Why is the sum of two odd numbers even?
a. think of the problem 11 + 15. draw dots to represent each number.
b. circle pairs of dots to determine if any of the dots are left over.
c. is this true every time two odd numbers are added together? why or why not?

a) and b) in photo.

there is no point left outside the circles, then 11+15 is even.

c) yes it is, because if both numbers are odd, they will have one point outside the circles each, and those two points will form an other pair.

odd number: 2n+1

even number: 2m

(m and n represent any natural number)

2n+1 + 2k+1= the sum of two odd numbers

=2(n+k) +2

=2(n+k) +2 =2(n+k+1)

the sum of 3 natural numbers will be a natural number too. let be m=n+k+1 with m n and k natural numbers.

=2m even number

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."