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.