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Unit 1, Activity 1, Extending Number and Picture Patterns
Geometry
Blackline Masters, Geometry
Page 1-1
Unit 1, Activity 1, Extending Number and Picture Patterns
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)
Blackline Masters, Geometry
Page 1-1
Unit 1, Activity 1, Extending Number and Picture Patterns with Answers
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__
even numbers or +2
2) -1, 0, 1, 2, 3, … __4__, __5__
add 1 to each
3) 4, 7, 10, 13, 16, … _19_, _22_
add 3
4) 1, 4, 9, 16, 25, … __36_, __49_
perfect squares
5) 1, 3, 6, 10, 15, … __21_. _28__
add 2, then 3, then 4, etc.
6) 1, 3, 7, 15, 31, 63, … _127_, _255_
add 2, then 4, then 8, then 16, etc.
7) 1, 1, 2, 3, 5, 8, … __13_, _21__
add the preceding two terms
Fibonacci Sequence
8) 3, 5, 9, 15, 23, … _33__, _45__
add 2, then 4, then 6, then 8, etc.
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)
The student should draw a shaded triangle,
then an unshaded square.
10)
The student should draw two
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.
Blackline Masters, Geometry
Page 1-2
Unit 1, Activity 1, Linear or Non-linear
“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
1)
2,5,8,11,14,...
2)
1, 2, 4,8,16,...
3)
3n 1
4)
x
1
2
3
4
5
y
18
15
12
9
?
5)
n2  1
6)
15, 10, 6, 3, 1,...
7)
8)
x
1
2
3
4
5
y
100
50
25
12.5
?
Non-linear
n4
2
Blackline Masters, Geometry
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Unit 1, Activity 1, Linear or Non-linear with Answers
“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)
2,5,8,11,14,...
2)
1, 2, 4,8,16,...
3)
3n 1
4)
x
1
2
3
4
5
y
18
15
12
9
?



5)
n2  1

6)
15, 10, 6, 3, 1,...

7)
8)
x
1
2
3
4
5
y
100
50
25
12.5
?
n4
2
Blackline Masters, Geometry


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Unit 1, Activity 1, Using Rules to Generate a Sequence
Linear versus Non-linear Relationships
Linear data are data that ____________________________
Consider a few different patterns.
1)
Term n
Value n-3
2)
3)
4)
5)
2
-1
3
0
4
5
6
7
8
Term n
1
Value 2n+3 5
2
7
3
9
4
5
6
7
8
Term n
1
Value 3n+1 4
2
7
3
10
4
5
6
7
8
Term n
Value n2
1
1
2
4
3
9
4
5
6
7
8
Term n
Value n3
1
1
2
8
3
27
4
5
6
7
8
Blackline Masters, Geometry
1
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Unit 1, Activity 1, Using Rules to Generate a Sequence
Questions to answer:
6) Which patterns had common differences (the same number added over and over)? Does that
number appear in the rule?
7) Recall from Algebra I: y = mx + b. What did the m stand for? If the rule for each pattern were
rewritten in this form, how should m be interpreted?
Graph each of the sequences above on a sheet of graph paper to determine if they are linear or
not linear.
8) Which sequences produced a line? What did these sequences have in common?
9) Which sequences did not produce a line? What did these sequences have in common?
10) Write a conjecture about all linear relationships and all non-linear relationships based on
your examples above.
Are the following sequences linear or non-linear?
11) -1.5, -1, -0.5, 0, 0.5, …
12) 4, 10, 18, 28, 40, …
13) 2, 1, 2/3, ½, 2/5, …
14) 1, 4, 7, 10, 13, …
Blackline Masters, Geometry
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Unit 1, Activity 1, Using Rules to Generate a Sequence with Answers
Linear versus Non-linear relationships
Linear data are data that _forms a line when graphed__
Consider a few different patterns.
1)
Term n
Value n-3
1
-2
2
-1
3
0
4
1
5
2
6
3
7
4
8
5
4
11
5
13
6
15
7
17
8
19
4
13
5
16
6
19
7
22
8
25
4
16
5
25
6
36
7
49
8
64
6
216
7
343
8
512
Difference between the terms is 1
2)
Term n
1
Value 2n+3 5
2
7
3
9
Difference between the terms is 2
3)
Term n
1
Value 3n+1 4
2
7
3
10
Difference between the terms is 3
4)
Term n
Value n2
1
1
2
4
3
9
There is no common difference between terms
5)
Term n
Value n3
1
1
2
8
3
27
4
64
5
125
There is no common difference between terms
Questions to answer:
Blackline Masters, Geometry
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Unit 1, Activity 1, Using Rules to Generate a Sequence with Answers
6) Which patterns had common differences (the same number added over and over)? Does that
number appear in the rule?
Patterns 1, 2, and 3 had common differences. These numbers are the coefficients of n.
7) Recall from Algebra I: y = mx + b. What did the m stand for? If the rule for each pattern were
rewritten in this form, how should m be interpreted?
The m stands for slope. If I rewrote the rule in the slope-intercept form it would tell me
the slope of the line which is the rate of change—how much y changes when x changes.
Graph each of the sequences above on a sheet of graph paper to determine if they are linear or
non-linear.
8) Which sequences produced a line? What did these sequences have in common?
Patterns 1, 2, and 3; each of these patterns had a common difference which is the
coefficient of n.
9) Which sequences did not produce a line? What did these sequences have in common?
Patterns 4 and 5; these patterns did not have a common difference.
10) Write a conjecture about all linear relationships and all non-linear relationships based on
your examples above.
Patterns which represent linear relationships will have a common difference between
terms. Patterns which are non-linear will not have a common difference between terms.
Are the following sequences linear or non-linear?
11) -1.5, -1, -0.5, 0, 0.5, …
Linear
13) 2, 1, 2/3, ½, 2/5, …
Non-linear
Blackline Masters, Geometry
12) 4, 10, 18, 28, 40, …
Non-linear
14) 1, 4, 7, 10, 13, …
Linear
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Unit 1, Activity 2, Generating the nth Term for Picture Patterns
Date ___________
Name ________________________
Directions: Find the indicated term for each of the patterns below.
1)
How many sides will the 15th term have?
2)
What will the 23rd figure look like?
3)
What is the 50th term of the sequence above?
4)
What is the 103rd term of the sequence?
Blackline Masters, Geometry
Page 1-9
Unit 1, Activity 2, Generating the nth Term for Picture Patterns with Answers
Date ___________
Name ________________________
Directions: Find the indicated term for each of the patterns below.
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.
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.
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.
4)
What is the 103rd term of the sequence?
Solution: The pattern repeats after five terms. The 100 th term is the fifth figure, so the 103rd term is the third
figure.
Blackline Masters, Geometry
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Unit 1, Activity 3, Square Figurate Numbers
Date _____________
Name ________________________
Square Numbers
Consider the following sequence:
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?
Blackline Masters, Geometry
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Unit 1, Activity 3, Square Figurate Numbers with Answers
Date _____________
Name ________________________
Square Numbers
Consider the following sequence:
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: n 2 ; the 25th term is 625.
4) How does each number relate to the area of a square?
The area of a square is s 2 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, … n 2 .
Blackline Masters, Geometry
Page 1-12
Unit 1, Activity 3, Rectangular Figurate Numbers
Date _____________
Name ________________________
Rectangular Numbers
Consider the following:
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?
Blackline Masters, Geometry
Page 1-13
Unit 1, Activity 3, Rectangular Figurate Numbers with Answers
Date _____________
Name ________________________
Rectangular Numbers
Consider the following:
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: n 2  n or n  n  1 ; 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.
Blackline Masters, Geometry
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Unit 1, Activity 3, Triangular Figurate Numbers
Date _____________
Name ________________________
Triangular Numbers
Consider the following:
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?
Blackline Masters, Geometry
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Unit 1, Activity 3, Triangular Figurate Numbers
Date _____________
Name ________________________
Triangular Numbers
Consider the following:
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:
n  n  1
n2  n
or
or 0.5n2  0.5n ; the 25th term is 325.
2
2
4) How does each number relate to the area of a triangle?
1
The area of a triangle is half the area of a rectangle, A  bh , so if we take the formula
2
for rectangular numbers, we can divide it by 2 to get the area of a triangle with the same
base as its corresponding rectangle.
Blackline Masters, Geometry
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