Colin's Teaching Resources
The joys of timetabling instrumental lessons
This page was last edited on 13/08/05
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| Background | Latin Squares | A small rota |
| Larger Rotas | Problems & issues |
Background
If your school is anything like mine, you have visiting peripatetic staff who give instrumental lessons. Pupils need to attend instrumental lessons on a rota basis so that the impact upon their academic lessons is kept to a minimum. So, every September, some Heads of Music, in Secondary Schools up and down the country, devise and display timetables in their departments which tell pupils when their lessons will be.
I say SOME Heads of Music, because sensible folk delegate this task to the peripatetic staff concerned. So Mrs. Flobsax devises the Woodwind timetable while Mr. Hornpet does the Brass rota, and so on.
In common with other schools, not all our visiting staff are with us for a whole day. Moreover, some Year 10 and 11 pupils are not keen, willing or able to attend music lessons which clash with other GCSE subjects. In short, so many factors conspire against peripatetic staff working out their own timetables that I have foolishly taken on this duty myself each year. At least this means that, if a member of staff feels pupils are leaving their lessons too often or for too long, I can be approached directly.
I have, therefore, become very experienced (I will not say 'skilled') in producing rotas over the years, and this web page contains some of the ploys I have used in the past. As I now expect my assistant to devise her own timetables I thought she might find this page helpful. (She'll probably be better at it than me!)
Latin Squares
The task became somewhat easier when I discovered Latin Squares. A Latin Square is a square of numbers in which each number occurs once in each row and once in each column. For example, a Latin Square of Size 5 and Step 1 (i.e. each number from left to right adds one to the previous number) would be:-
LS 5/1 (This is my own method of labelling)
| 1 | 2 | 3 | 4 | 5 |
| 2 | 3 | 4 | 5 | 1 |
| 3 | 4 | 5 | 1 | 2 |
| 4 | 5 | 1 | 2 | 3 |
| 5 | 1 | 2 | 3 | 4 |
For another Latin Square of Size 5 we could use a Step of 2 (i.e. each number from left to right adds two to the previous number) we would then have:-
LS 5/2
| 1 | 3 | 5 | 2 | 4 |
| 2 | 4 | 1 | 3 | 5 |
| 3 | 5 | 2 | 4 | 1 |
| 4 | 1 | 3 | 5 | 2 |
| 5 | 2 | 4 | 1 | 3 |
Question: Which square does one use? Let's consider an example.
A small rota
A school day might be organised like the one below in which many of the lessons might be 'doubles' (i.e. Lessons 2 and 3 might be one subject)
A TYPICAL SCHOOL DAY
| 8.45 | Registration and Form time |
| 9.00 | Assembly or PSE (Called Lesson 1) |
| 9.25 | Lesson 2 |
| 10.00 | Lesson 3 |
| 10.35 | Break |
| 10.55 | Lesson 4 |
| 11.25 | Lesson 5 |
| 12.00 | Lunch |
| 1.00 | Registration |
| 1.05 | Lesson 6 |
| 1.40 | Lesson 7 |
| 2.15 | Movement time |
| 2.20 | Lesson 8 |
| 2.55-3.30 | Lesson 9 |
Suppose you have a visiting String teacher, Mrs. Niloiv who arrives on Tuesdays at 10.00, hot foot from the Primary School up the road. She is with you for an hour and 40 minutes, so she leaves at 11.40, and she teaches 5 groups of pupils as follows:-
Group 1 |
Alison Bow (Year 7) |
Grade 3 Violin |
Group 2 |
Adam Resin (Year 9) and Peter Nut (Year 8) |
Grade 1 Viola |
Group 3 |
Sally Hair (Year 7) and Jenny Scroll (Year 7) |
Grade 2 Violin |
Group 4 |
David Bridge (Year 9) and Dean Dee (Year 9) |
Grade 4 Violin |
Group 5 |
Susan Peg (Year 11) |
Grade 6 Viola |
Below, I've added the lesson times to the two Latin Squares (LS 5/1 and LS 5/2) you saw earlier to give two possible timetables. I've also added the details of the subjects Susan Peg has on Tuesdays. Although none of the String lesson slots is entirely at break, two do occur during part of break. One assumes that the French teacher will have made all his or her teaching points in the early part of the lesson and a friend can jot down homework if it is issued at the end of the lesson (but that's another story!). Hence, 10.20-10.40 is not too bad a slot to have a music lesson and 10.40-11.00 is quite good too. [Were you ever to have this situation in real life and there was no way to change it, it is essential to explain the constraints under which you have to work to your colleagues. After all, it is not your fault the County have specified that Mrs. Niloiv visits at awkward times.]
Timetable One LS 5/1
| Week 1 | Week 2 | Week 3 | Week 4 | Week 5 | Susan's lessons | |
| 10.00-10.20 | 1 | 2 | 3 | 4 | 5 | French |
| 10.20-10.40 | 2 | 3 | 4 | 5 | 1 | French (15mins) |
| 10.40-11.00 | 3 | 4 | 5 | 1 | 2 | History (5 mins) |
| 11.00-11.20 | 4 | 5 | 1 | 2 | 3 | History |
| 11.20-11.40 | 5 | 1 | 2 | 3 | 4 | History |
Timetable Two LS 5/2
| Week 1 | Week 2 | Week 3 | Week 4 | Week 5 | Susan's lessons | |
| 10.00-10.20 | 1 | 3 | 5 | 2 | 4 | French |
| 10.20-10.40 | 2 | 4 | 1 | 3 | 5 | French (15mins) |
| 10.40-11.00 | 3 | 5 | 2 | 4 | 1 | History (5 mins) |
| 11.00-11.20 | 4 | 1 | 3 | 5 | 2 | History |
| 11.20-11.40 | 5 | 2 | 4 | 1 | 3 | History |
Obviously, Susan is going to miss a lot of French and History whatever you do. But I hope you will agree that Timetable Two is better. I expect you can see why. So the answer to the question posed earlier is, "The Latin Square which avoids clashes on consecutive weeks as far as possible".
I have tried solutions such as the following, but they do not seem to work. They might work for you, but you would need to have an arrangement with the History teacher if Susan is regularly going to arrive late for that subject. Is such a solution fair on the other groups?
Possible 3rd solution.
| Week 1 | Week 2 | Week 3 | Week 4 | |
| 10.00-10.18 | 1 | 4 | 2 | 3 |
| 10.18-10.36 | 2 | 3 | 1 | 4 |
| 10.36-11.00 | 5 | 5 | 5 | 5 |
| 11.00-11.20 | 3 | 1 | 4 | 2 |
| 11.20-11.40 | 4 | 2 | 3 | 1 |
You may think that 5 groups is easy to cope with. Consider, however, that there are 120 possible arrangements of 5 numbers. Ask any mathematician what '5-Factorial' is, or talk to any self-respecting bellringer. (OK that's my main hobby). Now for bigger numbers.
Larger Rotas
If your brass 'peri' teaches for 5 hours and you want to send 15 groups you'll need a better method than trial and error. Here's what you do.
We'll stick with the arrangement for the school day outlined above. If your Brass teacher visits from 10.30 to 4.00 (allow 30 minutes for their lunch break) you can produce a pretty good rota. (Luckily, peripatetic staff are often willing to work 'after the bell' and some have to do so to clock up sufficient hours.)
How many of the possible Latin Squares are useful? I use a simple method to make them. You saw earlier that I describe Latin Squares with two numbers. The size and the step. Not all number combinations work. If the step is a factor of the size (other than 1) then you will get odd results. If you try Size 10 and step 2 the top line is 1 3 5 7 9 1 3 5 7 9. Not all the numbers appear. You will also get odd results if the size and step share factors.
Downloads
Try this out by downloading and playing around with either the Excel Spreadsheet or the Lotus 1-2-3 Spreadsheet (Right click and then save either file). Mess about with the Step and Size numbers in the top row. This basic spreadsheet is for squares as large as 22 as it is unlikely you will need any more than that. Don't worry if your size is smaller than 22, just look at the number of columns and rows you actually need. Row 2 of the spreadsheet is just the heading for each column and not part of the table.
Returning to our problem of 15 groups. Either continue with the general version of the spreadsheet (making sure you insert 15 in the size cell) or download the Latin15 version Excel or Lotus. Try values of 1 (default) 2, 4, 7, 8, 11, 13 and 14 in the step cell. (I'm sorry that I have put these cells in reverse order on the spreadsheets, which I devised some time ago)
When you prepare your blank version of the Brass timetable you need to mark in where the lesson changes occur. I have done this with colours. Red slots mean pupils will miss very little academic work. I've used two squares to solve this timetable problem. I have shown how this works for Group 1 and I hope you will agree that the second solution works better. LS 15/4 makes the lessons jump around the day rather more. In fact, this is about as good a solution as one could hope for.
1st solution: Timetable using LS 15/8
| WEEK >>> | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O |
| 10.30 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 |
| 10.50 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 |
| 11.10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 |
| 11.30 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 |
| 11.50 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 |
| 12.10-12.30 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 |
| 1.00 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 |
| 1.20 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 |
| 1.40 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 |
| 2.00 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 |
| 2.20 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 |
| 2.40 | 12 | 5 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 |
| 3.00 | 13 | 6 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 |
| 3.20 | 14 | 7 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 |
| 3.40-4.00 | 15 | 8 | 1 | 9 | 2 | 10 | 3 | 11 | 4 | 12 | 5 | 13 | 6 | 14 | 7 |
2nd solution: Timetable using LS 15/4
| WEEK >>> | A | B | C | D | E | F | G | H | I | J | K | L | M | N | O |
| 10.30 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 |
| 10.50 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 |
| 11.10 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 |
| 11.30 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 |
| 11.50 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 |
| 12.10-12.30 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 |
| 1.00 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 |
| 1.20 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 |
| 1.40 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 |
| 2.00 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 |
| 2.20 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 |
| 2.40 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 |
| 3.00 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 |
| 3.20 | 14 | 3 | 7 | 11 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 |
| 3.40-4.00 | 15 | 4 | 8 | 12 | 1 | 5 | 9 | 13 | 2 | 6 | 10 | 14 | 3 | 7 | 11 |
But what if you had an 'A' level student who wanted to attend lessons in private study sessions to avoid missing any academic subjects altogether? My answer is to write in his/her lessons first. Let's say this pupil occupies Group 15 and is free just after lunch. As the 1.00 slot hits part of registration, I'd keep that for any group to use and gain the small amount of benefit. I'd slip Group 15 into the 1.20 slot every week and then use a Size 14 Latin Square (such as LS 14/5) to solve the timetable. It would look like the one below on which I have again shown the Group 1 lessons more clearly. You will see that the pupil misses the subject from 10.50-12.00 only 4 times in 14 weeks.
| WEEK >>> | A | B | C | D | E | F | G | H | I | J | K | L | M | N |
| 10.30 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 |
| 10.50 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 |
| 11.10 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 |
| 11.30 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 |
| 11.50 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 |
| 12.10-12.30 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 |
| 1.00 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 |
| 1.20 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15 |
| 1.40 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 |
| 2.00 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 |
| 2.20 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 |
| 2.40 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 |
| 3.00 | 12 | 3 | 8 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 |
| 3.20 | 13 | 4 | 9 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 |
| 3.40-4.00 | 14 | 5 | 10 | 1 | 6 | 11 | 2 | 7 | 12 | 3 | 8 | 13 | 4 | 9 |
Problems and issues
So now you can produce your own timetables, safe in the knowledge that you can avoid pupils missing the same subjects too regularly. Of course, life is never simple. Here are a few of the things which can go wrong.
- Pupils do not read the timetable properly.
- They 'forget' their instruments.
- They lose track of time.
- To address the issue of missed lessons peripatetic staff fill in a form for each visit listing the pupils who failed to attend and any reasons given. I transfer this information to a tally chart and interview pupils when say 4 lessons have been missed, or 3 consecutive ones. If the problem persists a standard letter is sent home explaining that, if the pupil continues to miss lessons they will be removed from the timetable.
- Pupils arrange swaps without your consent (put up a notice telling them NOT to do that)
- Your peripatetic staff arrange swaps (hopefully you can trust them to look at the consequences)
- A number of pupils have examinations and your peripatetic staff try to cram too much into a lesson and overrun. Pupils are asked to wait outside until the teacher is ready. Of course a 20 minute lesson does not mean pupils miss only 20 minutes of Maths, or whatever. If they have arrived a couple of minutes early and take a couple of minutes to pack away after their lesson that's 4 extra minutes of academic work they have missed. So, if a 'peri' overruns by 5 minutes your timetable can lose all credibility. I have, in the past, given copies of the timetable to all teaching staff so they know who should be leaving lessons and when they are to return. If you ever need to change your timetables you'll have to tell all the school staff.
- It can take a few weeks for the timetable to 'settle'. Your Year 7 pupils may have had different peripatetic teachers at their last school. The staff who visit you will want to get to know them. Only after a few weeks will they be able to tell you into which group they will best fit.
- Often several pupils are in the same class. I try to avoid more than 2 instrumentalists leaving the same lesson at the same time to go for tuition.
- Pupils do not consult the timetable soon enough and sometimes say "I can't come after school I've got to go to the dentist", "I've got to do a paper-round" or something.
In closing...
I hope all this has been useful. If you have a better system please let me know! I dream of being able to punch all the information into a computer and getting that to churn out the timetable. Now there's a challenge for someone.