history of the earth. Significant periods and events are shown along the line with numbers reflecting the number of years since the earth's beginning.

12. Develop number sense necessary for estimation.

• Students imagine collecting a million of something. They discuss what objects would be reasonable to collect (such as toothpicks, punched holes from fan-folded computer paper or pages in telephone books to be recycled), how much space this collection would take up, and how much it would weigh.

• Students make estimates to answer the question: How much drinking water do you think Columbus' ships carried with them on their trip across the ocean? Then they gather the data they need to make more informed estimates. (Addenda Series Grade 4 Book.)

• Students determine the number of decimal places in a simple decimal multiplication product, not by mechanically adding the number of places in the factors, but by estimating a reasonable range for the product and placing the decimal point so that their computed product falls within that range.

• Students investigate the question: What size room would be needed to hold one million ping-pong balls?

• Students read Counting on Frank and estimate how many dogs would fill their classroom.

• Students use estimates to compare fractions. For example, 3/7 < 9/16 since 3/7 is less than half and 9/16 is more than half.

13. Expand the sense of magnitudes of different number types to include integers, rational numbers, and roots.

• Students challenge each other to find target numbers on a number line. First one student asks another to find 3.2. The second target number then must be between 0 and 3.2, say 1.74. The third must then be between 0 and 1.74 and so on.

• Students use their calculators to explore the types of decimal expansions for common fractions. They discover that some such decimals terminate, some repeat, and some appear to do neither. (Actually, if the calculators could exhibit more digits, each such decimal would either terminate or repeat.)

• Students use calculators to come as close as they can to an answer to: What number multiplied by itself gives an answer of 2?

14. Understand and apply ratios, proportions, and percents in a variety of situations.

• Students begin to see a ratio as both the comparison of two quantities and as a number in its own right. They are challenged to find ratios that they frequently use like $0.65 per pound, 55 miles per hour, and so on.

• In a social studies unit, students use population and area data for countries in South America to compute population densities, and then compare their results to those for other areas of the world.

• Students use two different sizes of grid paper to copy a simple drawing of a house from the smaller grid to the larger grid, investigating and discussing the change from one to the other and exploring ways to represent it numerically. They then copy the same drawing onto a third grid, smaller than the second but larger than the first.

• Students are challenged to use any combination of the digits 3, 4, 5, and 9 to make a ratio as close as possible to 90%. As follow-up, they invent other closeness problems for each other.

• Students search for as many uses of percent as they can find over the course of a week. The sources for the uses, however, are to be exclusively within the school setting. Likely entries in the resulting list are: grades on tests, foul shot success of the basketball team, a measure of how close the PTA is to their fund-raising goal for the new playground equipment, and so on. For each use found, the students explain what 100% would represent and whether percentages above 100% would make any sense in the given context.

15. Develop and use order relations for integers and rational numbers.

• Students use a deck of fraction cards for a variety of tasks. A deck consists of one card containing each of a number of fractions, for example, 1/4, 9/10, 2/19, 4/7, 7/9, 1/3, 12/15, 2/5, and 5/8. They are asked to: Find the smallest fraction in the set. Sort into two groups more than 1/2 and less than 1/2. Determine which pairs have a value close to 1.

• Students use a number line, including both positive and negative integers, to graph inequalities stated verbally. For example: Show all of the numbers larger than -2.

• Students gain understanding of the order relationships among fractions and integers by comparing them with similar ones for whole numbers. How is the comparison of 4/7 to 5/7 like the comparison of 4 to 5? How is comparing 4/7 to 4/8 like comparing 7 to 8? How is comparing -4 to -7 like comparing 4 to 7? They answer similar questions on their test.

• Students use 4 digits, say 2, 3, 4, and 5 to construct as many true fraction sentences as they can. For example, 2/3 < 5/4.

16. Recognize and describe patterns in both finite and infinite number sequences involving wholenumbers, rational numbers, and integers.

• Given the first four rows, students formulate a rule for generating succeeding rows of Pascal's triangle. They look for other patterns in the triangle.

• Students explore the well-known problem of taking a long walk by first doing half of it, then half of what remains, half again of what remains, and so on. They write the series as 1/2 + 1/4 + 1/8 + ... . What happens to the walker?

• Students solve this classic problem: Which would you choose as the method for getting your allowance next month: $1.00 every day; or 1 cent the first day, 2 cents the second, 4 cents the third, 8 cents the fourth, and so on?

17. Develop and apply number theory concepts, such as, primes, factors, and multiples, in real-world and mathematical problem situations.

• Students build rectangular arrays with square tiles to determine which of the first fifty counting numbers are rectangular (composite) and which are non-rectangular (prime).

• Students use the Sieve of Eratosthenes to generate a list of all the primes in the first 100 counting numbers.

• Students use common multiples to solve problems like this: Hot dog buns come in packages of 8. Hot dogs come in packages of 6. What is the smallest number of packages of each that can be bought so that there are no extra buns or hot dogs?

18. Investigate the relationships among fractions, decimals, and percents, and use all of them appropriately.

• Students address the questions How are 0.50 and 40/100 alike? and How are they different? Answers can be written in their math journals.

• Students use shadings on a ten by ten grid to discuss all of the different equivalences. For example, the same shading can be named 3/10, 30/100, 0.3, 0.30, .3, .30, and 30%. Thinking of the grid as $1.00 leads to some interesting insights about two-place decimals.

• Students explore the density property of numbers by addressing problems like: Find 4 decimals between 0.456 and 0.457. Find 3 fractions between 3/5 and 4/5.

19. Identify, derive, and compare properties of numbers.

• Students use Venn diagrams to explore the multiple sets to which particular number belong. For example, a Venn diagram is created for these three sets of numbers less than 25: multiples of 3, factors of 24, primes; the Venn diagram is used to answer questions like: How many numbers are in exactly two of these sets? A similar question is used on their test.

• Students explore the property of closure for a variety of sets of numbers under various operations. For example: Using subtraction, is there always an answer within the set of positive whole numbers for any member of the set minus any other? (no); Is there always an answer within the set of integers? (yes); Is there always an answer within the set of even integers? (yes); within the set of odd integers? (no).

• Students explore the properties of odd and even numbers under various operations. For instance: What can always be said about the sum of two even numbers? of two odd numbers? of an even and an odd number?

• Students create a book about zero for second-graders.

• Students explore the concepts of place value and zero by learning about other number systems. For example, they might use the computer program Maya Math to learn about the Mayan number system.

References

Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth Stevens Publishing, 1991.

Juster, Norton. The Phantom Tollbooth. New York: Random House, 1961.

National Council of Teachers of Mathematics. Addenda Series Grade 4 Book. Reston, VA, 1993.

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

Software

Maya Math. Sunburst Communications.

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.