BEYOND PYTHAGORAS | |
The numbers 3, 4 and 5 satisfy the condition
3² + 4² = 5²
(smallest number)² + (middle number)² = (largest number)²
The numbers 3, 4 and 5 can be the lengths – in appropriate units – of the sides of a right-angled triangle.
By observation:
(3, 4, 5), (5, 12, 13) and (7, 24, 25) are all called Pythagorean triples because they satisfy the condition.
a² + b² = c² in a right angled triangle
With the family of right-angled triangles for which all the lengths are positive integers and the shortest is an odd number.
THE TASK:
aaaa aaaaaaaaaa aaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaa aaaaaaa aaaaaaa aaaaa aaaaaa aa aaaaaaaaa aaaa aaaa aa aaaa aaaaa aaaaaaa
aaaaaaaa aaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaa aaaa
Part 1
Investigate the family of Pythagorean triples linked to the triangles (3, 4, 5), (5, 12, 13) and (7, 24, 25) where the shortest side is an odd number, all 3 sides are positive integers and the two largest sides differ by one.
Part 2
Investigate other families of Pythagorean triples.
The Investigation should include three strands:
1. Making and monitoring decisions to solve problems
Find & explain patterns in the ‘triples’. Explain & set your own questioning agenda.
2. Communicating mathematically
Find formulae which ‘explain’ the above. Define your variables & be precise with your ‘mathematics’.
3. Developing skills of mathematical reasoning
‘Prove’ all/any formulae which you may have found.
This is to be written up & brought to the next G & T session on 19th Feb.
-----------------------
4
3
5
12
5
13
24
7
25
The number 5, 12 and 13 can also be the length – in appropriate units – of a right-angled triangle
This is also true for the numbers 7, 24 and 25
c
b
a
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa...