New Jersey Mathematics Curriculum Framework

## STANDARD 15 - CONCEPTUAL BUILDING BLOCKS OF CALCULUS

 All students will develop an understanding of the conceptual building blocks of calculus and will use them to model and analyze natural phenomena.

## Standard 15 - Conceptual Building Blocks of Calculus - Grades K-2

### Overview

Students in the early primary grades bring to the classroom intuitive notions of the meaning of such terms as biggest, largest, change, and so forth. While they may not know the names of large numbers, they certainly have a sense of "largeness." The cumulative process indicators related to this standard for grades K-2 deal primarily with investigating patterns of growth and change over time.

Students in grades K-2 should investigate many different types of patterns. For some of these patterns, such as 2, 4, 6, 8, ... , the same number is added (or subtracted) to each number to get the next number in the sequence. When these patterns are represented with a bar graph, the tops of the bars can be connected by a straight line, so the pattern represents linear growth. 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. In this situation, when the tops of the bars on a graph are connected, they do not form a straight line. 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 a constant addend (e.g., 2 + 2 = = = ) or a constant multiplier (e.g., 2 x 2 = = = ). (Note that some calculators require different keystrokes to achieve this effect.) 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 need 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 a limiting value for young children. For example, the size of a dinosaur footprint might be measured by covering 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 the chapters discussing these other standards.

## Standard 15 - Conceptual Building Blocks of Calculus - Grades K-2

### Indicators and Activities

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 kindergarten and grades 1 and 2.

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

1. 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. each time the = key is pressed. (Some calculators may need to have the pattern entered twice: 10 + 1 = + 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 larger 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 Unifix cubes. They begin with one cube and then "win" another cube. Then they have two cubes and "win" two more. They continue this pattern, each time "winning" as many cubes as they already have. Repeating this process, they begin to see how quickly the number of cubes grows. They investigate this further using a calculator.

• Students start with a rectangular sheet of paper that represents a cake. They simulate eating half of the cake by cutting the sheet in half and removing one of the halves. They eat half of what is left and continue this process. They describe the pattern, noting that after they repeat this about ten times, the cake is essentially gone.

2. Investigate and describe how certain quantities change over time.

• Students keep a daily record of the temperature both inside and outside the classroom. They graph these temperatures and look at the patterns.

• Students keep a monthly record of their height and record the data collected on a bulletin board. At the end of the school year, they describe what happened over time.

• Students play catch with a ball in the school playground. One person counts out the number of times the ball is thrown, the other counts out the distance that it travels, a third person adds that distance to the total, and a fourth person records the totals. Afterwards they discuss how the total distance changes over time; they recognize that the sameamount is added repeatedly.

• 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 at regular time intervals. They discuss how changes in time result in changes in the height of the plant. They also talk about how other factors might affect the growth of the plant, such as light and water.

3. Experiment with approximating length, area, and volume, using informal measurement instruments.

• 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 they can get a better estimate of length when they 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 bar 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, transferring water from one container to another. They note that it takes two cups of water to fill a small milk carton. A pitcher holds three milk cartons of water, but four milk cartons overflow the pitcher. Then they find that it takes seven cups to fill the pitcher even though three milk cartons is only six cups. They decide that the smaller container gives a better idea of how much the pitcher will hold.

• Students find the area of huge dinosaur footprints that they find taped to the classroom floor. They first try to fit as many green 4" tiles as possible into a footprint without any overlapping, and without any tiles sticking out of the footprint. Before removing the green tiles, they cover them with blue 2" tiles, and count the number of blue 2" tiles used. Then they remove the green tiles and try to fit more blue 2" tiles into the footprint without overlapping; they discover that they can fit more and discuss why that is the case. They repeat this, using red 1" tiles. They notice that with smaller tiles, less of the footprint is uncovered, so that the smaller tiles provide a better estimate of the footprint's size.

### 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.