Birth Date Values

One of the great things about my role(s) this year is that I have had the opportunity to meet some fantastic Maths teachers and educationalists and last week I hosted Mike Ollerton for two separate events.  Mike has made many significant contributions to Mathematics education over the years and he has kindly permitted me to write about the ideas he shared with us last week.

This is a simple activity that feels quite fun and personal but could lead to some rich discussions. Mike’s description of it is here:

After going round the class, asking several children for their BDVs, there are many questions which might present themselves. Can you ask children to work out someone else’s birthday given their BDV?  Mike suggests lots more questions:

• Which BDVs only have one birth date?
• What are the minimum and the maximum BDVs in a class?
• Which BDVs have the most dates?
• What is the smallest BDV which cannot be made?
• What is the largest unique BDV?
• Which dates are square BDVs?
• Which dates are triangular BDVs?
• In a group of people who has the average BDV?

What other problems can you devise based upon BDVs?

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Multiplying fractions – showing why

It was a long time ago, so I can’t be certain, but when I first learned to multiply fractions, it was a procedure that involved turning mixed numbers into improper fractions, multiplying numerators and denominators with maybe some cancelling down along the way. It was a procedure with no understanding.

You could just apply that procedure to these questions. But there is scope for a greater depth of understanding not to mention some creativity in showing why these work.  Bar models are one way to demonstrate and calculate. Here are two examples:

A worthwhile exercise is to go through each of these questions attempting a drawing to show why (squared paper is a must).

Depending on your class, you will probably need to show some examples first. Or maybe you would prefer to give the completed statements so the focus is on drawing the representation rather doing the calculations.

Here is the lyx file for these questions and the pdf.

Formal methods – the devil’s work!

I’ve been thinking a lot recently about “formal methods”, e.g. Column Addition or Long Multiplication.

As secondary maths teachers we don’t pay much attention to these.  I assess their ability to carry out the procedure and have sometimes attempted to teach long multiplication explicitly. But more often than not it is assumed that these methods have been learnt at primary school have been pretty well-practised and are therefore secure.

We have been doing a lot of number work with Year 7 so far this term including exploring in some detail the Laws of Arithmetic.

Here is an example of the type of question we have been looking at:

127 + 54 + 73

Most of my students’ first instinct with a question like this is to draw up a nice column addition and solve it.  Would you agree that most students do this?

I want them to look at the structure of the numbers first.  To realise that addition is commutative and that it’s much easier if we add the 127 to the 73 first because that’s a number bond to 200.

I am having 2 issues with this approach:

1. My students are not convinced that this is a valid exercise.  Basically they think it is contrived (which, of course all exercises are). They think I have chosen numbers to make this work, it will only work for those certain numbers and basically they feel like I have tricked them.
2. Some students perceive the message that the formal methods they spent hours practising at primary school are now not the way we do things at secondary school. I have said that this is not the case, they are not the devil’s work, and that those methods still have a place. But we need to have smarter ways to work. The fact that we have spent many lessons not solving calculations using formal methods leads me to believe that they are confused about what I want them to do.

Fundamentally I want my students to rely less on the authority of the teacher and rely more on their own understanding.  I want them to see the structure in everything they do, to be certain that what they have done is valid because the mathematics tells them it is so, not because I tell them that they are right.

How should we bridge the gap between informal methods which require (and develop) depth of understanding and formal methods which are efficient and accurate?

I have an idea using manipulatives.  I am not sure if it will work, but if I get chance to try it with the class, I will write about it again later.

In the meantime, does this resonate with anyone else? How do you bridge the gap? I’d love to read your comments.

 

Negative number grids

It might be good to introduce (or re-introduce) the idea of the multiplication grid first:

before adding some negative numbers into the mix:

Here are some more to practice:

Some more ideas for this topic on Resourceaholic here.

 

Multiplication methods – introducing formal long multiplication

Today I was with a group and we were trying to formalise long multiplication methods.  They were fairly secure on 2 digit by 1 digit using formal method (not grid/box method).  i.e. they were happy doing this.

So after a while, I tried to introduce them to 2 digit by 2 digit.  It was quite a leap, so I’ve been thinking about the microsteps in between.

Before moving on, however, I really like some of these ideas on Don Steward’s Median (still my favourite maths website!)  They make for good extension work as they go deeper and involve problem solving before moving on the “next” thing.

2-digit x 1-digit to 2-digit x 2-digit

It’s a big leap, so I’m thinking about the microsteps.

1. Multiples of 10.  We can start with a discussion about 12 x 20.  Some might well be able to do that mentally, and writing out their explanations should eventually lead to:
   We can discuss the practice of “putting a zero in the ones column” which then enables us to just focus on the number in the tens column and multiply that by the number above.  Here is a worksheet of these that I created using Math-Aids.
   multi_digit_power_ten
2. I’m assuming that the idea of partitioning a number into its tens and ones is pretty secure by now, so I might then look at this.
   
   Here is a worksheet that I have created to practise these. It’s a spreadsheet that you can change if you want to alter the questions.  If you’d prefer just simple pdfs to print, here are the questions and these are the solutions.
3. So, now we are ready to combine it all together and introduce the efficient method. Effectively doing the same calculations but with less repetition in the writing. Some suggested language to help solidify the steps:
   • always start with the ones – this is what we use the first row for
   • put in a zero when you are ready to multiply the tens – the answer goes on the second row
   • keep your columns in order, think about the place value of each number you write down
   • use the final row to do your column addition

If you are looking for some worksheets, here are some which I like because they are on squared paper.  The early ones have the hint of putting the zero in, the latter ones don’t. If you don’t like those, a quick Google images search on long multiplication worksheet will soon get you the one you want.

I’m sure there are lots of nuances I’ve skipped over here, but if this triggers any thoughts about how you would teach it, please leave a comment below.

Box Method for multiplication – why do we teach it?

I’m lucky enough to be doing some work in Primary this term.  It’s a great experience and in an ideal world this would be something all Secondary Maths teachers get the opportunity to do at some time.  I’m working with Years 3-6 and have had some really interesting small group sessions with the children looking at multiplication.

In Year 3, we progressed from multiples of 10 (e.g. 2 x 30, 6 x 60) one day to full two digit multiplication (e.g. 5 x 24) the next day.   I felt like this was a fairly rapid progression for Year 3.  Although some of them coped fine, it was a lot for some.  Later I got chance to do the same with some low attaining Year 5 students. Maybe unsurprisingly, the methods hadn’t stuck.  Unsurprising because I have taught plenty of Year 7s who would struggle with this.

60 x 4

First, let’s start with multiplying multiples of 10.  The standard way of teaching this to Year 3 seemed to result in this:

Steps are:

1. Cross out the zero
2. Do the multiplication of the other digits
3. Then multiply that by 10 to get the answer.

Or, in some cases this was shortened to:

1. Cross out the zero
2. Do the multiplication
3. Put the zero back.

The problem I had with this is that the end result didn’t make sense, i.e. what does this mean?So I tried to encourage them to just “cross out the zero in your head”.  It felt awkward to me as I was saying it!

We are trying to get a balance here between developing a deep understanding of what is going on and learning an efficient method.  I think we want students to move quite quickly to doing this mentally.  After all, if you know 6×8, you know 6×80 and once you understand what is going on, more writing (i.e. writing the steps above) doesn’t really help.  I think some Diennes blocks would really help at this point with the understanding.

Or you could try virtual manipulatives here or here.

Personally, I would have moved onto hundreds next (e.g. 3 x 500) to show that this is a general principle before attempting full 2 digit multiplication and spent some more time on that before moving onto the Box Method.

24 x 3 – The Box Method (also called Grid Method)

So, next day, next piece of learning and we now attempt two digit multiplications using the box method.  At secondary we always try to wean students off the box method in Year 7 and get them doing formal long division, i.e. this.I was really pleased to see that the Year 5 students that I helped were already using the formal method.  However, I have worked with some Year 10s that still use it.  When you are doing big numbers you get some fairly ridiculous column addition at the end and many more Opportunities for Error.

Having said that, let’s not forget that it can be a useful as Another Way to Show algebraic multiplication, and some students find it useful for multiplication with decimals.

                

The key piece of understanding at the beginning is that we are breaking down the 2-digit number into its tens and its ones, i.e. 24 can be broken down into 20 and 4 and the distributive law of multiplication allows us to multiply each part separately.  So the children are taught to fill in the squares in a box, i.e. this.Now I have no real evidence for the next statement, but personally I don’t see the point in the box method.  Why not write out the two multiplications? i.e. this:It’s not much more writing and I think it is clearer.  Specific problems I saw with the Box Method included:

1. Drawing the boxes in the first place.  Fine if someone does it for you, but when that scaffolding is removed…
2. Remembering which numbers go where when setting it up
3. Understanding the array structure, i.e. multiply the number at the top of the column by the number at the left of the row.
4. Remembering which numbers to then use for your column addition.

So, why do we teach the Box Method?  Especially as we then ditch it as soon as possible to teach formal long multiplication.  I’m not sure why.  My son in Year 6 isn’t sure why.  I’d love to hear your thoughts.