Students in grades K-2 should investigate many different types of patterns. Some of these patterns should be repeating patterns, such as 2, 4, 6, 8, ... These patterns involve linear growth since the same number is added (or subtracted) to a number to get the next number in the series. Older students should also see patterns that grow more rapidly, such as 2, 4, 8, ... These growing patterns involve exponential growth; each number in the series is multiplied (or divided) by the same number to get the next one. These types of patterns can be investigated very easily by using calculators to do the computation; students enjoy making the numbers bigger and bigger by using the constant addend (e.g., 2 + 2 = = = ) or the constant multiplier (e.g., 2 x 2 = = = ). By relating these problems to concrete situations, such as the growth of a plant, students begin to develop a sense of change over time.

Students also begin to develop a sense of change with respect to measurement. Students begin to measure the length of objects by using informal units such as paperclips or Unifix cubes; they should note that it takes more small objects to measure a given length than large ones. By the end of second grade, they begin to describe the area of objects by counting the number of squares that cover a figure. Again, they should note that it takes more small squares to cover an object than it does large ones. They should also begin to investigate what happens to the area of a square when each side is doubled. Students also begin to develop volume concepts by filling containers of different sizes. They might use two circular cans, one of which is twice as high and twice as wide as the other, to find that the large one holds eight times as much as the small one. Measurement may also lead to the beginnings of the idea of limiting value for young children. For example, the size of a dinosaur footprint might be measured by filling it with base ten blocks. If only the 100 blocks are used, then one estimate of the size of the footprint is found; if unit blocks are used, a more precise estimate of the size of the footprint can be found.

Students in grades K-2 should also begin to look at concepts involving infinity. As they learn to count to higher numbers, they begin to understand that, no matter how high they count, there is always a bigger number. By using calculators, they can also begin to see that they can continue to add two to a number forever and the result will just keep getting bigger.

The conceptual underpinnings of calculus for students in grades K-2 are closely tied to their developing understanding of number sense, measurement, and pattern. Additional activities relating to this standard can be found in these other standards.

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

A.investigate and describe patterns that continue indefinitely.

• Students model repeating patterns with counters or pennies. For example, they repeatedly add two pennies to their collection and describe the results.
• Students create repeating patterns with the calculator. They enter any number such as 10, and then add 1 for 10+1= ,=, =. The calculator will automatically repeat the function and display 11, 12, 13, 14, etc. (Some calculators may need to have the pattern entered twice: 10+1=11 +1=, =, =, etc. Others may use a key sequence such as 1++10=, =, =.) Students may repeatedly add (or subtract) any number.
• Second graders create a pattern with color tiles. They start with one square and then make a square that is two squares long on each side; they note that they need four tiles to do this. Then they make a square that is three squares long on each side; they need nine tiles to do this. They make a table of their results and describe the pattern they have found.
• Students investigate a doubling (growing) pattern with Unfix cubes. They begin with 1 cube and then "win" as many as they already have. Repeating this process, they begin to see how quickly the number of cubes grows. They investigate this further using the calculator.
• Students draw trees or bushes and study the effects repeated "branching" has on the final result. They discuss whether there is an end to this process, either in reality or in theory.
• Students start with a sheet of paper that represents a pizza. They eat half of the pizza by tearing the sheet in half. They eat half of what is left and continue this process. They describe the pattern, noting that they are getting close to eating all of the pizza but will always have just a bit left.

• Students keep a daily record of the temperature both inside and outside the classroom. They graph these temperatures and look at the patterns.
• Students study the changes in the direction and length of the shadow of a paper groundhog at different times of the day. They relate these observations to the position of the sun (e.g., as the sun gets higher, the shadow gets shorter).
• Students discuss how ice changes to water as it gets hotter. They talk about how it snows in January or February but rains in April or May.
• Students plant seeds and watch them grow. They write about what they see and measure the height of their plants as time passes. They discuss how changes in time bring about changes in the height of the plant. They also talk about how other factors might affect the plant, such as light and water.

• Students measure the width of a bookcase using the 10-rods from a base ten blocks set. They record this length (perhaps as 6 rods or 60 units). Then they measure the bookcase using ones cubes; some of the students decide that it is easier just to add some ones cubes to the 10-rods that they have already used. They find that the bookcase is actually closer to 66 units long. They decide that you can get a better estimate of length when you use smaller units.
• Students use pattern blocks to cover a picture of a turtle. They count how many of each type of block (green triangle, yellow hexagon, etc.) they used. They make a graph that shows how many blocks each student used. They discuss why some students used more blocks than others and what they could do to increase or decrease the number of blocks used.
• Students play with containers of various sizes and water, noting that it takes two cups to fill a small milk carton. They find out that a pitcher holds 3 milk cartons of water. (Four milk cartons overflow.) This would be enough for 6 cups of juice. Then they find that it takes seven cups to fill the pitcher. They decide that the smaller container gives a better idea of how much the pitcher will hold.