|
Core Curriculum Objective(s):
Identify the pattern and predict the missing numeral in an arithmetic and/or geometric sequence including rational numbers and integers; describe number patterns presented in tables. (7PF 1-1)
Demonstrate and represent relationships using materials such as patterns, tables, charts, graphs, and computer spreadsheets to formulate a rule for given pattern. (7PF1-5)
Overview: Students will study number sequences and patterns. They
will learn about Fibonacci's sequence and how these numbers are often found in nature. They will discover patterns in Pascal's Triangle and learn how the probabilities of tossing coins are related to the triangle. They will create number patterns and present them to their classmates. They will determine future terms that can be added to sequences and correctly identify the rule for finding the next term.
Focus/Essential Question(s): What is a number pattern? How can we find the next few numbers in a pattern? What is Fibonacci's Sequence? What is Pascal's Triangle? What are some number patterns which can be found in Pascal's Triangle?
Time Frame: Four one-hour class periods
Resources/Materials:
Pictures of objects which show patterns (tiles, flowers, wallpaper, etc)
Various objects from nature which demonstrate Fibonacci numbers (sea
shells, pine cones, bananas, apples, broccoli, cauliflower
Computer
Overhead
Overhead pens
Paper
Pencil
Overhead display of Fibonacci's Sequence
Real flowers or pictures of flowers, such as lily, iris, buttercup,
Delphinium, ragwort, corn marigold, aster, black-eyed susan
Sharp knife
4 pennies per student
Sticky notes
http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
Culminating Assessment: Students will present an original number pattern with five terms to the class. Classmates will be asked to add three terms to the presented sequence. Once several correct terms have been added to the pattern, presenters will identify the rule to the class. Students will be assessed using the following rubric:
| POINTS |
CRITERIA |
| 0 |
No pattern presented |
| 1 |
Pattern presented with numerous incorrect terms and unable to apply rule |
| 2 |
Pattern presented with no more than 2 incorrect terms and rule applied correctly |
| 3 |
Pattern presented with all terms correct and rule correctly applied |
Instructional Activities:
Activity One:
Display several simple number patterns on the board or overhead and ask students to identify what number comes next.
Some examples are: 1, 4, 7,10 (13), (16) 1,2, 4, 7, 11 (16), (22)
Ask students to explain how they determined the next term in each pattern. (Apply all suggested rules to the patterns and do not label them. Some students may suggest a rule which fits the pattern but is not the one which is being studied.)
Write the following numbers on the board: 1, 4, 9, 16, and 25. Draw dots on the board or overhead to show that these numbers are square numbers. For example, 1 is represented with one dot, 4 is represented by a 2x2 arrangement of dots, 9 is represented by a 3x3 arrangement of dots, and 25 is represented by a 5x5 arrangement. Ask students to supply the next two terms in the sequence and to indicate how they figured out the pattern. Tell the students that these numbers are called square numbers.
Write the numbers 1, 3, 6, and 10 on the board or overhead. Draw three dots with one dot on top and two dots in a row below so that all three form a triangle. Cover the bottom dots and show students that one dot is showing. Uncover the bottom dots and show students that three dots are showing. Add a row of three dots under the first two so that the dots resemble a setup of bowling pins. For example:
After students have counted the total dots, add a row of four dots and then a row of five dots. Tell the students that this pattern of numbers shows triangular numbers. Ask students to add two more terms to the pattern and explain how they found the next two numbers.
Write the following pattern on the overhead or board: 1,1,2,3,5. Allow students to study the pattern to determine the next term in the sequence. Distribute sticky notes to students and have them record their guesses on the sticky note. Select one student to collect the sticky notes. Call on a student to record the votes on the board as you call them out. (At this time, you can conduct a quick statistics review about mean, median, mode, and range.) Add the number 8 to the sequence and call on students to furnish the next term. Once a student has correctly given the next term (13), call on students to again furnish the next term. (21) Ask students to identify the rule (add the two previous numbers together to get the next term). This sequence is called the Fibonacci sequence.
Show the students the following web site: http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
which is called the "Fibonacci Home Site." This site has information about the Fibonacci sequence as well as a link to Fibonacci puzzles.
Activity Two:
Remind students of the Fibonacci sequence discussed in Activity One. Show students a spiral seashell and ask how the shell is related to the Fibonacci sequence. (There are usually knobs on the outside of a spiral shell and when counted will be one of the Fibonacci numbers.) Distribute or display the flowers and/or pictures to the students and ask them to determine how flowers are related to the Fibonacci sequence. Tell students that the number of petal on some flowers are Fibonacci numbers-daisies have 34, 55, or even 89 petals, buttercups have five petals, iris have 3 petals, corn marigolds have 13 petals, and some asters have 21 petals. (Note: You may not always find the Fibonnaci numbers in flower petals, although they usually come close to those numbers.)
Display a pinecone to the students and ask them how pinecones are related to the Fibonnaci sequence? Show students that pinecones are arranged in spirals in two different directions and that when the spirals are counted, they will be a Fibonacci number. Show the students cauliflower and broccoli heads. Again, illicit from students how these vegetables are related to the Fibonacci numbers. Cut off the bottom floret parallel to the main stem. Find the next one up and cut it off the same way. Repeat this with both the broccoli and the cauliflower. Show the students that they florets are arranged in a spiral, like a pinecone. Allow a student to count the cut florets up the stem and you will again come up with a Fibonacci number. Display a banana and ask the students how many faces (flat surfaces) it has, which is a Fibonacci number. Cut the banana in half (as if breaking it in half) and there is another Fibonacci number. Finally, show the students the apple. Cut the apple in half along the "equator" and there is once again another Fibonacci number.
Return to the web site mentioned in Activity One and allow students to work together in groups to complete any of the puzzles not finished previously.
http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
Activity Three:
Display a copy of Pascal's Triangle with four lines completed on the board or overhead. Ask students to work in groups to add two more lines to the triangle. Monitor and assist groups as needed. After several minutes, call on groups to furnish the next row and the second row. Call on a group to explain how the numbers in the next two rows were determined. (add the two numbers above to get the bottom number) The triangle will now look like this:
Allow groups time to complete 11 rows of numbers in the triangles at their desks. Monitor and assist as needed. Call on students, one from each group, to add a row of numbers to the example. Point out to students that each row of the triangle begins and ends with the integer 1. Ask students what the twenty-fifth number in the twenty-fifth row would be. (1) Show students that the second and next to last number in each row is one number less than the row. (5 is the second number in the sixth row.) Ask students what the next to last number in the fiftieth row would be. (49) Ask students to add the numbers together in the first 6 rows and write the sum off to the side of the row. Call on students to give the sums of the each of the first six rows and write them near the triangle. Allow groups time to determine the pattern that the sums of the numbers in each row forms (2 raised to the exponent of that row). Ask each group to share the discovered pattern and to supply the sum of the next three rows based on the pattern. Allow groups time to find the sum of the next three rows and share the results. (You may want to cover up or remove Pascal's Triangle at this time.)
Display a penny and remind students that the chance of flipping the coin and getting heads or tails are equally likely. Display the odds as 1:1. Test this prediction by having each student toss one coin. Count the numbers of heads and tails. Explain to the students that experimental results do not always math the theoretical probability. Now ask groups to list all the possible outcomes of tossing two coins. (They should come up with two tails, two heads, and one head/one tail.) Explain to the students that order do make a difference, so that HT and TH add together to double the odds. The odds for TT HT/TH HH are 1:2:1. Test this prediction by having each student toss two coins. Record the results in a chart on the board and discuss how the experimental results turned out. (Do the experimental results match the theoretical probability?)
Have groups figure out all the possible outcomes for tossing three coins. (They are TTT, TTH, THT, HTT, THH, HTH, and HHH.) Write down the theoretical odds 1:3:3:1 and test the prediction by having students toss three coins. Record the results in a chart and compare the experimental results to the theoretical probability. Allow groups to figure out the possible outcomes for tossing four coins. The theoretical odds are 1:4:6:4:1. (TTTT, TTTH, TTHT, HTTT, TTHH, THTH, THHT, HTTH, HTHT, HHTT, THHH, HTHH, HHTH, HHHT, HHHH) Write the odds in a triangular form as follows:
Ask students where they have seen this pattern before and they should recognize it as Pascal's Triangle.
Activity Four:
Students will work individually to create five sequences with five terms in each sequence. Teacher will monitor and assist as needed. Display the assessment rubric and discuss the scoring with the class. Students will return to their groups and present their sequences to the group. Groups will select one sequence for each student to present to the class. Each student will record his/her sequence on the board or overhead and the other students will be asked to furnish the next three terms in the sequence. Once three terms have been correctly added, presenters will state the rule used to find the next term in the sequence. Students will be assessed using the culminating assessment rubric.
|
