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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nRich games to practice key skills

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.

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.

Reinforcing Place Value at every opportunity

So, I learnt something today from a kid in Year 3.  He gave a perfect explanation of a column addition that made me stop and think.  How would you explain this?

His explanation:

“4 plus 1  is 5, so put a 5 in the ones column. Then twenty plus ninety is one hundred and ten, so we put a 1 in the hundreds column and a 1 in the tens column.  Then one hundred plus three hundred is four hundred so put a 4 in the hundreds column”

This has got me thinking about where else I can reinforce place value when discussing procedures.

 

 

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.