We will understand the basic concepts of Place Value, Carrying and Borrowing.
Dr. Maya Saran
We will use numbers less than 100 throughout.
STAGE 1: UNDERSTANDING NUMBERS AS GROUPS OF 10’s PLUS SOME UNITS
How to get there: through games, drills, activities. (Getting away from mechanical repetition without understanding.)
Suggested game for 2-‐4 players: Show me a numbe
· A cup of small tiles with 1 shown on each tile, or beads, or other small objects
· Cards with numbers written on them
· Pencil and paper
The game: One card is put face up. Children try to take that number of tiles from the cup. Whoever does it the fastest wins. (If there are several players in the group, you might have to provide one cup per child to make it easier.)
Start with numbers less than 10. Move on the numbers upto 20, then 30. Now the game will become harder.
Now change the game. Add a cup of tiles to the table with a “10” marked on each tile. Tell students that one of these can be used in place of 10 of the 1-‐tiles (or beads, whatever you’re using.) Don’t give any other instructions. Now see what happens -‐-‐ children will try to save time by taking the 10-‐ tiles. Now you can give cards with larger numbers, upto 99.
Another way to play the game, with 2 players: Start with the unit tiles only. The players are playing as a team, not against each other. Player 1 picks up a card, doesn’t show it to Player 2. Player 1 puts that number of tiles on the table. Player 2 has to count the tiles and write down that number. Then Player 1 shows the original card – the number should match. If the number match, the team is successful.
When you move on to larger numbers, upto 30, the game become harder. The team will make mistakes. Now introduce the cup of 10-‐tiles on the table. Tell students that one of these can be used in place of 10 of the 1-‐tiles (or beads, whatever you’re using.) Don’t give any other instructions. Now see what happens -‐-‐ suddenly the game will become easy again.
Note: if some students are not getting the idea of replacing groups of ten 1-‐tiles with a single 10-‐tile, then you can control the process by placing on the table 2 cups that contain, respectively, exactly 9 unit-‐tiles and exactly 9 ten-‐tiles. Then they will be forced to use the ten-‐tiles to show numbers greater than 9.
STAGE 2: GETTING READY TO CARRY
Idea: recognize groups of ten.
How to get there: Again, we use games and activities. Some suggested games:
Game 1: Khana time!
Groups of 3-‐4 students.
· 3 dice and a cup of beads or other small objects
The game: One player will roll the dice. For each number shown, they will take out that many beads from the cup (if the dice say 1, 4, 2 then they will take out 1 plus 4 plus 2 beads.) the other players watch. The other players watch. When the total crosses more than 10, they yell Khana time! and take away ten beads and put them back in the cup. After one round, when everyone has had one turn, the player with the most beads wins.
Notes: Instead of dice, you can use number cards, or non-‐standard dice, or just use 2 dice if the kids are just new to addition. Also a good game to practice addition for beginners.
Game 2: Too many mice (taken from mathpickle.com) 2 players
· 6 cards showing the numbers 1 to 6. (Can use playing cards.)
· 21 small objects (these are mice.)
The game: Place the mice in groups of 1, 2, 3, 4, 5 and 6 on the table. Lay the cards face up, with 1 on the bottom, going upwards to 6, which will be face up on the top of the pile. Players take turns to take the top card. When you take a card, let us say 6, then you pick up the pile of 6 mice and you have a choice: you can keep the whole pile or you can give it to the other player. The second player will pick up the card that says 5. Again, he or she will pick up the pile of 5 mice and make a choice: either keep the whole pile or give it to the other player. The catch is: as soon as your mice cross ten, then the cat will come and eat up ten mice! After the last card has gone, the player with the most remaining mice wins.
Notes: too many mice is a two player game, but you can add an additional role: one student can play the part of the cat.
1) If your students find the game is always won by the person who goes first, then increase the number of cards from 6 to 9.
2) Is a tie possible with 6 cards? With 7? 8? 9? 10?
3) How many different ways are there to play the game with 2 cards? With 3? 4? 5? 6?
4) If both players play very well, who will win -‐ the first player, the second player -‐ or will it end in a tie? With 1 card? 2 cards? 3? 4? 5? 6?
(Taken from mathpickle.com)
STAGE 3: CARRYING
How to get there: use a grid showing a tens column and a ones column to carry out addition using physical tiles
This game is just one possibility – depending on your resources, your particular class, your mood, their mood – you can come up with all kinds of other games and activities! The main idea is that the games whould help the children understand that a group of ten units/beads/objects can be replaced with a single token/tile/bundle representing TEN.
Set up for activity: Place on each table 2 cups that contain, respectively, a large number of unit-‐tiles and ten-‐tiles. Place both cups on a paper marked BANK. Give each child a grid of their own.
Step 1: get students comfortable showing numbers on this grid using 10-‐tiles and 1-‐tiles. Ask students to show, for example, 42, by placing 4 10-‐tiles in the 10’s column and 2 1-‐tiles in the unit column. You can do this as a whole-‐class activity: let each student have their own pesonal grid in front of them, along with a bank on each table. Call out the numbers one by one, and let all the children show the numbers on their own grids.
Let them take their time and get really comfortable with this. You can make a game out of it too… anything you like. IMPORTANT: The process should go both ways: they should take a number and show it on the grid, and if they see a number shown in the grid with tiles, they should be able to write down the number. You can make a game out of this: pair up the students. On each table place a number of cards with numbers shown on them, face down. The first student takes a card and looks at it without showing it to the other student. He or she shows the number on the grid. The second student looks at the grid and writes down the number shown. The team is successful if the number is matching with the original card. Then the students switch roles.
Step 2: try addition on this grid without carrying. Simple sums like:
You can make cards with one sum each. They pick up the cards and do the addition on their grid. Get them to write down the answers (maybe on the card itself, if you have enough copies.) Again, you can make a game out if it (e.g., timed contest.)
Step 3: Now try addition with carrying, very very very simple sums like this: 14+7
The problem is; where to put the extra units? DON’T TELL THEM! Let them think about it. Some students will figure this out themselves: they have to go to the bank to exchange the groups of ten tiles. Other students will copy those who figured it out. You can give a hint to the class like this:
· Can you go to the bank?
· Can you make an exchange?
· If you have too many unit tiles, can the bank help you by making an exchange?
Your hints should be as small as possible! Let them have the pleasure of figuring it out.
Step 4: addition with two 2-‐digit numbers. The sum should be less than 100 in all cases.
Give lots of practice with this (several sessions) before moving on the column-‐wise addition in the standard way.
STAGE 4: BORROWING
Again we use the same format of the grid, and the same process. Set up the table in the same way as for carrying, with a bank and a grid in front of each child.
Step 1: Start with simple subtraction without borrowing. Sums like 26-‐3
You can make activites out of this with 2 children: they can take a card that says 26-‐12, then one of them will show 23 on the grid and the second one will take away 12. They should write down the answer.
Step 2: Moving on to simple sums that need borrowing, the idea is the same as before: you need to go to the bank! Start with a simple sum where you only have to subtract a single digit number from a two-‐digit number, like this:
You have to take away 4 units. How can you do it? You have to go to the bank. Give them time to think about it, to figure it out by themselves. DON’T TELL THEM WHAT TO DO! Let them struggle. Give small hints only when needed (don’t be in a hurry to give hints.) The hints are the same as for carrying.
Step 3: Finally, move on to subtraction using two 2-‐digit numbers.
GOING FORWARD: THREE DIGIT NUMBERS
The basic ideas are all present when you work with 2 digit numbers. There should be no sush to go on to 3 digits. Once students are really comfortable with 2 digits, then 3 digits will be easy. The entire sequence of activities can be repeated with 3 columns – the activites will be much quicker as they will already be familiar with the processes. Again, give lots of time, and remember to encourage thinking and to keep a playful atmosphere in your classroom!