GANSBAAI ACADEMIA
MATHEMATICSGrade 10 / /
INVESTIGATION
March 2014
Total: 55Time: 1 week
EXAMINATOR
MODERATOR / L. Havenga
G. Edwards
INSTRUCTIONS
- This document consists of 2 pages and an information page.
- Write neat.
- Enjoy!
TASK 1[13]
The shape above is made up of 7 triangles. It uses 15 matches and has a perimeter of 9.
Investigate the connection between the number of triangles, the number of matches used and the perimeter of the shape.
Copy and complete the table below to help you.(6)
Number of triangles / 1 / 2 / 3 / 4 / 7 / 20 / nNumber of matches / 15
Perimeter / 9
Use the table to answer the following questions:
a)Write down in words the ‘rule’ you have used to find the number of matches for n triangles.(2)
b)Calculate the number or matches needed to form 45 triangles in the same arrangement as the
given sketch.(2)
c)How many triangles will be formed if 75 matches are used?(3)
TASK 2[6]
The next shape is made up of 7 squares (four-sided polygon). The 7 squares shape uses 22 matches and has a perimeter of 16. Create a table similar to the one above to help you find a connection between the number of squares, the number of matches and the perimeter of the shape.
Number of squares / 1 / 2 / 3 / 4 / 7 / 20 / nNumber of matches / 22
Perimeter / 16
(6)
TASK 3[34]
Extend your investigation to include regular polygons with 5 (pentagon), 6 (hexagon) and 8 (octagon) sides. For each case:
a)draw a shape with at least 7 connecting polygons.(3 x (2) = 6)
b)copy and complete a table similar to the one in Task 1.(3 x (7) = 21)
c)Complete the summary table below and write down the rule in words you used to find the total
number of matches needed for npolygons for ANY polygon.(7)
Number of sides of polygon / 3 / 4 / 5 / 6 / 8Number of matches for n polygons
TASK 4[2]
Use your findings from tasks 1 to 3 to answer the following:
Use , and to write down a general rule which can be used to calculate the number of matches which will be used to create any figure in which several identical polygons have been drawn, with one side attached if:
= the total number of matches (for 2 attached squares )
= the total number of sides of the polygon (for a square, )
= the total number of polygons (if we draw 6 squares, )
TOTAL: 55
1
INFO SHEET WITH EXAMPLES:
NUMBER PATTERNS
1
First find the nth term:
Type 1: The linear pattern:
There is a common difference between consecutive numbers in a sequence; add or subtract the common difference to get to the next number.
1. Calculate the general term of the pattern: 4; 7; 10; 13
Answer:
Use the formula:
The general difference between the numbers are = 3.
Therefore: a = 3
T1 = 3(1) + b
4 = 3 + b
Therefore: b = 1
Tn = 3n + 1
2. Given the pattern 11; 8; 5; ... Calculate the 64th term in the pattern.
Answer:
T1 = 11
A = T2 – T1 = T3 – T2
A = 8 – 11 = 5 – 8 = -3
Tn = an + b, T1 = 11; n = 1; a = -3
11 = -3(1) + b
Therefore 14 = b
T64 = -3(64) + 14
Therefore: T64 = -178
Type 2
Tn = a + (n - 1)d
3. Write the following number pattern: 2; 5; 8; 11; 14; ... Calculate :
a)The nth termb) the 50th term
Answer
a)The difference is = 3.
a = 2 en d = 3
Tn = a + (n – 1)d
Tn = 2 + (n – 1)3
Tn = 2 + 3n – 3
Tn = 3n – 1
b)T50 = ? a = 2 en d = 3
n = 50
T = 2 + (50 – 1)3
T50 = 2 + 49 x 3
= 149
T50 = 149
1