Depth 1 · 36 s
Two triangles make a rectangle
1, 3, 6, 10 - the numbers that make triangles of dots. Two copies make a rectangle, which is how to add 1 to 100 in a moment.
About this video
The ten pins at a bowling alley stand in a triangle: rows of 1, 2, 3 and 4. Numbers that can be laid out like that are the triangular numbers: 1, 3, 6, 10, 15, 21 and on, each one the last plus the next counting number. This short animation builds the triangle of 10 row by row.
Then it copies the triangle, turns the copy over and pushes the two together. They make a rectangle of 4 rows of 5 dots, 20 in all, so one triangle is half of it: 10. The same trick works for every triangle, and it gives a shortcut for adding 1 + 2 + 3 all the way to 100: two such triangles make a rectangle of 100 × 101 = 10,100 dots, so the sum is 5,050. A famous story says Carl Friedrich Gauss found it as a schoolboy. The video ends with one more surprise: two triangular numbers next to each other make a square, like 6 + 10 = 16 = 4 × 4. This video goes with the Triangular Numbers lesson of Number Theory Level 1 on Knaph.
This video shows depth 1 of Triangular numbers.
Learn Triangular numbersTranscript
- 0:00Which numbers can make a triangle of dots?
- 0:04A row of one, then two, then three, then four. One, three, six, ten.
- 0:11Copy it, turn it over, push them together.
- 0:14A rectangle, four by five, twenty. One triangle is half, ten.
- 0:19One to a hundred? Half of a hundred times a hundred and one. Five thousand and fifty.
- 0:26Two triangles next to each other make a square. Six and ten are sixteen.
- 0:31Triangles, rectangles and squares, all made of dots.