Meaning and Importance

When solving difficult problems, we frequently suggest to students that they try to solve a simpler problem, observe what happens in a few specific cases (that is, look for a pattern) and proceed from there. This pattern-based thinking, in which patterns are used to analyze and solve problems, is an extremely powerful tool for doing mathematics. Students who are comfortable looking for patterns and then analyzing those patterns to solve problems can also develop understanding of new concepts in the same way. Most of the major principles of algebra and geometry emerge as generalizations of patterns in number and shape. For example, one important idea in geometry is that: For a given perimeter, the figure with the largest possible area that can be constructed is a circle. This idea can easily be discovered by students in the middle grades by examining the pattern that comes from a series of constructions and measurements. Students can be given a length to use as the perimeter of all figures to be created, say 24 centimeters. Then they can construct and measure or compute the areas of a series of regular polygons: an equilateral triangle, a square, and a regular hexagon, octagon, and dodecagon (12 sides). The pattern that clearly emerges is that as the number of sides of the polygon increases (that is, as the polygon becomes more "circular"), the area increases.

All of the content standards have close interconnections, but this is one that is very closely tied to all of the others since an understanding of patterns can be either content or process. When the patterns themselves and their rules for generation are the objects of study, they represent the content being learned. However, when pattern-based thinking, or the search for patterns, is the approach taken to the discovery of some other mathematical principles, then patterning is a process, and the approach can easily be applied to content in numeration, geometry, operations, or the fundamentals of calculus. There is a very special relationship, though, between patterns and algebra. Algebra provides the language in which we communicate the patterns in mathematics. Early on in their mathematical careers, children must begin to make generalizations about patterns that they find and try to express those generalizations in mathematical terms. Examples of the different kinds of expressions they might use are given in the discussion below.

K-12 Development and Emphases

Young students should frequently play games which ask them to follow a sequence of rules (arithmetic or otherwise) or to discover a rule for a given pattern. Sequences which begin as counting patterns soon develop into rules involving arithmetic operations. Kindergartners, for example, will make the transition from 2, 4, 6, 8 ... as a counting by twos pattern to the rule "Add 2" or "+2." The calculator is a very useful tool for making this connection. Through the use of the calculator's constant function feature, any calculator can represent counting up or counting down by any constant amount. Students can be challenged to guess the number that will come up next in the calculator's display and then to explain to the class what the pattern, or rule, is.

At a slightly higher level, input-output activities which require recognition of relationships between one set of numbers (the "IN" values) and a second set (the "OUT" values) provide an early introduction to functions. One of these kinds of activities, the Function Machine games are a favorite among first through fifth graders. In these, one student has a rule in mind to transforms any number which is suggested by another student. The first number inserted into the imaginary Function Machine and another number comes out the other side. The rule might be plus 7, or, times 4 then minus 3, or even the number times itself. The class's job is to discover the rule by an examination of the input-output pairs.

In the middle grades, students begin to work with patterns that can be used to solve problems within the domain of mathematics as well as from the real world. There should also be a more obvious focus on relationships involving two variables. An exploration of the relationship between the number of teams in a round robin tournament and the number of games that must be played, or between a number of coins to be flipped and the number of possible resulting configurations, provides a real-world context for pattern-based thinking and informal work with functions. Graphing software is extremely valuable at this level to help students visualize the relationships they discover.

At the secondary level, students are able to bring more of the tools of algebra to the problem of analyzing and representing patterns and relationships. Thus we expect all students to be able to construct as well as to recognize symbolic representations such as y = f(x) = 4x+1. They should also develop an understanding of the many other representations and applications of functions as well as of a greater variety of functional relationships. Their work should extend to quadratic, polynomial, trigonometric, and exponential functions in addition to the primarily linear functions they worked with in earlier grades. They should be comfortable with the symbol f(x), both as the application of a rule of correspondence and as a "value" corresponding to x, in much the same way that elementary students have to view 3+2 both as a quantity and as a process.

The use of functions in modeling real-life observations also plays a central role in the high school mathematics experience. Line- and curve-fitting as approaches to the explanation of a set of experimental data go a long way toward making mathematics come alive for students. Technology must also play an important role in this process, as students are now able to graphically explore relationships more easily than ever before. Graphing calculators and computers should be made available to all students for use in this type of investigation.

In summary, an important task for every teacher of mathematics is to help students recognize, generalize, and use patterns that exist in number, shape, and the world around them. Students who have such skills are better problem solvers, have a better sense of the uses of mathematics, and are better prepared for work with algebraic functions than those who do not.

"Looking for patterns trains the mind to search out and discover the similarities that bind seemingly unrelated information together in a whole. . . . A child who expects things to 'make sense' looks for the sense in things and from this sen develops understanding. A child who does not see patterns often does not expect things to make sense and sees all events as discrete, separate, and unrelated."
- Mary Baratta-Lorton(1)

Children should construct, recognize and extend patterns with pattern blocks, cubes, toothpicks, beans, buttons and other concrete objects. Children in kindergarten can recognize patterns in motion, color, designs, sound, rhythm, music, position, sizes and quantities. They are very aware of sound and rhythm, and can clap out patterns that repeat, such as clap-clap-clap-pause, clap-clap-clap-pause, etc. They can sit in a circle and wear colored hats which make a pattern, such as red-white-blue, red-white-blue, etc. One child can walk around the circle and tap successive children in an arm-shoulder-head pattern. The teacher may ask the class who the next person to be tapped on the head would be if the pattern were to be continued. In addition to repeating patterns, students should have experiences with growing patterns. They can indicate such a pattern by using motion: skip-jump-turn around, skip-jump-jump-turn around, skip-jump-jump-jump-turn around, and so on. Songs are excellent examples of repetition of melody or of words as well as of rhythmic patterns. Children's literature abounds with stories which rely on rhythm, rhyming, repetition and sequencing. As students move on to first and second grade, they should start to create their own patterns and transform patterns using concrete objects into pictorial and symbolic representations of those patterns. The transition will be from working with patterns using physical objects to using pictures, letters, numbers, and geometric figures in two and three dimensions to using symbols (words and numbers) to represent patterns.

Also important for students in the primary grades are categorization and classification. Students in kindergarten should have numerous opportunities to sort, classify, describe, and order collections of many different types of objects. For example, students might be asked to sort attribute shapes, buttons, or boxes into two groups and explain why they sorted them as they did. This area offers an excellent opportunity for students to integrate learning in mathematics and science as they sort naturally-occurring shapes such as shells or leaves.

Primary-grade students should also use patterns to discover a rule. Kindergartners might look at Anno's Counting House(2) to see if they can figure out the pattern that is used in moving from one set of pages to the next. (The people in this book move, one by one, from one house to another.) Older students might try to find the pattern in a series of numbers, like 1, 3, 5, 7, 9, ...

Often, students will be looking for patterns in input-output situations. For example, they might look at the pictures in Anno's Math Games II(3) to find out what happens to objects as the elves put them into the magic machine. (Sometimes the number of objects doubles, sometimes the objects grow eyes, and sometimes each object turns into a circle.)

Establishing the habit of looking for patterns is exceedingly important in the primary grades. By studying patterns, young children can learn to become better learners of mathematics as well as better problem solvers. In addition, patterns help students to appreciate the beauty of mathematics and to make connections within mathematics and among mathematics and other subject areas.

Experiences will be such that all students in grades K-2:

A. reproduce, extend, create, and describe patterns and sequences using a variety of materials.

• Students make a collage with examples of patterns in nature.
• Students create visual patterns with objects, colors, or shapes using materials such as buttons, macaroni, pattern blocks, links, cubes, attrilinks or attribute blocks, toothpicks, beans, or teddy bear counters. They challenge other students to describe or extend their patterns.
• Students sort objects such as leaves, buttons, animal pictures, and blocks, using categories corresponding to characteristics like number of holes, number of sides, or thickness.
• One child walks around the outside of a circle and taps successive children in a head-shoulder-shoulder-head pattern. The teacher asks who the next person to be tapped on the head would be if the pattern were to be continued. The children sing and act out the song, "Head, shoulders, knees and toes."
• Students describe patterns made from circles, triangles, and squares and select the next shape in the pattern.
• Students make patterns with letters and extend the sequence.
• Students use letters to identify patterns they have created with objects -- for example, RRBRRB for red-red-blue-red-red-blue.
• Students connect the dots to make a picture by following a number sequence, such as 2, 4, 6, 8.
• Students create one more and one less patterns.
• Students create patterns with the calculator. They enter any number such as 10, and then add 1 for 10+1= ,=, =. The calculator will automatically repeat the function and display 11, 12, 13, 14, etc. Some calculators may need to have the pattern entered twice: 10+1=11 +1=, =, =, etc. Other calculators will need 1++, 10=, =, =, etc. Students may repeatedly add (or subtract) any number.
• Students use squares placed in two rows to reproduce odd and even number patterns. They also describe odd and even by using the sum of the dots on dominoes.
• Students name things that come in pairs (or 4s or 5s): eyes, ears, hands, arms, legs, mittens, shoes, bicycle wheels, etc. They work in pairs to find how many people there are if there are 20 eyes.
• Students count by 2, 5, and 10 using counting objects or creating color patterns with Unifix or linker cubes; they repeat this counting on a number line.
• Students use skip counting or calculators to color multiples of numbers on the hundreds chart. Linking cubes or Unifix cubes can be used to build towers or trains with every other cube or every third cube, a certain color to illustrate, recognize and practice skip counting patterns.
• Students write their first name repeatedly on a 10x10 grid then color the first letter of their name to create a pattern. They discuss the patterns formed.
• Students identify the same pattern in a variety of contexts. For example, black-white-black-white is like sit-stand-sit-stand and ABAB and up-down-up-down and straight-curve-straight-curve.
• Students identify patterns on a calendar using pictures or numerals. For example, in November, even dates might be marked with a snowflake, and odd dates with a picture of a turkey. Or, they might identify the day(s) of the week.
• Students describe a pattern made by using various stamp blocks or picture designs.
• Students use or create patterns with geometric figures (circles, triangles, squares, pentagons, hexagons, etc.) and record how many of each shape exist after each repeating cluster.
• Students create a mosaic design (tessellation) made of different shapes using objects such as pattern blocks. They color congruent shapes of a mosaic design with the same color.

• Students fill in a table given several starting numbers and a verbal rule.
• Students describe the pattern illustrated by the numbers in a table by using words (e.g., one more than), and then the teacher helps them to represent it with symbols in an open sentence ([] = * + 1).
• Students use colored squares to make a graph showing the multiples of 3 and relate this to a table and an expression involving a variable, such as 3 x [].
• Students describe how the distance from school changes as they are walking to school and draw a picture that illustrates this situation.

• Students put numbers into Max the Magic Number Machine and read what comes out. (The teacher acts as Max.) Then they describe what Max is doing to each number.
• Students investigate a hole-making machine that puts 4 holes in buttons. They make a table that shows the number of buttons and the number of holes for different numbers of buttons. Then they write a sentence that describes how the total number of holes changes as new buttons are added to the pattern.
• Students play "Guess my Rule." The teacher gives them a starting number and the result after using the rule. She continues giving examples until students discover the rule.
• Students count the number of pennies (or nickels) in 1 dime, 2 dimes, 3 dimes and record their results in chart form. They study the patterns and discuss the "rules" observed.
• Students consider the cost of two or three candies if one candy costs one dime. They make a chart using the information.
• Students count the number of lifesavers in an assorted pack. They make a table showing the number of each color and the total number in one pack. Then they look at the number of each color and the total for two, three, or more packs.

• Students discuss how ice changes to water as it gets hotter. They talk about how it snows in January or February but rains in April or May.
• Students plant seeds and watch them grow. They write about what they see and measure the height of their plants as time passes. They discuss how changes in time bring about changes
• in the height of the plant. They also talk about how other factors might affect the plant, such as light and water.

• Students go on a pattern hunt around the classroom and the school, discussing the patterns they see.
• Students sing and act out songs like "Rattlin' Bog" (Bird on the leaf, and the leaf on the tree, and the tree in the hole, and the hole in the ground, . . .) and "Old MacDonald Had a Farm."
• In reading, students recognize patterns in rhythm, in rhyming, in syllables and in sequencing. Stories such as Ten Black Dots by Donald Crews, Five Little Monkeys Jumping on a Bed by Eileen Christelow, Little Red Hen, Jump, Frog, Jump by Robert Kalan, Six Dinner Sid by Inge Moore, and Dr. Seuss books offer such opportunities. Visual patterns can be shown using picture representations for children's books such as 1 Hunter by Pat Hutchins, Rooster's Off to See the World, The Patchwork Quilt by Valerie Flournoy, and The Keeping Quilt by Patricia Polacco.
• Students identify every third letter of the alphabet; every fourth letter, etc.
• Students choose a day. They identify the name of the next day; of the previous day; and also the name of the day two days (or more) before and after. They select a date, and give the date of the next day and of the previous day; the name of the month, of the next month, and of the previous month. They give the name of the date 2 days before and after, 3 days (or more) before and after.
• Students graph daily weather patterns, showing sunny, cloudy, rainy or snowy days. Then they discuss monthly or seasonal patterns.
• In social studies, students identify traffic patterns such as how many cars, trucks, or buses pass the front of the school during five minutes at different times of the day. They keep records for five days, organizing the information in chart form.
• In art, students observe patterns in pictures, mosaics, tessellations, Escher-like drawings as well as in wallpaper, fabric and floor tile designs. They learn that it is important to match patterns when a part is used for the whole, such as in sewing or wallpapering.

• Students draw pictures of faces and make a table that shows the number of faces and the number of eyes. The teacher writes a sentence on the board that the class composes, describing the patterns that they find.
• Students observe that there are 12 eggs in a carton of eggs. These are called a dozen. They explain how to find the number of eggs in 2 cartons (dozen).
• Students write a sentence or more telling about the patterns they have observed in a particular activity. They may use pictures to describe or generalize what they have observed. For example, after students have colored multiples of a certain number on the hundreds chart, they write about the pattern they observe on the chart.

Students in grades 3-4 continue to construct, recognize, and extend patterns. At these grade levels, pictorial or symbolic representations of patterns are used much more extensively than in grades K-2. In addition to studying patterns observed in the environment, students at these grade levels should use manipulatives to investigate what happens in a pattern as the number of terms is extended or as the beginning number is changed. Students should also study patterns that involve multiplication and division more extensively than in grades K-2. Students can continue to investigate what happens with patterns involving money, measurement, time, and geometric shapes. They should use calculators to explore patterns.

Students in these grades also continue to categorize and classify objects. Now categories can become more complex, however, with students using two (or more) attributes to sort objects. For example, attribute shapes can be described as red, large, red and large, or neither red nor large. Classification of naturally-occurring objects, such as insects or trees, continues to offer an opportunity for linking the study of mathematics and science.

Students in grades 3 and 4 are more successful in playing discover a rule games than younger students and can work with a greater variety of operations. Most students will still be most comfortable, however, with one-step rules, such as "multiplying by 3" or "dividing by 4."

Third and fourth graders also continue to work with input-output situations. While they still enjoy putting these activities in a story setting (such as Max the Magic Math Machine who takes in numbers and hands out numbers according to certain rules), they are also able to consider these situations in more abstract contexts. Students at this age often enjoy playing the machine themselves and making up rules for each other.

In grades 3 and 4, then, students expand their study of patterns to include more complex patterns based on a greater variety of numerical operations and geometric shapes. They also work to organize their study of patterns more carefully and systematically, learning to use tables more effectively. In addition, they begin to apply their understanding of patterns to learning about new mathematics concepts, such as multiplication and division.

Building upon K-2 expectations, experiences in grades 3-4 will be such that all students:

A. reproduce, extend, create, and describe patterns and sequences using a variety of materials.

• Students make a pattern book that shows examples of patterns in the world around them.
• Students use pattern blocks, attribute blocks, cubes, links, buttons, beans, toothpicks, counters, crayons, magic markers, leaves and other objects to create and extend patterns. They might describe a pattern involving the number of holes in buttons, the number of sides in a geometric figure, or the thickness of objects.
• Students use sequences of letters or numbers to identify the patterns they have created.
• Students investigate the sum of the dots on opposite faces of an ordinary die and find they always add up to 7.
• Students count by 2, 3, 4, 5, 6, 10 and 12 on a number line, on a number grid, and on a circle design.
• Students begin with numbers from 50 - 100 and count backwards by 2, 3, 5, or 10.
• Students create patterns with the calculator: They enter any number such as 50, and then add or subtract a 1 (or 2, etc.): for example 50+1 = , =, =. The calculator will automatically repeat the function and display 51, 52, 53, 54, etc. Some calculators may need to have the pattern entered twice: 50+1 = 51+1 =, =, =, etc. Others may need 1++, 50=, =, =, etc.
• Students begin with a number less than 10, double it, and repeat the doubling at least five times. They record the results of each doubling in a table and summarize their observations in a sentence.
• Students supply the missing numbers on a picture of a ruler which has some blanks: using inches, then centimeters. Then they explore how to find the missing numbers between any two given numbers on a number line. They extend this to larger numbers; they might label each of five intervals from 200-300 or each of four intervals from 1,000-2,000.
• Students investigate number patterns using their calculators. For example, they might begin at 30, repeatedly add 6, and record the first 10 answers, making a prediction about what the calculator will show before they hit the equals key. Or they might begin at 90 and repeatedly subtract 9.

• Students complete a table when given one number in each row and a verbal rule that tells the relationship between the two numbers in each row.
• Students describe the pattern illustrated by the numbers in a table by using words (e.g., twice as much as) and then represent it with symbols in an open sentence ([] = 2 x *).
• Students plot the multiples of 3 on a coordinate grid and join them with a line, making a line graph. They relate this to a table and write the rule as an expression involving a variable, such as 3 x [].
• Students make up a story to match a graph that shows the water level in a bathtub at different times of the day.
• Students repeatedly add (or subtract) multiples of 10 to (from) a 3-digit starting number. They describe the pattern orally and write it symbolically.
• Students work in groups to solve problems that involve organizing information in a table and looking for a pattern. For example, "If you have 12 wheels, how many bicycles can you make? How many tricycles? How many bicycles and tricycles together?" Using objects or pictures, children make models and organize the information in a table. They discuss whether they have looked at all of the possibilities systematically and describe in words the patterns they have found. They write about the patterns in their journals and, with some assistance, develop some symbolic notation (e.g., 2 wheels for each bike and 3 wheels for each trike to get 12 wheels all together might become 2 x W + 3 x T = 12).

• Students use buttons with two or four holes and describe how the total number of holes depends upon the number of buttons.
• Students use multilink cubes or base ten blocks to build rectangular solids. They count how many cubes tall their structure is, how many cubes long it is, and how many cubes wide it is. Then they count the total number of cubes in their structure. They record all of this information in a table and look for patterns.
• Students take turns putting numbers into Max the Magic Math Machine, reading what comes out, and finding the rule that tells what Max is doing to each number. A student acts as Max each time.
• Students play "Guess my Rule." The teacher gives them a starting number and the result after using the rule. She continues giving examples until students discover the rule.

• Students use cubes to build a one-story "house" and count the number of cubes used. They add a story and observe how the total number of cubes used changes. They explain how changing the number of stories changes the number of cubes used to build the house.
• Students measure the temperature of a cup of water with ice cubes in it every fifteen minutes over the course of a day. They record their results (time passed and temperature) in a table and plot this information on a coordinate grid to make a line graph. They discuss how the temperature changes over time and why.
• Students plant seeds in vermiculite and in soil. They observe the plants as they grow, measuring their height each week and recording their data in tables. They examine not only how the height of each plant changes as time passes but also whether the seeds in vermiculite or soil grow faster.

• Students go on a scavenger hunt for patterns around the classroom and the school. They are given a list of verbal descriptions of specific patterns to look for, such as "a pattern using squares" or "an ABAB pattern." They use cameras to make photographs of the patterns that they find.
• Students learn about the different time zones across the country. They describe the number patterns found in moving from the east to west, and vice versa.
• Students read books such as Six Dinner Sid by Inge Moore, The Greedy Triangle by Marilyn Burns, Counting on Frank, or Gulliver's Travels by Jonathan Swift. They explore the patterns and relationships found in these books.
• Students study patterns in television programming. For example, they might look at the number of commercials on TV in an hour or how many cartoon shows are on at different times of the day. They discuss the patterns that they find as well as possible reasons for those patterns.
• Students examine populations (people, plants, or animals) within the school and community.
• In art, students observe patterns in pictures, mosaics, tessellations, Escher-like drawings, wallpaper, fabric and floor tile designs. They create their own potato-print patterns.

• Students measure the length of one side of a square in inches. They find the perimeter of one square, two squares (not joined), three squares, and so on. They make a table of values and describe a rule which relates the perimeter to the number of squares. They predict the perimeter of ten squares.
• Students use their calculators to find the answers to a number of problems in which they multiply a two-digit number by 10, 100, or 1000. Looking at their answers, they develop a "rule" that they think will help them do this type of multiplication without the calculator. They test their rule on some new problems and check whether their rule works by multiplying the numbers on the calculator.

Patterns, relationships and functions will become a powerful problem solving strategy. In many routine problem solving activities, the student is taught an algorithm which will lead to a solution. Thus, when faced with a problem of the same type they just use that algorithm to get the solution. In the real world, however, real problems are not usually packaged as nicely as textbook problems. The information is vague or fuzzy, and some of the information needed to solve the problem might be missing, or there might be extraneous information on hand. In fact, an algorithm might not exist to solve the problem. These problems are generally referred to as non-routine problems.

When students are faced with non-routine problems and have no algorithms upon which to draw, many simply give up because they do not know how to get started. The ability to discover and analyze patterns becomes an important tool to help students move forward. When the students start to collect data and look for a pattern in order to solve a problem, they are often uncertain about what they are looking for. As they organize their information into charts or tables and start to analyze their data, sometimes, almost like magic, patterns begin to appear, and students can use this knowledge to solve the problem.

Patterns help students develop an understanding of mathematics. Whenever possible, students in grades 5 and 6 should be encouraged to use manipulatives to create, explore, discover, analyze, extend and generalize patterns as they encounter new topics throughout mathematics. By dealing with more sophisticated patterns in numerical form, they begin to lay a foundation for more abstract algebraic concepts. Looking for patterns helps students to tie together concepts, gain a greater conceptual understanding of the world of mathematics and become better problem solvers.

Students in the middle grades also continue to work with categorization and classification, although the emphasis on these activities is much less. Now, the categorization and classification will often be applied to new mathematical topics. For example, as students learn about fractions and mixed numbers, they must identify fractions as being less than one or more than one, as being in lowest terms or not. They also apply these skills in geometry, as they distinguish between different types of geometrical figures and learn more about the properties of these figures.

Students in grades 5 and 6 should begin using letters to represent variables as they do activities in which they are asked to discover a rule. Students should also begin working with rules that involve more than one operation. Students at this level describe patterns that they see by looking at diagrams and pictorial representations of a mathematical relationship; some students will be more comfortable using manipulatives first and then moving to a pictorial representation. Students should record their findings in words, in tables, and in symbolic equations.

Students in grades 5 and 6 should begin thinking of input/output situations as functions. They should recognize that a function machine takes a number (or shape) in and operates with a consistent rule with a predictable outcome. They should begin to use letters to represent the number going in and the number coming out, although considerable assistance from the teacher will often be needed.

Throughout their work with patterns, students in grades 5 and 6 should use the calculator as a powerful tool which greatly facilitates computation and promotes higher level thinking. Teachers should explore its capabilities with the students and encourage its use as a tool, so that students become proficient in its use for problem solving.

Building upon the K-4 expectations, experiences in grades 5-6 will be such that all students:

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

• Students let the length of one side of a square be one unit. They then find the perimeter of one square, two squares connected along an edge, three squares connected along their edges, and so forth, as shown below.

       [Graphic Not Available]
  

  They make a table of values and use it to determine a function rule which describes the pattern. They understand the rule P = 2 x s + 2, and they predict the perimeter of ten squares.
• Students use a geoboard to model squares with sides of 1, 2, 3, and 4 units. They determine the area and discover the rule for the area, A = s^2, given the length of the side of a square. They repeat this for rectangles of varying sizes, recording the length and width and corresponding area in a table. Students discover the pattern and develop the formula for the area of a rectangle, A = l x w, by inspecting the numbers in the table.
• Students use multilink cubes or base ten blocks to build rectangular solids. They find the volume of the rectangular solids (either by counting the cubes or by developing their own shortcuts), record the length, width, and height of each solid along with its volume in a table, and use this information to find a rule (formula) for finding the volume of a rectangular solid.
• Students find the number of primes between 1 and 100 using a hundreds chart and applying the process of the Sieve of Eratosthenes (i.e., first crossing out all multiples of 2, then all multiples of 3, etc.).
• Students play the Secret Number calculator game. One student enters a secret number into the calculator by dividing the number by itself (e.g., 17/17). She then asks her opponent to guess the number. Each guess is entered into the calculator and then the equals sign is pressed; the calculator shows the result of dividing the guess by the secret number. For example, if 17 is the secret number and 34 is guessed, then the student enters 34 = and sees 2 on the calculator. Play continues until 1 is shown on the calculator, so that the opponent has guessed the secret number. (Some calculators will need to have different keys pressed for the same result, such as //17 , 34 =, etc.)
• Students look for the numbers which are palindromes (have the same value if the digits are reversed) between any given pair of numbers. They decide which years in the 21st century will be the first 5 palindromes.
• Students look at how much money is earned hourly for a job mowing lawns or babysitting. They find the amount earned for working different numbers of hours. They organize the data in chart or table form. They look for a pattern and write simple equations; "for babysitting or mowing lawns, I get $5 per hour" translates into an equation of E = 5 x h (Earnings equal five times the number of hours worked).
• Students use patterns to help them find the value of a point on a number line between two whole numbers when the number line is divided into fractional or decimal parts.
• Students investigate patterns involving arithmetic operations that can be generalized to a mathematical expression with a variable. For example:
  

       1.  Choose any number             *                                  n
       2.  Multiply your number by 6     * * * * * *                        6 x n
       3.  Add 12 to the result          * * * * * *  || || || || || ||     6 x n + 12
       4.  Take half                     * * *  || || ||                    3 x n + 6
       5.  Subtract 6                    * * *                              3 x n
       6.  Divide by 3                   *                                  n
       7.  Write your answer.
  

• Students use lima beans or other counters to create trinumbers as shown below. They try to predict what the tenth trinumber will be and then use this result to develop an expression for the nth trinumber (3 x n).
  

• Using a 4x4 geoboard or dot paper, students create various sized squares. For each square, they record the length of the side and the area in a table. Then they show their results as a bar graph (side vs. area), as ordered pairs (side, area), as a verbal rule ("the area of a square is the length of its side times itself"), and as an equation (A = s x s).
• Students determine the number of collisions possible between two, three and four bumper cars at an amusement park and record the information in a table. They investigate the number of possible handshakes between 2, 3, and 4 people, and record the information in a table. They discuss the number of diagonals possible in a triangle, 4-sided figure (a quadrilateral), 5-sided figure (pentagon), and discover that the information is similar. They discuss what other problems might be related; such as, playing games with teams if each team plays every other team only once, or connecting telephone lines to houses, etc. They work toward understanding that all of the problems can be solved in the same way and that there is a formula that can be used to find the answer. For example, each of four people shakes hands with the other three (4x3), but this count is twice as big as it should be since that counts John shaking with Mary and Mary shaking with John as two different handshakes when it is really only one. Therefore, the formula is h = (4x3)/2, or, in the general case, n(n-1)/2.
• Students cut out squares from graph paper, recording the length of the side of the square and the number of squares around the border of the square. They look for a pattern that will allow them to predict the number of squares in a border of a 10 x 10 square and then a 100 x 100 square. They describe their pattern in words. The teacher then helps them to develop a formula for finding the number of squares in the border of an n x n square
  (4 x n - 4).
• Students explore patterns involving the sums of the odd integers (1, 1 + 3, 1 + 3 + 5, ...) by using squares to make Ls to represent each number and nesting the Ls.
  

  Then they make a table that shows how many Ls are nested and the total number of squares used. They look for a pattern that will help them predict how many squares will be needed if 10 Ls are nested (i.e., if the first 10 odd numbers are added together). They make a prediction and describe how they found their prediction (e.g., when you added the first 3 odd numbers, it made a square that was three units on a side, so when you add the first 10 odd numbers, it should make a square that is ten units on a side and you will need 100 squares). They share their solution strategies with each other and develop one (or more) formulas that can be used to find the sum of the first n odd numbers (e.g., n x n).

• Students use calculators to study the patterns of the digits in the decimal expansions of repeating fractions, predicting what the digit is in any given place.
• Students look at what happens when a ball is hit at a 45 degree angle on a rectangular pool table with pockets in each corner. They make a table that shows the dimensions of the table (using integers) and which corner the ball eventually goes into. They look for patterns which will help them predict which corner the ball will go into for a given table.
• Students work in groups to decide how to make a supermarket display of boxes of SuperCrunch cereal. The boss wants the boxes to be in a triangular display 20 rows high. Each box is 12 inches high and 8 inches wide. The students use patterns to help them decide how many boxes they would need and whether this is a practical way to display the cereal.
• Students look at the fee structure for crossing the Walt Whitman bridge into Pennsylvania and use patterns to help them decide whether it makes sense for someone who works in Pennsylvania 12 days a month to buy a commuter sticker.

• 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 there will be enough money to retire.
• Students consider what happens if they start with two bacteria and the number of bacteria doubles every hour. They make a table showing the number of hours that have passed and the number of bacteria and then plot their results on a coordinate graph.
• 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. Then they write a sentence that explains how they are finding the cost. 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 for this rate structure, the students look for which rides would cost less under the new rate structure and which would cost more.

• 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.

  Input	Output
  3	11
  5	17
  0	2

• 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.
• 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 and as increasing, decreasing, or mixed.

• Students start with a single triangle with side of length 1 and find its perimeter. Then they add a second triangle, matching sides exactly. to make a train or a wall and find its perimeter. They continue adding triangles and finding the perimeters. The students first try to predict the perimeter of ten triangles in a wall and then look for a function rule which describes the pattern (e.g., if n is the number of triangles, then P = n + 2).
• Students look for as many different ways to make change for 50 cents as they can find. They make a table showing their results listed in an organized fashion and explain why they think they have found all of the possibilities.
• Students investigate what happens when you do arithmetic on a 12-hour clock. They find that 3 + 11 = 2 and that 4 - 6 = 10. They understand that 5, 17, and 29 all wind up at 5 and connect this to the remainder obtained when dividing these numbers by 12.
• Students develop a patchwork quilt design using squares and isosceles right triangles to make a 12 inch by 12 inch block. They use patterns to help them decide how many pieces of each size they need to make in order to complete a 3 foot by 5 foot quilt.
• Students use a table or graph to help them find out who will win a 100-meter race between Pat and his older sister, Terry. Pat runs at an average of 3 meters per second, while Terry runs at an average of 5 meters per second. Since Pat is slower, he gets a 40-meter head start. The students explain how they solved the problem in their math journals.

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 918? The pattern 91, 92, 93, . . .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=x2, y = 3x2, y = x2 + 1, y=x3, 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 (39) and twentieth (319) stages.

       [Graphic Not Available]
  

• 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.

Students in high school construct, recognize, and extend patterns as they encounter new areas of the mathematics curriculum. For example, students in algebra look at the patterns that they find when multiplying binomials and students in geometry look for the patterns that they find in similar triangles. Students in high school should also analyze a variety of different types of sequences, including both arithmetic and geometric ones.

High school students also continue to categorize and classify objects, especially in the context of learning new mathematics. In studying geometry, they classify objects in circles as chords or secants or tangents, for example. In studying algebra, it is critical that students differentiate linear relationships from non-linear relationships.

The function concept is one of the most fundamental unifying ideas of modern mathematics and yet is also one of the most misunderstood concepts at virtually all grade levels. Students begin their study of functions in the primary grades, as they observe and study patterns in nature and create patterns using concrete models. As students grow and their ability to abstract matures, students investigate patterns using concrete models, and then abstract them to form rules, display information in a table or chart, and write equations which express the relationships they have observed. In high school, students move to looking at functions as a natural outcome of the earlier discussion of patterns and relationships. Concepts such as domain and range is formalized and the f(x) notation is introduced as natural extensions of initial informal experiences.

Perhaps one of the difficulties with the function concept is its many possible meanings. The formal ordered-pair definition of a function, while perhaps the most familiar to many teachers, is also the least understood and possibly most abstract way of approaching functions (Wagner & Parker, 1993). Looking at functions as correspondences between two sets seems to be more easily grasped while facilitating the introduction of the concepts of domain and range. Graphs (the good old vertical line test) provide an extremely accessible way of representing functions, especially when graphing calculators and computers are used. Students entering high school should already have encountered functions as a process in the guise of function machines and functions expressed as formulas involving dependent variables, although the correspondence between these two meanings may not be very clear to them. Students need additional experiences in discovering a rule to help them better understand the notion of a dependent variable. Those students moving on to calculus also need to view functions as objects of study in themselves.

High school students should spend considerable time in analyzing relationships among two variables. Beginning with concrete situations (perhaps involving science concepts), students should collect and graph data (perhaps using graphing calculators or computers), discover the relationship between the two variables, and express this relationship symbolically. Students need to have experiences with situations involving linear, quadratic, polynomial, trigonometric, exponential, and rational functions as well as piecewise-defined functions and relationships that are not functions at all.

High school students should further use functions extensively in solving problems. They should frequently be asked to analyze a real-world situation by using patterns and functions. They should extend their understanding of relationships among two variables to using functions with several dependent variables in mathematical modeling.

Throughout high school, students continue to work with patterns by collecting and organizing data in tables, by graphing the relationships among variables, and by discovering and describing these relationships in written and symbolic form.

Wagner, S., & Parker, S. "Advancing Algebra" in P. Wilson (Ed.), Research Ideas for the Classroom: High School Mathematics. New York: Macmillan Publishing Company, 1993.

Building upon the K-8 expectations, experiences in grades 9-12 will be such that all students:

M. analyze and describe how a change in the independent variable can produce a change in a dependent variable.

• 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 graph the values they have generated, note that the relationship is linear, and find an equation that fits those values, using a graphing calculator to check how well their equation fits the data.
• Students investigate the relationship between stopping distance and speed of travel in a car. The students gather data from the driver's education manual, graph the values they have found, note that the relationship is linear, and look for an equation that fits the data.
• Students investigate the effect of changing the radius of a circle upon its circumference by measuring the radius and the circumference of circular objects. They graph the values they have generated, notice that it is close to a straight line, and develop an equation 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 in writing and symbolically.
• Students investigate how the area of a parallelogram changes as the length of the base is doubled, or the height is doubled, or both are doubled. They repeat the experiment for tripling and quadrupling each measurement. They discuss their findings and represent them symbolically.
• Students make models of cubes using blocks or other manipulatives, and investigate how the volume changes if the length, width, and height are all multiplied by the same constant.
• Students compare two fare structures for taxis: one in which the taxi charges $2.75 for the first 1/4 mile and $.50 for each additional 1/4 mile and one in which $4.25 is charged for the first 1/4 mile and $.20 for each additional 1/8 mile. They develop tables, graph specific points, and generate equations to describe each situation. They find which trips cost more for each system and when both fare structures will cost the same amount.

• Students model population growth, decline and decay of people, animals, bacteria and radioactive materials, using the appropriate exponential functions.
• Students use a sound probe and a graphing calculator or computer to collect data on sound waves or voice patterns, noting that these are trigonometric (wave or periodic) functions.
• Students use M&Ms to model decay. They spill a package of M&Ms on a paper plate and remove those with the M showing, recording the number of M&Ms removed. They put the remaining M&Ms in a cup, shake, and repeat the process until all of the M&Ms are gone. They plot the trial number versus the number of M&Ms removed and note that the graph represents an exponential function. Some of the students try out different equations until they find one that they think fits pretty well. They verify their results using a graphing calculator.
• Students use a graphing calculator, together with a light probe, to develop the relationship between brightness of a light and distance from it. They do this by collecting data with the probe on the brightness of a light bulb at increasing distances and analyzing the graph generated on the calculator.
• Students learn about the Richter Scale for measuring earthquakes, focussing on its representation as an exponential function.
• Students work in groups to investigate what size square to cut from each corner of a rectangular piece of cardboard in order to make the largest possible open-top box. They make models, record the size of the square and the volume for each model, and plot the points on a graph. They note that the relationship is not linear and make a conjecture about the maximum volume, based on the graph. The students also generate a symbolic expression describing this situation and check to see if it matches their data by using a graphing calculator.

• Different groups of students work on problems with different settings but identical structures. For example, one group determines the number of collisions possible between two, three and four bumper cars at an amusement park and develops an equation to represent the number of collisions among n bumper cars (ignoring the limitations of physical reality). Another group investigates the number of possible handshakes between 2, 3, and 4 people, and develops an equation to represent the number of handshakes for n people. A third group discusses the number of diagonals possible in a triangle, a quadrilateral, and a pentagon, and develops an equation that gives the number of diagonals for an n-sided plane figure. A fourth group looks at the number of games required for a tournament if each team plays every other team only once, while a fifth considers connecting telephone lines to houses. Each group presents its problem, its approach to solving the problem, and its solution. Then the teacher leads the class in a discussion of the similarities and differences among the problems. Students note that all of the groups used a similar approach and came up with the general equation n(n-1)/2.
• Students investigate a number of situations involving the equation y = 2x. They look at how much money you might earn by starting out with a penny on the first day and doubling the amount on each successive day. They discuss what happens if you start with two bacteria and the number of bacteria doubles every half hour. They consider the total number of pizzas possible as more and more toppings are added. They consider the number of subsets for a given set. They fold a sheet of paper repeatedly in half and look at how many sections are created after each fold.
• Students look for connections among problem situations involving temperature in Celsius and Fahrenheit, the relationship of the circumference of a circle to its diameter, the relationship between stopping distance and car speed, money earned for hours worked, the relationship between distance and time if the rate is kept constant, and profit with respect to price per ticket and ticket sales.

• Students look at a graph that shows the height of a flag on its pole at various times and explain what was happening to the flag. They use the computer program Interpreting Graphs to match up other graphs with appropriate problem situations.
• Students make up graphs to represent specific problem situations, such as the cost of pencils that sell at two for a dime, the temperature of an oven as a function of the length of time since it was turned on, the height of your head from the ground as you ride a ferris wheel as a function of the amount of time since you got on, the time it takes to travel 100 miles as a function of average speed, the cost of mailing a first-class letter based on its weight in ounces.
• Students can use a 30 inch string (constant length) and list all possible lengths and widths of rectangles with integral sides which have this size perimeter. They determine the perimeter and area for each rectangle. Then they make three graphs from their data: length vs. width, length vs. perimeter, and length vs. area. They look for equations to match each graph, determine an appropriate range of values for each variable, and then graph the functions using graphing calculators or computers. The maximum area, a square, does not have integral values but can be found using the trace function or algebraic procedures. Students also investigate what happens if a circle is made with the string rather than a rectangle.
• Students determine the maximum volume of an open box with various sizes of uniform squares (by half inch intervals from 1/2 to 4 inches) cut from the corners of a 9" x 12" rectangular piece of paper. They begin by actually cutting and folding the paper into a box. Once the data and/or the equation is graphed, students trace the function to find the maximum. They also look for boxes with specific volumes, such as 40 cubic inches, finding two dramatically different boxes with this volume.
• Students take on the role of "forensic mathematicians," trying to determine how tall a person was whose femur is 17 inches long. They measure their own femurs and their heights, entering this data into a graphing calculator or computer and creating a scatterplot. They note that the data are approximately linear, so they use the built-in linear regression procedures to find the line of best fit.

• Students look at the effects of changing the coefficients of a quadratic equation on the graph. For example, how is the graph of y = 4x2 different from that of y = x2? How is y = .2x2 different from y = x2? How are y = x2 + 4, y = x2 - 4, y = x2 - 4x, and y = x2 -4x + 4 each different from y = x2? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.
• Students investigate the characteristics of the linear functions. For example, in y = kx, how does a change in "k" affect the graph? In y = mx + b, what does "b" do? Does "k" in the first equation serve the same purpose as "m" in the second? Students use the graphing calculator to investigate and verify their conclusions.
• Students investigate the effects of a dilation and/or a horizontal or vertical shift on the coefficients of a quadratic function. For example, how does moving the graph up 3 units affect the equation?
• Students study the behavior of functions of the form y = axn. They investigate the effect of "a" on the curve and the characteristics of the graph when n is even or odd. They use the graphing calculator to assist them and write a sentence summarizing their discoveries.

• Students in all mathematics classes use functions, making explicit connections to what they have previously learned about functions. As students encounter a new use or meaning for functions, they relate it to their previous understanding.
• Students use recursive definitions of functions in both geometry and algebra. For example, they define n! recursively as n! = n (n-1)! They use recursion to generate fractals in studying geometry. They may use patterns such as spirolaterals, the Koch Snowflake, the Monkey's Tree curve, the Chaos Game, or the Sierpinski Triangle. They may use Logo or other computer programs to iterate patterns, or they may use the graphing calculator. In studying algebra, students consider the equation y = .1x + .6, starting with an x value of .6, and find the resulting y-value. Using this y-value as the new x-value, they then calculate its corresponding y-value, and so on. (The resulting values are .6, .66, .666, .6666, etc.--an approximation to the decimal value of 2/3!) Students investigate using other starting values for the same function; the results are surprising! They use other equations to repeat the procedure. They graph the results and investigate the behavior of the resulting functions, using a calculator to reduce the computational burden.