{"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":"2026-03-21T11:01:30","modified_gmt":"2026-03-21T11:01:30","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

In grades 1\u20132, we began 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

Children set to work building odd numbers using link cubes, alternating colours to form L patterns \u20141 cube, then 3, then 5, and so on. Soon the tables were covered with bright, striped L towers of odd-number models.<\/p>\n\n\n\n

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

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

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

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

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

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

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

We explored this discovery together:<\/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 was astonishingly simple\u2014and beautifully powerful.<\/p>\n\n\n\n

    Next, we counted 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 rediscovered one of the most elegant ideas 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 asked,
      \u201cIf odd numbers hide in squares\u2026 where do even numbers hide?\u201d<\/strong><\/p>\n\n\n\n

      This question launched a new wave of building, arranging, testing, and noticing\u2014another 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

      Working in pairs, students built two identical sets of the first four natural numbers.<\/p>\n\n\n\n

      They experimented with combining these sets\u2014and to their delight, discovered that they could 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 their secret structure.<\/p>\n\n\n\n

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

      And then the room exploded with creativity.<\/p>\n\n\n\n

      \n\n\n\n

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

      Students began rearranging number models into patterns, pictures, and playful designs.
      Mathematical structure became artistic structure.
      The classroom became a gallery of joyful \u201cmart\u201d.<\/p>\n\n\n\n

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

      Visiting Their Buddy Class<\/strong><\/h2>\n\n\n\n

      These younger students brought their discoveries to their older buddy class in grade 6, 7 or 8.<\/p>\n\n\n\n

      With great pride, they demonstrated how odd numbers form squares, how natural numbers fit together, and how patterns emerge when you simply look closely enough.<\/p>\n\n\n\n

      In return, the older students showed them how 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

      For both groups, the models became bridges between arithmetic, geometry, and algebra.<\/p>\n\n\n\n

      \n\n\n\n

      A Surprise in Grade 11<\/strong><\/h2>\n\n\n\n
      \n

      Later, when an education leader visited the grade 2 classroom, they watched the students confidently exploring ideas usually reserved for high school courses.<\/p>\n<\/div>\n\n\n\n

      They were stunned:<\/p>\n\n\n\n

      \u201cWe teach these concepts in grade 11.\u201d<\/strong><\/p>\n\n\n\n

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

      Then added:<\/p>\n\n\n\n

      \u201cBut I\u2019ve never seen these physical representations.\u201d<\/strong><\/p>\n\n\n\n

      And with that one sentence, the room fell quiet.
      What else might be possible if students across the grades were 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 did exactly that.<\/p>\n\n\n\n

      They became tools for seeing-as<\/strong>\u2014
      seeing number as shape,
      seeing pattern as structure,
      seeing mathematics as something you can hold, build, rotate, and understand.<\/p>\n\n\n\n

      They turned ideas into objects, and objects into insight.<\/p>\n\n\n\n

      And the children, without fear or hesitation, stepped right into the heart of mathematical thinking.<\/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\u2014but they\u2019re about as sneaky as a giraffe in sunglasses.<\/p>\n\n\n\n

      Odd numbers, for example, 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\u2014\u201cmore elbow room,\u201d they say.
      They line up neatly inside the sides, trying to look very geometric<\/em> and official,
      but they giggle whenever someone walks by yelling, \u201cReady or not!\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\u2014evidence of their journey through the puzzling, delightful world of odd and even numbers.<\/p>\n\n\n\n