Richland Parish School Board



Date ___________

Name ________________________

Extending Patterns and Sequences

When presented with a sequence and asked to find the next term, inductive reasoning is applied. Analyze the specific examples provided, determine a pattern, and then find the missing term. Making a prediction about missing terms is called making a conjecture.

Examples: For each of the following, write the next two terms and describe the pattern.

1) 2, 4, 6, 8, 10, … _____, _____ 2) -1, 0, 1, 2, 3, … _____, _____

3) 4, 7, 10, 13, 16, … _____, _____ 4) 1, 4, 9, 16, 25, … _____, _____

5) 1, 3, 6, 10, 15, … _____. _____ 6) 1, 3, 7, 15, 31, 63, … _____, _____

7) 1, 1, 2, 3, 5, 8, … _____, _____ 8) 3, 5, 9, 15, 23, … _____, _____

Inductive reasoning can also be used to find missing terms in sequences and patterns dealing with pictures. Draw the next two figures for each of the following and describe the pattern.

9) 10)

11)

Date ___________

Name ________________________

Extending Patterns and Sequences

When presented with a sequence and asked to find the next term, inductive reasoning is applied. Analyze the specific examples provided, determine a pattern, and then find the missing term. Making a prediction about missing terms is called making a conjecture.

Examples: For each of the following, write the next two terms and describe the pattern.

1) 2, 4, 6, 8, 10, … __12_, _14__ 2) -1, 0, 1, 2, 3, … __4__, __5__

even numbers or +2 add 1 to each

3) 4, 7, 10, 13, 16, … _19_, _22_ 4) 1, 4, 9, 16, 25, … __36_, __49_

add 3 perfect squares

5) 1, 3, 6, 10, 15, … __21_. _28__ 6) 1, 3, 7, 15, 31, 63, … _127_, _255_

add 2, then 3, then 4, etc. add 2, then 4, then 8, then 16, etc.

7) 1, 1, 2, 3, 5, 8, … __13_, _21__ 8) 3, 5, 9, 15, 23, … _33__, _45__

add the preceding two terms add 2, then 4, then 6, then 8, etc.

Fibonacci Sequence

Inductive reasoning can also be used to find missing terms in sequences and patterns dealing with pictures. Draw the next two figures for each of the following and describe the pattern.

9) 10)

The student should draw a shaded triangle, The student should draw two

then an unshaded square. shaded pentagons.

11)

The student should draw a circle with an inscribed pentagon. The points on the circles increase by one in each picture, which are connected to make polygons.

“Tis Linear or Not linear; That is the Question”

Directions: Decide whether each of the given rules, sequences, or tables represents a linear or non-linear pattern. Place a check under the column which corresponds to your decision. Be prepared to explain why you made your decision.

|Is the given pattern |Linear |Non-linear |

|1) [pic] | | |

|2) [pic] | | |

|3) [pic] | | |

|4) |x |1 |

|6) [pic] | | |

|7) |x |1 |

“Tis Linear or Not linear; That is the Question”

Directions: Decide whether each of the given rules, sequences, or tables represents a linear or non-linear pattern. Place a check under the column which corresponds to your decision. Be prepared to explain why you made your decision.

|Is the given pattern |Linear |Non-linear |

|1) [pic] |( | |

|2) [pic] | |( |

|3) [pic] |( | |

|4) |x |1 |

|6) [pic] | |( |

|7) |x |1 |

Linear versus Non-linear Relationships

Linear data are data that ____________________________

Consider a few different patterns.

|Term |n |

| |How many sides will the 15th term have? |

| | |

| | |

| |[pic] |

|2) | |

| |What will the 23rd figure look like? |

| | |

| | |

| |[pic] |

|3) | |

| |What is the 50th term of the sequence above? |

| | |

| | |

| |[pic] |

|4) | |

| |What is the 103rd term of the sequence? |

Date ___________

Name ________________________

Directions: Find the indicated term for each of the patterns below.

| |[pic] |

|1) | |

| |How many sides will the 15th term have? |

| |Solution: n + 2; 17 sides Add two to the figure number, to determine the number of sides. For example, the 3rd figure has 5 sides. |

| |[pic] |

|2) | |

| |What will the 23rd figure look like? |

| |Solution: Since the pattern repeats after four figures, students should realize that every term that is a multiple of four will look |

| |like the fourth figure. The nearest multiple to 23 is 20; the students should then continue the pattern—it is the 3rd figure. |

| |[pic] |

|3) | |

| |What is the 50th term of the sequence above? |

| |Solution: The shapes repeat after 3 terms so 48 is the closest multiple of 3 to 50, so the shape is a square. The square is not shaded |

| |because the even terms are not shaded. |

| |[pic] |

|4) | |

| |What is the 103rd term of the sequence? |

| |Solution: The pattern repeats after five terms. The 100th term is the fifth figure, so the 103rd term is the third figure. |

Date _____________

Name ________________________

Square Numbers

Consider the following sequence:

[pic]

1) What is the number pattern?

2) Is it linear? Why?

3) What is the formula to find the nth term in this set? What would be the 25th term?

4) How does each number relate to the area of a square?

Date _____________

Name ________________________

Square Numbers

Consider the following sequence:

[pic]

1) What is the number pattern?

1, 4, 9, 16, 25

2) Is it linear? Why?

It is not linear because the difference between consecutive terms is not constant.

3) What is the formula to find the nth term in this set? What would be the 25th term?

Formula: [pic]; the 25th term is 625.

4) How does each number relate to the area of a square?

The area of a square is [pic] where s is the measure of the side. In each of the squares, the measure of the sides are the same, and they increase by one each time.

Therefore the area is 22, 32, 42, …[pic].

Date _____________

Name ________________________

Rectangular Numbers

Consider the following:

[pic]

1) What is the number pattern?

2) Is it linear? Why?

3) What is the formula to find the nth term in this set? What would be the 25th term?

4) How does each number relate to the area of a rectangle?

Date _____________

Name ________________________

Rectangular Numbers

Consider the following:

[pic]

1) What is the number pattern?

2, 6, 12, 20, 30

2) Is it linear? Why?

It is not linear because the difference between consecutive terms is not constant.

3) What is the formula to find the nth term in this set? What would be the 25th term?

Formula: [pic] or [pic]; the 25th term is 650.

4) How does each number relate to the area of a rectangle?

Each rectangle has a height the same as the figure number and a base which is one greater than the height; therefore, the number of dots needed for any figure is the same as the area of the rectangle, n(n+1), where n is the height and the base is one more than the height.

Date _____________

Name ________________________

Triangular Numbers

Consider the following:

[pic]

1) What is the number pattern?

2) Is it linear? Why?

3) What is the formula to find the nth term in this set? What would be the 25th term?

4) How does each number relate to the area of a triangle?

Date _____________

Name ________________________

Triangular Numbers

Consider the following:

[pic]

1) What is the number pattern?

1, 3, 6, 10, 15

2) Is it linear? Why?

It is not linear because the difference between consecutive terms is not constant.

3) What is the formula to find the nth term in this set? What would be the 25th term?

Formula: [pic]; the 25th term is 325.

4) How does each number relate to the area of a triangle?

The area of a triangle is half the area of a rectangle, [pic], so if we take the formula for rectangular numbers, we can divide it by 2 to get the area of a triangle with the same base as its corresponding rectangle.

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Geometry

[pic]

[pic]

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