NAME
_________________________
CLASS
_________
0
INTRODUCTION
Number patterns and Geometric patterns are presented as a list of numbers, formed
according to a mathematical rule.
They can be found in various forms; including pictures, tables, sequences, spidergrams
and flow diagrams.
A term is a number in a pattern or sequence.
Consecutive terms are terms that follow each other according to the rule of the number
pattern.
1.
NUMBER SEQUENCES
If the same number is added to or subtracted to get the next term in a sequence, this is
called a constant difference.
If each term in a sequence is multiplied or divided by the same number to get the next
term, this is called a constant ratio.
Investigating number patterns with a constant difference.
1.1
Find the next three terms in each of the following sequences. Write the rule
of the number pattern in the brackets.
1.1.1
45; 43; 41; 39; _______; _______; _______
[
]
1.1.2
42; 48; 54; _______; _______; _______
[
]
1.1.3
–4; –9; –14; –19; _______; _______; _______
[
]
1.1.4
–4; –1; 2; 5; 8; _______; _______; _______
[
]
1.1.5
20,5; 21; 21,5; 22; 22,5; _______; _______; _______
[
]
1.2
Write the first five numbers in the following number sequences.
1.2.1
The first number is 5. Add four each time. ____________________________
1.2.2
Add ten to each consecutive term, starting at 34. ______________________
1.2.3
The third number is 36. Subtract six each time. _______________________
1.2.4
The first number is –3. Add –2 each time. ____________________________
1.2.5
The second number of a sequence is 1 000. Add four each time.
_____________________________________________________________
1
1.3
William has been reading a riveting book. He reads 11 pages on Monday, 22
pages on Tuesday and 33 pages on Wednesday. If he continues reading in
this pattern, how many pages will he read on Thursday, Friday and
Saturday?
Complete the table below to show the solution.
Day
No. of pages
1.4
Mon
Tues
Wed
11
22
33
Thurs
Fri
Sat
Kate is on a training programme for the Gun Run. On day 1 she runs 5,5 km;
day 2 she runs 7,5 km and day 3 she runs 9,5 km. What distances will she
run on days 4 and 5?
Complete the table below to show the solution for the problem.
Day
Distance run
Investigating number patterns with a constant ratio.
1.5.
Find the missing terms in each of the following sequences. Write the rule of
the number pattern in the brackets.
1.5.1
5; 20; 80; 320; _______; _______; _______
[
]
1.5.2
1; 3; 9; 27; _______; _______; _______
[
]
1.5.3
3200; 1600; 800; _______; _______; _______
[
]
1.5.4
15; _______; 60; 120; _______; _______; 960
[
]
1.5.5
41; 4,1; 0,41; 0,041; ________; _________; _________ [
]
2
Investigating number patterns with neither constant differences nor constant
ratios.
1.6
Find the missing terms in each of the following sequences. Write the rule of
the number pattern in the brackets.
1.6.1
1; 4; 9; 16; 25; _______; _______; _______
[
]
1.6.2
1; 8; 27; 64; _______; _______; _______
[
]
1.6.3
1; 2; 5; 10; 17; _______; _______; _______
[
]
1.6.4
1; 1; 2; 3; 5; 8; 13; 21; _______; _______; _______
[
]
2.
SPIDERGRAMS AND FLOW DIAGRAMS
2.1.1 Find all the missing values in this spidergram below.
Input / Term Number
Answer / Output
n
Tn
1
10
2
3
4
13
×
+7
16
31
12
2.1.2
The rule for the spidergram written as:
Tn = n × 3 + 7
or
2.1.3
Tn = 3n + 7
What is the output value (Tn) when the input value (n) is 102 ?
T102 =
2.1.4
What is the input value (n) when the output value (Tn) is 52?
If Tn = 52
then
3
2.2.1
Complete the following spidergram:
Input / Term Number
Answer / Output
n
Tn
1
3
5
7
×3
+5
9
11
2.2.2
Write down the first 6 terms of the sequence you have produced above:
2.2.3
Write the rule for the spidergram:
Tn =
2.2.4
What is the output value (Tn) when the input value (n) is 43 ?
T43 =
2.2.5
What is the input value (n) when the output value (Tn) is 155?
If Tn = 155
then
4
2.3.1
Fill in the missing values on the spidergram:
Term number
n
Output
Tn
1
2
×2
+3
3
4
12
20
2.3.2
Write down the first 6 terms of the sequence you have produced above.
2.3.3
What is the rule for making this sequence?
Tn =
2.3.4
Complete the following:
T50 =
T100 =
2.3.5
What is the input value (n) when the output value (Tn) is 52?
If Tn = 52
2.3.6
then
The above spidergram can be summarized in the following flow diagram:
n
+3
×2
Tn = (n + 3) × 2
5
2.4
Study the flow diagram.
–2
n
2.4.1
3
Tn
Use the above flow diagram to find
T11
T53
2.4.2
Which term in the sequence is equal to 6?
2.4.3
Write the rule for the number pattern.
2.5.1
Complete the spidergram:
1
15
2
3
4
5
20
25
+2
95
80
2.5.2
Write the rule:
Tn =
2.5.3
Use the rule to complete:
T15 =
Tn = 135
2.6
Study the flow diagram
n
+1
×5
Tn
2.6.1
Write down the rule represented by the above flow diagram.
2.6.2
Write down the first 6 terms of the sequence.
2.6.3
Which term will give a value of 100?
6
3.
GEOMETRIC PATTERNS
Geometric patterns are number patterns represented in diagrams.
3.1.1
Complete the pattern by drawing Necklace 4.
Necklace 1
3.1.2
Necklace 2
Necklace 3
Necklace 4
Complete the table to illustrate this number pattern.
Necklace number (n)
1
2
3
4
5
6
7
8
20
Total number of beads (Tn)
3.1.3
Write down the first 8 terms of the sequence you have produced above.
3.1.4.
Complete the spidergram for the number sequence.
Pattern number
Number of beads
n
Tn
1
2
3
4
5
3.1.5
Complete the flow diagram for this number sequence.
n
3.1.6
Write down the rule for this number sequence
Tn =
3.1.7
How many beads will be in Necklace 55?
3.1.8
What is the necklace number with 1 000 beads?
3.1.9
Find the output value for the following
terms using rule for this pattern.
3.1.10
Find the input values that result
in the following output values.
T43 =
Tn = 144
T85 =
Tn = 496
7
3.2.1
Complete the pattern by drawing Pattern 4
Pattern 1
3.2.2
Pattern 2
Pattern 3
Pattern 4
Complete the table to illustrate this number pattern.
Pattern number (n)
1
2
3
4
5
6
7
8
20
Total number of lines (Tn)
3.2.3
Write down the first 5 terms of the sequence you have produced above.
3.2.4.
Complete the spidergram for the number sequence.
Pattern Number
n
Total number of lines
Tn
1
2
3
4
5
3.2.5
Complete the flow diagram for this number sequence.
n
3.2.6
Write down the rule for this number sequence.
Tn =
3.2.7
How many lines will be in Pattern 55?
3.2.8
What is the number of the pattern with 999 lines?
3.2.9
Find the output value for the following
terms using rule for this pattern.
3.2.10
Find the input values that result
in the following output values.
T12 =
Tn = 55
T35 =
Tn = 497
8
3.3
Matchsticks are used to make the following diagrams:
Pattern 1
3.3.1
Pattern 2
Pattern 3
Complete the following table to illustrate this number pattern.
Pattern Number (n)
1
2
3
4
5
Number of matchsticks (Tn)
3.3.2
Complete the spidergram to find the rule:
Pattern Number
n
Total number of matchsticks
Tn
1
2
3
4
5
3.3.3
Complete the flow diagram for this number sequence.
n
3.3.4
Write down the rule for this number sequence.
Tn =
3.3.5
How many matchsticks are needed to make Pattern 10?
3.3.6
Which pattern can be formed with 129 matchsticks?
3.3.7
Use the rule to find:
3.3.8
Use the rule to find:
T10 =
Tn = 75
T25 =
Tn = 210
9
3.4
Matchsticks are used to make the following pattern.
3.4.1
Use the matchstick pattern to complete the table
Number of squares (n)
1
2
3
4
5
6
Number of matchsticks (Tn)
3.4.2
What is the rule for making this pattern? Use a spidergram to help you.
Tn =
3.4.3
Complete the following:
T50 =
T201 =
Tn = 76
*3.5
The numbers in the sequence 1; 7; 19; 37; are called hexagonal numbers,
because they can be arranged as hexagons.
3.5.1
Draw in the next pattern.
3.5.2
Complete the table
Pattern Number (n)
Total number of dots (Tn)
3.5.3
What is the rule for making this pattern? You may use the internet to help
you.
Tn =
10
4.
NUMERIC PATTERNS
4.1
Complete the table. The nth term is the rule for the number sequence.
Term Number (n)
1
2
3
4
5
Term (Tn)
6
11
16
21
26
4.1.2
Use a spidergram to help you find the rule.
Term number
Term
n
Tn
1
2
3
4
5
4.1.3
Which term will be 126?
4.1.4
What will the 35th term be?
11
6
30
nth term
4.2
Complete the last row of the following tables without extending the pattern
and find the nth term (i.e. the rule) for each number pattern.
Use the spidergrams to help you.
4.2.1
Position in the sequence (n)
1
2
3
4
Term (Tn)
7
9
11
13
Position in the sequence
Term
n
Tn
10
nth term
16
nth term
1
2
3
4
5
4.2.2
Position in the sequence (n)
1
2
3
4
Term (Tn)
3
7
11
15
Position in the sequence
Term
n
Tn
1
2
3
4
5
4.2.3
Position in the sequence (n)
1
2
3
4
Term (Tn)
6
12
18
24
Position in the sequence
Term
n
Tn
1
2
3
4
5
12
nth term
72
4.2.4
Position in the sequence (n)
1
2
3
4
Term (Tn)
1
4
9
16
Position in the sequence
Term
n
Tn
nth term
169
1
2
3
4
5
4.3
Provide a rule to describe the relationship between the following numbers
and then find the given term (use a spidergram to help you).
4.3.1
2; 5; 8; 11; ……..…
1
2
Tn =
3
T20 =
4
5
4.3.2
25; 30; 35; 40; …..…
1
Tn =
2
T15 =
3
4
5
4.3.3
9; 15; 21; 27; ………
1
Tn =
2
T15 =
3
4
5
13
5.
ALGEBRAIC EXPRESSIONS
Mathematical language and symbols
We can translate word problems into mathematical language using the mathematical
symbols.
Word
Sum / Increase / More than
Difference / Decrease/ Less than
Product / Times / Of
Quotient
Symbol
Meaning
Operation (symbol)
Addition
Subtraction
Multiplication
Division
Symbol
(+)
(–)
(×)
(÷)
Meaning
Add
Subtract
Multiply
Divide
Meaning
=
equals
≠
not equal to
<
less than
≤
less than or equal to
>
greater than
≥
greater than or equal to
( )2
3
( )
squared
√
square root
cubed
√
cube root
5.1
Write these word problems in mathematical language.
5.1.1
The sum of seven and eight
5.1.2
The product of 12 and 13
5.1.3
54 less than 65
5.1.4
42 increased by 17
5.1.5
The difference between 19 and 12
5.1.6
The quotient when 15 is divided by 5
5.1.7
The square root of 25 is greater than the square of 2
5.1.8
The quotient of 93 and 31
5.1.9
The cube of 3 is greater than the square root of 16
14
Variables in algebraic language
An algebraic expression is a number sentence involving letters, numbers and
operators (+, –, × and ÷).
In the algebraic expression,
the is known as the variable (its value can vary
depending on the question) and 6 is known as the constant (always a number).
By substituting different values for the variable, you will get different values for an
expression:
Example:
Find the different values for the expression
5.2
if
then the expression is
if
then the expression is
Find the values of the following expressions:
if
5.2.1
–
5.2.2
–
5.2.3
5.2.4
5.2.5
5.2.6
if
if
√
if
√
if
and
if
;
15
and
An algebraic expression is made up of terms.
An expression with two terms and may look like this:
and 6 separated by the plus sign.
𝑥
coefficient
variable
constant
𝑥
terms
[ 𝑥
]
expression
Other examples of algebraic expressions are:
In algebra we usually write the number before the variable in each term.
5.3
and not
Write down the constant terms in each of the following expressions.
5.3.1
5.3.2
5.3.3
5.3.4
16
5.4
Write down the coefficient of
in each of the following expressions.
5.4.1
5.4.2
5.4.3
5.4.4
5.5
Write these word problems in mathematical language (write the unknown
number as ).
5.5.1
The sum of a number and 7
5.5.2
A number divided by 7
5.5.3
A number multiplied by 7
5.5.4
15 less than a number
5.5.5
The product of a certain number and 3
5.5.6
The quotient of a number and 2
5.5.7
Double a number increased by 8
5.5.8
A number multiplied by 3 and subtracted from the same number cubed
5.5.9
The sum of the square root of 64 and twice a number
5.5.10
A number decreased by the cube root of 8
17
6.
ALGEBRAIC EQUATIONS
Relationships between numbers can be expressed in words or as a number sentence.
For example, nine plus eight is equal to seventeen and be written
as 9 + 8 = 17. Another name for a number sentence is an equation.
An algebraic equation looks like this:
𝑥
coefficient
= 18
variable
𝑥
constant
constant
= 18
terms
[ 𝑥
8]
equation or number sentence
When you solve an equation you are have to find the value of the variable that will make
the number sentence true. Both sides of the equation have to balance.
You can solve equations by inspection. You do not need to show any of your working.
You can simply use inverse operations as you did when you solved a spidergram or
flow diagram problems.
Example:
means
To solve this equation work backwards doing the reverse operations as you did in flow
diagrams when you worked out the term number (n) if you had the answer T(n).
You just have to replace the symbol n with an x.
x
×2
+6
20
(20 – 6) ÷ 2 = 7 Therefore the value of x is 7. We now see that when we replace
the x with 7, both the left and right hand sides of the equation equal 20.
18
6.1.
By inspection, solve for
in the following equations:
6.1.1
6.1.2
6.1.3
6.1.4
6.1.5
6.1.6
6.1.7
6.1.8
6.1.9
6.1.10
8
8
6.1.11
19
6.2
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
6.2.6
6.2.7
6.2.8
6.2.9
6.2.10
6.2.11
Determine the value of x in each of the following equations.
x 46
x 
x  4  15
x 
x  4  3  16
x 
5  x  1  24
x 
4  x  3  23
x 
x  4
7
x 
x 2 5
7
x 
24  3  0
x
x 
4 x  3  23
x 
2 x  3  13
x 
3x  2  5
x 
20
6.3
Solve the following equations:
6.3.1
5x  13  38
6.3.2
14  2
h
6.3.3
k  5
4
6.3.4
n  11  6  10
6.3.5
75  15
r
6.3.6
x 4
7
6.3.7
125  5  30
x
6.3.8
3t  t  6
6.3.9
5w  4w  8
6.3.10
2 x  3  13
6.3.11
5  2n  17
21
7.
SPECIAL NUMBER PATTERNS
7.1
PRIME NUMBERS
7.1.1
List all the prime numbers up to 100.
______________________________________________________________
______________________________________________________________
7.2
SQUARE NUMBERS
7.2.1
List the first 10 square numbers.
______________________________________________________________
7.2.2
List the first 10 square numbers in index form.
______________________________________________________________
7.2.3
Write the general rule for the square number sequence.
________________________
7.3
CUBE NUMBERS
7.3.1
List the first 10 cube numbers.
______________________________________________________________
7.3.2
List the first 10 cube numbers in index form.
______________________________________________________________
7.3.3
Write the general rule for the cube number sequence.
________________________
22
7.4
DOUBLING UP SEQUENCE
7.4.1
List the first 10 terms of the doubling up number sequence starting
with 2, 4, …. etc.
______________________________________________________________
7.4.2
List the first of 10 terms of the doubling up sequence in index form.
______________________________________________________________
7.4.3
Write the general rule for the doubling up number sequence.
________________________
7.5
TRIANGULAR NUMBERS
7.5.1
List the first 10 terms of the triangular number sequence starting
with 1, 3, 6; , 10… etc.
______________________________________________________________
*7.5.2
Write the general rule for the triangular numbers.
________________________
7.6
FIBONACCI NUMBER SEQUENCE
7.6.1
Continue the following number sequence
1; 1; 2; 3; 5; 8; 13;
7.6.2
Describe in words how this number pattern is formed.
______________________________________________________________
______________________________________________________________
23
7.7
PASCAL’S TRIANGLE
7.7.1
Can you predict the numbers in the empty rows?
1
1
1
1
1
1
1
1
5
6
28
6
15
1
4
10
20
35
56
1
3
10
21
7
8
3
4
1
1
2
1
5
21
56
1
6
15
35
70
1
7
28
1
8
1
7.7.2
Do any numbers repeat?
7.7.3
What number patterns can you find? Highlight them using different colours.
7.7.4
Is there a pattern in the sums of the numbers in the rows?
24
8.
8.1
MIXED EXAMPLES
Complete the spidergram:
1
15
2
3
4
5
20
25
+2
95
80
8.1.1
Write the rule:
Tn =
8.1.2
Use the rule to find the 15th term:
8.1.3
Which term will equal 135?
8.2
A pattern has the following flow diagram:
n
8.2.1
÷2
–8
Write the rule:
Tn =
8.2.2
Find:
T24 =
T102 =
8.2.3
Complete:
Tn = 20
Tn= 168
25
8.3
Study the flow diagram.
n
8.3.1
–2
3
Tn
Use the above flow diagram to find
T11
T53
8.3.2
Which term in the sequence is equal to 6?
8.3.3
Write the rule for the number pattern.
8.4.1
Complete the spidergram:
11
3
14
4
17
80
–2
5
72
8.4.2
Write down the nth term for the above number sequence:
Tn =
26
8.5
The owner of a stud farm fences off his paddocks with wooden fencing as
illustrated below. He wants to calculate how many crossbars he will need.
8.5.1
Use the illustration of the fence above to complete the table:
8.5.2
Number of uprights (n)
1
Number of crossbars (Tn)
0
2
3
4
5
80
nth term
122
Use the above table to complete the spidergram below, and then write the
rule for this number pattern.
Tn =
1
0
2
3
4
5
122
80
27
8.6
Study the flow diagram
n
+
4
Tn
8.6.1
Give the rule to represent the above flow diagram.
8.6.2
Complete:
T77 =
Tn = 110
n=
8.6.3
Which term is equal to 62?
8.6.4
What is the 52nd term?
8.7.1
You need 4 matchsticks to build the first pattern, 12 matchsticks to build the
second pattern and so on.
8.7.2
The table below represents information from the patterns above. Fill in the
blank spaces:
8.7.3
Pattern Number (n)
1
2
3
Number of Matchsticks (Tn)
4
12
20
4
Use the spidergram to find the rule for the sequence.
Tn =
28
20
100
8.8
You need 5 matches to make a ladder with 1 step and you need 8 matches to
make a ladder with two steps and so on.
8.8.1
Continue the pattern and draw another ladder with 4 steps below the three
given ladders.
8.8.2
The table below represents information taken from the pattern of ladders
above. Fill in the blank spaces.
Number of steps on ladder (n)
1
2
3
Number of matches (Tn)
5
8
11
4
5
15
8.8.3
What is the rule for this sequence of numbers? (You may need to draw a
spidergram to help you)
8.8.4
Find:
T69 =
Tn = 65
n=
29
296
8.9
List the next two terms in each sequence below:
2; 5; 10; 17; __________ ;
______
1; 2; 4; 8;
______
__________ ;
1; 1; 2; 3; 5; __________ ;
______
10 000; 1 000; 100; 10; ______ ;
______
1; 3; 9; __________ ; ________
4; 16; 64; __________ ; ________
8.10.1
Give the next two terms of the following sequence:
1; 4; 9; 16; ________ ; _________
8.10.1
Write this sequence in index form.
8.10.3
What is the name of this number sequence?
8.11
List the first 5 triangular numbers, starting at 1.
8.12
List the first 6 square numbers, starting at 1.
8.13.1
List the first 10 numbers of the doubling up sequence starting at 1.
8.13.2
What is the 21st term of this number sequence?
30
8.14.1
Give the rule represented by the flow diagram:
n
8.14.2
2
+3
Tn
Complete:
T50 =
Tn = 60,5
n=
8.15
Look pattern of squares created below.
8.15.1
Continue the pattern and draw the next picture in the sequence
8.15.2
Complete the table below
Tower
1
2
3
Number of black blocks
3
6
10
Number of white blocks
1
3
6
Total number of blocks
4
9
16
4
5
6
8.15.3
What is the name given to the sequence of white blocks?
8.15.4
What is the name given to the sequence of total blocks?
8.15.5
Use the pattern in your table to calculate the total number of blocks needed to
make the 100th tower.
31
8.16
A local deli is having a winter special on all their fruit and nuts: For every
kilogram bought the first 1 kg is free. The usual price for chocolate coated
2
nuts is R8 per kilogram. The deli made the following spidergram to work out
how much to charge for 1 kg, 2 kg, 3 kg and so on.
Number or kilograms
Cost in Rands
n
Tn
1
4
2
12
3
92
12
144
8.16.1
Complete the spidergram above.
8.16.2
How much would you pay if you bought 20 kg of chocolate coated nuts?
8.16.3
Give the rule for determining the cost of buying any number of kilograms of
chocolate coated nuts.
Cost =
8.16.4
How many kilograms did you buy if you paid R180?
32
8.17
Write these word problems in mathematical language (write the unknown
number as ).
8.17.1
The sum of 15 and a number.
8.17.2
The quotient of a number and 4.
8.17.3
The product of a number and 7.
8.17.4
The difference between a number and 17.
8.17.5
The product of a certain number squared and 13.
8.17.6
A number cubed increased by 8.
8.17.7
A number multiplied by 3 and subtracted from the square of the same
number.
8.17.8
The difference of the square root of 36 and double a number.
8.17.9
A number decreased by the cube root of 27.
33
8.18
Solve the following equations:
8.18.1
5x  11  36
8.18.2
18  3
h
8.18.3
k  6
4
8.18.4
n  14  4  10
8.18.5
75  25
r
8.18.6
x  49
7
34
8.19
Find the values of the following expressions:
8.19.1
2x2
if x 
if b  9
–
8.19.2
1
2
8.19.3
√
if m  3
8.19.4
√
if
p  25
m  1 and n  2
8.19.5
√
if
8.19.6
√
if x  2, y  3 and w  9
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