Q) A while ago you sent me your poem about teaching algebra using envelopes, which I had seen in a previous column. I used this successfully with my adult learners (I work in several community centres in the Midlands). Do you have any poems about the three averages?
A) I have one for mode, another for median and, to complete the package, have written one for mean.
The Middle Median Average
Numbers written here and there
Roughly recorded from the paper stare.
The values are written all jumbled
As from the experiment they tumbled.
Central tendency is required
Which one though, you enquired.
Feeling full of doubt
On how to work it out?
Read on for complete coverage
Of calculating median average.
Take each data value
Now arrange them anew,
Order them by their size
So you see the values rise.
The task's diminished
when this is finished.
Cross out equally from either end
Towards the centre you will tend.
14, 15, 16, 16, 20, 20, 20
As other items are shed
The median average is read
A value left in the middle!
Isn't this a diddle!
As you see from this sample
Sixteen is the example.
But what median ensues
When there are two values
Sitting in the middle?
How do we fiddle
To solve this riddle?
The two numbers you add,
Divide this by two and, aren't you glad,
You've a single value coverage
Of this the median average.
Means' Doing Sums
For the average that is really mean,
there's lots of calculations to be seen.
First add all the values of the data
(dividing this answer comes later
using the size of the data sample).
(Have a look at my example).
Total the number of items, that is,
With this, divide, for the mean
Here is an example working out the mean average number of letters per word in the poem "Means Doing Sums". First the number of letters for each word are written down.
5 5 4
3 3 7 4 2 6 4
6 4 2 12 2 2 4
5 3 3 3 6 2 3 4
8 4 6 5 5
5 3 4 2 3 4 6
4 1 4 2 2 7
5 3 6 2 5 4 2 4 5
4 4 6 3 3 4 7 6
To find the total number of letters in the poem, all the values of the words are added together and come to 252. To get the total number of words in the poem, including the title, just count the words, which gives 60.
So the mean average is found by dividing the total number of letters by the total number of words: 252Z60 = 4.2 So on average this poem has about 4.2 letters per word. This doesn't make much sense, because you can't have part of a letter in a word. A better average, in this case, would be the mode.
This can be easily found by creating a stem-leaf plot and looking for the longest "leaf"; in this case 4. So the modal average is 4 letters per word.
The median can also be calculated from this graph as the data is now arranged in order. Counting from either end leaves us with a median of 4 letters per word.
I found this quite interesting when I looked at the overall picture of word length. Just one of 8 letters and one of 12. I wonder if certain types of poems elicit similar patterns, or whether a particular poet has a signature pattern? I don't think so, but it could be a cross-curricular investigation. It seems that I tend to favour four-letter words.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.
Email your questions to Mathagony Aunt at firstname.lastname@example.org Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX
SOLUTIONS A curious number (Ages 14 to 16). The number is 102,564. You can find this curious number by starting with a dividend of 4 and dividing by 4, adding the digits of the answer to the dividend as you go along. Another curious number is the 18-digit number 105,263,157,894,736,842. When the 2 is moved from the end to the beginning, the new number is twice the original number. The same method can be used to find a 22-digit number ending in 7, such that the new number formed by moving the 7 from the end to the beginning is 7 times the original number.