NCTCA 2017: Edmonton

Increasing Algebra and Number Sense through Fun Puzzles

Presented by:Rosalind Carson

B.Ed., B.Sc., M.Sc.

Mathematics Education Consultant

Materials: Bags of Pebbles and Wood Cubes

  1. Odd and Even Numbers
  1. Sums of Even Numbers, and Odd Numbers
  1. Multiples of Even Numbers, Odd Numbers
  1. What is the product of two consecutive even numbers plus one? (Blocks)
  1. What pattern is formed from summing consecutive Natural Numbers?
  1. What is the pattern connection between triangle numbers and square numbers?
  1. Figurative Numbers: Pythagoreans: How to Build these and explore the connections:

Table 1.1

Term Number / Triangular Numbers / Square Numbers / Pentagonal Numbers / Hexagonal Numbers
1 / 1 / 1 / 1 / 1
2 / 3 / 4 / 5 / 6
3 / 6 / 9 / 12 / 15
4 / 10 / 16 / 22 / 28
5 / 15 / 25 / 35 / 45
6 / 21 / 36 / 51 / 66
7 / 28 / 49 / 70 / 91
8 / 36 / 64 / 92 / 120
9 / 45 / 81 / 117 / 153
10 / 55 / 100 / 145 / 190
11 / 66 / 121 / 176 / 231
12 / 78 / 144 / 210 / 276
13 / 91 / 169 / 247 / 325
14 / 105 / 196 / 287 / 378
  1. What are the formulas for heptagonal, octagonal, decagonal numbers, etc.?
  1. Oblong Numbers: are numbers that form a rectangular array by n(n+1)
  1. Pentagonal Numbers as arrays:
  1. Use the pebbles to solve the following puzzles: deductively and inductively.
  1. Every pentagonal number is the sum of the square number and one previous triangle number.
  1. Hexagonal numbers: look at these squished into arrays:
  1. How are hexagonal numbers formed from other figurative numbers?
  1. An oblong number is the sum of consecutive even numbers:
  1. An oblong number is twice a triangle number:
  1. The sum of a number and the square of that number is an oblong number:
  1. The sum of two consecutive oblong numbers and twice the square number between them results in a square number:
  1. The sum of an oblong number and the next square number is a triangle number:
  1. The sum of a square number and an oblong number is a triangle number:
  1. The sum of two consecutive square numbers and the square of the oblong number between them results in a square number:
  1. Encourage students to look for more patterns and prove these algebraically.
  1. Every odd square number is the sum of eight times a triangle number plus one:
  1. Every pentagonal number is the sum of three triangle numbers:
  1. Hexagonal numbers are equal to the odd-numbered triangular numbers:
  1. Pythagorean Theorem with pebbles:
  1. Trapezoid proof of Pythagorean Theorem:
  1. Use blocks to find a clever way to sum 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1=
  1. What is the pattern when you sum(difference) a two digit number with its reverse digits?
  1. Let us square two digit numbers ending in 5. What is the pattern? Prove it!
  1. Take any two digit number ending in 9 – it can be expressed as the sum of the digits plus the product of the digits. (2 digit # ending in 9) = (sum of digits) + (product of digits). Inductive and deductive proof…play!
  1. Select any three digit number with all digits different from one another. Write all possible 2 – digit numbers that can be formed from the three digits selected earlier. Then divide their sum by the sum of the digits in the original 3-digit number:
  1. One plus the sum of squares of any three consecutive odd numbers is always divisible by 12:
  1. Polynomial Sudoku (created and shared by Thomas Wood, Calgary Board of Education)
  1. I hope you have fun working through all of these puzzles. If you want more professional development on playing with number to improve number sense: please email me at

Rosalind Carson

Rosalind Carson / 1