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Algebra Foundations

You will apply the process of mathematical problem solving to solve a real-life problem about adding up the integers from 1 to n.

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After completing this tutorial, you will be able to complete the following:

- Find the sum of a series of integers from 1 to n by finding the sums of pairs of numbers (Gauss’ pattern).

In this Activity Object, students will use problem solving strategies to notice patterns and figure out how to add up integers from 1 to n.

This same problem was given to the German mathematician, Carl Gauss, when he was only 7 years old. His teacher asked the class to add up the numbers from 1 to 100. He quickly spotted a pattern to find the correct answer. Instead of computing the answer the 'long way,' Gauss added up 100 pairs of numbers, each pair totaling 101.

For example,

Gauss might have listed the numbers from 1-100:

1 + 2 + 3 + ..........+ 98 + 99 +100

Then listed the numbers backwards:

100 + 99 + 98.....+ 3 + 2 + 1

And added each pair of numbers:

1 + 100 = 101, 2 + 99=101, 3 + 98 = 101, etc.

He noticed that each pair added up to 101, so he could multiply 101 by the total amount of numbers.

101 ´ 100

Since he added each number twice, he divided by 2 to find the answer.

Here is the formula for the pattern:

Other important vocabulary includes:

Gauss - a German mathematician who is sometimes called the "Prince of Mathematics"

pattern - a set of numbers or objects in which all of the members in that set are related to each other by a specific rule

consecutive integers - whole numbers that follow each other in order.

For example: 1, 2, 3, 4, 5, 6...

Be sure that students are familiar with the concept of adding consecutive integers together.

For example:

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

Approximate Time | 30 Minutes |

Pre-requisite Concepts | Learners should be familiar with the area of quadrilaterals. |

Course | Algebra Foundations |

Type of Tutorial | Problem Solving & Reasoning |

Key Vocabulary | problem solving, Gauss, patterns |