Number patterns, Strategic thinking
The Gauss Pairing Trick
A small change in how you look at numbers can reveal a much simpler way to add them.
THE KEY IDEA
Different pairs. The same total.
When adding consecutive numbers, pair the smallest with the largest. Move inwards, and each pair has the same total.
For the numbers 1 to 100, there are 50 pairs, each adding to 101. So the total is 50 × 101 = 5,050.
The important part is explaining why the pairing works, not simply remembering a formula.
YOUR TURN
Try it yourself
Find the sum of all the whole numbers from 1 to 20.
1 + 2 + 3 + ... + 20 = ?
How many pairs can you make, and what does each pair add to?
Need a hint?
Start with 1 + 20, then 2 + 19. Notice what stays the same.
Show the worked answer
There are 10 pairs, and every pair adds to 21.
10 x 21 = 210.
Check that each of the 20 numbers appears in exactly one pair.
A SHARED TEACHING REFERENCE
Teaching with this video
Use the video as a common starting point, then ask students to explain the structure in their own words.
Discussion prompts & teaching notes
Notice: What stays the same when the smallest and largest remaining numbers are paired?
Explain: Why are there 50 pairs when we use 100 numbers?
Check: Has every number been counted exactly once?
Extend: What changes if there is an odd number of terms? For 1 to 21, ten pairs total 22 each, with 11 left in the middle: 231 altogether.
Shared vocabulary: consecutive numbers, pair, pair total, number of pairs.