The key components of pattern-based thinking, as identified in the K-12 Overview, involve exploring, analyzing, and generalizing patterns, and viewing rules and input/output situations as functions. In grades 7-8, the importance of studying patterns continues with an emphasis on representing and describing relationships with tables and graphs and on the development of rules using variables. Patterns also become important now in the analysis of statistics and the development of geometric relationships. Graphing calculators and computers are helpful in illustrating the usefulness of symbols and in making symbolic relationships more tangible. Although the symbolism and notation used become more algebraic at these grade levels (e.g., A = 4s instead of A = 4 x s), students should still be encouraged to model many patterns with concrete materials. Engineers, scientists, architects, and other researchers all build working models of projects for analysis and demonstration.

Students in these grades should also be given ample opportunity to analyze patterns, to discover the relevant features of the patterns, and to construct understandings of the concepts and relationships involved in the patterns. From these investigations, students should develop the language necessary to communicate their ideas about the patterns and should learn to differentiate among the variety of patterns they have studied (that is, to categorize and to classify them). They will apply their understanding of patterns as they learn about such topics as exponents, rational numbers, measurement, geometry, probability, and functions.

Seventh and eighth graders continue to discover rules for mathematical relationships and for quantifiable situations from other subject areas. In particular, students should focus on relationships involving two variables. Students should analyze how a change in one quantity results in a change in another. They further need to develop their understanding of the general behavior of functions and use these to model a variety of phenomena.

Students should be encouraged to solve problems by looking for patterns that involve words, pictures, manipulatives, and number descriptions. These situations naturally lead to the use of variables and informal algebra in solving problems.

As seen in prior grades, computers and graphing calculators provide many benefits for students in investigating mathematical concepts and problems. These tools make mathematics accessible to more students because they enable the students to analyze what they can see rather than requiring them to develop mental images or manipulate situations symbolically from the outset. Furthermore, technology enables students to calculate rapidly and to investigate conclusions immediately, freeing them from the limitations imposed by cumbersome and timeconsuming computations.

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

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

7. Represent and describe mathematical relationships with tables, rules, simple equations, and graphs.

• Students use calculators to investigate which fractions have decimal equivalents that terminate and which repeat. They summarize their findings in their math journals.

• Students study patterns made by the units digit in the expansion of powers of a number. For example, what is the units digit of 9¹⁸? The pattern 9¹, 9², 9³, . . . yields either a 1 or a 9 in the units place. Students record their findings in a table or as a graph on rectangular coordinate paper. They write a paragraph justifying their answer. They then similarly investigate the patterns made by the units digit in the expansion of the powers of other one-digit numbers.

• Students consider what happens if you start with two bacteria on a kitchen counter and the number of bacteria doubles every hour. They make a table and graph their results, noting that the graph is not linear.

8. Understand and describe the relationships among various representations of patterns and functions.

• Students arrange bowling pins in the shape of equilateral triangles of various sizes, as shown in the diagram below. They make a table showing the number n of rows in each triangle and the number b of bowling pins in each triangle. The numbers in the second column - 1, 3, 6, 10, ... - are called the triangular numbers. They find a rule expressing this relationship p = n(n + 1) /2, by putting two triangles of the same size side by side, counting the total number of bowling pins, and dividing by 2.
  

• Using a 5x5 geoboard or dot paper, students create various sized parallelograms. For each parallelogram, they record the length of the base (b), the height of the parallelogram (h), and the area of the parallelogram (A), found by counting squares. The students look for a relationship among the numbers in the three columns of their table, express this relationship as a verbal rule, and then write the rule in symbolic form.

• Students investigate how many stools with three legs and how many chairs with four legs can be made using 48 legs. They may use objects or draw pictures to make models of the solutions. They look for patterns in the numbers and display their results in a table, as ordered pairs graphed on the rectangular coordinate plane, as a rule like 3s + 4c = 48, and as an equation like s = 16 - 4c/3, which gives the number of stools as a function of the number of chairs. They describe the pattern and how they found it in writing.

• Students create their own designs using iteration. They may use patterns such as spirolaterals or write a program in Logo on the computer. They use 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 and use this to predict other values. Then students use a slightly different equation, y = .1x + .3. Again, starting with an x value of 0 they find the resulting y value of .3. Using this as the new x value gives a value of .33 for y. Repeating this process yields the series of y values .3, .33, .333, ... which get closer and closer to 1/3.

9. Use patterns, relationships, and functions to model situations and to solve problems in mathematics and in other subject areas.

• Students analyze a given series of terms and fill in the missing terms. Patterns include various arithmetic (repeating patterns) and geometric (growing patterns) sequences and other number and picture patterns. Students develop an awareness of the assumptions they are making. For example, given the sequence 0, 10, 20, 30, 40, 50, one might expect 60 to be next; but not on a football field, where the numbers now decrease!

• 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 supply missing fractions between any two given numbers on a number line. They might label each of eight intervals between 1 and 2, or they might label the next 16 intervals from 23 1/2 to 24. They extend this to decimals, labeling each missing number in increments of .1 or .01. For example, students might label each of five intervals between 59.34 and 59.35.

• Students decide how many different double-dip ice cream cones can be made from two flavors, three flavors, and so on up to Baskin and Robbins' 31 flavors. 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 would change their results. They also discuss a similar problem (see Standard 14 and the 5-6 Vignette Pizza Possibilities in the First Four Standards): How many different types of pizzas can be made using different toppings?

• 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. They 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 inwriting 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 ten generations ago and also to the problem of telling a secret to two people who each tell two people, etc.

10. Analyze functional relationships to explain how a change in one quantity results in a change in another.

• Students investigate how increasing the temperature measured in degrees Celsius affects the temperature measured in degrees Fahrenheit and vice versa. They collect data using water, ice, and a burner. They use their data to develop a formula relating Celsius to Fahrenheit, summarize the formula in a sentence, and graph the values they have generated.

• Students investigate how the temperature affects the number of chirps a cricket makes in a minute.

• Students investigate the effect of changing the radius or diameter of a circle upon its circumference by measuring the radius (or diameter) and the circumference of circular objects. They graph the values they have generated, notice that it is close to a straight line, and describe the relationship they have found in a paragraph. Then they develop a symbolic expression that describes that relationship.

• Students investigate the effect on the perimeters of given shapes if each side is doubled or tripled. They summarize their findings.

• Students investigate how the areas of rectangles change as the length is doubled, or the width is doubled, or both are doubled. They discuss their findings.

• Students work on problems like this one from the New Jersey Department of Education's Mathematics Instructional Guide (p. 7-69): Two of the opposite sides of a square are increased by 20% and the other two sides are decreased by 10%. What is the percent of change in the area of the original square to the area of the newly formed rectangle? Explain the process you used to solve the problem.

• Students investigate how the areas of triangles change if the base is kept the same, but the height is repeatedly increased by one unit.

• Students stack a given number of unit cubes in various ways and find the surface areas of the structures they have built. They sketch their figures and discuss which of the figures has the largest surface area and which has the smallest, and justify their conclusions.

• Students make models of cubes using blocks or other manipulatives, and investigate how the volume changes if the length, width, and height are all doubled.

• Using a spreadsheet, students investigate how adding (or subtracting) values to given data can affect the mean, median, mode, or range of the data. They discuss how various other changes to the data would affect the mean, the median, the mode, or the range.

11. Understand and describe the general behavior of functions.

• Students investigate graphs without numbers. For example, they may study a graph that shows how far Olivia has walked on a trip from home to the store and back, where time isshown on the horizontal axis and the distance covered is on the vertical axis. Students tell a story about her trip, noting that where the graph is horizontal, she has stopped for some reason. In addition, their stories account for those parts of the graph that are steeper, by explaining why Olivia is walking faster (e.g., she is running from a dog), and those parts of the graph that are not as steep, by explaining why Olivia is walking slower (e.g., she is going up a hill).

• 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 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.

• Given several nonlinear functions, such as y = x², y = 3x², y = x² + 1, y = x³, or y = 16/x, students create a table of values for each and use graphing calculators to graph them.

12. Use patterns, relationships, and linear functions to model situations in mathematics and in other areas.

• Groups of students pretend that they work for construction companies bidding on a federal project to build a monument. The monument is to be built from marble cubes, with each cube being one cubic foot. The monument is to have a "triangular" shape, with one cube on top, then two cubes in the row below, then three cubes, four cubes, and so on. The monument is to be 100 feet high. The students make a chart and look for a pattern to help them predict how many cubes they will need to buy so that they can include the cost of the cubes in their bid.

• Students look at the Sierpinski triangle as an example of a fractal. Stage 0 is an unshaded triangle. To get Stage 1, 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 2, you repeat this process for each of the unshaded triangles in Stage 1. 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 (3⁹) and twentieth (3¹⁹) stages.
  

• Students use the constant function on the calculator to determine when an item will be on sale for half price. If the price goes down by a constant dollar amount each week, then they record successive prices, such as 95 - 15 = = = . . . (or 15 - - 95 = = = on other calculators). If the price is reduced by a certain percent each week, then they use the constant function on the calculator to obtain successive discounts as percents bymultiplying. For example, if a $95 item is reduced by 10% each week, they key in 95 x .9 = = = . . . (or as .9 x x 95 = = = . . . on other calculators).

• Using a temperature probe and a graphing calculator or computer, students measure the temperature of boiling water in a 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 or 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 use coins to simulate boys (tails) and girls (heads) in a family with five children. They make a list of all of the possible combinations, use patterns to help them organize all of the possibilities, and find the probability that all five children are girls or that exactly three are girls. As a question on a test, they are asked to react to an argument between Pam and Jerry, a couple who want to have four children. Jerry thinks that they will probably end up with two boys and two girls, while Pam thinks that they will probably wind up with an unequal number of boys and girls.

• Students make Ferris wheel models from paper plates, with notches representing the cars. They use the models to make a table showing the height above the ground of a person on a ferris wheel at specified time intervals, determined by the time needed for the 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.

13. Develop, analyze, and explain arithmetic sequences.

• Students use the following chart of postal rate history to make a graph of the increases and then to try to predict what the cost will be to mail a oneounce letter in the year 2001.

  Cost to Mail a One-Ounce Letter Since 1917
  Date Cost 1917 3 cents 1919 2 cents 1932 3 cents 1958 4 cents 1963 5 cents 1968 6 cents 1971 8 cents 1974 10 cents		Date Cost 1975 13 cents 1978 15 cents 1981 18 cents 1981 20 cents 1985 22 cents 1988 25 cents 1991 29 cents 1995 32 cents

• Students describe, analyze, and extend the Fibonacci sequence 1, 1, 2, 3, 5, 8, ... , where each term is the sum of the two preceding terms. They investigate 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 read Isaac Asimov's short story Endlessness and write book reports to convey their reactions.

References

Asimov, Isaac. Endlessness. (in Literature: Bronze, 2nd Ed.) Englewood Cliffs, NJ: Prentice Hall, 1991.

New Jersey Department of Education. Mathematics Instruction Guide. D. Varygiannes, Coord. January, 1996.

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.