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Transcript
Educator and Tagging Information:
Learning Area:
Maths
Resource Name:
Maths
Assessment Exemplar Number:
M9.46
Item:
46
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 generalising 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.
Learning Space:
Assessment
Hyperlinks:
To be completed later.
Number of questions for exemplar:
2
Rating:
Easy questions:
Question 1 and 2
Medium questions:
Difficult questions:
Assessment Task
Questions:
1.
2.
a)
Write down the sequence of Fibonacci numbers up to the 10th term.
b)
Peter has read somewhere that “When you write out any 10 consecutive numbers in a
Fibonacci sequence, multiplying the seventh number by 11 gives the total sum of all 10
numbers”. Use your numbers in (a) to help show that this is true.
c)
Now using the sequence where T1  x and T2  y , write down the first 10 terms in the
Fibonacci sequence in terms of x and y.
d)
Use your working in (c) to show that the rule in (b) holds for any sequence of 10
consecutive Fibonacci numbers.
Below the incomplete input and output table has been given:
Input
Output
0
1
1
2
2
5
3
10
4
7
11
25
a)
Complete the table, showing your calculations.
b)
Find a general rule that represents the pattern in the number sequence.
c)
If the output started at 5 and kept the same structure, how will your rule change?
Solution:
1.
a)
1, 1, 2, 3, 5, 8; 13 ; 21 ; 34; 55
b)
7th term = 13. So 13.11 = 143 and the sum of the ten terms = 143
c)
T1  x
T2  y
T3  x  y
T4  x  2 y
T5  2 x  3 y
T6  3 x  5 y
T7  5 x  8 y
T8  8 x  13 y
T9  13 x  21 y
T10  21x  34 y
d)
2.
Now the sum of this is: 55x + 88y and this is equal to 11 5x  8 y   11 T7
a)
Input
Output
0
1
1
2
2
5
3
10
4
17
7
50
11
122
25
577
b)
Output = (input)2 + 1
c)
Since the pattern moves to the third term that is now term 1, we have a rule that adjusts
to : Output = (input + 2)2 + 1
The focus in this activity is the idea of proof and reasoning and the use of algebra to argue
generalities.
This question can be asked at grade 8 level too with no adjustment necessary.
Appendix of Assignment Tools
Learners use their knowledge about the Fibonacci sequence structure to write it down
Algebraic structures are investigated
Factoring and what factors represent