Meaning and Importance

K-12 Development and Emphases

Middle school students should be moving beyond the concrete and pictorial representations used in the elementary grades to more symbolic ones, involving functions and equations. They should use graphing calculators and computers to develop and analyze graphical representations of the changes represented in the tables, and to produce linear and quadratic regression models of the data. They should apply their knowledge of decimals to solving problems involving interest, making use of a calculator to determine for example the yield of a given investment or the length of time it would take for an investment to double. In high school, students can apply their knowledge of exponents, algebra, and functions to solve these and other more difficult interest problems algebraically and graphically.

Throughout their school years, students should be examining a variety of situations where populations and other quantities change over time, and use the mathematical tools at their disposal to describe and analyze this change. As they progress, the situations considered should become more complex; students who experiment with constant motion in their early years will be able to understand the motion of projectiles (a ball thrown into the air, for example) by the time they complete high school.

Similarly, students should be aware of the effect of change on measurement, such as the effect that changes in the linear dimensions of an object have on its area and volume. In the early years children should learn through appropriate hands-on experiments; for example, they might find that doubling the diameter of a circular can increases the volume four-fold by filling the smaller can with water and emptying it into the larger one. By the time students are familiar with variables, this intuition will provide them with the information they need to understand formulas such as those involving volume.

In many settings, the kind of change that takes place over time is repetitive and an important question that should be discussed is what happens in the long run. The principal tool for understanding and discussing such questions is the concept of infinite sequences and the types of patterns that emerge from them. Thus a second central theme is that of "infinity".

There is nothing more fascinating to all students than the mysteries of the infinitely large and, later, the infinitely small. Children are excited about large numbers and "infinity," and that excitement should be nourished and be used, as with other "teachable moments," to motivate the learning of more mathematics. Primary students enjoy naming their "largest" number or proudly declaring that there is no largest! In the early years, large numbers and their significance should be discussed, as should the idea that you can extend simple processes forever (e.g., keep adding 2, keep multiplying by 3).

Once students have familiarity with fractions and decimals, these notions can be extended. What happens when you keep dividing by 2? By 10? Can you find a fraction between 0.499 and 1? What decimal comes just before 1? Such applications can be related to compound interest and population explosion. Students should explore and experiment with infinite repeating decimals and other infinite series, where they can make tables and look for patterns. They should learn that by repeated iteration of simple processes you can get better and better answers in both arithmetic (with increased decimal accuracy) and in geometry (with more accurate estimates of the area and volume of irregular objects).

Although the concept of a limiting value (or a limit) may appear inaccessible to K-8 students, this basic notion of calculus can be explored through the process of measuring the area of a region. Students can be provided with diagrams of a large circular (or irregular) region, say a foot in diameter, and a large supply of tiles of different square sizes. By covering the space inside the region (with no protrusions!) with 4" tiles, then with 2" tiles, then with 1" tiles, then with .5" tiles, students can get an appreciation that the smaller the unit, the larger the area. They will recognize that the space cannot be filled completely with small tiles, yet, at the same time, the sum of the areas of the small tiles gets closer and closer to that of the region.

In summary, these kinds of experiences will provide a good foundation for the notions of limits, infinity, and changes in quantities over time. Such concepts find many applications in both science and mathematics, and students will feel much more comfortable with them if we begin their development in the early grades.

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.

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.

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.

• 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.

• 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.

Students in grades 5 and 6 should begin to distinguish between patterns involving linear growth (where a constant is added or subtracted to each number to get the next one) and exponential growth (where each term is multiplied or divided by the same number each time to get the next number). Students should recognize that linear growth patterns change at the same rate. For example, a plant may grow one inch every day. They should also begin to see that if these patterns are graphed, then the graph looks like a straight line. They may model this line by using a piece of spaghetti and use their graph to make predictions and answer questions about points that are not included in their data tables. Exponential growth patterns, however, change at an increasingly rapid rate; if you start with one penny and double that amount each day, you receive more and more pennies each day as time goes on. Students should note that the graphs of these situations are not straight lines.

Many of the examples used should come from other subject areas, such as science and social studies. Students might look at such linear relationships as profit as a function of selling price, but they should also consider nonlinear relationships such as the amount of rainfall over time. Students should also look at functions which have "holes" or jumps in their graphs. For example, if students make a table of the amount of income tax paid for various incomes and then plot the results, they will find that they cannot just connect the points; instead there are jumps in the graph where the tax rate goes up. A similar situation exists for graphs of the price of a postage stamp or the minimum wage over time. Many of the situations investigated by students should involve such changes over time. Students might, for example, look at the speed of a fly on a spinning disk; as the fly moves away from the center of the disk, he spins faster and faster. Students might be asked to write a short narrative about the fly on the disk and draw a graph of the fly's speed over time that matches their story.

As students begin to explore the decimal equivalents for fractions, they encounter non-terminating decimals for the first time. Students should recognize that calculators often use approximations for fractions (such as .33 for 1/3). They should look for patterns involving decimal representations of fractions, such as recognizing which fractions have terminating decimal equivalents and which do not. 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 these approximations are fairly close to the actual value of pi and can usually be used for computational purposes. The examination of decimals extends students' understanding of infinity to very small numbers.

Students in grades 5 and 6 continue to develop a better understanding of the approximate nature of measurement. Students are able to measure objects with increasing degrees of accuracy and begin to consider significant digits by looking at the range of possible values that might result from computations with approximate measures. For example, if the length of a rectangle to the nearest centimeter is 10 cm and its width (to the nearest centimeter) is 5 cm, then the area is about 50 square centimeters. However, the rectangle might really be as small as 9.5 cm x 4.5 cm, in which case the area would only be 42.75 square centimeters, or it might be as large as 10.5 cm x 5.5 cm, with an area of 57.75 square centimeters. Students should continue to explore how to determine the surface area of a irregular figures; they might, for example, be asked to develop a strategy for finding the area of their hand or foot. They should do similar activities involving volume, perhaps looking for the volume of air in a car. Most of their work in this area in fifth and sixth grade will involve using squares or cubes to approximate these areas or volume.

Building on the K-4 expectations, experiences in grades 5-6 will be such that all students:

D. recognize and express the difference between linear and exponential growth.

• Students develop a table showing the sales tax paid on different amounts of purchases, graph their results, note that the graph is a straight line, and recognize that this situation represents a constant rate of change, or linear growth.
• Students make a table showing the value of a computer over time (depreciation). They note that the graph of their data is not a straight line and thus represents exponential "growth."

• 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 rabbits ever 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 patterns in the figure below:

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

  They discuss their patterns and use them to find the next two rows of the table. The teacher tells the students that this is called Pascal's Triangle and is used in many different situations in mathematics.

• Students look for the number of different ways to walk from the lower left corner (A) to each other point on a rectangular grid, as shown below. They find that there is one way to walk to each point along the left side and the bottom. They find that the other points make a pattern like Pascal's Triangle.

  1	5	15	35	70
  1	4	10	20	35
  1	3	6	10	15
  1	2	3	4	5
  A	1	1	1	1

• Students use their calculators to find the decimal equivalent for 2/3 by dividing 2 by 3. Some of the students get an answer of 0.66667, while others get 0.6666667. They do the problem by hand to try to understand what is happening. They decide that different calculators round off the answer after different numbers of decimal places. The teacher explains that the decimal for 2/3 can be written exactly as .6666...
• Students have been looking for the number of different squares that can be made on a 5 x 5 geoboard and have come up with 1x1, 2x2, 3x3, 4x4, and 5x5 squares. One student finds a different square, however, whose area is 2 square units. The students wonder how long the side of the square is. Since they know that the area is the length of the side times itself, they try out different numbers, multiplying 1.4 x 1.4 on their calculators to get 1.89 and 1.5 x 1.5 to get 2.25. They keep adding decimal places, trying to get the exact answer of 2, but find that they cannot, no matter how many places they try!

• Students study which is the better way to cool down a 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 and the class summarizes the predictions. Then the teacher collects the data, using probes and graphing calculators or computers and displaying 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 their results in science class.
• Students make a graph that shows the price of mailing a letter from 1850 through 1995. Some of the students begin by simply plotting points and connecting them but soon realize that the price of a stamp is constant for a time and then abruptly jumps up. They decide that parts of this graph are like horizontal lines. The teacher tells them that mathematicians call this a "step function" or a piecewise linear graph.
• Students discuss Mark's trip home from school on his bike. They suggest that Mark might spend the first few minutes after school getting his books and talking with friends. He might leave the school grounds about five minutes after school was over. He might race with Ted to Ted's house and stop for ten minutes to talk about their math project. Then he might ride on home. The students draw a graph showing the distance covered by Mark with respect to time. Then, with the teacher's help, the class constructs a graph showing the speed at which Mark traveled with respect to time. Each student then writes their own story and generates a graph of distance vs. time and speed vs. time.

• Students find the volume of a cookie jar by first using Multilink cubes (which are 2 cm on a side) and then by using centimeter cubes. They realize that the second measurement is more accurate than the first.
• Students measure the circumference and diameter of a paper plate to the nearest inch and then divide the circumference by the diameter. They repeat this process, using more accurate measures each time (to the nearest half-inch, to the nearest quarter-inch, etc.). They see that the number they get after dividing gets closer and closer to .

• Students measure the length and width of a rectangle in centimeters and find its area. Then they measure its length and width 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 only to the nearest centimeter, then the result can be correct only to the nearest square centimeter, while the second measurements are correct to the nearest square millimeter.
• 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.

• Students trace around their hand on graph paper and count squares to find the area of their hand.
• Students work in groups to find the surface area of a leaf. They describe the different methods they have used to accomplish this task. Some groups are asked to go back and reexamine their results. When the class is convinced that all of the results are reasonably accurate, they consider how the surface area of the leaf might be related to the growth of the tree and its needs for carbon dioxide, sunshine, and water.
• Each group of students is given a mixing bowl and asked to find its volume. One group decides to fill the bowl with centimeter cubes, packing them as tightly as they can and then to add a little. Another group decides to turn the bowl upside down and try to build the same shape next to it by making layers of centimeter cubes. Still another group decides to fill the hollow 1000-centimeter cube with water and empty it into the bowl as many times as they can to fill it; they find that doing this three times almost fills the bowl and add 24 centimeter cubes to bring the water level up to the top of the bowl.

• While learning about area, the students became curious about how many square inches there are in a square foot. Some students thought it would be 12, while others thought it might be more. They explore this question using square-inch tiles to make a square that is one foot on each side. They decide that there are 144 square inches in a square foot; some of the students think this is because 144 is 12 x 12 and there are 12 inches in a foot.
• Having measured the length, width, and height of the classroom in feet, the students now must find how many cubic yards of air there are. Some of the students convert their measurements to yards and then multiply to find the volume. Others multiply first, but find that dividing by 3 does not give a reasonable answer. They make a model using cubes that shows that there are 27 cubic feet in a cubic yard and divide their answer by 27, getting the same result as the other students.

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 postage stamps 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 introduced to the idea of a step function. Students in these grades should approximate irrational numbers, such as square roots, by using decimals; they should recognize the size of the error introduced by using these approximations. Students should take care to note that is a nonterminating, nonrepeating decimal; it is not exactly equal to 22/7 or 3.14, but these approximations are fairly close to the actual value of 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 Chapter 10 for more information.)

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 apply 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 cm^3. They need to understand that this answer should be rounded off to 660.09 cm^3 (five significant digits). They also should understand that the true volume might be as low as pi(8.25/2)^2(12.15) is about 649.49 cm^3 or as high as pi(8.35/2)^2(12.25) is about 670.81 cm^3.

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 of water displaced by the object. They might find surface areas by laying out patterns (nets) of the objects and then placing a grid on the net, noting that the finer the grid the more accurate the estimate of the area.

Explorations involving 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 those chapters.

Building on the K-6 expectations, experiences in grades 7-8 will be such that all students:

D. recognize and express the difference between linear and exponential growth.

• Students develop a sales tax table, graph their results, note that the graph is a straight line, and recognize that this situation represents a constant rate of change, or linear 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 make a table showing the depreciated value of a car over time. They note that the graph of their data is not a straight line and thus represents exponential "growth."
• Students investigate growing patterns (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 (if none is withdrawn) after one quarter, two quarters and so on for ten years, for example. They represent their findings graphically, note that this is not a linear relationship (although simple interest is linear), and write an equation describing the relationship between the amount deposited initially (P), the interest rate (r), the number of times that interest is paid each year (n), the number of years (y), and the total (T) available at the end of that time period:
  T = (1 + r/n)^ny(P).
• 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 create a table comparing the pay for different numbers of hours worked 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 generate a rule for finding 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 problem of telling a secret to 2 people who each tell two people, etc.
• Students decide how many different double-dip ice cream cones can be made from two flavors, three flavors, etc. They arrange the information in a table. They discuss whether one flavor on top and another on the bottom is a different arrangement from the other way around, and how that changes their results. They also discuss a similar problem (see Chapter 10): how many different types of pizzas can be made using different toppings.

• Students describe, analyze, and extend the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). They research applications of this sequence in nature, such as sunflower seeds, the fruit of the pineapple, and the rabbit problem. They create their own Fibonacci-like sequences, using different starting numbers.
• Students look at Sierpinski's Triangle as an example of a fractal. Stage 1 is an unshaded triangle. To get Stage 2, you take the three midpoints of the sides of the unshaded triangle, connect them, and shade the new triangle in the middle. To get Stage 3, you repeat this process for each of the unshaded triangles in Stage 2. This process continues an infinite number of times. The students make a table that records the number of unshaded triangles at each stage, look for a pattern, and use their results to predict the number of unshaded triangles there will be at the tenth (3^9) and twentieth (3^19) stages.
• Students look for patterns in Pascal's Triangle:

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

  The students look not only for how each row is generated but also for other patterns, such as the pattern involving the sum of the numbers in each row (1, 2, 4, 8, ...). They relate Pascal's triangle to coin tossing and heredity. They also generate other triangles like this one by starting with numbers other than one (e.g., 2 or 1/2).

• Students look for the number of different ways to walk from the lower left corner (A) to each other point on a rectangular grid, as shown below. They find that there is one way to walk to each point along the left side and the bottom. They find that the other points make a pattern like Pascal's Triangle.

  1	5	15	35	70
  1	4	10	20	35
  1	3	6	10	15
  1	2	3	4	5
  A	1	1	1	1

• Students explore the question of which fractions have terminating decimal equivalents and which have repeating decimal equivalents. One group of students first looks at proper fractions with denominators of 2, 3, 5, and 7, while other groups consider denominators of 11, 13, 17, and 19. Each group presents its findings to the class. They then use these results to consider proper fractions with denominators that are composite numbers. Finally, they write up their results in their journals.
• 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 value yields a 2 for y. Using this as the next x gives a 3, and so on. The related values can be organized in a table and the ordered pairs graphed on a rectangular coordinate system. Students note that the graph is a straight line. Then students use a slightly different equation, y = .1x + .6, starting with an x value of .6 and finding the resulting y-value. Repeating this process yields the series of y-values .6, .66, .666, .6666, ..., which approximates the decimal value of 2/3.

• 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 in tabular 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 in a Styrofoam cup as it cools. 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 study which is the better way to cool down a 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 and the class summarizes the predictions. Then the teacher collects the data, using probes and graphing calculators or computers and displaying 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 their results in science class.
• Students compute the speed of a toy car as it travels down an inclined plane by measuring the distance it travels and the time. They try different angles for the plane, recording their results. They make a graph of angle vs. speed and note that the data are generally in a straight line.
• 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. The teacher tells them that mathematicians call this a "step function" or a piecewise linear graph.

• Students are given a circle drawn on graph paper. They find the largest rectangle that fits entirely within the circle and record its length, width, and area. They also find the smallest rectangle that encloses the circle and record its length, width, and area. They note that the area of the circle must lie between these two numbers. Next they look for two rectangles with the same width that lie entirely within the circle and two rectangles of the same width that together enclose the circle, again recording the lengths, widths, and areas of each. They again note that the area of the circle must lie between these two numbers; they also note that using two rectangles gives them a smaller range for the possible area of the circle. They repeat this process using 4, 8, and 16 congruent rectangles. After discussing their results, they also discuss other ways to find the area of the circle.
• 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 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 only to the nearest centimeter, then the result can be correct only to the nearest square centimeter, while the second measurements are correct to the nearest square millimeter.
• 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.
• Students explore the different answers that they get by using different values for 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.

• 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 a sphere, and their neck, arms, and legs are 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, fill it with water and note how much water is in the cylinder, submerge the balloon, and then read off how much water is left after the balloon is taken out. Since they know that 1 ml of water is 1 cm^3, 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.

• 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.
• Students analyze cylindrical cans to find out how the dimensions of the can affect the volume of food contained in the can and how the amount of metal used to make the can varies. Students measure actual cans and also make their own cans by varying each dimension one at a time.

Instruction should be highly exploratory, based on numerical and geometric experiences that capitalize on both calculator and computer technology. Activities should be aimed at providing students with firm conceptual underpinnings of calculus rather that at developing manipulative techniques.

The development of calculus is one of the great intellectual achievements in history, especially with respect to its use in physics. Calculus is also increasingly being used in the social and biological sciences and in business. As students explore this area, they should develop an awareness of and appreciation for the historical origins and cultural contributions of calculus.

Students earlier study of patterns leads in high school to the study of finite and infinite processes. Students continue to look at linear growth patterns as they develop procedures for finding the sums of arithmetic series (e.g., the sum of the numbers from 1 to 100). They may consider this sum in many different ways, building different types of models. Some students may look at 1 + 2 + 3 + ... + 100 geometrically by putting together two "staircases" to form a rectangle that is 100 by 101. Other students may look at the sum arithmetically by adding 1 + 2 + 3 + ... + 100 to 100 + 99 + 98 + ... + 1 and getting 100 pairs of numbers that add up to 101. Still others may look at the sum by finding the limit of the sequence of partial sums. Students also look at exponential growth as they develop procedures for finding the sum of finite and infinite geometric series (e.g., 2 + 4 + 8 + 16 + 32 or 6 + 3 + 3/2 + ... or finding the total distance traveled by a bouncing ball). Students' work with patterns and infinity also includes elaborating on the intuitive notion of limit that has been addressed in the earlier grades.

High school students further develop their understanding of change over time through informal activities that focus on the understanding of interrelationships. Students need to collect data, generate graphs, and analyze the results for real-world situations that can be described by linear, quadratic, trigonometric, and exponential models. Some of the types of situations that should be analyzed include motion, epidemics, carbon dating, pendulums, and biological and economic growth. Students need to recognize the basic models (y = mx + b, y = ax^2 + bx + c, y=sin(x), and y = 2^x) and be able to relate geometric transformations to the equations of these models. Students need to develop a thorough understanding of the idea of slope; for example, they need to be able compare the steepness of two graphs at various points on the graph. They also need to be able explain what the slope means in terms of the real-world situation described by a graph. For example, what information does the slope give for a graph of the levels of medicine in the bloodstream over time? Students also extend their understanding of the behavior of functions to include the concept of the continuity of a function, considering features such as removable discontinuities (holes or jumps), asymptotes, and corners.

Students in high school apply their understanding of approximation techniques not only with respect to numbers in the context of using initial portions of nonrepeating, nonterminating decimals but also with respect to measurement situations. Students further develop their understanding of significant digits and the arithmetic of approximate values. They also use repeated approximations to find the areas of irregular figures, including experimenting with situations in which they need to find the area under a curve.

Looking at the conceptual underpinnings of calculus provides an opportunity for high school students to pull together their experiences with data analysis, patterns, algebra, measurement, number sense, and numerical operations. It also provides the opportunity to apply technology to real-world situations and to gain experience with mathematics as a dynamic human endeavor.

Building on the K-8 expectations, experiences in grades 9-12 will be such that all students:

L. develop and use models based on sequences and series.

• Students work in groups to collect data about a bouncing ball. They first decide how to measure the height of each bounce and then record their data in a table. They notice the pattern of the heights and make two graphs, one of height vs. bounce and the other of total distance traveled vs. bounce. They describe the general behavior of each graph and decide to have their graphing calculators compute regression lines. They try several different types of regressions until they find the one that fits best. In their report, they describe what they did, their results, and why they think that the type of function they used to describe each graph is reasonable.
• Students use M&Ms to model decay. They spill a package of M&Ms on a paper plate and remove those with the M showing, recording the number of M&Ms removed. They put the remaining M&Ms in a cup, shake, and repeat the process until all of the M&Ms are gone. They plot the trial number versus the number of M&Ms removed and note that the graph represents an exponential function. Some of the students try out different equations until they find one that they think fits pretty well.

• Students investigate a situation in which a contractor is fined $400 if he is one day late completing a project, $475 more if he is two days late, $550 more if he is three days late, and so on. They want to find out how much he will lose if he is two weeks late finishing the job. Some of the students solve the problem by first deciding that he will owe $400 + 14($75) = $1450 on the fourteenth day and then writing the following:

             400 +  475 +  550 + ... + 1300 + 1375 + 1450
            1450 + 1375 + 1300 + ... +  550 +  475 +  400
            ---------------------------------------------
            1850 + 1850 + 1850 + ... + 1850 + 1850 + 1850
  

  They decide that they have 14 pairs of numbers, each of which adds up to 1850. This gives them $25,900 which they divide in half (since they added up two sequences) to find the answer, $12,950. Another group decides that they will multiply 14 x 400, since he would owe the basic $400 each day, and then add on 75 (1 + 2 + 3 + ... + 13 + 14).

• Students investigate patterns in finding the sum 9 + 3 + 1 + 1/3 + 1/9 + 1/27. Most start by computing the sum (364/27 or about 13.815) on their calculators. One of the students notices that 364 is half of one less than 27^2. They make a conjecture that you might be able to find the sum by taking the reciprocal of the last term, squaring it, subtracting one, and then dividing by two. They decide to test this conjecture on another problem, 3 + 1/3 + 1/27, however, and it does not work. One group of students decides to try something like Gauss's approach to arithmetic series; they multiply the whole sequence by 3 (since that is the ratio between any two terms) and write:

              3 series = 27 + 9 + 3 +   1 + 1/3 +  1/9
            - 1 series =      9 + 3 + 1/3 + 1/9 + 1/27
            ------------------------------------------
              2 series = 27 - 1/27
  

  They make a conjecture that if you take the ratio, r, of any two terms and multiply that by the first term in the series and then subtract the last term in the series, you will have (r-1) times the sum. Each group shares its approach and its results with the class. The teacher develops the conjecture of the group that tried the Gauss-type approach into the standard formula,
  S(n) = a (1 - r^n)/(1 - r).

• After investigating how to find the sum of a finite geometric sequence, students begin looking at infinite geometric sequences. They think about what will happen if r is larger than one as n gets bigger and decide that the sum will keep increasing. Then they think about what will happen if r is less than one. As n gets bigger, r^n will get closer to zero, so the sum will get closer to a/(1 - r). They confirm their conjecture by checking out the partial sums of some sequences.

• Students look for the fractional equivalent of .9999... by first considering that 1/3 = .3333... They decide that .9999... is three times as large and so should be equal to 3/3, or 1. This does not seem quite right to them, however, so they decide to look at .999... as a sequence:
  .9, .09, .009, .0009, ...
  They compute the partial sums and see that, indeed, the series keeps getting closer to 1. They decide that 1 is the limit of the sequence.
• Students consider the sequence 1/2, 1/4, 1/8, ... in different contexts. First, they look at it as representing a situation in which someone eats half of a pizza, then half of what is left, then half of what is left, etc. They decide that, while there will always be some of pizza left, when you consider it all together, the sum of the sequence must be the whole pizza. Then they look at tearing a sheet of paper in half repeatedly and decide that the limit there must also be the whole sheet, or 1.

• Students use a graphing calculator, together with a light probe, to develop the relationship between brightness of a light and distance from it. They do this by collecting data with the probe on the brightness of a light bulb at increasing distances and analyzing the graph generated on the calculator.
• Students learn about the Richter Scale for measuring earthquakes, focussing on its representation as an exponential function.
• Students use recursive definitions of functions in both geometry and algebra. For example, they define n! recursively as n! = n (n-1)! They use recursion to generate fractals in studying geometry. They may use patterns such as spirolaterals, the Koch Snowflake, the Monkey's Tree curve, the Chaos Game, or the Sierpinski Triangle. They may use Logo or other computer programs to iterate patterns, or they may use the graphing calculator. In study algebra, students consider the equation y = .1x + .6, starting with an x value of .6, and find the resulting y-value. Using this y-value as the new x-value, they then calculate its corresponding y-value, and so on. (The resulting values are .6, .66, .666, .6666, etc.--an approximation to the decimal value of 2/3!) Students investigate using other starting values for the same function; the results are surprising! They use other equations to repeat the procedure. They graph the results and investigate the behavior of the resulting functions, using a calculator to reduce the computational burden.
• Students grow mold and collect data on the area of a pie plate covered by the mold. They make a graph showing the percent of increase in the area vs. the days, as in the vignette, "Breaking the Mold." The students graph their data and find an equation that fits the data to their satisfaction.

• Students work in groups to investigate what size square to cut from each corner of a rectangular piece of cardboard in order to make the largest possible open-top box. They make models, record the size of the square and the volume for each model, and plot the points on a graph. They note that the relationship seems to be a polynomial function and make a conjecture about the maximum volume, based on the graph. The students also generate a symbolic expression describing this situation and check to see if it matches their data by using a graphing calculator.
• Students look at the effects of changing the coefficients of a trigonometric equation on the graph. For example, how is the graph of y = 4 sin x different from that of y = sin x? How is y = .2 sin x different from y = sin x? How are y = sin x + 4, y = sin x - 4, y = sin (x - 4), and y = sin (x + 4) each different from y = sin x? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.
• Students study the behavior of functions of the form y = ax^n. They investigate the effect of "a" on the curve and the characteristics of the graph when n is even or odd. They use the graphing calculator to assist them and write a sentence summarizing their discoveries.
• Students begin with the graph of y = 2^x. They shift the graph up one unit and try to find the equation of the resulting curve. They shift the original graph one unit to the right and try to find the equation of that curve. They reflect the original graph across the x-axis and try to find the equation of that curve. Finally, they reflect the original graph across the y-axis and try to find the equation of the resulting curve. They describe what they have learned in their journals.

• Students collect data about the height of a ball that is thrown in the air and make a scatterplot of their data. They note that the points lie on a quadratic function and use their graphing calculators to find the curve of best fit. Then they make some conjectures about the speed at which the ball is traveling. They think that the ball is slowing down as it rises, stopping at the maximum point, and speeding up again as it falls.
• Students take on the role of "forensic mathematicians," trying to determine how tall a person would be whose femur is 17 inches long. They measure their own femurs and their heights, entering this data into a graphing calculator or computer and creating a scatterplot. They note that the data are approximately linear, so they find the y-intercept and slope from the graph and generate an equation that they think will fit the data. They graph their equation and check its fit. They also use the built-in linear regression procedure to find the line of best fit and compare that equation to the one they generated.
• Students plot the data from a table that gives the amount of alcohol in the bloodstream after drinking two beers over time. Different groups use different techniques to generate an equation for the graph; after some discussion, the class decides which equation they think is best. The students consider the following questions: What information does the slope give for this situation? Would that be important to know? Why or why not?
• Students investigate the effect of changing the radius of a circle upon its circumference by measuring the radius and the circumference of circular objects. They graph the values they have generated, notice that it is close to a straight line, and use the slope to develop an equation that describes that relationship. Then they discuss the meaning of the slope in this situation.

• Students play the game described in the vignette, "On the Boardwalk," in which a quarter is thrown onto a grid. If the quarter does not touch a line, then you win. The students collect data and make a graph of their results (size of squares vs. number of wins out of 100 tosses). They note that, although the graph looks pretty much like a straight line, it actually has a bit of curve in it. No matter how large the square is, the most you could possibly win would be 100 games out of 100, so there is an asymptote at y=100.
• The school store sells pencils for 15 cents each, but it has some bulk pricing available if you need more pencils. Ten pencils sell for $1, and twenty-five pencils sell for $2. The students make a table showing the cost of different numbers of pencils and then generate a graph of number of pencils vs. cost. The students note that the graph has discontinuities at ten and twenty-five, since these are the jump points for pricing. They also note that if you need at least seven pencils, it is better to buy the package of ten and if you need 17 or more, you should get the package of 25.
• Students make a table, plot a graph (number of people vs. cost), and look for a function to describe a situation in which the Student Council is sponsoring a Valentine's Day dance and must pay $300 to the band, no matter how many people come. They also must pay $4 per person for refreshments, with a minimum of 50 people. The students note that the cost will be $500 for anywhere from 0-50 people and then increase at a rate of $4 per person. They decide that this is a function with a corner and needs to be defined in pieces:
  

            f(x) = 500 for x=<50
            f(x) = 500 + 4x for x>50
  

• Students investigate finding the area under the curve y = x^2 + 1 between -1 and 1. They approximate the area geometrically by dividing it into rectangles 0.5 units wide. They find the height of each rectangle that fits under the curve and use it to find the areas. Then they find the height of each rectangle that contains the curve and use these measurements to find the areas. They realize that this gives them a range of values for the area under the curve. They refine this approximation by using narrower rectangles, such as 0.1.
• After some experience with collecting data about balls thrown into the air, students are given a table of data about a model rocket and its height at different times. They plot the data, find an equation that fits the data, and use the trace functions on their graphing calculators to find the maximum height.