I’ve been teaching for 3 years, I’ve learnt a lot, but also appreciated that the learning never stops. CPD doesn’t mean going on courses, it means spending a reasonable portion of your time looking at new ways of doing things, never just accepting that you are going to teach things the same this year as you did last year.
This wonderful tome written by a group of fabulous Amercian teachers is something that I think might become one of those sources I go back to again and again.
The basic idea is to replace “Tricks” with a proper teaching of deep understanding. There is a wonderful array of unhelpful mnemonics, cute stories and memorised procedures in here including such gems as “Ball to the Wall” and “Make Mixed Numbers MAD”. For each one, the authors suggest a better approach to instil deep thinking.
On reading this, I was relieved that I don’t use a lot of these bad tricks in my teaching, but there were a few that made me stop and think about my own practice. Including:
- 2.8 BIDMAS (usually PEDMAS in US schools). The first problem is that Divide doesn’t necessarily come before Multiply. The authors suggest an alternative, GEMA, but I’m not sure that this is all that helpful either. I’ve tended to teach this early, in Year 7 and 8, but the problem is that the second item you come in the list, call it Indices, Powers or Exponents, is not something known at this stage. So I just say, ditch the acronym until much later. Just start with saying multiply/divide happens before add/subtract unless we use brackets to indicate otherwise.
- 3.5 Dividing fractions. Whilst I have never said “Ours is not to reason why; just invert and multiply” I must confess that I do use Keep Flip Change (KFC) a lot! So I like the idea of getting common denominators first and then dividing the numerators. The idea being that once students practice a few they will discover the short cut. This takes a confident teacher, though. I can think of some students who would just feel that I have wasted their time showing them a long method and might even think I didn’t know the short-cut!
- 4.5 Pythagoras. Instead of a² + b² = c², leg² + leg² = hypotenuse². Interesting, but I’m looking for a better word than “leg”…
Anyway, that really is just scratching the surface of this wonderful resource. It should just make you stop and think about the fundamental way you explain maths concepts. Because it doesn’t matter how many wonderful resources and activities you plan into your lessons, a significant part of students’ learning will still come from your explanations.
I like to use this as a sort of a crescendo to teaching prime factor decomposition which is itself a very satisfying experience.
Although it sometimes feels a bit procedural it’s a nice way of:
- Practicing times tables
- Getting to know your primes
- Appreciating the commutativity of multiplication.
Anyway, here’s the trick (everyone needs a calculator in front of them)
- Chose any three digit number. Write it down somewhere.
- Type your number into your calculator and divide by 7.
- Hands up if you got an integer answer. Opportunity here for a nice discussion that we might expect 1 in 7 hands to be up at this point.
- Press clear and divide the same number by 11. Repeat again with 13. Right, now they’ve appreciated that not many numbers are divisible exactly by 7, 11 and 13. Time to blow their minds…
- This time type your 3 digit number into your calculator twice so you have a 6-digit number. e.g.
- Divide by 7. Hands up if you have a whole number. Wow, everyone. Now don’t press clear, but divide by 11. And then 13. Wow. Gets you exactly back to your original 3-digit number.
How much you chose to explain this will depend on the ability of the class, but the points are:
- Whatever 3-digit number you chose, the 6-digit number is 1001 times the 3-digit.
- 1001=7x11x13. Weird but true. And this is why it works.
If your students seem to like it, I always ask them to try it out on their family when they get home. I love the idea that I just might have created a discussion about maths around the dinner table – you never know…
This is a classic that I’m sure most maths teachers will know. I’m just hoping my Year 9 class won’t have seen it when I try it on them next week as part of a probability lesson.
In looking for some images I came across an on-line survey that Transum had done following an enquiry from a TV show. The results were interesting and worthy of a few minutes of discussion in class once the “trick” itself has been revealed.
If you haven’t seen this before, you essentially ask the following questions (respondents keep the answers in their heads):
The first part uses the fact that the sum of digits of multiples of nine is always 9, so that you always end up with a 4 which then turns into D. Most people think of Denmark and then Elephant, but not everyone does as demonstrated by the results of Transum’s online survey.
This is a nice intro to Tree Diagrams but also gives some scope for discussion of these figures, i.e.
- What’s the overall % of people that this trick works on?
- 68% seems low – would it be higher if it was a UK only audience? (rather than a global Internet audience)
- What were the percentages in our class?
- What percentage of people does it need to work on to have the desired effect in the room?
You could even follow this up with a homework to survey 10 people. Just don’t call it the “Elephant in Denmark trick” as my son did. Oh dear…
I’ve always felt that secure knowledge of times tables at Year 7 is so important simply because it gives kids the confidence to engage in so many maths topics covered in that year. As such any opportunity to practice is good even when it is in a simple game like this.
A Simple Factors and Multiples Team Game for 3-4 players
I came up with this idea whilst playing the traditional Happy Families card game with my family when on holiday. Kids seem to love this game – could I create a maths game as engaging?
I’ve tried this several times with Year 7 classes, playing in teams of 3 or 4 and they love it.
It takes very little preparation or explanation – in fact the students make the resources themselves!
You need a set of 36 blank cards for each team. Anything will do. I spent about 10 minutes furious chopping on the guillotine for 7 teams, getting 12 cards out of each A4 sheet, so 3 sheets per team, 21 sheets in all.
The learning starts by getting the teams to create their cards using the following instructions:
1, Arrange your cards into 4 columns by 9 rows
2, You need to write the first 4 multiples of each number 2 to 10 so that every card has a number on it.
I put the 36 blank cards and the above on a slip of paper in an envelope and gave an envelope to each team. With a bit of discussion within the teams, they worked out what they needed to do, but if you feel the task needs a bit more scaffolding you could use this diagram:
Once each team has their cards laid out on the table, they can start playing.
- Shuffle the cards and deal them all out.
- The objective is to collect “families” of numbers, e.g. 3,6,9,12 is the 3 family. The player with the most families wins.
- Play starts with the first player asking one of the other players (they decide who) for a particular card, e.g. “Natasha, do you have a 5?” If Natasha has that card, she must hand it over. The first player can ask again (again, they can chose any player). If the answer is no, play moves on to the next player.
- When a player has a family they must lay it face up on the table.
- Play continues until all the cards are gone – it’s that simple!