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.