Tens and ones - teaching place value

How to teach the concept of place value systematically to those kids who don't just 'see it' at the first glance?  When kids count, they basically just learn numbers as some kind of continuum that continues and continues.  With simple counting your child might not catch on to the inherent structure and how it goes in groups of tens and hundreds and thousands.

For children to understand place value, they first need to be able to name numbers, do simple additions and subtractions with small numbers (and hopefully memorize some of these 'basic facts'), and understand about groups in counting (or skip-counting).  To the latter end, have them count up and down in twos, threes, fives, tens, and hundreds.  Explain that if you have lots and lots of objects, the efficient way is to count them in groups, not individually. 

You can use matches or rocks for example.  Dump a bunch of them on a table and show how it is easier to count them in groups of ten.  First you make groups of ten, then count the ten-groups separately and the individual matches separately.  So you say, "I have here five ten-groups, and four individual matches." Then you can count another amount of matches by grouping them first into groups of tens, and counting the ten-groups and the ones separately. Introduce also the words twenty, thirty, etc.

Then you can make ten-groups by using rubber bands to band 10 matches together and practice with those. You can practice and do all this until the child understands the idea well. Counting in groups also paves way for multiplication concept.

Then would come the actual representation of this idea on paper, with numbers. The crucial point in place value is that a certain column represents a certain size group.  Then the digit in that column tells you how many of those certain size groups.  The difficulty for some children may be in that these columns are relational and quite abstract, depending on the positioning of the digits.

Here's an excellent idea to teach place value and this idea of representing certain size groups by something else (from the article The Concept and Teaching of Place Value).  You need different color plastic items; for example white, red, and blue poker chips are inexpensive and serve well the purpose.  The white are ones, the blue are tens, and the red are hundreds.  

In many manipulative sets the ten-blocks or whatever actually contain ten objects.  It is better to use same size objects so that children are forced to picture the color chip in their mind as representing a group, instead of actually being that group.  That is because in writing numbers, the symbol or digit '5' can just be representing fifty if it's written in a certain place, but it is not in reality fifty.

Now, do not use the word "represent" unless you're sure it wouldn't confuse.  Instead, talk about trading and exchanging and making: "You can trade ten white ones into one blue one."  Show them how to count ten blue ones, saying, "10, 20, 30, 40, .., 90, 100" and how you can trade ten blue ones into one red one.  Then start playing with the chips and asking questions: If you have 15 white ones, what can you trade? I have two blue ones, how many whites could I get? How could you make the number 54 with the chips, using the LEAST possible amount? I have two blue ones and four white ones, what number would this be? etc.

Continue until children can easily make numbers with the chips, or tell what number a certain combination of chips represents.  This way they will understand "group representation" - the concept that one entity (a chip) represents a group.  This is a prelude to understanding how a certain column represents a group.

It's very advisable also do some adding and subtracting with the chips.  For example, you have 2 blue ones and 7 white.   Add 8 whites.  What happens?  The example easily leads to the concept of trading the ones into a ten.  Similarly, if you have three blues and 2 whites, and you need to subtract 6, ask the students, what would they do - and you're on your way to explaining trading of a ten into 10 ones for the purpose of subtracting.

After doing math with the color chips, it will be easier for the children to grasp the idea that when numbers are written on paper, the columns are like the different color chips.  The first column is like white chips, telling you how many ones you have, and the second column is like blue chips, telling you how many tens you have, etc.

Another similar teaching idea is found in Instructional Sequence for Teaching Students Place Value, Addition and Subtraction to 1000. It uses packaging of candies into 10-rolls, 100-boxes, and 1000-boxes to introduce students to place value, and also uses certain symbols to represent these different packages.

Yet more ideas for teaching place value:

• Have a counting game using matches where the rules go that you can ONLY use words one to ten when you count.  In other words, don't use words like eleven, thirteen, twenty, etc.  In the game, each player adds one more match to the common pile on the table, and says the name of the number in the broken down form.  For example eleven is said as "ten and one", twelve is "ten and two", twenty is "two tens", twenty-five is "two tens and five", etc.  Whenever a whole ten is fulfilled, those ten matches are bundled together with a rubber band.
  You can modify the game so that on their turn, each player adds two matches to the pile instead of one.
  Another modification is to name the number both in the usual way and in the broken down form.
• See here some example exercises for learning ones and tens place value
• Check this link for games for teaching place value
• Check out Math Mammoth Place Value 1 (place value 0-100) and Place Value 2 (place value 0-1000) from Math Mammoth books.
• Use a 100-bead abacus. I've written about using such abacus here. My book Place Value 1 includes instruction that uses a 100-bead abacus.

67
+ 58

11 
15

125

67
+ 58

15
11 

125

857
+ 979

17  
12 
16

1836

857
+ 979

16
12 
 17  

1836