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Summing Consecutive Numbers

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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?

Your Number Is...

Stage: 3 Challenge Level: Challenge Level:1

Why do this problem?

This problem provides a great opportunity to introduce a very useful way of representing operations on unknown numbers visually and symbolically (algebraically).

Possible approach

Ask learners to each think of a number and read out the following instructions:

Look surprised, when all of them reveal they have ended up with the same number.
"Did you all start with the same number then?"
"Perhaps none of you started with a fraction, or decimal, or a negative number - shall we try it again with one of those?"
More surprise that they still end up with the same number.
Now show the interactivity, or alternatively draw the representation on the right hand side of the screenshot above, and ask learners to think about how the visual representation helps us to understand what is happening. Give them a couple of minutes to discuss in pairs, and then draw the class together to share their insights.
Next, suggest that the visual representation might be too time-consuming, particularly if we'd added and multiplied by larger numbers. Introduce a parallel representation using algebra:
$$x$$ $$x+3$$ $$2(x+3) \text{ or }2x+6$$ $$2x+10$$ $$x+5$$ $$5$$
Now invite three members of the class up to the board to each use a different representation simultaneously for some other "think of a number" sets of instructions.
For example: 
Think of a number
Multiply by 4
Add 2
Halve it
Add 7
Divide it by 2
Take away the number you first thought of.
This is what the board might look like when the three learners have finished:

 Photo of work on board
Finally, challenge the class to come up with their own examples of "Think of a number" instructions, which will lead to an anticipated solution. Encourage them to record their working using both the visual and the symbolic representation.

At the end of the lesson, a selection of students can read out their instructions for the class to try, and then reveal the anticipated solution (written in advance) with a flourish.

Key questions

How can I use the visual or symbolic representation to explain the "Think of a number" puzzle?
How can I use the visual or symbolic representation to create a "Think of a number" puzzle?

Possible extension

Your Number Was... is a suitable follow-up activity.
Think of Two Numbers is a challenging extension.

Possible support

Focus on the visual representations until learners are secure with this, before introducing symbolic notation.