Download 2014Wisk_gr10_KW1_Ondersoek

GANSBAAI ACADEMIA
INVESTIGATION
March 2014
Total: 55
Time: 1 week
MATHEMATICS
Grade 10
EXAMINATOR
MODERATOR
L. Havenga
G. Edwards
INSTRUCTIONS
1. This document consists of 2 pages and an information page.
2. Write neat.
3. 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.
Number of triangles
Number of matches
Perimeter
1
2
(6)
3
4
7
15
9
20
n
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.
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?
1
(2)
(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
Number of matches
Perimeter
1
2
3
4
7
22
16
20
n
(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.
b)
copy and complete a table similar to the one in Task 1.
c)
Complete the summary table below and write down the rule in words you used to find the total
number of matches needed for n polygons for ANY polygon.
(7)
Number of sides of polygon
Number of matches for n polygons
3
(3 x (2) = 6)
4
TASK 4
(3 x (7) = 21)
5
6
8
[2]
Use your findings from tasks 1 to 3 to answer the following:
Use s , m and n 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:
s = the total number of matches (for 2 attached squares m  7 )
m = the total number of sides of the polygon (for a square, s  4 )
n = the total number of polygons (if we draw 6 squares, n  6 )
TOTAL: 55
2
INFO SHEET WITH EXAMPLES:
NUMBER PATTERNS
A = 8 – 11 = 5 – 8 = -3
Tn = an + b, T1 = 11; n = 1; a = -3
11 = -3(1) + b
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.
Therefore 14 = b
T64 = -3(64) + 14
Therefore: T64 = -178
1. Calculate the general term of the pattern: 4; 7; 10; 13
Type 2
Answer:
Tn = a + (n - 1)d
Use the formula: 𝑇𝑛 = 𝑎𝑛 + 𝑏
3. Write the following number pattern: 2; 5; 8; 11; 14; ... Calculate :
The general difference between the numbers are = 3.
a) The nth term
Therefore: a = 3
Answer
T1 = 3(1) + b
4=3+b
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
Therefore: b = 1
Tn = 3n + 1
2. Given the pattern 11; 8; 5; ... Calculate the 64th term in the pattern.
b) T50 = ? a = 2 en d = 3
n = 50
T = 2 + (50 – 1)3
T50 = 2 + 49 x 3
= 149
T50 = 149
Answer:
T1 = 11
A = T2 – T1 = T3 – T2
3
b) the 50th term
4