Chapter 18 3-4

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 18: CONCEPTUAL UNDERPINNINGS OF CALCULUS

All students will develop their understanding of the conceptual underpinnings of calculus through experiences which enable them to describe and analyze how various quantities change, to build informal concepts of infinity and limits, and to use these concepts to model, describe, and analyze natural phenomena.

3-4 Overview

Students in grades 3 and 4 continue to develop the conceptual underpinnings 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. Some of the patterns investigated involve adding or subtracting a constant to a number to get the next number in the sequence; these are repeating patterns and show linear growth. Examples of such patterns include skip counting, starting the week with $5 and paying 75 cents each day for lunch, or 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. Looking at the areas of a series of squares whose sides increase by one each time is an example of this type of pattern, as is a situation in which 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 3.5 inches long. 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 you 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 pizza or a sheet of paper) repeatedly suggests that there are also infinitely small numbers (that get closer and 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 these other chapters.

STANDARD 18: CONCEPTUAL UNDERPINNINGS OF CALCULUS

3-4 Expectations and Activities

The expectations for these grade levels appear below in boldface type. Each expectation is followed by activities which illustrate how the expectation can be addressed in the classroom.

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

A. investigate and describe patterns that continue indefinitely.

Students investigate the growth patterns of sunflowers, pinecones, pineapples, or snails to study the natural occurrence of spirals.
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 start with a long piece of string. They fold it in half and cut off 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, but it never completely disappears.

B. investigate and describe how certain quantities change over time.

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.

C. 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.