Math Detours · Grade 5 · Around the corner · 2

Square numbers and triangle numbers

Some numbers make squares of dots, some make triangles. Two neighbouring triangles always make a square.

Name ____________________Date ____________

Tonight's idea

Some numbers of dots make a perfect square. Four rows of four is 16. We say "4 squared is 16". The square numbers start 1, 4, 9, 16, 25.

Other numbers make a triangle: one dot, then a row of two, then three, and so on. Four rows is 1 + 2 + 3 + 4 = 10. The triangle numbers start 1, 3, 6, 10, 15.

+ = triangle: 6 triangle: 10 square: 16
The 3rd and 4th triangle numbers fit together into the 4th square number.

Add two triangle numbers that are neighbours in the list. You always get a square number: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16.

The picture shows why. Cut a square of dots just above its diagonal. The bottom piece has rows of 1, 2, 3, 4. The top piece has rows of 3, 2, 1. Both are triangles. Any size of square splits the same way.

Watch one: the 5th of each

The 5th square number is 5 × 5 = 25.

The 5th triangle number is 1 + 2 + 3 + 4 + 5 = 15.

Cut the triangle out of the square: 25 − 15 = 10 dots are left. That is the 4th triangle number, as the picture promised. So 10 + 15 = 25.

Next year's sheet has a fast way to add 1 + 2 + 3 + … + 100: mathdetours.com/grade-6/sheets/b5-gauss.html.

Your turn

  1. Write the first eight square numbers. Then write the first eight triangle numbers.
  2. The 5th and 6th triangle numbers are 15 and 21. (a) Which square number do they add up to? (b) Draw that square in dots. Draw one line that cuts it into the two triangles.
  3. Two triangle numbers that are next to each other add up to 100. Which two are they?
  4. The 20th triangle number is 210. Do not add up twenty numbers. (a) Find the 21st triangle number. (b) Find the 19th triangle number by using a square.

Stretch: times 8, add 1

Take a triangle number. Multiply it by 8 and add 1. Try it with 1, 3, 6 and 10. What kind of number do you get every time? Guess what 15 will give, then check.

Answers

  1. Squares: 1, 4, 9, 16, 25, 36, 49, 64. Triangles: 1, 3, 6, 10, 15, 21, 28, 36. (36 is on both lists.)
  2. (a) \(15 + 21 = 36\), which is 6 × 6. (b) A 6 by 6 square of dots. The line runs just above the diagonal. One side has rows of 1, 2, 3, 4, 5, 6 (21 dots). The other has rows of 5, 4, 3, 2, 1 (15 dots).
  3. 100 is 10 × 10, so they are the 9th and 10th triangle numbers: 45 and 55.
  4. (a) Add a row of 21: \(210 + 21 = 231\). (b) The 19th and 20th together make the square 20 × 20 = 400. So the 19th is \(400 - 210 = 190\). Check: taking the row of 20 away from 210 also gives 190.

Stretch. You get 9, 25, 49, 81. These are square numbers, and the squares of odd numbers: 3 × 3, 5 × 5, 7 × 7, 9 × 9. For 15 the guess is 11 × 11 = 121, and \(8 \times 15 + 1 = 121\). It always works. Eight copies of a triangle can be fitted around one single dot to make a square.

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