# Key Eleven

## An Advanced Mathematics Lesson Starter Of The Day

Choose 4 keys on your calculator key pad that are positioned in the four corners of a rectangle. Use these keys to type in a 4-digit number going round your rectangle either clockwise or anticlockwise. Eg. 3146

Check that your 4-digit number is a multiple of eleven.

Can you prove that all 4-digit numbers formed this way are multiples of 11?

Is this strange result also true for numbers formed from keys on the corners of a parallelogram?

Note 1: The rectangles referred to above have horizontal or vertical sides.

Note 2: A divisibility test for eleven is to consider the alternating digit sum of the number. If this is divisible by eleven then the original number is also divisible by eleven. For example, if a four digit number has digits a, b, c and d then the alternating digit sum is a - b + c - d.

Share

Topics: Starter

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.

If you don't have the time to provide feedback we'd really appreciate it if you could give this page a score! We are constantly improving and adding to these starters so it would be really helpful to know which ones are most useful. Simply click on a button below:

Excellent, I would like to see more like this
Good, achieved the results I required
Satisfactory
Didn't really capture the interest of the students
Not for me! I wouldn't use this type of activity.

This starter has scored a mean of 2.0 out of 5 based on 1 votes.

Previous Day | This starter is for | Next Day

Let the rectangle have a width of $$w$$ key spacings and a height of $$h$$ key spacings. (The example in the diagram above has $$w = 2$$ and $$h = 1$$ )

Let the number be formed by going around the rectangle in a clockwise direction starting from the top left (unlike the example in the diagram above).

Let the first digit of the 4-digit number be $$x$$.

The alternating digit sum is $$x - (x+w) + (x+w+h) - (x+h)$$

$$= x - x - w + x + w + h - x - h$$

$$= 0$$ (which is a multiple of 11 so the 4-digit number must also be a multiple of 11)

Also any cyclic permutation of the 4-digit number will be divisible by 11 (accounting for the example in the diagram above)

This should also be applicable to parallelograms

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world. Click here to enter your comments.

Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon search box and some items chosen and recommended by Transum Mathematics to get you started.

## Numbers and the Making of Us

I initially heard this book described on the Grammar Girl podcast and immediately went to find out more about it. I now have it on my Christmas present wish list and am looking forward to receiving a copy (hint!).

"Caleb Everett provides a fascinating account of the development of human numeracy, from innate abilities to the complexities of agricultural and trading societies, all viewed against the general background of human cultural evolution. He successfully draws together insights from linguistics, cognitive psychology, anthropology, and archaeology in a way that is accessible to the general reader as well as to specialists." more...

For Students:

For All: