The development of pattern-based thinking, using patterns to analyze and solve problems, is an extremely powerful tool for doing mathematics, and leads in later grades to an appreciation of how functions are used to describe relationships. The key components of pattern-based thinking at the early grade levels, as identified in the K-12 Overview, are recognizing, constructing, and extending patterns, categorizing and classifying objects discovering rules, and working with input-output situations.

In grades 3 and 4, students begin to learn the importance of investigating a pattern in an organized and systematic way. Many of the activities at these grade levels focus on creating and using tables as a means of analyzing and reporting patterns. In addition, students in these grades begin to move from learning about patterns to learning with patterns, using patterns to help them make sense of the mathematics that they are learning.

Students in grades 34 continue to construct, recognize, and extend patterns. At these grade levels, pictorial or symbolic representations of patterns are used much more extensively than in grades K-2. In addition to studying patterns observed in the environment, students should use manipulatives to investigate what happens in a pattern as the number of terms is extended or as the beginning number is changed. Students should also study patterns that involve multiplication and division more extensively than in earlier grades. Students continue to investigate what happens with patterns involving money, measurement, time, and geometric shapes. They should use calculators to explore patterns.

Students in these grades continue to categorize and classify objects. Now categories can become more complex, however, with students using two (or more) attributes to sort objects. For example, attribute shapes can be described as red, large, red and large, or neither red nor large. Classification of naturally-occurring objects, such as insects or trees, continues to offer an opportunity for linking the study of mathematics and science.

Students in grades 3 and 4 are more successful in playing discover a rule games than younger students and can work with a greater variety of operations. Most students will still be most comfortable, however, with one-step rules, such as multiplying by 3 or dividing by 4.

Third and fourth graders also continue to work with input-output situations. While they still enjoy putting these activities in a story setting (such as Max the Magic Math Machine which takes in numbers and hands out numbers according to certain rules), they are now able to consider these situations in more abstract contexts. Students at this age often enjoy pretending to be the machine themselves and making up rules for each other.

In grades 3 and 4, then, students expand their study of patterns to include more complex patterns based on a greater variety of numerical operations and geometric shapes. They also work to organize their study of patterns more carefully and systematically, learning to use tables more effectively. In addition, they begin to apply their understanding of patterns to learning about new mathematics concepts, such as multiplication and division.

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. Reproduce, extend, create, and describe patterns and sequences using a variety of materials.

• Students make a pattern book that shows examples of patterns in the world around them.

• Students use pattern blocks, attribute blocks, cubes, links, buttons, beans, toothpicks, counters, crayons, magic markers, leaves, and other objects to create and extend patterns. They might describe a pattern involving the number of holes in buttons, the number of sides in a geometric figure, the shape or the thickness of objects.

• Students use sequences of letters or numbers to identify the patterns they have created.

• Students investigate the sum of the dots on opposite faces of an ordinary die and find they always add up to 7.

• Students solve twodimensional attribute block patterns where, for instance, each column is a different shape and each row is a different color. They should be able to choose the block that fills in the missing cell in such patterns.

• Students count by 2, 3, 4, 5, 6, 10 and 12 on a number line, on a number grid, and on a circle design.

• Students begin with numbers between 50 and 100 and count backwards by 2, 3, 5, or 10.

• Students create patterns with the calculator: They enter any number such as 50, and then repeatedly add or subtract 1 or 2 or 3 etc. If, for example, they enter 50+1=== ... , the calculator will automatically repeat the function and display 51, 52, 53, 54, ... . Some calculators may need to have the pattern entered twice: 50+1=+1=== ... . Others may need 1++ 50=== ... .

• Students begin with a number less than 10, double it, and repeat the doubling at least five times. They record the results of each doubling in a table and summarize their observations in a sentence.

• Students read Anno's Magic Seeds by Mitsumasa Anno. In it, a wizard gives Jack two seeds and tells him that if he eats one, he won't be hungry for a year and if he plants the other one, two new seeds will be formed. Jack continues in this way for awhile and then tries other schemes that produce even more new seeds. The students work in groups to make charts and tables to show how many seeds Jack has at given points in time. As an individual assessment assignment, students are asked to find how many seeds Jack has after ten years using one of the discovered patterns and to support their answers in writing and with tables.

• Students supply the missing numbers on a picture of a ruler which has some blanks. Then they explore how to find the missing numbers between any two given numbers on a number line. They extend this to larger numbers; they might label each of five intervals from 200 to 300 or each of four intervals from 1,000 to 2,000.

• Students investigate number patterns using their calculators. For example, they might begin at 30, repeatedly add 6, and record the first 10 answers, making a prediction about what the calculator will show before they hit the equals key. Or they might begin at 90 and repeatedly subtract 9.

2. Use tables, rules, variables, open sentences, and graphs to describe patterns and other relationships.

• As a regular assessment activity, done during the year whenever new numerical operations have been explored, students fill in guess my rule tables like those shown below. Sometimes they are given the rule and sometimes they are asked to find the rule.

  times 2 6 ? 9 ? 2 ? 7 ?

  plus 12 34 ? 58 ? ? 37 ? 12

  ????? 12 4 27 9 9 3 15 ?

• Students describe the pattern illustrated by the numbers in a table by using words (e.g., twice as much as) and then represent it with symbols in an open sentence ( = 2 x ).

• On a coordinate grid, students plot coordinate pairs consisting of a number and the product of the number times 3. They join them with a line, making a line graph. They relate this to a table, and write the rule as an expression involving a variable, such as 3 x .

• Students repeatedly add (or subtract) multiples of 10 to (from) a 3-digit starting number. They describe the pattern orally and write it symbolically as, for example, 357, 337, 317, 297, ... .

• Students work in groups to solve problems that involve organizing information in a table and looking for a pattern. For example, If you have 12 wheels, how many bicycles can you make? How many tricycles? How many bicycles and tricycles together? Using objects or pictures, children make models and organize the information in a table. They discuss whether they have looked at all of the possibilities systematically and describe in words the patterns they have found. They write about the patterns in their journals and, with some assistance, develop some symbolic notation (e.g., 2 wheels for each bike and 3 wheels for each trike to get 12 wheels all together might become 2xB + 3xT = 12).

3. Use concrete and pictorial models to explore the basic concept of a function.

• Students use buttons with two or four holes and describe how the total number of holes is related to the number of buttons.

• Students use multilink cubes or base ten blocks to build rectangular solids. They count how many cubes tall their structure is, how many cubes long it is, and how many cubeswide it is. Then they count the total number of cubes in their structure. They record all of this information in a table and look for patterns.

• Students take turns putting numbers into Max the Magic Math Machine, reading what comes out, and finding the rule that tells what Max is doing to each number. A student acts as Max each time. Appropriate rules to use in grades 3 and 4 involve multiplication and division.

4. Observe and explain how a change in one physical quantity can produce a corresponding change in another.

• Students use cubes to build a one-story "house" and count the number of cubes used. They add a story and observe how the total number of cubes used changes. They explain how changing the number of stories changes the number of cubes used to build the house.

• Students measure the temperature of a cup of water with ice cubes in it every fifteen minutes over the course of a day. They record their results (time passed and temperature) in a table and plot this information on a coordinate grid to make a broken line graph. They discuss how the temperature changes over time and why.

• Students plant seeds in vermiculite and in soil. They observe the plants as they grow, measuring their height each week and recording their data in tables. They examine not only how the height of each plant changes as time passes but also whether the seeds in vermiculite or soil grow faster.

5. Observe and recognize examples of patterns, relationships, and functions in other disciplines and contexts.

• Students go on a scavenger hunt for patterns around the classroom and the school. They are given a list of verbal descriptions of specific patterns to look for, such as a pattern using squares or an ABAB pattern. They use cameras to make photographs of the patterns that they find.

• Students read The Twelve Days of Summer by Elizabeth Lee O'Donnell and Karen Lee Schmidt. Using the same pattern as the song The Twelve Days of Christmas, the authors tell the story of a young girl on vacation by the ocean. On the first day, she sees "a little purple sea anemone," on the eighth, "eight crabs ascuttling," and so on. Since she sees everything that she has previously seen on every succeeding day, the book offers the obvious question How many things did the little girl see today?

• Students learn about the different time zones across the country. They describe the number patterns found in moving from east to west, and vice versa.

• Students read books such as Six Dinner Sid by Inga Moore or The Greedy Triangle by Marilyn Burns. They explore the patterns and relationships found in these books.

• Students study patterns in television programming. For example, they might look at the number of commercials on TV in an hour or how many cartoon shows are on at different times of the day. They discuss the patterns that they find as well as possible reasons for those patterns.

6. Form and verify generalizations based on observations of patterns and relationships.

• Students measure the length of one side of a square in inches. They find the perimeter of one square, two squares (not joined), three squares, and so on. They make a table of values and describe a rule which relates the perimeter to the number of squares. They predict the perimeter of ten squares.

• Students use their calculators to find the answers to a number of problems in which they multiply a two-digit number by 10, 100, or 1000. Looking at their answers, they develop a "rule" that they think will help them do this type of multiplication without the calculator. They test their rule on some new problems and check whether their rule works by multiplying the numbers on the calculator.

References

Anno, Mitsumasa. Anno's Magic Seed. New York: Philomel Books, 1995.

Burns, Marilyn. The Greedy Triangle. New York: Scholastic Inc., 1994.

O'Donnell, Elizabeth Lee, and Karen Lee Schmidt. The Twelve Days of Summer. Morrow Junior Books, 1991.

Moore, Inga. Six Dinner Sid. New York: Simon and Schuster, 1993.

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.