Grade 9: Integers
Updated 10 March 2013
Grade 9
Numeric and Geometric Patterns
Goals:
□
□
□
□
□
Identify patterns and their basic structure
Investigate and extend numeric and geometric patterns
Find general rules for patterns
Represent geometric and numeric patterns in tables, formulae, pictures, words, etc.
Solve problems involving patterns
Terminology





Numeric Pattern
Geometric Pattern
Rule
Term
Consecutive Terms
1
Grade 9: Integers
Numeric Patterns
Numeric Pattern (or Sequence) – A numeric pattern (or sequence) is an ordered list of
numbers which follows a particular rule. Individual numbers in the pattern are called terms.
The first number in a pattern is term 1, or 𝑡1 . The second number in a pattern is term 2, or
𝑡2 . The subsequent numbers are terms 3, 4, 5, etc., (i.e. 𝑡3 , 𝑡4 , 𝑡5 , …). Terms which are next
to each other are called consecutive terms.
Term 6
Term 1
𝑡1 ; 𝑡2 ; 𝑡3 ; 𝑡4 ; 𝑡5 ; 𝑡6 ; …
Consecutive Terms
When identifying number patterns, the key is to find the relationship between consecutive
terms. Patterns can be categorized according to the relationship between consecutive
terms:
1) Constant Difference: Patterns in which the same number is added or subtracted to
each term to get the next term.
2) Constant Ratio: Patterns in which each term is multiplied or divided by the same
number to get the next term. For example, the following pattern has a constant ratio
of × 2, or 2: 1
3; 6; 12; 24; 48; 96; …
×2 ×2 ×2 ×2 ×2
3) Neither: These are patterns which do not have a constant difference or ratio. For
example, the following pattern.
Though a pattern’s rule can be very complex, it’s helpful to first see if there’s a simple rule,
such as adding/subtracting 9 or multiplying/dividing by 3.
When writing down the rule for a pattern, it is very important to include 2 things:
1) The first number in the pattern
2) A rule for finding the next numbers in the sequence.
Examples:
98; 91; 84; 77; 70; 63; …
−7
−7
−7
Hence the rule is …
Start at 98 and subtract 7 from each term.
2
−7
−7
Grade 9: Integers
8; 4;
2;
1;
1
;
2
1
;…
4
÷2 ÷2 ÷2 ÷2 ÷2
Hence the rule is …
Start at 8 and divide each term by 2.
Exercise 1
1. Write down a general rule for each pattern. Use your work from exercise 1 question 2 to
help if necessary.
1.1. 0; 6; 12; 18; 24; 30; 36; …
1.2. 1; 3; 5; 7; 9; 11; …
1.3. –2; –3; –4; –5; –6; …
1.4. –3; 1; 5; 9; 13; 17; …
1.5. 3; 6; 12; 24; 48; 96; …
1.6. 1; 4; 7; 10; 13; 16; …
1.7. 20; 16; 12; 8; 4; …
1.8. 1/2; 1/4; 1/6; 1/8; 1/10; 1/12; …
1.9. 1; 4; 9; 16; 25; 36; 49; 64; …
1.10. 4; 2; 1; 1/2; 1/4; 1/8; …
1.11. 2; –4; 8; –16; 32; –64; …
1.12. 384; 192; 96; 48; 24; …
1.13. 71; 66; 61; 56; 51; 46; 41; 36; …
1.14. 10; 17; 24; 31; 38; 45; …
1.15. 20; 15; 10; 5; 0; …
1.16. 0; 4; 8; 12; 16; 20; …
2. Complete each table. Then write a rule for each pattern. For 2.6, make up your own
pattern and rule:
2.1
Term
1
2
3
Value
2
4
6
Term
1
Value
100
5
8
10
n
10
84
100
Rule:
2.2
2
3
4
9
80
70
3
60
0
– 100
100 − 10𝑛
Grade 9: Integers
Rule:
2.3
Term
1
2
3
Value
4
7
10
Term
1
2
3
Value
9
8
7
Term
1
2
3
Value
5
10
15
Term
1
2
3
4
50
16
31
5
10
n
301
Rule:
2.4
4
20
0
n
– 40
Rule:
2.5
4
5
10
50
n
75
500
Rule:
2.6
4
Value
Rule:
4
5
n
Grade 9: Integers
Pattern Representations
Pattern Representations – Patterns can be represented in many different ways. Often the
challenge in working with patterns is moving from one representation to another. Here’s a
list of representations, each of which show the same pattern in a different form.
1. In words:
“Start with the number 3 and then multiply by 2 to get the next number.”
2. Number Sequence:
3 ; 6 ; 12 ; 24 ; 48 ; ____ ; 192 ; …
3. Table:
Terms:
Value:
1
3
4. Geometrically:



→



2
6
→
3
12
4
24



5
→
6
96
7
192



5. Using a Formula:
𝑠𝑛 = 3 × 2𝑛−1
( 𝑛 can be any positive integer and 𝑠3 means “the 3rd term in the pattern.”)
Exercise 1
1. Identify the 5th term of each of the following patterns:
1.1. -2; -1; 0; 1; 2; 3; 4; 5; …
1.2. 17; 34; 51; 68; 85; …
1.3. 19; 16; 13; 10; 7; 4; 1; …
1.4. -2; 8; -16; 32; -64; 128
1.5. 2; 4; 6; 8; 10; 12; 14; …
2. Write down the next 3 terms of each of the following patterns:
2.1. 0; 6; 12; 18; 24; 30; 36; …
2.2. 1; 3; 5; 7; 9; 11; …
2.3. –2; –3; –4; –5; –6; …
2.4. –3; 1; 5; 9; 13; 17; …
2.5. 3; 6; 12; 24; 48; 96; …
2.6. 1; 4; 7; 10; 13; 16; …
2.7. 20; 16; 12; 8; 4; …
5
8
→
…
Grade 9: Integers
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.
2.14.
2.15.
2.16.
1/2; 1/4; 1/6; 1/8; 1/10; 1/12; …
1; 4; 9; 16; 25; 36; 49; 64; …
4; 2; 1; 1/2; 1/4; 1/8; …
2; –4; 8; –16; 32; –64; …
384; 192; 96; 48; 24; …
71; 66; 61; 56; 51; 46; 41; 36; …
10; 17; 24; 31; 38; 45; …
20; 15; 10; 5; 0; …
0; 4; 8; 12; 16; 20; …
3. Write out the first 5 numbers of each of the following patterns:
3.1. The multiples of 5
3.2. The multiples of 32
3.3. Start at 4 and count up by 3’s
3.4. Start at -2 and count up by 2’s
3.5. Start at 3 and count down by 1’s
3.6. Start at 11 and count down by 11’s
3.7. The powers of negative 1, starting at (−1)0
3.8. The powers of two, starting at 20
3.9. Start at 64 and divide by 2
3.10. Start at 243 and divide by 3
3.11. Start at 1 and divide by 2
3.12. Start at 0 and add 1, then 2, then 3, then 4, etc
6
Grade 9: Integers
Geometric Patterns
Exercise 3
Geometric patterns follow the same principles as numeric patterns, except they are
represented using pictures instead of numbers. For each of the following geometric
patterns, complete the chart and then write out a general rule for the pattern. Finally, draw
a picture of the next stage of the pattern.
1. Cube Pattern:
Stage 1
Stage 2
Stage (i.e. “term”)
1
Stage 3
2
Stage 4
3
4
5
6
n
Number of Cubes
Rule:
Drawing of stage 5:
2. Square Pattern:
Stage 1
Stage
Stage 2
1
2
Stage 3
3
Number of
Squares
Rule:
Drawing of stage 5:
7
Stage 4
4
5
6
n
Grade 9: Integers
3. Cube Pattern:
Stage 1
Stage
Stage 2
1
2
Stage 3
3
Stage 4
4
5
6
n
5
6
n
Number of
Cubes
Rule:
Drawing of stage 5:
4. Cube Pattern:
Stage 1
Stage
Stage 2
1
Stage 3
2
3
Number of
Cubes
Rule:
Drawing of stage 5:
8
Stage 4
4
Grade 9: Integers
5. Smiley Face Pattern:

Stage 1
Stage
1








Stage 2
Stage 3
2
3
4







Stage 4
5
Number of
Smiles
n
25
Rule:
Drawing of stage 5:
6. Dimensional Shapes: Points, Lines, and Squares
0 Dimensions
1 Dimension
2 Dimensions
3 Dimensions
Fill in the blanks as best you can. Boxes with exclamation marks are extra challenges which
you are strongly encouraged but not required to attempt.
Dimensions
Number of
Points
Number of
Lines
Number of
Squares
0
1
1
2
0
1
2
3
4
5
2𝑛
16
4
0
6
48
!
!
!
!
!
Write down any observations you have about the patterns above.
What would the 4th dimensional shape look like? (If you’re not sure, then draw what you
think it could look like, making your best effort.)
9
n
Grade 9: Integers
Pattern Problems
Exercise 4
1. Create your own geometric pattern: On a separate sheet of paper OR using
pens/pencils, blocks, matchsticks, etc., create your own geometric pattern. Write down
the rule for your pattern, and then show the pattern to someone else and see if they can
figure out the correct rule.
2. Square and Cube Patterns:
2.1. Using small square pieces of paper or cardboard, arrange the small squares to
make bigger squares, then compare the number of small squares on one side of
the big square to the total number of small squares used to make up the big
square. Record your findings below:
Small squares
on one side
Total number of
small squares
1
2
3
4
5
1
n
36
2.2.
What are numbers in the second row of the above table called? Why do you think
they are called by that name?
2.3.
This exercise can also be repeated using small cubes to make big cubes:
1 cube on a side
Small cubes
on one side
Total number
of small cubes
2.4.
2 cubes on a side
1
2
3
1
3 cubes on a side
4
5
n
216
What are the numbers in the second row called? Why do you think they are called
by that name?
10
Grade 9: Integers
3. Special Patterns: Each of the following patterns has a special name. Try to find the rule,
and/or next number for each pattern. Some of these are quite challenging!
3.1. Square Numbers
3.2. Cubic Numbers
3.3. 4th Degree Numbers
1; 4; 9; 16; 25; …
1; 8; 27; 64; 125; …
1; 16; 81; 256; 625; 2401; …
3.4. Triangular Numbers:
3.5. Tetrahedral Numbers:
3.6. Pentalope Numbers:
1; 3; 6; 10; 15; 21; 28; 35; …
1; 4; 10; 20; 35; 54; …
1; 5; 15; 35; 70; …
3.7. Fibonacci Numbers:
3.8. Tribonacci Numbers:
1; 1; 2; 3; 5; 8; 13; 21; 34; …
1; 1; 1; 3; 5; 9; 17; 31; 57; …
3.9. Harmonic Sequence:
1; 2 ; 4 ;
3.10. Harmonic Series:
1; 2 ; 4 ;
3.11. Factorials:
3.12. 𝜋
1; 2; 6; 24; 120; 720; 5 040 ; 40 320; …
3; 1; 4; 1; 5; 9; 2; 6; 5; 3; 5; 8; 9; 7; 9; 3; 5; 3; …
1
3
1
7
1
1
;
;
1
;
1
8 16 32 64
15 31 63
8
;
16
;
32
;…
;…
4. The Collatz Conjecture: Start with any positive integer and use the following rule:
If your number is EVEN, then divide it by 2.
If your number is ODD, then multiply it by 3 and add 1.
Repeat the process for the number you ended up with, and then continue repeating the
process until you have a number pattern. E.g., starting with the number 17 yields the
following pattern:
17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 …
4.1. What do you notice about the “end” of your pattern? (NB: Depending on your
starting number, it may take a long time to reach the “end” of your pattern.)
4.2. Start on a different number and use the same rule. How is this pattern similar or
different from the one above?
4.3. Make a conjecture based on the first two parts of this question.
5. Patterns in Nature: What patterns occur in nature? Describe a specific natural
occurrence which you think could be a described mathematically and describe why you
think it is a pattern.
11