Use the commutative and associative laws to add a series of numbers.

Name Section Date

Activity 7.7 Use the commutative and associative laws to add a series of numbers.

Focus Commutative and associative laws

Time 15-20 minutes

Group size 2-3

Background Legend has it that while still in grade school, the mathematician Carl Freidrich Gauss (1777-1855) was able to add the numbers from l to l00mentally. Gauss did not add them from left to right; instead, he paired the numbers I

and 99, 2 and 98, 3 and 97, and so on, such that each pair added to 100. In this activity, you will use a method similar to Gauss’s method to add a selies of numbers.

l. Follow the steps given below to simplify the expression. You will be pairing numbers as

Gauss did (see background).

1+2+3+4+5+6+7+8+9+10

Which numbers will you pair up?

What will each pair of numbers add to?

How many pairs will there be? Is there any number left over?

Calculate the sum using the pairs.

Check your answer by adding the numbers from left to right. Use a calc ulator if you wish.

Collaborative Learning Activities Developmental Mathematics 289

2. Explain how the associative and commutative laws of addition were used in your calculations above.

3. Next, use Gauss’s method to find the sum of the first 100 natural numbers:

1 + 2 + 3 + … + 48 + 49 + 50
+ 51 + 52 + … + 97 + 98 + 99 + 1 00

Sum of each pair

Number of pairs Leftover number Total sum

Compare your – group’s answers with those of another group.

4. Using a method similar to Gauss’, find the sum of the first 200 natural numbers.

Sum of each pair Number of pairs Leftover number Total sum

Conclusion As you saw in this activity, the properties of real numbers can help you quickly add long sums of numbers. Use this method whenever you need to add mentally. You might even impress your friends with your mathematical prowess!

290 Developmental Mathematics Collaborative Learning Activities