using mental methods to construct a standard written method for addition and subtraction

Using Speaking and Listening to “Think Together” about standard written methods for Addition and

Subtraction, in the Primary Numeracy Hour.

By Mr Mills and Year 4 2004

Simon Mills Jan 2004

We observed the patterns that are made when you double multiples of 10, 100 and

1000.• We found out that

to double a multiple of ten you can can divide by 10, double the digit, and then multiply by 10 again

So double 60 is the same as(60 ÷ 10) x 2 x 10= (6 x 2) x 10 = 12 x 10 = 120

Did this work for doublingMultiples of 100 or 1000?


To double multiples of 100 or 1000 we found out that you can divide by the multiple, double the digit you are left with, then multiply by the same multiple again.

So to double 200Divide by 100 200 becomes 2Double 2Becomes 4Multiply by 100Become 400

Does this work for all Multiples of 100?

To double 2000Divide by 1000 2000 becomes 2Double 2Becomes 4Multiply by 1000Become 4000

Does this work for all Multiples of 1000?

AND


What strategy could we use to total these numbers efficiently?

• 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9

• 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90

• 100 + 200 + 300 + 400 + 500 + 600 + 700 + 800 + 900

That was our challenge and to show how we did it.


Informal jottings + discussions = Thinking TogetherWe are beginning to reason about numbers!


Firstly you find pairs of numberswhich total 10

Then you count up the multiples of 10

There are 4 pairs of numbers whichTotal ten in the number sequence1+9, 2+8, 3+7, 4+6 and the 5 is leftOver

4 lots of 10 gives 40

Add the 5

And the total is

45

Can we use this strategyTo help us find the other totals?

For multiples of 10..Find pairs which total 100?

For multiples of 100Find pairs which total 1000?


Adding multiples of 10

10+90, 20+80, 30+70,40+60, total 100 each with 50 left over.

4 lots of 100 is 400

Add the 50

Total 450

Adding multiples of 100

100+900, 200+800, 300+700,400+600, total 1000 each with 500 left over.

4 lots of 1000 is 4000

Add the 500

Total 4500

What else is special about the totals we have made?

We thought about place value and what happens when we multiply numbers by 10, 100 or 1000!

Our total pattern… 45, 450, 4500


Can you use these strategies to carry out these calculations?

3 + 5 +7 + 6 + 4 80+ 70 + 20 + 60 + 30

Can we use what we know about place value to find aShorter way to calculate the solutions to these number sentences based

On the ones above?

30 + 50 +70 + 60 + 40 800+ 700 + 200 + 600 + 300


We used the round to the nearest 10 and adjust strategy, for

addition of 2 and 3 digit numbers

How can we mentally calculate 23 + 29?

Can we do this our heads?

Maybe, but will we remember every step?

Will a Jotting help, maybe but which one?


What about a blank number line

How do we mentally calculate 23 + 29?

23

+30

-1

53

52

=23 + 30 – 1=53 – 1=52

23 +29 is the same as 23+30-1SO…


Informal jottings + discussions = Thinking TogetherWorking a problem


How could we carry out this calculation?

= 54 + 38 What do we know?We know that “=“ means “the same as”So Is the same as or equal to 54 + 38SoThe calculation can also be written as

54 + 38 = as well.

What strategy might we use?

In Talking twos we discussed and jotted our methods.!!!


Sharing methods we know + discussions = Thinking TogetherWorking a problem


54

+40

-2

9492

54

+10 +6+20

9264 84

54 + 38

50 + 30 +4 +8

80 + 12

Which of these is the most efficient method or strategy?

Blank Number Lines

Partitioning

Decompose and count on

Round up and adjust

Our Methods…

90

+292


50 + 4+ 30 + 8 80 + 12

= 92

54 + 38 80 12 80 10 2 92

Is there a way we can use the strategies we know to develop a shorter jotting?

54 + 38 =

54 + 38 12 80 2 10 80 92

Addition can be done in any order

How about this way? Or this?

Which of the strategies we used beforeAre we using in these jottings?

Can we map it to our calculation?


“Joining up our thinking” Mapping the

partitioning strategy to our jottings.

54 + 38 80 12 80 10 2 92

50 + 4+ 30 + 8 80 +12

= 92

54 + 38

50 + 30 +4 +8

Partitioning:What is

happening in our mind

when we use this jotting?

54 + 38

50 + 30 +4 +8Partitioning:What is happening in our mind when we use this jotting?


Joining up our thinking

Practice + jottings Talking + our actions+ Partner + suggestion Thinking together


It’s good to talk, it helps us to share ideas, and join up our thinking with other people’s.

Group joined up thinking + Teacher Assessment = Small Group review + Whole Class Sweep + Teacher Review


Joining up our thinking. Using the mental methods we know to

create a written method.


How would you calculate the answer to 94 +73

In talking twos wediscussed strategies and then as individualsmade jottings to show how we wouldcarry out the calculation on whiteboards.

We checked each otherssolutions, and talked about what wehad done.


We had used a whole range of methods, to come up with the same solution, this was great because the

methods we chose were all methods and mind maps of our thinking from last week


We shared our methods, and tried to explain them to the class. We tried to find out if they had anything in common. We used Power Point to review our work from last week, to

see if this would help.


In all of our jottings we had used partitioning, or decomposition to help us break the

calculation down into manageable chunks.

90 + 4+ 70 + 3 3 + 4 = 7 90 + 70=160 160 + 7 = 167

94 + 73

90 + 70 +4 +3Partitioning:What is happening our mind when we use this jotting?

94

+3+70

167164

Decompose and count on94 + 73 is the same as 94 + 70 +3 is the same as164 + 3 is the same as167


In all of our jottings we had used partitioning, or decomposition to help us break the

calculation down into manageable chunks.

94+ 73 90+ 70 4+ 3 167

94 + 73

90 + 70 +4 +3Partitioning:What is happening our mind when we use this jotting?

3 + 4 =

90 + 70=

160 + 7 =In our heads!

On our think space

160

7


Mr Mills listened and watched as we worked. He asked questions to help us explain what

was going on in our minds. He joined in, and helped us to understand what was going on in our Class “think Space.” He said he wanted

to use our ideas to help us to make the written calculation shorter. He did this by showing us

a method and explaining how our ideas helped to make the method work.


1) This is our calculation

2) Partition or decomposethe numbers in thecalculation132 is the same as100 + 30 + 256 is the same as50 + 6

3) Write out you expanded numbers.

Make sure that you write them so 100’s, 10s and 1s are in the same columns.

4) Total the columns, 1s first, then the 10s, then the 100s. And so on, record you totals under the column you have totalled ie 2 + 6 = 8

5) Recombine your total eg 100 + 80 + 8 = 188

Can you find your method or our mental Strategiesin this written method?


We tried out the calculation method for ourselves using examples given to us to

practice with.Some of us were so good Mr Mills challenged us to see what would happen If we had to use bigger Numbers. Would our method still work? How

would the calculation change?

We worked through the calculations together in our class think space at the end, and marked and checked each other’s work. We discussed how the method had worked and what happened with big

numbers.


We explored how our written method for addition, might be used to carry out subtraction calculations Mr

Mills talked us through and we had a go using

examples given to us to practice with.We all found this a bit of a challenge. The big question was what to do with those addition signs?The last one was easy, but sometimes we forgot we had used them to help us partition or decompose our large number and instead of adding columns, this time we were subtracting them.

The method got easier though as we got used to ignoring the addition signs until the end of our calculation.

Can you find your method or our mental Strategiesin this written method? Simon Mills Jan 2004

using mental methods to construct a standard written method for addition and subtraction

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