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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 offers an interesting context in which to introduce and develop 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 understand why it happens". Ask students to work in pairs and challenge them to come up with 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 when you put x in a different end cell. "What does this mean? Do you think it matters where we place the x?"

Set students to work in pairs, placing $x$ in various different cells and confirming that the proof that the totals are equal always holds true. They could try other cross shapes or grid sizes if they have time. Placing x at the centre of the cross can help students appreciate why the opposite cells will always add to the same total.

Move on to comparing N+W with S+E.
Suggest that they begin algebraically, and just look at numerical cases to check that they do what they expect. Ask someone to select a grid, a cross, and the position of $x$. Ask students to then supply and justify the expressions for all the 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.

Does anything interesting happen with the products of the ends of the crosses?
 

Possible support

This problem may be more accessible when applied to a 1-100 square (a size 10 grid) in which the relationships between the end points may be clearer to see.

An easier problem, also using 100 squares and place value like this is Diagonal Sums