What do you notice about the sums in each of the following cases? Do you think it will be true for all pairs of such numbers? Explain your observations. You may use the paired arrangement to explain your thinking.
12 and 6 are a pair of even numbers. Choose 5 such pairs of even numbers. Add the numbers in each of the pairs.
Even + Even = Even. Examples: 4+6=10, 8+12=20, 14+16=30, 22+28=50, 100+200=300. All sums are even.
13 and 9 are a pair of odd numbers. Choose 5 such pairs of odd numbers. Add the numbers in each of the pairs.
Odd + Odd = Even. Examples: 3+5=8, 7+11=18, 15+17=32, 21+29=50, 99+101=200. All sums are even.
7 and 12 are a pair of odd and even numbers. Choose 5 such pairs of odd and even numbers. Add the numbers in each of the pairs.
Odd + Even = Odd. Examples: 3+4=7, 5+10=15, 11+20=31, 17+30=47, 25+50=75. All sums are odd.
Explanation
(a) When two even numbers are added, the sum is always even because both numbers have complete pairs. (b) When two odd numbers are added, the sum is always even because the two unpaired items form a pair. (c) When an odd and even number are added, the sum is always odd because the unpaired item from the odd number remains unpaired.