Students in grades 3-4 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 at these grade levels 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 grades K-2. Students can 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 also 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 who takes in numbers and hands out numbers according to certain rules), they are also able to consider these situations in more abstract contexts. Students at this age often enjoy playing 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.

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

A. 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, 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 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 from 50 - 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 add or subtract a 1 (or 2, etc.): for example 50+1 = , =, =. The calculator will automatically repeat the function and display 51, 52, 53, 54, etc. Some calculators may need to have the pattern entered twice: 50+1 = 51+1 =, =, =, etc. Others may need 1++, 50=, =, =, etc.
• 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 supply the missing numbers on a picture of a ruler which has some blanks: using inches, then centimeters. 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 200-300 or each of four intervals from 1,000-2,000.
• 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.

• Students complete a table when given one number in each row and a verbal rule that tells the relationship between the two numbers in each row.
• 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 *).
• Students plot the multiples of 3 on a coordinate grid and 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 make up a story to match a graph that shows the water level in a bathtub at different times of the day.
• Students repeatedly add (or subtract) multiples of 10 to (from) a 3-digit starting number. They describe the pattern orally and write it symbolically.
• 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 2 x W + 3 x T = 12).

• Students use buttons with two or four holes and describe how the total number of holes depends upon 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 cubes wide 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.
• Students play "Guess my Rule." The teacher gives them a starting number and the result after using the rule. She continues giving examples until students discover the rule.

• 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 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.

• 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 learn about the different time zones across the country. They describe the number patterns found in moving from the east to west, and vice versa.
• Students read books such as Six Dinner Sid by Inge Moore, The Greedy Triangle by Marilyn Burns, Counting on Frank, or Gulliver's Travels by Jonathan Swift. 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.
• Students examine populations (people, plants, or animals) within the school and community.
• In art, students observe patterns in pictures, mosaics, tessellations, Escher-like drawings, wallpaper, fabric and floor tile designs. They create their own potato-print patterns.

• 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.