The number 11.
I blame the 8-2 Bayern Munich victory rattling out last night for the reason that I went to bed with 11s in my head. But this isn’t really about the number 11, it’s about patterns of thought, consolidation, and elaboration.
Elaboration came up in a guidance report by the Education Endowment Foundation. While, like dual coding, it is nothing new, per se, it is a strategy that is currently being highlighted in supporting learners to reach their potential by providing the scaffolding needed for them to connect thoughts and, in a way, form a more complete picture to integrate information.
So how do we work with the number 11?
11 = 10 + 1.
It doesn’t take a mathematician to see that the ‘7’ appears once in the units’ column and once in the tens’ column.
This works beautifully up to 9 x 11. What happens after this?
10 x 11 =
The pattern here is nice and obvious but it continues:
21 x 11 =
This number may not be an obvious multiple of 11 and even the most committed high school students often don’t learn their time tables up into the 20s.
We can however use the method to both calculate and check for multiples of 11:
As the ‘1’ in 21 was shifted in the tens’ column when multiplied by 10, it has been added to the ‘2’ in 21 that remains in the tens’ column when multiplied by 1. Thus, the value in the tens’ column is the sum of the two values in the initial number being multiplied by 11.
Carrying this forward, 12 x 11 can be checked by summing the ‘1’ and the ‘2’ in 12 to get 3. This will sit in the tens’ column while the ‘1’ in 12 moves to the hundreds’ column when multiplied by ten and the ‘2’ in 12 remains in the units’ column when multiplied by one All values are accounted for so 12 x 11 must be 132 and indeed it is.
Does this work for greater values multiplied by 11?
Let’s try 24 x 11. Using the method above we simply sum the two digits in 24 (2 + 4 = 6) and this provides us the total for the tens’ column, which sits between the two other values giving 264.
As experienced learners we know that if the tens’ column gathers a value greater than 9, it will affect the hundreds’ column. So our method falls apart? Not quite:
65 x 11 =
We can see that 6+5 will be 11, so rather than the tens’ column being ‘11’, it provides the hundreds’ column with an extra ‘1’ and leaves ‘1’ in the tens’ column. This gives 6+1, 1, and 5, or 715.
Most of us regularly multiply by 11 in our head but can we use this to check whether or not values are multiples of 11? (Also, what does this have to do with elaboration? I’m coming to it I promise.)
We know that 561 is a multiple of 11. Because the central value is the sum of the two on the outside:
This also provides us with the outside values that reveal how many 11s are in 561: 51.
Let’s look at 825
8 + 5 would give 13. But the central number is 2. So is this game over? Can we now not determine whether or not 825 is a multiple of 11? Well looking closely we see that 13 is precisely 11 more than 2. This suggests that the central value donated a ten into the hundreds column. So let’s rewrite the column values in a way that is equivalent to 825:
Now treating these columns as the parts of their sum, 7 + 5 = 12. So our central value is equal to the sum of our outside values and 825 must be 75 x 11.
9 + 3 = 12
That’s (1 x) 11 away from the central value ‘1’. So let’s call it 8, 11, 3. Yep 8 + 3 = 11 so 83 x 11 = 913.
We can even extend this further. Is 1001 a multiple of 11?
1000 is effectively 10 hundreds. So we can say:
10 + 1 is 11. That is 1 x 11 greater than the central value (0). So rewrite as:
9 + 1 is 10, which is now in the centre, so 91 x 11 must be 1001.
2002 must be a multiple of 11… does it still work?
20 + 2 = 22 which is 2 x 11 greater than the central value (0).
If we reduced 20 by 1, we would get 19, 10, 2. Checking our values would give 19 + 2 = 21 which is still 1 x 11 away from the central value (which is now 10). So some of us may have skipped the step and gone right to reducing 20 by 2…
Note that the central value has gained 2 x 10 now rather than 1 so becomes 20. Checking, 18 + 2 = 20 and so 182 x 11 must be 2002. That figures given that it is twice 91, as 2002 is twice 1001.
You do: SLOP
This is a random ‘lesson’ based on number patterns in the base 10 so I will probably never use it and I have no idea if maths teachers do this kind of lesson, within a curriculum or outside of it, but I like mathematical modelling of teaching given the fundamental nature of numbers. Please do consider alternative number patterns… I have just selected my favourite today; multiples of 11 can be found in many ways!
At this part in my fantasy lesson I would provide shed loads of practice (SLOP). Some examples of how to implement this with increasing demand follow:
Let’s start with a few they’ve seen before. It’s familiar, they can check back and gain confidence:
21 x 11
24 x 11
Now a few that haven’t been seen but stick to sums of less than 10:
32 x 11
36 x 11
Now to increase the demand with components that sum to more than 10:
47 x 11
55 x 11
The reverse method – increasing the demand by changing the subject:
How many 11s are in 297?
How many 11s are in 209?
Rewording to increase the demand by interpretation of the question:
2222 divided by 11 =
6215 divided by 11 =
Finally we reach the point of the lesson for teachers.
My partner asked me this morning why I seemed to be awake tossing and turning all night. My response of ‘because I was thinking about the number 11’ did not cut it. He asked, so I showed. He became my student and I noticed several things:
- He spotted the pattern but could not immediately repeat it. So we practiced retrieval of the pattern to the method.
- He then made mistakes in applying said pattern, so we went through a stage of faded guidance.
- He was then able to do SLOP without too much trouble.
Then I caught him at the end of his solution to ‘how many 11s in 7150?’ and said ‘yes! Why?’
And he was stumped. He had no idea how to reply; he simply repeated back the method.
At this point I’m really intrigued as my intelligent, numerate partner is playing a very convincing role of typical student. At this point we had to elaborate. According to the previously mentioned guidance report by the EEF, elaboration ‘involves describing and explaining in detail something you have learnt’. *
So my bottom line here is that SLOP is an excellent strategy for learners, but we must go beyond this if we are going to allow all students to have the opportunity to properly assimilate facts, concepts in a meaningful way. We should help them to elaborate. From here on I would like to visit the GCSE physics curriculum for further examples of how we could, perhaps should, embed elaboration into some lessons to allow our learners to go from ‘able to apply’ to ‘understands why’.
Elaboration is often reduced down into ‘how?’ and ‘why?’ questions owing to the definition (according to the EEF) being the ability to describe and explain something. This is why elaboration is nothing new; we have been challenging students and checking their understanding using these questions for a long time. But analysing why these questions are useful moves us from the idea of using them to challenge the higher students, to using them to support all of our learners. Too often I’ve seen how and why questions written as an extension rather than embedded for the real purpose of elaboration: To connect facts, concepts and methods to provide a more comprehensive schema whereby the knowledge is appropriately linked to form a ‘bigger picture’.
Elaboration in physics
When learning to calculate acceleration and following an ‘I do, we do, you do/ SLOP’ type lesson structure, some students may be able to correctly determine an acceleration of 10 ms-2.
To elaborate here, we might ask ‘why are the units for acceleration ms-2?’
This is a good question because it allows us to assess whether the following facts or concepts have been appropriately arranged to allow the student to draw on all of them:
- What is acceleration?
- Where do ‘m’ and ‘s’ come from in the units?
- Why is the ‘s’ raised to the power of -2?
A student may have determined that a force at an angle to the horizontal provides 6N of force in the direction of the horizontal.
To elaborate here we could ask, ‘Why is the horizontal component less than the total force?’
We might be testing here to see if they know:
- That a force is comprised of the 2 component forces (that are at right angles to one another).
- That the size of the horizontal component is related to the angle between the force and the horizontal.
Perhaps a student has determined that adding an identical bulb in parallel to the one shown below
Will receive that same current as the one already shown. But again here we should elaborate. ‘Why is this the case?’
- Do they know that the additional bulb will draw its own current?
- Can they relate the ‘identical’ nature of the bulb to the size of the current drawn?
We could question and elaborate without exhausting possibilities.
Final thoughts on using elaboration
As a teacher we might be tempted to provide this question to stimulate thinking and I would argue that this is absolutely the right thing to do.
But don’t stop there; elaborate for them if they can’t. Leaving them hanging here is no better than playing ‘guess what’s in my head’. Take ideas from students and help others to criticise them, but absolutely make sure that all learners hear the expert’s elaboration so that those crucial pieces of knowledge can be organised. Do not underestimate the impact that this can have on ‘low attainers’ who may struggle to make these connections by themselves but make giant leaps in their ability when handed the tools to make their learning even more meaningful.
*At this stage I was confronted with the fact that despite by able to fluently apply the method, he could not explain why it worked. By elaborating for him, we moved towards the ‘lightbulb’ moment whereby he could begin to explain it in his own words, and (possibly) apply the method with even more speed and confidence. Then I had my own lightbulb moment on the significance of elaboration, and wrote this article.