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 3-4

### Overview

Students in grades 3 and 4 continue to develop the conceptual building blocks of calculus primarily through their work with patterns and changes over time. Students investigate a variety of patterns, using physical materials and calculators as well as pictures. In some of the patterns investigated, a constant is added to, or subtracted from, each number to get the next number in the sequence. These patterns, involving repeated operations, show linear growth; when these patterns are represented with a bar graph, the tops of the bars can be connected by a straight line. Examples of such patterns include skip counting, starting the week with \$5 and paying 75 cents each day for lunch, or enumerating the multiples of 9. Other patterns should involve multiplying or dividing a number by a constant to get the next number in the sequence. These growing patterns illustrate exponential growth, as is the pattern which results when you start with two guppies (one male and one female) and the number of guppies doubles each week. Patterns should also include looking at changes over time, since these types of patterns are extremely important not only in mathematics but also in science and social studies. Students might chart the height of plants over time, the number of teeth lost each month throughout the school year, or the temperature outside the classroom over the course of several months.

Students continue to develop their understanding of measurement, gaining a greater understanding of the approximate nature of measurement. Students can guess at the length of a stick that is between 3 and 4 inches long, saying it is about 3 1/2 inches long and recognizing when this is a better approximation than either 3 or 4. They can use grids of different sizes to approximate the area of a puddle, recognizing that the smaller the grid the more accurate the measurement. They can begin to consider how one might measure the amount of water in a puddle, coming up with alternative strategies and comparing them to see which would be more accurate. As they develop a better understanding of volume, they may use cubes to build a solid, build a second solid whose sides are all twice as long as the first, and then compare the number of cubes used to build each solid. The students may be surprised to find that it takes eight times as many cubes to build the larger solid!

Students continue to develop their understanding of infinity in grades 3 and 4. Additional work with counting sequences, skip counting, and calculators further reinforces the notion that there is always a bigger number. Taking half of something (like a rectangular cake or a sheet of paper) repeatedly suggests that there is always a number that is still closer to zero.

As students develop the conceptual underpinnings of calculus in third and fourth grades, they are also working to develop their understanding of numbers, patterns, measurement, data analysis, and mathematical connections. Additional ideas for activities relating to this standard can be found in the chapters discussing these other standards.

## Standard 15 - Conceptual Building Blocks of Calculus - Grades 3-4

### 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 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. Investigate and describe patterns that continue indefinitely.

• Using the constant multiplier feature of a calculator, students see how many times 1 must be doubled before one million is reached. They might first guess the number of steps to one million, and to half a million.

• Students start with a long piece of string. They fold it in half and cut it in two, setting aside one piece. Then they take the remaining half, fold it in half, cut it apart, and set aside half. They continue this process. They discuss how the length of the string keeps getting smaller, half as much each time, so that after about ten cuts, there is essentially no string left. Some students may understand that the process could keep going for several more steps, if we could only cut more carefully, and some may realize that in theory the process could continue forever.

• Students count out 1, 2, 3, 4, 5, 6, ... , and recognize that this pattern could continue forever. They also count out other patterns, like the even numbers, or the square numbers, or skip-counting by 3s starting with 2, and recognize that these patterns also could be continued indefinitely.

• Students investigate the growth patterns of sunflowers, pinecones, pineapples, or snails to study the natural occurrence of spirals.

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

• Students begin with a number like 5, add 3 to it, add 3 to it again, and repeat this five times. They record the results in a table and make a bar graph which represents the numbers that they have generated. They draw a straight line connecting the tops of the bars. They experiment with numbers other than 5 or 3 to see if the same thing occurs.

• 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 make a bar graph which represents the numbers they have generated. They discuss whether they can connect the tops of the bars with a straight line.

• 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. Initially the temperature will increase rapidly, but as the water warms up, its temperature will increase more slowlyuntil it essentially reaches room temperature.

• Students are given several examples of bar graphs with straight lines connecting the tops of the bars. They are asked to describe a motion scenario which reflects the data. For example, they might indicate that a graph reflects their running to an after-school activity, staying there for an hour, and then slowly walking home to do their chores.

• 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. They also find each month the average height of all the students in the class, and discuss how the change in average height over the year is similar to, and how it is different from, the change in height over the year of the individual students in the class.

• Students plant some seeds in vermiculite and some 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.

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

• Students measure the length of their classroom using their paces and compare their results. They discuss what would happen if the teacher measured the room with her pace.

• Students use pattern blocks to cover a drawing of a dinosaur with as few blocks as they can. They record the number of blocks of each type used in a table and then discuss their results, making a frequency chart or bar graph of the total number of blocks used by each pair of students. Then they try to cover the same drawing with as many blocks as they can. They again record and discuss their results and make a graph. They look for connections between the numbers and types of blocks used each time. Some students simply trade blocks (e.g., a hexagon is traded for six triangles), while other students try to use all tan parallelograms since that seems to be the smallest block. (It actually has the same area as the triangle, however.)

• Students compare the volumes of a half-gallon milk carton, a quart milk carton, a pint milk carton, and a half-pint milk carton. They also measure the length of the side of the square base of each carton and its height. They make a table of their results and look for patterns. The students notice that the difference between the height measurements is not the same as the difference between the volume. The differences in volume grow more quickly than the differences in the heights. They see how many small cubes or marbles fill up each of their containers, and they try to explain why more than twice as many fit into a quart container than a pint container.

### 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.