An idea for a mixed attainment class that came to me about 5 minutes before a lesson today:
- 3.4 – 3.04
- 5.2 – 5.02
- 7.8 – 7.08
- 8.2 – 8.02
Find other questions like this. (The “weakest” student in the class told me the pattern before I’d even finished writing the fourth question on the board.)
What do you notice? Why is the answer to Q2 the same as the answer to Q4?
Can you create a question with 0.54 as an answer? How many different answers are there to these types of questions?
- 5.7 – 5.007
- 8.3 – 8.003
- 6.4 – 6.004
These are more tricky and test the skills of column subtraction, something that should be secure by Year 7 but may not be. Maybe an opportunity for collaboration amongst students to show how.
And then finally, try these two calculations. Which is easier and why?
7 – 1.392
6.999 – 1.391
Show on a number line why this works and then try some more. I think these questions are interesting to explore. But I would hesitate to recommend it as a must-do method to solve e.g 8-2.5687. Whether or not it is easier to turn it into 7.999-2.5686 or not is an interesting discussion and one which I would want my students to form their own opinion on, not be too swayed by mine.
This was the question Mark McCourt was getting us to ponder in the first session of Maths Teacher Network. He started off by running through the classic 1089 “trick” which I have written about before here.
These types of activity are such a powerful way to get students to carry out repeated practice to build fluency. If you handed out a sheet with 20 column subtractions and 20 column additions, you would get groans and do no favours for the general popularity of maths as a subject. If those repeated calculations have a purpose as they do here, the dynamic and energy in the classroom is completely different.
I’d used this before but had never given it the time that it deserves, nor had I extended it:
- Try it with 4 digits, 5 digits, etc.
- Go back to three digits and try it with a, b, c. In other words prove it algebraically.
- Now try in a different base, e.g. Base 7!
I actually got a bit obsessed with this when I got home. I do love a good spreadsheet challenge, so I attempted to build something that would provide an algebraic proof of different numbers of digits and bases. Here is the result…
My reflection from this session was that even if I have “seen” something before, have I really investigated it deeply and could I use it more effectively with my classes? To which my answers were no and yes!
Another approach, back to 3 digits. The prime factorisation of 1089 is 9×11×11. Why is this? And what are the prime factorisations of the 4 digit magic numbers. Can these be explained?
There is a range of “take turns” dice games like this one on nRich.
I see this activity as a way of practicing key skills (in this case column addition) but in a much deeper way than repeated practice as you are working backwards to achieve a result. I would think of this as adding a significant degree of difficulty over simply doing a page of sums; it would be something you would only chose to do once the basic process of column addition is reasonably secure. However, because it is engaging (i.e. competitive) students are more likely to stick with it.
To avoid the need to provide students with 10-dice an interactive dice could be displayed on the board. nRich have this handy spinners tool here and there are lots of other options online.
All students would then have the same set of numbers and it would be a competition to get closest to 1000. You might need to note down the numbers as you go to prevent cheating!
Again, of course, the value in the activity comes from the discussion, both in pairs and in whole class. I like the idea of saying that the target is 1000 but actually rewarding good discussion and reasoning rather than just closest answer (i.e 5 points for a new “noticing”, 5 points for the closest answer)
Another way is to determine all the required random numbers at the outset and the students can fill in the grid with full knowledge of their options. Less luck involved and so probably less fun!
I would do one game whole-class, where students are playing individually. Then a second game where students are to work together in pairs competing against another pair so they can compare strategies once they have a degree of familiarilty with the problem and get some good discussion going on strategy.
These only cover adding and subtracting negative number, i.e. they can be used before going onto multiplying and dividing negative number.
They can be printed (here is the pdf), cut out and then stuck on A3 paper under the three headings with examples and counter-examples to explain why they have been put under that heading.
I should point out that two statements on here are deliberately vague, i.e. “two negatives make a positive” and “a positive and negative make a negative”. This is often how students remember them and this can lead to problems down the line (e.g. the misconception that -3-2=5). My idea with these is that the end up in the “Sometimes” column but ultimately we dismiss them as not being very useful.
I made this a while ago for supporting Year 7 students on directed number (i.e. positive and negative numbers). I think there is something more intuitive about a vertical number line – if you are adding you go up, subtracting you go down. Having said this, I have always had a horizontal number line across the top of my board!
If the number line is stuck on the inside back cover of the exercise book, it is always there whatever page the child is working on. It can then be folded safely away whether using large or small format books.
Doing some work in primary this week, I realised that the same idea could be useful for supporting younger children learning the essential skills of counting back and counting on when doing addition and subtraction of positive numbers. So I have made another version just with positive numbers.
The end points of these lines are arbitrary of course. I have deliberately gone for something a bit random to start a discussion, “Sir, why does it stop at 44?”. But if that offends your preference for order in life, then feel free to adjust it on the spreadsheet that I used to create the pdfs in the first place.
Vertical number line to stick in books.xls
Vertical number line to stick in books zero to 44.pdf
Vertical number line to stick in books minus22 to plus22.pdf
We are spending the first term with Year 7 on fundamental number skills to ensure solid foundations. This week, we are looking at subtraction. There won’t be any specific methods or techniques that they haven’t already seen at primary school, but I know the degree to which they have mastered these techniques will vary markedly within the class.
Rather than feeling daunted at the prospect of teaching them something they already know, I’m seeing this an opportunity to show them some (hopefully) cool maths that requires them to use and practise their skills.
My new school has a wealth of manipulatives which I have had little opportunity to use before. I’m looking forward to using dienes blocks to show why the column subtraction method works and why we cross out digits before embarking on the subtraction.
It’s not really “borrowing” it’s “re-distributing” one of the tens into the ones column.
Next up, Diffy, which I learned about from Don Steward’s blog post here. The post contains some great ideas on how to do it. I love a good spreadsheet so created this which does all the calculations for you! The real reason though is so I can give my students a chart to help them structure their Diffy calculations.
I really want to focus on clarity of instruction with this class. They are a typical Year 7 class who need instructions broken right down and economy of language from me. My measure of success will be the number of questions I get asked before they start!
I also really like these questions from Maths Mastery:
I shall insist on 100% silence, pens down and do a Think, Pair, Share on this. The “Think” will be 30 seconds in complete silence with a timer with pupils just looking at the calculations. I’m hoping for an excited rush of discussion once the 30 seconds is up and they are itching to tell their neighbour what they’ve spotted. Then they will check their hypothesis by actually completing the calculations in their books.
And finally the old classic 1089. Powerpoint slide here.