PATTERNS AND POSSIBILITIES
Grades 5-8

This lesson provides students with an understanding of the value and importance of patterns in mathematics. Through video, interaction and hands-on activities, students will identify concrete and abstract patterns incorporating logic and deductive and inductive reasoning. Video is used as a catalyst allowing individuals to employ conjecture to determine potential methods (answers) related to how a pattern is formed.

MATH WORKS
Problem Solving: Simplifying the Problem
#107
Problem Solving: Using Tables
#110
Problem Solving: Looking for a Pattern
#114

MATHEMEDIA
Logical Reasoning #103

Students will be able to:

• find patterns;
• correlate patterns with possible outcomes;
• show relationship of patterns to logical, deductive and in-ductive reasoning;
• establish a mutual relation between patterns and increased probability and pre-diction;
• use conjecture to discover possible methods/answers to how a pattern is formed.

(per class)

• 1 chalkboard and chalk
• 1 black dry-erase marker
• 1 each red, yellow and
• blue dry-erase marker
• 1 box facial tissues

(per group of four)

• 1 roll of clear tape

(per student)

• 1 sheet notebook paper
• 1 pencil
• 1 scissors
• 1 Activity Sheet (grid)

Have students stand and count-off one through six. Use the activity to form groups of six students each. Say, "You are going to participate in a handshake game. The object is to shake the hand of each person in your group. Be sure to shake each individual's hand only once." Allow time for each group to complete the task; monitor to ensure instructions have been followed. Ask, "How many hands were shaken within your group?" Allow students to interact. Say, "Did you include yourself in the count? Why?" Permit students to respond and qualify no. Ask, "What type computation did you use to solve the problem?" Accept all reasonable attempts to solve the problem. Ask, "Can you tell an easier way to find an answer to the question?" Elicit dis-cussion leading to conclusion there is an easier method.

Say, "You are going to see a video that will help you determine an easier method to resolve the question." To give students a specific responsibility while viewing say, "Watch and be prepared to compare the number of handshakers in the video to the number in our class."

MATH WORKS
Problem Solving: Simplifying the Problem #107
BEGIN tape with opening sequence; PAUSE tape on visual of host and student with a look of puzzlement on his face. Audio is, "Of course there are a number of ways to go about solving this problem or any other problem." Ask, "What was the number of participants at the ball?" (fifty) "How did this differ with the number in our class?" Survey students as you evaluate individual perceptions that reveal understanding there were more participants at the ball.

Ask, "Would it be easier if the large group of fifty were divided into smaller groups?" (yes) "How could you figure the total number if divided into small groups?" Allow students to share their beliefs related to the question. To give students a specific responsibility while viewing say, "Watch the next video and decide if your method for determining the total number of handshakers is better than the one on the video." RESUME tape. PAUSE tape with visual of host asking, "Do you know what the answer is?"

Encourage students to compare group sizes on video with size of groups the class was divided into. (class and video groups were same or six individuals each) Ask, "In your opinion, is it easier to solve the problem in a smaller group?" Allow students to confirm yes and qualify their opinion. Ask, "What did the students recognize about the method used to determine the number of handshakers in the groups of six?" Elicit discussion leading to conclusion that a pattern was established; a table was set up to show concrete evidence of the pattern.

Distribute a sheet of notebook paper and a pencil to each student. Say, "Set up a table similar to the one you saw on the video, then use it to prove the number of handshakes that occurred earlier in your group." Move among students and verify accuracy of tables and/or assist individuals as needed. Allow time for the task to be completed. Have a volunteer tell and demonstrate on chalkboard how they set up and used the table to arrive at the answer, fifteen. Encourage remaining students to evaluate accuracy of the chalk-board table as they endorse correctness or tell changes needed to make it correct. Say, "Give close attention to how it was necessary to recognize the pattern of the five totals before determining the answer to number six." Emphasize, "This is a pattern." For reinforcement, have someone again explain the pattern. Ask, "Is there a different pattern in each column?" Allow volunteers to tell there are differences as they are explained or contrasted among columns.

Refer to chalkboard as you explain the first column increases by one, the second column increases by one more, etc. Say, "This is similar yet different from the pattern shown on the video. Who will compare it as similar? Who will contrast the difference?" (sum of the two numbers in preceding column is the base needed to begin calculating the unknown number of handshakes in the next column) Ask, "Now that you understand the pattern, how might it be helpful for calculating the total number of handshakes involving fifty people?" Allow for responses. To give students a specific responsibility while viewing say, "Watch the next video to find out if your belief about how a pattern might be helpful is the same as the video host's."

RESUME tape. STOP tape on visual of the host seated beside the Math Works logo; audio is, "... and since we discovered a pattern, we might be able to find the answer to fifty people." Allow time for students to compare/contrast their beliefs with the video host's. Say, "Work together as a team and calculate the total number of handshakes when fifty people are involved. Decide as a group which of the two patterns you prefer, then use it to solve the problem." Allow time for each group to present and explain their computations. Make a mental note of individuals who need additional help or reteaching of the concepts. Schedule small group or individual assistance at a later time.

Say, "Becoming skillful in recognizing patterns isn't just a school activity. This ability is important as you attempt to solve everyday problems in your life and for many people, solving problems on the job. Use your knowledge about patterns and tables and imagine how these skills might help you solve a crime if you were a law enforcement officer." Encourage students to offer creative ways to use the skills as you list each on the chalkboard. Accept each without challenge. To give students a specific responsibility while viewing say, "Watch the next video of a police officer profiling characteristics of crimes that have been committed. See if your list includes ways patterns are used to help solve crimes as shown on the video."

MATH WORKS
Problem Solving: Using Tables #110
BEGIN tape on visual of officer walking toward a computer; audio is, "...at police headquarters in NYC..." STOP tape on audio, "...this method of interrogation really seems to work." Allow students to tell whether they had previously listed a use of patterns and tables as seen on video. Ask, "What ways of using patterns at the NYC police headquarters were shown?" (Patterns are used to show where more crimes are occurring and to predict when and where the next crime will occur. In addition, crimes are profiled with the information used to compare similar profiles of known criminals.) Ask, "What important qualification necessary to becoming a police officer was discussed in the video?" (ability to identify patterns in tables)

Engage students in a discussion on people patterns emphasizing that everyone has her/his own unique patterns. Have them contemplate, then share what they believe their personal people patterns might include. Ask, "Once you recognize and understand your own personal patterns, how can you use this knowledge to help you in the future?" Elicit discussion drawing conclusion that if you recognize a pattern that often or generally creates a problem for yourself, look for ways the pattern can be changed to benefit you. Ask, "Who has a personal pattern you're willing to share that always causes you trouble?" Allow volunteer to explain. Suggest they include how they could change this pattern so it will benefit them. Allow students to tell patterns they recognize in others. Ask, "How can skill in recognizing patterns be helpful in analyzing the past to predict the future?" Ask, "What are some of my patterns that help you predict how I would react to specific situations?" Allow students to be candid and have fun with patterns they have recognized.

Say, "Understanding patterns is important to all aspects of our lives." To give students a specific responsibility while viewing say, "Watch the next video and be prepared to name the occupation of a man with a great fondness for patterns."

MATH WORKS
Problem Solving: Looking for a Pattern #114
BEGIN tape immediately following opening credits; PAUSE tape after audio, "... he loved all kinds of patterns." Allow students to identify the man's occupation as a floor tile salesman. To give students a specific responsibility while viewing say, "Watch the next video and be able to answer: Exactly what makes a pattern?" RESUME tape; PAUSE tape after a few seconds on audio, "Here's one kind of pattern made up of numbers and shapes." Allow time for students to tell: a pattern is a sequence of shapes and numbers that can be predicted in an orderly way. Engage the class in discussion as you evaluate the level of comprehension related to easily predicting a direction the pattern's sequence is leading.

Write abstract and concrete on the chalkboard. Ask volunteers to contrast the terms. Discuss why people patterns are abstract, while (e.g.) patterns created by floor tiles are concrete. Have students name other abstract patterns (weather, personal habits, etc.) and other concrete patterns (city blocks, waves created by tossing a pebble in a pond, etc.). Ask, "How might you represent abstract patterns in a concrete way?" (charts, graphs, tables, etc.) Say, "Concrete patterns can be found everywhere in our environment if we would take time to observe them. Do you see examples of concrete patterns in the classroom?" (window panes, floor tiles, book shelves, arrangement of seats, etc.) To give students a specific responsibility while viewing, say "In the next video, three different patterns are shown. Watch and be ready to tell what they are." RESUME tape. PAUSE tape after all numbers have appeared in both sequencing patterns. Allow students to name patterns as floor designs, geometric designs and number patterns. Refer to the freeze-frame of numbers on the television monitor, then ask "Are all numbers on the monitor one large pattern?" (No; there are two sets of sequences.) Say, "Observe the top row of numbers. What number pattern do you recognize?" (Each number increases by three.) "Observe the second row of numbers. Is there a pattern?" Allow for observation/discovery, then accept the answer that each number doubles over the previous number.

Use the following or teacher-selected non-patterned numbers, then list in column form on the chalkboard: 4, 24, 64, 44, 84. Say, "Observe, then describe the pattern you identify in the column of numbers." (This task is not possible as there is no logical sequencing.) Allow time for students to discover and tell there is no pattern represented. Ask, "Who will come to the chalkboard and rearrange the numbers to create a pattern?" (A correct sequencing might be 4, 24, 44, 54, 84; left to right, each number increases by 20. An additional correct sequencing might be 84, 64, 44, 24, 4; left to right, each number decreases by 20.) Ask, "Is this an abstract or concrete pattern?" (concrete)

Say, "Patterns can be easily identified at times and more difficult at other times. In the next video, patterns are used to help treasure hunters find a hidden key that will unlock a safe." To give students a specific responsibility while viewing say, "Watch and find out how quickly you can discover the number pattern used in the treasure hunt. In addition, listen for the name of the method used to discover the pattern." RESUME tape. PAUSE tape after the narrator says, "... if it does, you simply carry out the pattern to solve the problem." Ask, "How soon did you discover the pattern used in sequencing the numbers?" Allow students to respond, then tell that each number increased by four. Ask, "What name was given to the method used to find the pattern?" (guess and check) Allow a volunteer to explain the guess and check method. Create a teacher-designed number pattern and write it on the chalk-board. Select a student to use the guess and check method for discovering the pattern as they explain each step verbally. Instruct all students to use the guess and check method to find a pattern for each of the following: 17, 34, 51, 68 (increases by 17); 144, 132, 120, 108 (decreases by 12).

Say, "Not all patterns are easily identified." To give students a specific responsibility while viewing say, "Watch the next video and look for clues to the pattern as I freeze frame the row of numbers." RESUME tape and FREEZE frame on visual showing a row of numbers; audio is, "The numbers seem to get larger and smaller in a random fashion." Provide a dry-erase marker as volunteers go to the television monitor and circle, then explain a clue they believe will lead to identifying the pattern. Use teacher discretion as activity is brought to closure. Use a facial tissue to clean the monitor's screen. RESUME tape. PAUSE tape on visual of students in a basement; audio is, "... and proceed that many steps." Allow time for students to discuss clues they identified that were helpful in discovering the pattern.

To give students a specific responsibility while viewing say, "Again, see how quickly you can find clues to the pattern as I freeze the frame." RESUME tape. PAUSE and freeze frame on visual of pattern diagram fully on monitor's screen. Follow the previous procedure as volunteers circle clues on the screen, then explain why it was chosen as a clue. Clean screen. RESUME tape and PAUSE on visual of two people walking in the dark; audio is, "Keep going Tim, don't stop!" Ask, "Did you find identifying this pattern to be more difficult?" Allow for response. Ask, "In what way was this pattern different from previous patterns?" Allow students to explain.

Write Leonardo Fibonacci on chalkboard. Explain, he designed a sequence of numbers that became known as the Fibonacci Sequence. To give students a specific responsibility while viewing say, "Watch as numbers listed on the next video for the Fibonacci Sequence are shown in the freeze frame mode. Copy the sequence on your notebook paper, then be prepared to explain the pattern you discover." RESUME tape. PAUSE tape and freeze frame on visual of numbers listed for the Fibonacci Sequence. Allow students to duplicate numbers on their notebook paper, then incorporate the guess and check method to discover the pattern of the Fibonacci Sequence. Provide time for volunteers to identify and explain how they solved the problem. To give students a specific responsibility while viewing say, "Watch and compare your explanation to one given on the video." RESUME tape. Allow students to compare explanations of the Fibonacci Sequence as you PAUSE tape on visual of pine cone; audio is, "...rows or scales are numbers in the Fibonacci Sequence."

Say, "In the next video, once again a table is used to identify a pattern." To give students a specific responsibility while viewing say, "Look for the pattern and raise your hand when you've found it." RESUME tape; PAUSE and freeze frame on visual of blue screen background and the empty table prior to numbers being provided; audio is, "Start looking for a pattern; do you see one?" Encourage volunteers to suggest potential answers for the pattern progression, then provide a dry-erase marker so information can be written on the television monitor. Following discussion say, "Watch to check your answers with the ones given on the next video segment." RESUME tape. PAUSE tape after numbers have been added to create the table. Have students compare their answers with those given on the video.

Explain there are ways other than numbers to create patterns. To give students a specific responsibility while viewing say, "As I freeze frame the diagram of a quilt, decide ways the colors might repeat to form an orderly sequence." RESUME tape. PAUSE tape and freeze frame on visual of gray background and the diagram of a quilt; audio is, "Let's see if the colors repeat themselves in an orderly way." Invite three volunteers to go to the monitor and solve the problem. Give a red dry-erase marker to one volunteer, a yellow to another volunteer and a blue marker to the third volunteer. Say, "Work as a team to determine the number of red, yellow and blue pieces needed to create a sequence of color pattern on the quilt." After numbers are determined, students should demonstrate the pattern using their dry-erase markers and the monitor screen. NOTE: The entire quilt is not shown on the screen. Assist as needed to compute the answer. Discuss, then erase the monitor's screen. Say, "Let's watch another video and consider another way to determine how many yellow pieces are needed." To give students a specific responsibility while viewing say, "Raise your hand when you recognize the alternate method for solving the problem." RESUME tape; STOP tape at end of video and prior to closing credits. Allow students to discuss/review use of tables for identifying patterns.

Explain that another video shows how a detective uses logical reasoning related to patterns for determining if a suspect could be responsible for other unsolved crimes. To give students a specific responsibility while viewing say, "Watch the video and be prepared to explain any patterns you discover."

MATHEMEDIA
Logical Reasoning #103
BEGIN tape immediately following opening credits. PAUSE tape and freeze frame after all X's have been added; audio is, "Did the person in custody commit any of the other robberies?" Have students observe, then decide if there are patterns that could be considered as links to any of the seven crimes. To give students a specific responsibility while viewing say, "Watch to see if you identified the correct links." RESUME tape. PAUSE tape on visual of black background with chart in view; audio is, "...with their hands tucked under their shirts." Allow students to validate their identification of logical links.

Say, "The logical reasoning approach utilized patterns to help solve some very important cases. Although you may have been unable to follow every strand of logic used in the video, the fact that patterning can be much more complex and diverse than simply studying numbers and floor tiles was well made." Write inductive reasoning on the chalkboard. Say, "Whenever you search for a pattern that will apply every time, you are using inductive reasoning." Write conjecture on the chalkboard. To give students a specific responsibility while viewing say, "Watch the next video and be prepared to explain the term conjecture." RESUME tape. PAUSE tape on visual of boy pondering the game; audio is the girl, "... maybe you'll figure it out." Allow students to explain conjecture (a conclusion deduced by surmise or guesswork). Ask, "Has anyone figured out the trick? Are you sure you understand rules to the game?" Accept responses. Ask, "Does it matter who goes first in the game?" (yes) To give students a specific responsibility while viewing say, "Watch the video to learn the rules and be ready to explain the method of winning." RESUME tape. STOP tape at end of game; audio is the boy, "... whatever they do, you can always leave them with the last coin." Elicit discussion as students explain rules to the game and the method of winning.

Attach column headings numbered 1-36 to the classroom walls. Divide the class into groups of four students each. Distribute 1 scissors and 1 Activity Sheet (grid) to each student and a roll of clear tape to each group. Assign each group two small and two larger numbers that fall between 1-36.

Say, "Patterns can be found in factors of numbers. The pattern will lead to discovery of whether a number is prime or composite, whether it is a square and what its area and perimeter is." Instruct groups to carefully consider all rectangles that can be made from each assigned number. (e.g.) Use the number 4. 2 x 2 = 4; cut out two squares by two squares, creating a 2 x 2 square. REMIND: A square meets all criteria of a rectangle. Say, "The factor of 4 x 1 is also applicable to the number 4." Cut out a rectangle four squares long by one square wide. Tape both grid models to the wall under the heading numbered four.

Allow time for groups to complete the assigned task. Move around the classroom and assist as needed. Additional grid sheets will be needed by some groups. After all groups have finished, instruct students to list all composite numbers from 1-36 and all prime numbers from 1-36. Initiate discussion as students are encouraged to identify a pattern applicable to prime numbers they listed. (All prime numbers will model as a single long rectangle because their only factor is one and that number.) Ask for identification of the pattern in all composite numbers. (All have the one times that number pattern as with prime numbers; however, they have other factors, sometimes several others.) Ask, "Do you see a pattern with odd and even numbers?" (No. Odd numbers are not necessarily prime numbers.) "Does the size of the number necessarily mean it has more factors than the preceding number?" (No. Difference is determined by whether the number is prime or composite.) Prompt students toward discovery that all square numbers form a square with that factor as you ask, "What other important pattern do you see?" (e.g.) Two squared equals four; three squared is nine; four squared is sixteen, etc. Numbers with a whole number as a square root (thus making them squared) are: 4, 9, 16, 25, 36. Discuss.

Make arrangements for a field trip to a local television station. Request to meet with the meteorologist. Ask her/him to demonstrate how patterns are used to forecast and report weather. Have students pre-determine questions about patterns that help in preparing for hurricanes, tornadoes and other types of extreme weather.

Invite a law enforcement officer to visit your classroom and discuss actual cases where patterns played an integral part in identifying suspects related to a crime. Ask him/her to discuss steps generally taken when investigating a crime or suspected criminal activity.

Science
Have students research genetics and heredity with a focus on Punnet squares, the method of finding trait possibilities concentrating on dominant and recessive genes.

Art
Encourage students to draw on their personal creativity and design a pattern to be used in making camouflage garments for military personnel serving in a jungle environment. Display the designs in your classroom.

Master Teachers: James Parsons and Sharon Braden