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Why do this problem?
provides self-checking practice of two digit addition.
However, fundamentally this problem is about introducing and
developing students' understanding of algebraic notation.
Display a number grid and select a 'square cross'. Ask one side of
the class to add the top and bottom numbers, and the other side to
add the far left and far right numbers. "Surprised?" Repeat with a
'square cross' of a different size.
Hand out some number
and ask students to check what happens with other square
crosses. As students become convinced that the totals will always
be equal, suggest they move on to 'rectangular crosses'.
"It's not enough to know what happens, we need to know why"
. Stop students
calculating in order to work in pairs trying to phrase a really
clear explanation of why the totals always match. Bring the class
together to compare explanations. If it is not suggested, select a
grid and introduce the idea of labelling the top cell $x$ and ask
how the other three cells at the end of the cross could be
labelled. Then add the expressions in the opposite cells. As often
as possible, ask students to relate specific expressions to the
numerical context, "what does it mean here? would it still be true
Ask for a volunteer to demonstrate how the technique would work on
a different grid or cross shape. Do the two totals still match? Ask
for a volunteer to fill in the ends of a cross where you have put x
in a different end cell. "What does this mean? Do you think it
never matters where we put the x?"
Set students to work in pairs, placing $x$ in various different
cells and confirming that the proof that the totals are
equal still holds
true. They could
try other cross shapes or grid sizes if they have time..
Move on to comparing N+W with S+E.
Suggest that we begin
algebraically, and just
look at numerical cases to check that they do what we expect.. Ask
someone to select a grid, a cross and the position of $x$. Students
supply and justify the expressions for the other end cells.
"How do the totals compare?" (expect comments like not equal, a
fixed difference). Ask students to pick their own cross of the same
size and on the same grid, and confirm that the algebraic
predictions are correct. Ask students to repeat this process for
their choices of grid, cross and position of $x$.
How are the numbers on the grid related to each other?
How can I represent this algebraically?
If I have a cross and tell you what (S+E)-(N+W) comes to, can you
work out what my cross looks like?
Some students may like to generalise across the grid sizes.
What happens with the products of the ends of these crosses?
This problem may be more accessible when only applied to a 100 square
as the place value
emphasises the structure. An easier problem, also using 100 squares
and place value like this is Diagonal