The following is a rough outline of how I might introduce place value in the classroom. It begins, as all good learning does, with a story:
Let me tell you about when I used to work on a farm. We used to grow all sorts of vegetables and fruit. We sow seed in spring and tend the young crops all summer and then in the autumn we would harvest the vegetables.
Now sometimes when we dug up the root vegetables they would be damp and muddy, so we would need to lay them somewhere to dry. Well for the potatoes we had a drying rack:
And as we harvested the potatoes we would put them on the rack. The first day I harvested two potatoes:
The second day I harvested three potatoes:
So how many potatoes did I have all together? – that’s right, five potatoes.
Well, the next day I harvested four potatoes:
So how many potatoes did I have all together? – that’s right, nine potatoes.
And the next day was not a good day for harvesting. I only found one potato:
So how many potatoes did I have all together? – that’s right, ten potatoes. But there was a problem. I had no more room on my rack. Well, not to worry because when I had 10 potatoes, I could fill up a bag to take to market as we often sold bags of ten potatoes.
And I could hang my sack on a hook on the wall.
And carry on harvesting potatoes as before. Well the next day was a good day and I harvested nine potatoes all at once. So where can I put my nine potatoes? On the rack!
Great, so next day I harvest just one potato again.
Where does it go? No room on the rack. OK, so I have enough for another bag, great:
And I can put my bag on a hook:
OK, so now, how many bags of potatoes do I have? Two. How many potatoes are in each bag? Ten. So how many potatoes do I have all together? Twenty. OK, so I have 2 bags of 10 on my hooks. I have twenty all together.
Well the next day was a super harvest day and I dug up 12 potatoes!
What am I going to do with all these potatoes? They won’t all fit on my rack.
Ah, so I can use a bag. How many can I fit in a bag? Ten. OK lets put 10 in a bag.
So I can put the bag of 10 potatoes on my hook and the spare potatoes can go on my rack
Now how many potatoes do I have all together? I have three bags of potatoes on my rack. How many potatoes in each bag? 10. Ok, so three bags of 10 makes … 30. And I got two spare ones. So I have 32 all together.
At this point we can play around with different numbers, I can present different amounts of potatoes and bags of potatoes and ask the children to tell me how many potatoes we have all together, or I can say a number and ask the children how many bags and spares I will need. I can even bring in a little bit of mental addition at this point saying, OK, If I have 63 potatoes and then I harvest 4 more, what will I end up with? How many bags and how many spares?
Stage 1: Demo from front: Using large manipulatives, I could if I was enterprising use real potatoes etc, but maybe magnetic manipulatives on a board would suffice – A visualiser and projector would work if tech was an option.
Stage 2 I would have the children play with their own manipulatives as well, so I could print out a sheet of a rack and hooks to give them each with counters and cards for potatoes and sacks. They could work in pairs initially.
Stage 3: Children can draw this in their main lesson books.
Once I felt the majority were confident with this method I would move onto the hundreds:
Begin with a recall about the story, ask the children to tell me how we arranged the potatoes etc.
Then show them this and ask them how many potatoes I have, 99:
And ask what would happen if I dug up one more potato? I would have 100 potatoes!
Well, it can’t go in my rack because that is full. But I can fill a bag because I have 10 loose potatoes.
But where to put the bag? I have no more hooks for bags of potatoes. How many bags of 10 potatoes do we have? 10. 10 bags of 10 makes 100.
Well, sometimes, we sell sacks of potatoes to shops, so I can put my 10 bags of potatoes into a big sack to sell to the greengrocer:
At this point we can have another play with numbers up to 999, again moving from a whole class demonstration at the front to paired work and then individual manipulation, call and response for number recognition, asking “how many?” from an array and then asking “what would XXX look like?”.
Potential extension would be the crate of 10 sacks to sell to the wholesaler.
Children can draw this representation of the potato rack and bags and sacks in their main lesson book.
Recall. Numbers and images up to 999.
Introducing vertical addition
By extending the story and keeping it visual, we can introduce addition:
I had collected so many potatoes and my friend had collected so many, how many do we have altogether?
I would begin with sums which do not require any carrying to begin with. We will naturally at this stage begin to need names for the columns so the names, 1s, 10s, 100s and perhaps 1000s (represented by a crate) will arise.
Again I would like to move from whole class demonstration to paired and then individual work with manipulatives with the children drawing one or more representation of addition in this way in their books.
Recall simple addition.
The introduction of sums which require carrying
Whole class demo – Paired manipulation – individual representation in main lesson books
Subtraction. Beginning with differences which don’t require any ‘borrowing’. Having considered this, I would not represent the subtrahend with images, but with the number as this would distinguish it from the images for addition. It also would not work in relation to the story, because the customer does not turn up with a quantity to be matched by the vendor, but turns up with a request:
Mrs O’Riley just wants 2 potatoes for her dinner:
Or Mr Walsh is entertaining guests and wants 12 potatoes etc:
Whole class – paired – individual
Recall simple subtraction.
Subtraction 2: ‘borrowing’. With the explanation that if a customer wants to buy 16 potatoes when I have 24, I will need to open up a bag for the customer:
Whole class – paired – individual
Recall addition and subtraction.
After a week or more working with these images, the children will, on the whole be ready to move onto the more abstract forms of written column addition and subtraction. I would keep all the problems contextual so that the children are using the mathematical tools to represent and model real life problems, rather than just practicing isolated algorithms.
Day 8, 9 and 10:
Extend and deepen the exploration of column addition and column subtraction with a variety of problems to solve, investigations and/or games such as dice games, exploring consecutive number addition and games involving making certain totals etc. The NRich website has plenty of material to keep a class busy for three days exploring addition and subtraction.
After consideration and with a helpful comment from Kieran Mackle (@Kieran_M_Ed) I decided I would arrange the rack on the right, with 9 potatoes in a 2-2-2-1 formation rather than a straight line as this would make counting/seeing how many loose potatoes I have much easier.
Also, the schedule of days represented here may well be too fast and I would certainly consider slowing down the process more with more practice of the key skills and concepts before moving on.