{"id":54,"date":"2026-01-25T01:15:21","date_gmt":"2026-01-25T01:15:21","guid":{"rendered":"https:\/\/learnx.ca\/um\/?page_id=54"},"modified":"2026-04-24T07:41:05","modified_gmt":"2026-04-24T07:41:05","slug":"odds-evens","status":"publish","type":"page","link":"https:\/\/imaginethis.ca\/u\/odds-evens\/","title":{"rendered":"Odds & evens"},"content":{"rendered":"\n

Where do odd numbers hide?<\/h2>\n\n\n\n

Let’s begin with a simple, playful question:<\/p>\n\n\n\n

\u201cWhere do odd numbers hide?\u201d<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Young children may work building odd numbers using link cubes, alternating colours to form L patterns…1 cube, then 3, then 5, and so on. Soon the tables are covered with bright, striped L towers of odd-number models.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Then something magical happens.<\/p>\n\n\n\n

Using nothing but their eyes and their sense of pattern, children start arranging the models to fit like spoons. As the shapes click together, they notice a surprising fit.<\/p>\n\n\n\n

\n
\"\"<\/figure>\n<\/figure>\n\n\n\n

The odd numbers form perfect squares.<\/strong><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

\u201cOdd numbers hide in squares!\u201d a few children exclaim, wide\u2011eyed.<\/p>\n\n\n\n

They explore this discovery further:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
    \n
  • The first odd number makes a 1 \u00d7 1<\/strong> square.<\/li>\n\n\n\n
  • The first two odd numbers make a 2 \u00d7 2<\/strong> square.<\/li>\n\n\n\n
  • The first three odd numbers make a 3 \u00d7 3<\/strong> square.<\/li>\n\n\n\n
  • The first four odd numbers make a 4 \u00d7 4<\/strong> square.<\/li>\n\n\n\n
  • The first five odd numbers make a 5 \u00d7 5<\/strong> square.<\/li>\n\n\n\n
  • The first ten make a 10 \u00d7 10<\/strong> square.<\/li>\n<\/ul>\n\n\n\n

    The pattern is astonishingly simple, yet beautifully powerful.<\/p>\n\n\n\n

    Let’s count the total number of blocks:<\/p>\n\n\n\n

    \"\"<\/figure>\n\n\n\n
      \n
    • The first odd number needs 1<\/strong> block.<\/li>\n\n\n\n
    • The first two need 4<\/strong> blocks.<\/li>\n\n\n\n
    • The first three need 9<\/strong> blocks.<\/li>\n\n\n\n
    • The first four need 16<\/strong> blocks.<\/li>\n\n\n\n
    • The first five need 25<\/strong> blocks.<\/li>\n\n\n\n
    • The first ten need 100<\/strong> blocks.<\/li>\n<\/ul>\n\n\n\n

      And so, through touch and sight and play, children rediscover an elegant idea in mathematics:<\/p>\n\n\n\n

      The sum of the first n<\/em> odd numbers is n\u00b2<\/em>.<\/strong><\/p>\n\n\n\n

      \n\n\n\n

      Where Do Even Numbers Hide?<\/strong><\/h2>\n\n\n\n

      Naturally, someone asks,
      \u201cIf odd numbers hide in squares\u2026 where do even numbers hide?\u201d<\/strong><\/p>\n\n\n\n

      This question launches a new wave of building, arranging, testing, and noticing. Another journey of mathematical discovery.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Where Do Natural Numbers Hide?<\/strong><\/h2>\n\n\n\n

      Children built two identical sets of the first four natural numbers.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      They experiment with combining these sets and discover that they may be arranged to form the exact shapes made by the first four even numbers.<\/p>\n\n\n\n

      In their hands, the natural numbers revealed a secret structure.<\/p>\n\n\n\n

      \n\n\n\n

      Math Art (or \u201cMart\u201d)<\/strong><\/h2>\n\n\n\n

      Children rearrange number models into patterns, pictures, and playful designs.
      Mathematical structure becomes artistic structure.
      The classroom fills with joyful \u201cm-art\u201d.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n
      \n\n\n\n

      As they travel through grades<\/strong><\/h2>\n\n\n\n

      These same physical models can be used to derive algebraic formulas for the sums of odd, even, and natural numbers.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      These models become bridges across arithmetic, geometry, and algebra.<\/p>\n\n\n\n

      \n\n\n\n

      A Surprise in High School<\/strong><\/h2>\n\n\n\n

      We teach these concepts in grade 11.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n


      What else might be possible if students across the grades are engaged in seeing<\/em> and understanding<\/em> mathematics in such powerful and accessible ways?<\/p>\n\n\n\n

      \n\n\n\n

      Seeing-As<\/strong><\/h2>\n\n\n\n

      The Canadian poet\u2011philosopher Jan Zwicky writes that metaphors and representations invite us to \u201clook at things like this.\u201d<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      These classroom models do exactly that.<\/p>\n\n\n\n

      They become tools for seeing-as<\/strong>…
      seeing number as shape,
      seeing pattern as structure,
      seeing mathematics as something you can hold, build, manipulate, and understand.<\/p>\n\n\n\n

      They turn ideas into objects to think with, and into insights.<\/p>\n\n\n\n

      So, where do numbers hide when they play hide and seek?<\/strong><\/p>\n\n\n\n

      Well\u2026 numbers aren\u2019t exactly subtle.
      They think<\/em> they are, but they\u2019re about as sneaky as a giraffe in sunglasses.<\/p>\n\n\n\n

      Odd numbers are convinced they\u2019re invisible if they stand inside perfect squares.
      You can still see them peeking out of the corners, counting themselves over and over:
      \u201c1, 3, 5\u2026 shh!! Don\u2019t give us away!\u201d<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Even numbers prefer rectangles, with \u201cmore elbow room\u201d.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      And natural numbers?
      Oh, they’re mischievous!
      They hide inside the even numbers, who find this extremely<\/em> annoying.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      What do you see?<\/strong><\/h4>\n\n\n\n

      How many odd numbers are hiding in the blue square?<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n
      \n\n\n\n

      Odds are tricky<\/h2>\n\n\n\n

      It started during a math\u2011for\u2011teachers course that the group eventually nicknamed the math therapy<\/em> course. <\/p>\n\n\n\n

      In one of the sessions, what began as a routine investigation of odd an even numbers turned into a moment of clarity\u2014one of those small mathematical awakenings that makes everyone sit back and say, \u201cWait\u2026 why didn\u2019t anyone show it to me like this before?\u201d<\/p>\n\n\n\n

      The lyrics in the song below are based on the teachers\u2019 comments\u2014collected, treasured, and performed through song and animation, as evidence of their journey through the puzzling, delightful world of odd and even numbers.<\/p>\n\n\n\n