Students in grades 5 and 6 extend and clarify their understanding of patterns, measurement, data analysis, number sense, and algebra as they further develop the conceptual building blocks of calculus. Many of the basic ideas of calculus can be examined in a very concrete and intuitive way in the middle grades.

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 a constant 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. In contrast, exponential growth patterns 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. At this grade level, students should also begin to imagine processes that could in theory continue forever even though they could not be carried out in practice; for example, although in practice a cake can be repeatedly divided in half only about ten times, nevertheless it is possible to imagine continuing to divide it into smaller and smaller pieces.

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 look at functions which have "holes" or jumps in their graphs. For example, if students make a table of the parking fees paid for various amounts of time and then plot the results, they will find that they cannot just connect the points; instead there are jumps in the graph where the parking fee goes up. A similar situation exists for graphs of the price of a postage stamp or the minimum wage over the course of the years. Many of the situations investigated by students should involve such changes over time. Students might, for example, consider 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 cmand 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 almost 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 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 volumes.

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 5 and 6.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 5-6 will be such that all students:

4. 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 how much money they would have at the end of each of eight years if $100 was invested at the beginning and the investment grew by 10% each year. They note that the graph of their data is not a straight line; this graph represents exponential growth.

• Students make a table showing the value of a car as it depreciates over time. They note that the graph of their data is not a straight line; this graph represents "exponential decay."

• Students are presented stories which represent real life occurrences of linear and exponential growth and decay over time, and are asked to construct graphs which represent the situation and indicate whether the change is linear, exponential, or neither.

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

• Students make equilateral triangles of different sizes out of small equilateral triangles and record the number of small triangles used for each larger triangle. These numbers are called triangular numbers. If the following triangular pattern is continued indefinitely, then the number of 1s in the first row, the number of 1s in the first two rows, the number of 1s in the first three rows, etc. form the sequence of triangular numbers. The triangular numbers also emerge from the handshake problem: If each two people in a room shake hands exactly once, how many handshakes take place altogether? If the answers are listed for 2, 3, 4, 5, 6, 7, ... people, the numbers are again the triangular numbers 1, 3, 6, 10, 15, 21, ... .

  1
  1 1
  1 1 1
  1 1 1 1
  1 1 1 1 1
  1 1 1 1 1 1

• Students imagine cutting a sheet of paper into half, cutting the two pieces into half, cutting the four pieces into half, and continuing this over and over again, for about 25 times. Then they imagine taking all of the little pieces of paper and stacking them on top of one another. Finally, they estimate how tall that stack would be.

• Students describe, analyze, and extend the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). They research occurrences 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.

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

• 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 .666... .

• 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!

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

• 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 class 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 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 period of 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"; another name for this kind of graph is a piecewise linear graph because the graph consists of linear pieces.

• Students review Mark's trip home from school on his bike. Mark spent the first few minutes after school getting his books and talking with friends, and left the school grounds about five minutes after school was over. He raced with Ted to Ted's house and stopped for ten minutes to talk about their math project. Then he went straight 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. The students then write their own stories and generate graphs of distance vs. time and graphs of speed vs. time.

8. Approximate quantities with increasing degrees of accuracy.

• 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 quotients get closer and closer to pi.

• Using a ruler, students draw an irregularly shaped pentagon on square-grid paper, taking care to locate the vertices of the pentagon at grid points. They estimate the area of the pentagon by counting the number of squares completely inside the pentagon and adding to it an estimate of the number of full square that the partial squares inside the pentagon would add up to. Then they divide the pentagon into triangles and rectangles and find the area of the pentagon as a sum of the areas of the triangles and rectangles. They compare the results and explain any difference.

9. Understand and use the concept of significant digits.

• 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 to the nearest centimeter, then the result would be correct to the nearest square centimeter, while the second measurements would be correct to the nearest square millimeter. However, when they experiment with different rectangles, for example, one whose dimensions are 3.2 by 5.2 centimeters, they find that the area of 15 square centimeters is not correct to the nearest centimeter.

• 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.

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

• Students trace around their hand on graph paper and count squares to find an approximate value of the area of their hand. They use graph paper with smaller squares to find a better approximation.

• 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.

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

• 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; they make the connection with multiplication, noticing that 144 is 12 x 12 and that there are 12 inches in a foot. They realize that the square numbers have that name because they are the areas of squares whose sides are the whole numbers.

• 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.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.