concepts and methodology in mathematics education - Nelson ...

Practical ActivityAs an investigation, ask your learners to find the triangular number that has 7 dots on a side. What about195 dots on each side? Is 57 a triangle number?17. Perfect NumbersA perfect number is defined as a number that equals the sum of its factors, apart from itself. Examplesinclude: 1 + 2 + 3 = 6; 1 + 2 + 3 + 4 + 7 + 14 = 28.You might be interested to know that perfect numbers are very rare and the first four perfect numbers are 6;28; 496 and 8128. The tenth perfect number is more than 50 digits long! Note that these are all evenperfect numbers. Note too that they are also triangular numbers. All known perfect numbers end in either6 or 8. No one knows whether there are any odd perfect numbers. All known perfect numbers aredifferences of certain powers of 2.Examples are: 6 = 2³ - 2¹ (8 – 2= 6) 28 = 2 5 – 2² (32 – 4 = 28) 496 = 2 9 – 2 4 (256 – 16 = 496)18. Squares and Square Roots; Cubes and Cube RootsInteresting investigation and discussion:Legend has it that there was a Chinese emperor who wanted to reward one of his subjects. Foolishly, theEmperor asked the man what reward he wanted. The man pointed to a chessboard and asked for a singlegrain of rice on the first day; two grains on the second square; double that on the third square and so on. Askyour learners to work out on which square the man would have 1 million grains of rice. On which squarewould he have 10 million grains?An investigation could be to compare the wealth of two subjects, if the other chose 1 million grains on eachsquare of the chessboard. On which square would they have almost equal totals of rice? Interestingly theEmperor would need 26 3 (why not 26 4 ?) grains on the last square. This equates to more than 9 000 000 000000 000 000 grains on the last square. This could lead into a discussion on how big are big numbers – and anintroduction to scientific notation.We have dealt with square numbers earlier in this unit. They can be depicted as squares of dots, linking inlearners’ minds the concept of the area of a square being S 2 . Cubed numbers are written as 2 × 2 × 2 = 2 3 = 8and can be linked in learners’ minds with the concept of the volume of a cube being S 3 .3 × 3 × = 3 2 = 9 2 × 2 × 2 = 2 3 = 8Area of square = 9 Volume of cube = 838