Where do you see patterns? How are they connected to mathematics and science? We will use the series Math Vantage to explore the concepts of patterns used in magic tricks. Participants will have the chance to investigate and create patterns. After viewing a magician's amazing card trick, students will have an opportunity to create and perform "magic" tricks of their own!
"Math Vantage segment: Patterns #101"
Students will be able to:
1 per student
Ask students to, "Predict what the next numbers are in this pattern '2, 4, 6,...'". After students answer, ask them "How did you know the next numbers were 8, 10, 12?" Discuss that the pattern is even numbers and that two was added each time. Say, "Now finish this pattern, 1,3,5,7...". Discuss the students' answers, and what patterns they see. Say, "We've looked at two patterns. Now I would like you and your group to create a pattern for another group to try." Direct students to work in small groups to create number patterns. Have groups exchange patterns, and try to complete each others patterns. Class discussion follows. Ask students, "Why do we use patterns?" Direct the class to brainstorm why we use patterns. One student will record the responses on the overhead while the other students record the responses in their journals.
To give students specific responsibility while viewing, introduce the video with the following directions: "Let's see if you predicted all the functions of patterns. If you see one of the classes' responses during the video, put a star by it. If you see a use for a pattern that you don't have, add it to your journal."
START tape at the beginning when the commentator says, "Ever notice how every morning the sun rises in the east, and sets in the west?"
To underscore the point of why we use patterns, PAUSE the tape when the commentator says "In fact, math has been called the science of patterns. We use patterns to explain, create and predict." Ask the students, "Why do we use patterns? " Class discussion follows. Emphasize the uses of patterns: to explain, create and predict.
Give each student a calculator. Say, "I've been working on my magic skills, and I want to show you a magic trick that I've discovered. Everybody pick a number and write it down. Now I would like you to add 6 to the number you picked. Next, multiply the sum you got when you added your number and six, by 2. Take the answer that you now have and subtract 12 from it." Check to make sure everyone is on task. "Take the new answer and divide it by 2. Next, subtract your original number. Now look at the final answer and don't say anything while I read your minds... (The teacher exaggerates, acts as if she is reading the students' minds) Wait, I'm getting a picture...I see the number zero!"
Wait while the students express their amazement (or skepticism), then say, "Let's try another trick! Pick another number and write it down. Add 10 to your number. Has everyone picked a number and added 10 to it? Now I would like you to multiply the answer that you have by 3. Does everyone have that product? Next step, please subtract 30. Now, divide that answer by 3. Next, subtract your original number. Ok let me get a reading, oh, I am getting a strong reading of ZERO!"
Ask the class, "Do you think I'm really a magician?" Pause while the students discuss the "magic" among themselves. Then say, "Maybe I just know a pattern!" Turn on the overhead and say, "Lets look for a pattern." On the overhead place a rabbit, and say, "Let's let the rabbit represent the number you picked. Next, we added the number 6 to the rabbit, (place 6 carrots on the overhead). Then, we multiplied this sum by 2 . What should we have on the overhead now? (put another rabbit and 6 more carrots on the overhead). We then subtracted 12, which is the same as 6x2 (take the 12 carrots off the overhead) and then, we divided by 2 ("divide by 2", take one rabbit away, leave one rabbit on the overhead). Next, we subtracted a rabbit (take the last rabbit away). What do I have left? Do you see a pattern?" The students discuss the pattern in their groups. A class discussion will follow about patterns that were found in the problem. Say, "That's how I came up with zero, no magic involved!".
Direct students to work in groups to discover a pattern for the second problem, using the worksheet, "Mostly Magic?" "Work in your groups and see if there is a similar pattern for the other "magic trick". Give groups five minutes to find the pattern in the second magic trick using the hand out "Mostly Magic?" Have the class disc uss what patterns their groups found.
To give students specific responsibility while viewing, ask the class, "Has anyone ever seen a magician? Was it really magic? Now we're going to see a magician perform a magic trick. When he performs the magic trick I want you to watch closely to see if it's magic or if the trick is really a pattern."
RESUME the video, so that students can observe the magician and look for patterns. To enhance students' observation skills, talk over the tape when the magician appears and ask, "Do you think he's really a magician?" When the magician starts laying out the cards, talk over the tape and say,"Watch carefully how he lays out the cards." When the arrow points to the ace of hearts, say over the tape, "We're picking the ace of hearts. Notice there are three rows with four cards in each row. How many cards are there? Watch how he picks up the cards!"
PAUSE the tape after the card trick, when the commentator says, "Hmmm I think I see a ma thematical pattern!", to allow time for students to predict a pattern. Ask the students, "Did anyone see a pattern?" Class discussion follows.
REWIND the tape to the beginning of the magic trick to provide time for students to check their theories. Say, "Let's watch the card trick one more time and see if your hypothesis works." Watch the trick again. Remind students to watch how the magician picks up the cards.
PAUSE the tape after the card trick, when the commentator says, "Hmmm I think I see a mathematical pattern!", to give students a chance to test their theories. Give each group a deck of playing cards. Tell students, "You will now have 5 minutes to experiment with playing cards to test your theories and look for a pattern. Work in your group to discover the pattern used in the card trick. Make sure you keep a record of which theories work, as well as which theories don't work."
After students have had a chance to experiment using the playing cards, have grou ps discuss with the class what patterns they found.
RESUME the tape to validate students' ideas and to allow students to see the pattern. Say, "Lets watch the rest of the magic segment to see if you're right."
Continue the tape through the explanation. PAUSE the tape to emphasize the word explain when the speaker says "We see how patterns explain things". Say, "In the card trick the pattern was used to explain the magic!"
STOP the Video.
Working in groups of two, students will create a "magic trick" using a pattern. The students will perform their magic trick for the other groups, and see if the other students can discover their pattern. Students will write a description of the "magic trick" they created, and explain in their description how the pattern works. A book will be made of the classes' different magic tricks, which will be placed in the school library.
Students will take their "magic show" on the road! The class will travel to a nearby elementary school and nursing home, where they will perform (and explain) their magic! Also, students will perform magic tricks at "parent nights", and during the annual Math Fair.
MathematicsHave students create games which use patterns, like the game NIM. Using calculators, have students create a game using patterns. Use a "Hundreds Chart" to look for patterns. Have students create number tricks, and use algebra to solve them. For example, the solution to the first "pick a number" problem could be described as: [2( x+6)-12]/2-x
ScienceInvestigate patterns and shapes in nature. Students will create a news report which uses patterns to describe weather. Investigate patterns found in science, such as DNA. Students can explore patterns found in their finger prints. Students can study the Fibonacci sequence, and discover how rabbits "multiply".
Language Arts Write a story that uses a pattern to solve a mystery. For example, a detective might discover that the numbers of all missing math books are multiples of three.
ArtInvestigate patterns used in Hopi Baskets and Navajo Rugs. Have students create Hopi Baskets and Navajo Rugs using materials s uch as construction paper or yarn. Use a computer to design patterns. Use patterns to create tessellations.
MusicHave students use patterns and items found in the classroom to create a musical composition.
ResourcesThe Nebraska Mathematics and Science Coalition's "Math Vantage" segment: Patterns, contains segments which demonstrate patterns used in music, sports, art, weather and games.Zaslavsky, Claudia. Multicultural Mathematics, J. Weston Walch,Publisher P.O. Box 658, Portland, Maine 04104-0658. Contains lessons on the patterns found in Hopi baskets and Navajo rugs.
Reimer, Wilbert and Luetta. Historical Connections in Mathematics Volumes I and II, 1993 AIMS Education Foundation, P.O. Box 8120 Fresno, CA. 93747-8120. Contains "Number Tricks" and patterns such as square and triangular numbers.
Family Math Contains lessons which use patterns, including the game "Balloon Ride", a game similar to NIM.
MASTER TEACHER: Diane Weaver
Click here to view the worksheet associated with this lesson.
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