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South Carolina ETV
1101 George Rogers Boulevard
Columbia, SC 29201-4761
Phone: 803-737-3545

2, 4, 6, 8 . . .What Do We Appreciate-Patterns, Patterns, Patterns (Grade 7-8)

Master Teacher

Brian Kennedy

Time Allotment

Two 50-minute class periods.

Overview

In this lesson students will discover and predict patterns in a numerical sequence. Through a series of exercises including the use of video and the Internet, students will begin with basic number patterns and extend their knowledge into higher math skills. The students will test their predictions of a pattern against teachers who are also learning about patterns by the way of video. They will also be introduced to a famous German Mathematician named Carl Gauss presented in a BBC mini-lesson on video. Afterwards the students will learn and practice various arithmetic progressions on a BBC Web site.

Subject Matter

Pre-algebra

Learning Objectives

Students will be able to:

• Recognize patterns in a numerical sequence;
• Create new patterns and sequences;
• Predict new outcomes to established sequences.

South Carolina Standards

Visit SC Mathematics Standards Web site

Grades 6-8 Algebra

BBC - Numbers Math. This Web site is highly interactive. Students go through a series of screens demonstrating arithmetic progressions.

Annenberg/CPB Web Site. This Web site shows teachers learning how to think algebraically. There is a segment on seeing a variety of patterns and reveals to students that even teachers see different things when they learn math patterns.

Materials

• chalkboard or overhead transparency with Wet Erase Markers
• Activity Sheet 1
• Activity Sheet 2

Equipment

• computers with Internet Accessor Laptop Lab with wireless cards
• Gateway Destination for Large Group Viewing
• overhead and blank transparency
• TV and VCR

Prep for Teachers

Videotape Note: Mathematical Investigations, Lesson 3: "Arithmetic Progressions" is a 1985 British Broadcasting Company Production. The segment is called MI 10 because each segment in the series is suppose to be 10 minutes long, however this segment is only 7 minutes long. There is a PAUSE FOR THOUGHT screen that will appear after a short introduction. When you see this message turn down the audio and press PAUSE for comments and discussion from the students. If you do not PAUSE the tape, you will have to tolerate a loud menacing tone that counts down 10 seconds for "thinking" time, a potential distraction to all students. After the PAUSE for appropriate discussion, press play and increase the volume after the screen has changed back to video.

A word of caution: This is a BBC program and the opening sequence asks students to guess the sequence as it is slowly revealed on the screen. The numbers are actually taken from a London phone directory. The students may think they know a pattern initially but then are befuddled when they see that no one has correctly guessed the sequence!

Web Site Notes: Before you use the Websites, create bookmarks under Add Favorites on your Microsoft Explorer menu or Netscape Navigator. Load the Macromedia plug in "Flash" ( onto each computer as well.

Each lesson in the videotape series challenges students to ask questions or to find out about the principle being discussed.

For the Annenberg/CPB Channel Online Video Web site you will need an up to date Windows Media Player version 7.You can watch the segment from the site if you have a fast network connection or broadband.

Otherwise I would recommend buying the videotape for any teacher with a slow Internet connection. If you choose to buy the videotape, you should visit the Web site. Also, all the videos are broadcast free: For information about broadcast times, please check the Web site.

When using media, provide students with a Focus for Media Interaction, a specific task to complete and/or information to identify during or after viewing of video segments, Web sites or other multimedia elements.

Introductory Activity

Step 1: Using the chalkboard or an overhead transparency, write out a numerical sequence first using natural numbers 1, 2, 3 and leaving 3 blanks afterwards. Have students predict what comes next.

Step 2: Then using even and odd numbers, again leaving 3 blanks, again asking the students which 3 numbers come next.

Step 3: Next list the letters o, t, t, f on the board, again leaving 3 blanks, and ask the students to guess what comes next. The letters that come next are f, s, s, respectively. The sequence is numerical but is actually a code for the ordinal numbers: o-ne, t-wo, t-here, f-our, f-ive, s-ix, s-even, etc.

Learning Activities

Step 1: Tell the students that they are going to be introduced to a famous German mathematician by the name of Carl Gauss (1777-1855).

Say," When he was ten years old, his teacher asked the class to find the sums from 1 to 100. The teacher thought the problem would keep his class busy for a long time, but Carl got the answer almost immediately!"

Step 2: Now tell the students they will see the secret revealed on the videotape and should be prepared to copy the quick answer that Carl Gauss discovered.

Step 3: BEGIN the videotape. The program begins with sequences that are mathematical and quickly there is a PAUSE FOR THOUGHT message, which shows a static screen but clicks away from 10 to 1. (Note to Teacher: Remember, I recommend turning down the sound and pressing the pause button while taking comments from the students.) Ask the students what they think comes next.

Step 4: Hand out Activity Sheet 1. Provide students with a Focus for Media Interaction by telling them to be prepared to write down a math "trick" they will be shown in the upcoming segment.

Press PLAY and turn the sound up after the screen has changed back to motion video. A picture of the famous mathematician Gauss will show up. The answer to the trick is discussed in a series of animated screens showing the algorithm that Gauss created.

Here is a paraphrase of what is reviewed on the videotape. This is how he obtained the answer:

Think of writing the numbers from 1 to 100, but only write the largest and smallest: 1 + 2 + 3 + . . . + 99 + 100. Then write the numbers a second time, under the first list, but this time from largest to smallest:

1 + 2 + 3 + . . . + 99 + 100

100 + 99 + 98 + . . . + 2 + 1

Add the two numbers that are paired vertically, and the sum of 101 results over and over again; it has to, since each time one number increases by one,

the other decreases by one.

1 + 2 + 3 + . . . + 99 + 100

100 + 99 + 98 + . . . + 2 + 1

101 + 101 + 101 + . . . + 101 + 101

You end with the same number (101) a hundred times, so multiply 100 times

101. But since each number was used twice, divide the result by two. Thus, the famous summation formula for summing numbers from 1 to n:

n ( n + 1 ) / 2.

Step 5: Challenge students to develop a method of summing numbers from one to any number. They may start with the numbers one through ten, or twenty, where they can try different methods and then easily verify the answer.

Step 6: The content of this session centers on patterns. Go over the processes of finding, describing, explaining, and predicting with patterns. Tell your students, "Finding, describing, explaining, and using patterns to make predictions are among the most important skills in mathematics. These skills allow users of mathematics to impose order, meaning, and understanding on situations that at first seem like collections of random facts."

Continue saying, " Different people notice different things, so what one person sees is often different from another. That is why it is so important to describe patterns in language that everyone understands-so others can see what you see. Algebra is a tool for describing patterns, and there are many others. It's important to keep in mind, however, that algebra is much more than a language. In fact, 'making sense' is what doing mathematics is all about."

Step 7: Hand out Activity Sheet 2 and tell the students that once they have completed their patterns, they will compare their patterns with those of the teachers who did the same activity on the video.

Step 8: Provide students with a Focus for Media Interaction by telling them: "Watch this video segment and describe in what ways do people see different patterns?"

Culminating Activity

Step 1: Have students take the "Test" on Revise Wise Maths Section on patterns at the BBC Web site.

Step 2: Demonstrate to students the patterns found in music. Go to Number 7: Aesthetics of Prime Sequence. Scroll down to Primes and music and click on the different music files: rain.mid, walking.mid, etc. Provide students with a Focus for Media Interaction by asking them if they detect harmony or a melody in each song.

(Note to Teacher: Explain to students that Midi files are files that play music. You must have audio speakers in order to use this link!)

Assessment

The test the students took on the BBC Web site can serve as the Assessment.