Download Grade 9 Patterns: number

Educator and Tagging Information:
Learning Area:
Maths
Resource Name:
Maths
Assessment Exemplar Number:
M9.47
Item:
47
Phase:
Senior
Grade:
9
Learning Outcome(s) and Assessment Standard(s):
Learning Outcome 2: Patterns, Functions and Algebra
Assessment Standard: We know this when the learner
9.2.1 Investigates, in different ways, a variety of numeric and geometric patterns and relationships
by representing and generalizing them, and by explaining and justifying the rules that generate them
(including patterns found in natural and cultural forms and patterns of the learner’s own creation).
9.2.3 Represents and uses relationships between variables in order to determine input and/or output
values in a variety of ways using: verbal descriptions; flow diagrams; tables; formulae and
equations.
9.2.7 Determines, analyses and interprets the equivalence of different descriptions of the same
relationship or rule presented: verbally; in flow diagrams; in tables; by equations or expressions; by
graphs on the Cartesian plane in order to select the most useful representation for a given situation.
9.2.11 Uses factorisation to simplify algebraic expressions and solve equations.
Learning Space:
Assessment
Hyperlinks:
To be completed later.
Number of questions for exemplar:
1
Rating:
Easy questions:
Medium questions:
Question 1
Difficult questions:
Assessment Task
Questions:
1.
Consider the array of dots in the following diagram:
These numbers are known as the triangular numbers.
a)
For this sequence of numbers, complete the table below:
Picture number
1
1
Number of dots
2
3
3
6
4
5
6
9
b)
Show clearly how you got to T9 in this sequence.
c)
The general rule for the triangular number sequence 1 ; 3; 6; ... is Tn 
n  n  1
2
.
If the
sequence started at 3 and continued as normal, show how you will adjust this rule to
accommodate this change?
d)
Use the rule Tn 
n  n  1
2
up to a square number.
to prove that two consecutive triangular numbers always add
Solution:
1.
a)
Picture number
1
1
Number of dots
2
3
3
6
4
10
5
15
6
21
9
28
b)
T2 = T1 + 2;
T3 = T2 + 3;
T4 = T3 + 4;
T5 = T4 + 5;
T6 = T5 + 6;
T7 = T6 + 7; so T9 = T8 + 9 or in general the recursive pattern is Tn 1  Tn  (n  1) .
c)
This will be a horizontal translation of 1 unit to the left for the sequence. It thus
becomes:
Picture number
1
3
Number of dots
d)
Tn 
2
6
3
10
 n  1 n  1  1   n  1 n  2  .
2
2
For two consecutive numbers that are triangular:
 n  1 n  2  .
n  n  1
and Tn1 
Tn 
2
2
So:
Tn  Tn 1 



n  n  1  n  1 n  2 

2
2
 n  1 n  n  2 
2
 n  1 2n  2 
2
2  n  1 n  1
  n  1
2
2
This is clearly a square number.
Learners learn about sequences by looking at them recursively – that is what happens to the
previous term to obtain the next term.
The translation is one unit to the left for the sequence, so that term 1 is now 3. Thus we need to
change the n in the sequence to n + 1 to accommodate this translation. Another way to look at this is
realise that n is still one and that what we change in the generalised number must have an output of
3.
Appendix of Assignment Tools
Number patterns
Function
Transformation of functions
Recursive expressions