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Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Crossed Ends

Age 11 to 14 Challenge Level:

Why do this problem?

This problem provides self-checking practice of two digit addition. However, fundamentally this problem is about introducing and developing students' understanding of algebraic notation.

Possible approach

This printable worksheet may be useful: CrossedEnds.pdf 

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 grids 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 over here?"

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 $x$.

Key questions

How are the numbers on the grid related to each other?
How can I represent this algebraically?

Possible extension

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?

Possible support

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 Sums