Monday, June 2, 2014

Always True/ Sometimes True/ Never True

I was reminded about how much I love this question today when I took part in a crowdsourcing of questions courtesy of Tracy Zager (@TracyZager) via Twitter and Google docs. I have given this type of questions to students from all grades  (K to 12) and what is great is that it gets them reasoning and proving from a very early age. Consider the following:

Addition makes a number bigger
Most students (especially in the primary, junior grades) will say 'Always true' and back it up with examples. Yet there will be some who wonder about what happens when you add zero? Does this make the number bigger? And intermediate students will then begin to reason that adding a negative number actually makes the number smaller. So the answer is 'Sometimes true'
Sometimes a question helps them broaden their understanding of math terminology:

Two identical triangles can be put together to make a parallelogram

I know some students will say 'Sometimes true' offering the case where two right-angled isosceles triangles join to make a square (in green below) which, they think, is not a parallelogram. Others might make what they think is a more obvious parallelogram (in blue below).
This gives us a great opportunity to learn why all squares are parallelograms (quadrilaterals with two pairs of parallel sides). This leads into an understanding of why the area of a triangle (½×base×height) is simply half the area of a parallelogram (base×height)

Perhaps my two favourite questions I got today were:
A solid that has a square shadow is a cube
A solid that has a circular shadow is a sphere
It immediately got me thinking about other solids that might have these shadows. Or what about if I reverse the order of each statement?:
A cube has a square shadow
A sphere has a circular shadow

So here, for your delight, are some other Always true/Sometimes true/ Never true questions:

  • A rectangle is a square
  • When you cut a piece off a shape, you reduce its area
  • When you cut a piece off a shape, you reduce its perimeter
  • Bigger objects are heavier than smaller ones
  • The diagonals of a parallelogram are unequal in length
  • Multiplication makes numbers bigger
  • Division makes numbers smaller
  • The sum of four consecutive numbers is a multiple of 4
  • The sum of three consecutive numbers is a multiple of 3
  • The more you roll a dice, the more likely you are to get a 6.
  • The sum of two odd numbers is an odd number
  • The product of an even number and an odd number is an odd number.


  1. I really like the cutting a peice out of a shape questions.If you cut a peice off of a shape you reduce it's perimeter. Couldn't you cut a V shape (triangle) out of a rectangle? The sum of the two sides now in the perimeter must be larger than the third side which was formerly in the perimeter. But,if you cut the corner off the rectangle you have reduced the perimeter for the same reason: the two legs of the right triangle cut out had a larger sum that hypotenuse that is now in the perimeter.

    1. That's right so 'cutting a piece off a shape reduces its perimeter' is sometimes true; I like to see and hear students argue with their examples and counter examples.

  2. Beautiful Mike... I do keep asking similar questions to my students... But you have really not just extended but enriched my bank... :-)

    @Carisa, Thanks for posting your comment.. Learnt from that as well :)

    1. Thanks Rupesh, I'm glad you found this useful. Tracy Zager's Google doc which inspired this post can be found at