What you should learn
To extend sequences NCTM Curriculm Standards 2, 6 - 10
To extend sequences
NCTM Curriculm Standards 2, 6 - 10
In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:
Sequence Terms Inductive Reasoning Deductive Reasoning
Sequence
Terms
Inductive Reasoning
Deductive Reasoning
Introduction: This lesson is about looking for patterns
Exercise 1: Study the pattern below.
a. Draw the next three figures in the pattern. The pattern consists of squares with one corner shaded. The corner that is shaded is rotated in a clockwise direction. The next three figures are drawn below.
a. Draw the next three figures in the pattern.
The pattern consists of squares with one corner shaded. The corner that is shaded is rotated in a clockwise direction. The next three figures are drawn below.
You may also find a different pattern, such as the seventh square may be the same as the third square, etc. b. Draw the 35th square in the pattern.
You may also find a different pattern, such as the seventh square may be the same as the third square, etc.
b. Draw the 35th square in the pattern.
The numbers 2, 4, 6, 8, 10, and 12 form a pattern called a sequence. A sequence is a set of numbers in a specific order. The numbers in the sequence are called terms.
Exercise 2: Find the next three terms in each sequence.
a. 7, 13, 19, 25, ... Study the pattern in the sequence Each term is 6 more than the term before it.
a. 7, 13, 19, 25, ...
Study the pattern in the sequence Each term is 6 more than the term before it.
Study the pattern in the sequence
Each term is 6 more than the term before it.
The next three terms are 31, 37, and 43. b. 243, 81, 27, 9, ...
The next three terms are 31, 37, and 43.
b. 243, 81, 27, 9, ...
One of the most-used strategies in problem soling is look for a pattern. When using this strategy, you will often need to make a table to organize the information.
Exercise 3: What is the number of diagonals in a 10-sided polygon?
Drawing a 10-sided polygon and all its diagonals would be difficult. Another way to solve the problem is to study the number of diagonals for polygons with fewer sides and then look for a pattern.
Use a chart to see the pattern:
Now use the pattern to complete the above chart.
Activity: Looking for patterns
Materials: String and scissors If you use a pair of scissors to cut a piece of string in the normal way, you will have 3 piecces of string. What happens if you loop the string around one of the cutting edges of the scissors and cut? Your Turn a. Make a piece of string loop once around your scissors as shown above. Cut the string. How many pieces do you have? b. Make 2 loops and cut. How many pieces do you have? c. Continue making loops and cutting until you see a pattern. Describe the pattern and write the sequence. d. How many pieces would you have if you made 20 loops? Explain whether you used inductive or deductive resasoning to determine your answer. e. Now tie the ends of the string together before you loop the string around the scissors. Investigate to determine how many pieces you would have if you made 10 loops with this string.
Materials: String and scissors
If you use a pair of scissors to cut a piece of string in the normal way, you will have 3 piecces of string. What happens if you loop the string around one of the cutting edges of the scissors and cut?
Your Turn
a. Make a piece of string loop once around your scissors as shown above. Cut the string. How many pieces do you have? b. Make 2 loops and cut. How many pieces do you have? c. Continue making loops and cutting until you see a pattern. Describe the pattern and write the sequence. d. How many pieces would you have if you made 20 loops? Explain whether you used inductive or deductive resasoning to determine your answer. e. Now tie the ends of the string together before you loop the string around the scissors. Investigate to determine how many pieces you would have if you made 10 loops with this string.
a. Make a piece of string loop once around your scissors as shown above. Cut the string. How many pieces do you have?
b. Make 2 loops and cut. How many pieces do you have?
c. Continue making loops and cutting until you see a pattern. Describe the pattern and write the sequence.
d. How many pieces would you have if you made 20 loops? Explain whether you used inductive or deductive resasoning to determine your answer.
e. Now tie the ends of the string together before you loop the string around the scissors. Investigate to determine how many pieces you would have if you made 10 loops with this string.
Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.
Homework: The homework to be assigned for tonight would be: 11 - 25 odd, 26, 27, 29, 30 - 35
Alternative Homework: Enriched: 10 - 24 even, 26 - 35
Extra Practice: Students book page 756 Lesson 1-2
Extra Practice Worksheet: Click Here.
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