Students in grades 7 and 8 should be encouraged to work through a process of analyzing patterns that involves discovering the relevant features of the pattern, constructing understanding of the concepts and relationships involved in the pattern, developing a language that can be used to talk about the pattern, relating the pattern to other patterns which have been studied in the past, and differentiating this new pattern from those previously studied (categorization and classification). 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 situations and for situations from other subject areas that involve quantitative relationships. In particular, students should focus on situations and relationships involving two variables. In these situations, students analyze how a change in one quantity results in a change in another. They further develop their understanding of the general behavior of functions and use these to model situations in mathematics and in other areas.

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

In these grades in particular, students derive much benefit from the use of computers and graphing calculators. These make mathematics available to many more students because they enable students to see with their eyes what is actually occurring, rather than requiring them to imagine it in their minds. These tools also permit students to make calculations rapidly and to investigate conclusions immediately, freeing students from the limitations imposed by weak computational skills.

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

G. represent and describe mathematical relationships with tables, rules, simple equations, and graphs.

• Students use calculators to investigate which fractions have decimal equivalents that are terminating decimals and which are repeating decimals. They summarize their findings in writing 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 for 9^18? The pattern 9^1, 9^2, 9^3, . . .yields either a 1 or a 9 in the one's place. Students record their findings in a table or as a graph on rectangular coordinate paper. They write a paragraph justifying their answer.
• Students investigate growing patterns 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 look for how many stools with three legs and how many chairs with four legs can be made out of 48 legs. They may use objects or draw pictures to make models of the solutions. They look for patterns in the numbers and make a table to organize this data. They state one or more rules: 3s = 48 or 4c = 48, or 3s + 4c = 48. Students display their results in a table, as an equation, and/or as ordered pairs graphed on the rectangular coordinate plane. They describe the pattern and how they found it in writing.
• Students represent triangular numbers (1, 3, 6, 10, ...) as an arrangement of bowling pins using dots. They make a table showing the number of rows in the triangle (n) and the number of dots used for the triangle. They find a rule expressing this relationship, noticing that twice the number of dots is equal to the number of rows times one more than the number of rows. They express this rule symbolically: n x (n + 1)/2.
• Using a 4x4 geoboard or dot paper, students create various sized parallelograms. For each parallelogram, the 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 three columns of their table, express this relationship as a verbal rule, and then write the rule in symbolic form.
• Students create their own designs using iteration. They may use patterns such as spirolaterals or write a program in Logo on the computer. They 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 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 decrease again!
• 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 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 predict what size container is needed to hold pennies if on the first day of a 30 day month you put in 1 penny and double the number of pennies each day. They discuss whether the amount of money this equals will be enough to retire.
• Students consider what happens if you start with two bacteria and the number of bacteria doubles every hour by making a table and graphing their results. They note that the graph is not linear.
• 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 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 investigate how increasing the temperature measured in degrees Celsius affects the temperature measured in degrees Fahrenheit and vice versa. Students 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 relationship between stopping distance and speed of travel in a car. The students gather data from the driver's education manual for discussion in class. Then they use the data to develop a formula, summarize the discovery in a sentence, and graph the values they have found.
• 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 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 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.
• 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.
• Students investigate how adding (or subtracting) values to given data can affect the mean, median, mode or range of data. They discuss what values might be changed or could be added to a given set of values to affect the mean, the median, the mode or range.
• Students make a chart that helps them to understand the charges for a taxi ride when the taxi charges $2.75 for the first 1/4 mile and $.50 for each additional 1/4 mile. They look at rides of different lengths and figure out how much each trip would cost. They write a sentence that explains how they are finding the cost and then translate this into an equation. Then they look at a new rate structure in which $4.25 is charged for the first 1/4 mile and $.20 for each additional 1/8 mile. After making a similar table and equation for this rate structure, the students look for which rides would cost less under the new rate structure and which would cost more.

• Students investigate graphs without numbers. For example, they may study a graph that shows how far Fran has walked on a trip from home to the store and back, where time is shown 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 Fran 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 Fran 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 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 use and create function machines with an input, an output, and a rule. For example, in the table below, the rule is to multiply the number that goes into the machine by 3 and then add 2. The function is expressed as y = 3x + 2, where y is the output and x is the input. In addition to finding the rule, students express the rule as a sequence of two function machines, one which multiplies by 3 and then one which adds 2.

  Input	Output
  3	11
  5	17
  0	2

• Students examine "faulty" input/output machines. For example, if a machine follows the rule that the output is some number less than the input, then you might put in a 7 and have 3 come out. The next time you use the machine, you might put in a 7 again but have a 4 come out! This "faulty" machine is not a function machine. Function machines always give you the same, predictable answer when you put in a given number.
• Given several non-linear functions, such as y=x^2, y = 3x^2, y = x^2 + 1, y=x^3, or y = 16/x, students create a table of values for each and graph them on a coordinate plane.

• Groups of students pretend that they are construction companies bidding on a federal project to build a monument to our former presidents. 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 that will help them to predict how many cubes they will need to buy so that they can figure the cost of the cubes into their bid.
• 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 =, =, =, depending on the type of calculator). 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 by multiplying. For example, if an item is reduced by 10% each week, they key in $95 x .9 =, =, = . . . .(On some calculators, this is entered as $95 x 90% =, =, = . . . or as .9 x x, 95 =, =, =, ...).
• 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.

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• 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 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, using patterns to help them organize all of the possibilities, and find the probability of having all girls or exactly three girls. For assessment, 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 have three girls and one boy.
• 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.