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.

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  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.