Window Frames
Materials
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Task 4 ... Years 4 - 10
Summary
Students explore the structure of a 0-99 number
chart using frames which isolate particular cells of
the chart. What emerges is that no matter where
the frame is placed on the grid, special
relationships exist between the numbers in the
window. At one level the problem is about
searching for and finding these relationships. At
another it is about explaining their existence.
One 0-99 number
chart and a range of
plastic window
frames like the ones
in the photo
Content
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basic number facts
looking for
relationships between
numbers
explaining
relationships
algebraic
representation
Iceberg
A task is the tip of a learning
iceberg. There is always more to a
task than is recorded on the card.
Part of the iceberg is built into this task. For
example, in the photo, the numbers from 44 to 48
are showing in the window. A student might notice
that twice the middle number equals the sum of the
two end numbers. But what happens if we move
the frame?
The answer is that this property remains true. Our
hypothesis becomes: Wherever the frame is put,
twice the middle number equals the sum of the end
numbers. But how do we test this hypothesis?
Perhaps we reason that the number at one end is
the middle minus 2 and the number at the other
end is the middle plus 2. Therefore when the end
numbers are added the minus and plus 2 will
cancel leaving twice the middle.
Or perhaps we might reason that the left end
number is a, the middle is (a + 2) and the right end
number is (a + 4). Then:
Left + Right = a + (a + 4) = 2a + 4 = 2(a + 2)
But what happens if...
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we change the shape of the window to 3
cells long, or 7 cells, or 4 cells...?
we use our window frames to explore the
times table chart?
Note: There is a strong link between Window
Frames and Task 7, Consecutive Sums.
Whole Class Investigation
Tasks are an invitation for two
students to work like a
mathematician. Tasks can also be
modified to become whole class
investigations which model how a
mathematician works.
To convert this task to a whole class investigation
it is useful to have an overhead projector with a
transparent copy of the number chart and the
window frames as in the task. Students have a
paper copy of the chart and as you move the frame
on the OHP from place to place you ask particular
students to mark this frame on their paper grid.
Every student is then searching for relationships in
a different set of numbers governed by the same
window frame. If any student thinks they see a
relationship, it can be checked by other students.
Any hypothesis which results can be recorded on
the board so the discussion can shift to the notion
of proof.
At this stage, Window Frames does not have a
matching lesson on Maths300.