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- Grade 7-8

### Overview

In grades 7 and 8, students begin to develop a more detailed and formal notion of the concepts of approximation, rates of change for various quantities, infinitely repeating processes, and limits. Activities should continue to emerge from concrete, physical situations, often involving the collection of data.

Students in grades 7 and 8 continue to develop their understanding of linear growth, exponential growth, infinity, and change over time. By collecting data in many different situations, they come to see the commonalities and differences in these types of situations. They should recognize that, in linear situations, the rate of change is constant and the graph is a straight line, as in plotting distance vs. time at constant speeds or plotting the height of a candle vs. time as it burns. In situations involving exponential growth, the graph is not a straight line and the rate of change increases or decreases over time. For example, in a situation in which a population of fish triples every year, the number of fish added each year is more than it was in the previous year. Students should also have some experience with graphs with holes or jumps (discontinuities) in them. For example, students may look at how the price of mailing a letter has changed over the last hundred years, first making a table and then generating a graph. They should recognize that plotting points and then connecting them with straight lines is inappropriate, since the cost of mailing a letter stayed constant over a period of several years and then abruptly increased. They should be aware of other examples of "step functions" whose graphs look like a sequence of steps.

Students in these grades should approximate irrational numbers, such as square roots, by using decimals; they should recognize the size of the error when they use these approximations. Students should take care to note that pi is a nonterminating, nonrepeating decimal; it is not exactly equal to 22/7 or 3.14, but lies between these values. These approximations are fairly close to the actual value of pi and can usually be used for computational purposes. Students may also consider sequences involving rational numbers such as 1/2, 2/3, 3/4, 4/5, ... . They should recognize that this sequence goes on forever, getting very close to a limit of one. Students should also consider sequences in the context of learning about fractals. (See Standard 7 or 14. )

Seventh and eighth graders continue to benefit from activities that physically model the process of approximating measurement results with increasing accuracy. Students should develop a clearer understanding of the concept of significant digits as they begin to use scientific notation. They should be able to use these ideas as they develop and apply the formulas for finding the areas of such figures as parallelograms and trapezoids. Students should understand, for example, that if they are measuring the height and diameter of a cylinder in order to find its volume, then some error is introduced from each of these measurements. If they measure the height as 12.2 cm and the diameter as 8.3 cm, then they will get a volume of pi(8.3/2)2(12.2), which their calculator may compute as being 660.09417 cm3. They need to understand that this answer should be rounded off to 660.09 cm3 (five significant digits). They also should understand that the true volume might be as low as pi(8.25/2)2(12.15) ÷ 649.49 cm3 or almost as high as pi(8.35/2)2(12.25) ÷ 670.81 cm3.

Students in these grades should continue to build a repertoire of strategies for finding the surface area and volume of irregularly shaped objects. For example, they might find volume not only by approximating irregular shapes with familiar solids but also by submerging objects in water and finding the amount ofwater displaced by the object. They might find surface areas by first laying out patterns of the objects called "nets"; for example, the net of a cube consists of six squares connected in the shape of a cross - when creased along the edges of the squares, this "net" can be folded to form a cube. Then they would place a grid on the net and count the small squares, noting that the finer the grid the more accurate the estimate of the area.

Explorations developing the conceptual underpinnings of calculus in grades 7 and 8 should continue to take advantage of students' intrinsic interest in infinite, iterative patterns. They should also build connections between number sense, estimation, measurement, patterns, data analysis, and algebra. More information about activities related to these areas can be found in the chapters discussing those standards.

## Standard 15 - Conceptual Building Blocks of Calculus- Grades 7-8

### Indicators and Activities

The cumulative progress indicators for grade 8 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 7 and 8.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

4. Recognize and express the difference between linear and exponential growth.

• Students measure the height of water in a beaker at five second intervals as it is being filled, being careful to leave the faucet on so that the water runs at a constant rate. They make a table of their results and generate a graph. They note that this is a linear function.

• Students investigate patterns of exponential growth with the calculator, such as compound interest or bacterial growth. They make a table showing how much money is in a savings account after one quarter, two quarters and so on for ten years, if \$1000 is deposited at 5% interest and there are no further deposits or withdrawals. They represent their findings graphically, noting that this is not a linear relationship, although in the case of simple interest, where the interest does not earn interest, the graph is linear.

• Students obtain a table showing the depreciated value of a car over time. They graph the data in the table and observe that it is not a straight line. The value of the car exhibits "exponential decay."

• Students compare different pay scales, deciding which is a better deal. For example, is it better to be paid a salary of \$250 per week or to be paid \$6 per hour? They realize that the answer to this question depends on the number of hours worked, so they create a table comparing the pay for different numbers of hours worked. They make a graph and decide at what point the hourly rate becomes a better deal.

• Students predict how many times they will be able to fold a piece of paper in half. Then they fold a paper in half repeatedly, recording the number of sections formed each time in a table. Students find that the number of folds physically possible is surprisingly small (about 7). The students try different kinds of paper: tissue paper, foil, etc. They describe in writing any patterns they discover and try to find a rule for the number of sections after 10, 20, or n folds. They also graph the data on a rectangular coordinate plane using integral values. They extend this problem to a new situation by finding the number of ancestors each person had perhaps ten generations ago and also to the situation of telling a secret to 2 people who each tell two people, etc.

5. Develop an understanding of infinite sequences that arise in natural situations.

• Students discuss how the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ... ) is related to the following problem: begin with two rabbits (one male and one female), each adult pair of rabbits produces two babies (one male and one female) each month, the babies themselves become adults (and start having their own babies) after one month, and none of the rabbitsever die. The students decide that the Fibonacci sequence shows how many pairs of rabbits there are each month. The students explore other patterns in this sequence, noting that each term is the sum of the two preceding terms.

• Students look for infinite sequences in Pascal's triangle. Starting at the top 1 and moving diagonally to the left, there is a constant infinite sequence 1, 1, 1, 1, ... . Starting at the next 1 and moving diagonally to the left there is the sequence 1, 2, 3, 4, 5, ... of whole numbers. Starting at the next 1 and moving diagonally to the left, there is the sequence 1, 3, 6, 10, 15, ... of triangular numbers, which records the solutions to all handshake problems. Also the sum of the numbers in each row yield the exponential sequence 1, 2, 4, 8, 16, ... .

 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

6. Investigate, represent, and use non-terminating decimals.

• Students investigate using simple equations to iterate patterns. For example, they use the equation y = x + 1 and start with any x-value, say 0. The resulting y-value is 1. Using this as the new x yields a 2 for y. Using this as the next x gives a 3, and so on, resulting in the sequence 1, 2, 3, 4, ... . Then students use a slightly different equation, y = .1x + .6, starting with an x-value of .6 and finding the resulting yvalue. Repeating this process yields the sequence of y-values .6, .66, .666, .6666, ..., which approximates the decimal value of 2/3.

• Students explore the question of which fractions have terminating decimal equivalents and which have repeating decimal equivalents. They discover that the only fractions in lowest terms which correspond to terminating decimals are those whose denominators have only 2 and 5 as prime factors.

• Students explore the question of which fractions have decimal equivalents where one digit repeats and learn that these are the fractions 1/9, 2/9, 3/9, ... . They generalize this to find the fractions whose decimal equivalents have two digits repeating like .171717 ... .

7. Represent, analyze, and predict relations between quantities, especially quantities changing over time.

• Students describe what happens when a ball is tossed into the air, experimenting with a ball as needed. They make a graph that shows the height of the ball at different times and discuss what makes the ball come back down. They also consider the speed of the ball: when is it going fastest? slowest? With some help from the teacher, they make a graph showing the speed of the ball over time.

• Students use probes and graphing calculators or computers to collect data involving two variables for several different science experiments (such as measuring the time and distance that a toy car rolls down an inclined plane or measuring the brightness of a light bulb as the distance from the light bulb increases or measuring the temperature of a beaker of water when ice cubes are added). They look at the data that has been collected intabular form and as a graph on a coordinate grid. They classify the graphs as straight or curved lines and as increasing (direct variation), decreasing (inverse variation), or mixed. For those graphs that are straight lines, the students try to match the graph by entering and graphing a suitable equation.

• Students measure the temperature of boiling water as it cools in a cup. They make a table showing the temperature at five-minute intervals for an hour. Then they graph the results and make observations about the shape of the graph, such as "the temperature went down the most in the first few minutes," "it cooled more slowly after more time had passed," or "it's not a linear relationship." The students also predict what the graph would look like if they continued to collect data for another twelve hours.

• Students make Ferris wheel models from paper plates (with notches cut to represent the cars). They use the models to make a table showing the height above the ground (desk) of a person on a Ferris wheel at specified time intervals (time needed for next chair to move to loading position). After collecting data through two or three complete turns of the wheel, they make a graph of time versus height. In their math notebooks, they respond to questions about their graphs: Why doesn't the graph start at zero? What is the maximum height? Why does the shape of the graph repeat? The students learn that this graph represents a periodic function.

• Students compare two ways of cooling a glass of soda, adding lots of ice at the beginning or adding one cube at a time at one minute intervals. Each student first makes a prediction about which cools the soda faster, and the class summarizes the predictions. Then the teacher collects the data, using probes and graphing calculators or computers and displays the results in table and graph form on the overhead. The students compare the graphs and write their conclusions in their math notebooks. They discuss the reasons for any difference between these two methods with their science teacher.

• Students compute the average speed of a toy car as it travels down a ramp by dividing the length of the ramp by the time the car takes to travel the ramp. They try different angles for the ramp, recording their results. They make a graph of average speed vs. angle and discuss whether this graph is linear.

• Students make a graph that shows the minimum wage from the time it was first instituted until the present day. Some of the students begin by simply plotting points and connecting them but soon realize that the minimum wage was constant for a time and then abruptly jumped up. They decide that parts of this graph are like horizontal lines. They look for other examples of "step functions."

8. Approximate quantities with increasing degrees of accuracy.

• Students measure the speed of cars using different strategies and instruments and compare the accuracy of each. For example, they first determine the speed of a car by using a stopwatch to find out how long it takes to travel a specific distance. They note that the speed of the car actually changes over the time interval, however. They decide that they can get a better idea of how fast the car is moving at a specific time by shortening the distance. They collect data for shorter and shorter distances. Finally, they ask a police officer to bring a radar gun to their class to help them collect data about the speed of the cars going past the school.

• Students find the area of a "blob" using a square grid. First, they count the number of squares that fit entirely within the blob (no parts hanging outside). They say that this is the least that the area could be. Then they count the number of squares that have any part of the blob in them. They say that this is the most that the area could be. They note that the actual area is somewhere between these two numbers. Finally, the students put together parts of squares to try to get a more accurate estimate of the area of the blob.

9. Understand and use the concept of significant digits.

• Students measure the radius of a circle in centimeters and find its area. Then they measure its radius in millimeters and find the area. They note the difference between these two results and discuss the reasons for such a difference. Some of the students think that, since the original measurements were correct to the nearest centimeter, then the result would be correct to the nearest square centimeter, while using the second measurements would give a value for the area which is correct to the nearest square millimeter. However, after experimenting with circles of different sizes, they find that if the radius is measured to the nearest centimeter.

• Students explore the different answers that they get by using different values for pi when finding the area of a circle. They discuss why these answers vary and how to decide what value to use.

• Students estimate the amount of wallpaper, paint, or carpet needed for a room, recognizing that measurements that are accurate to several decimal places are unnecessary for this purpose.

10. Develop informal ways of approximating the surface area and volume of familiar objects, and discuss whether the approximations make sense.

• In conjunction with a science project, students need to find the surface area of their bodies. Some of the students decide to approximate their bodies with geometric solids; for example, their head is approximately a sphere, and their neck, arms, and legs are approximately cylinders. They then take the needed measurements and compute the surface areas of the relevant solids. Other students decide to use newspaper to wrap their bodies and then measure the dimensions of the sheets of newspaper used.

• Students estimate the volume of air in a balloon as a way of looking at lung capacity. Some of the students decide that the balloon is approximately the shape of a cylinder, measure its length and diameter, and compute the volume. Other students think the balloon is shaped more like a cylinder with cones at the ends; they measure the diameter of the balloon at its widest part, the length of the cylinder part, and the height of each cone and then compute the volume of each shape. Some other students decide that they would like to check their work another way; they place a large graduated cylinder in the sink, and fill it with water. They submerge the balloon, and read off how much water is left after the balloon is taken out. Since they know that 1 ml of water is 1 cm3, they know that the volume of the water that was displaced is the same as that of the balloon.

• Students develop different strategies for finding the volume of water in a puddle.

11. Express mathematically and explain the impact of the change of an object's linear dimensions on its surface area and volume.

• Students analyze cardboard milk containers to determine how the dimensions of the container affect the volume of milk contained in the carton and how the amount of cardboard used varies. In addition to measuring actual cartons, students make their own cartons of different sizes by varying the length, width, and height one at a time. They write up their results and share them with the class.

• Placing a number of identical cereal boxes next to or on top of one another, students learn that doubling one of an object's length, width, or height doubles its volume, that doubling two of these dimensions increases the volume by a factor of 4, and that doubling all three dimensions increases the volume by a factor of 8.

• Students sketch a 3-dimensional object such as a box or a cylindrical trash can. They then make a sketch twice as large in all dimensions. How much larger is the volume of the larger object? How much larger would it be if the dimensions all increased by a factor of 3? Square grid paper might be helpful for this exercise.

### On-Line Resources

http://dimacs.rutgers.edu/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.