# Ideas for teaching place value with two-digit numbers - tens and ones

When children count, they basically learn numbers as a kind of "continuum" that goes on and on.  With simple counting your child might not catch on to the inherent structure of our number system, and how it is built with groups of tens, hundreds, thousands, and so on.

For children to understand place value (typically in 1st grade), they need to first be able to name small numbers, do simple additions and subtractions with small numbers, and understand about groups in counting (or skip-counting). Explain that if you have lots and lots of objects, the efficient way is to count them in groups, not individually.

## An activity to help children understand tens and ones

Use beans or rocks. Place a pile of them on a table and show that it is easier to count them in groups of ten. First make groups of ten, then count the ten-groups and the individual beans separately. Say, "I have here five ten-groups, and four individual beans."

Continue in a similar way. Take a different amount of beans. Group them into groups of tens (and some left-overs). Count the ten-groups and the ones separately.

As you are doing that, you can introduce the words twenty, thirty, forty, etc.

Another way to practice this is to use bags or rubber bands so you can bag or band 10 of your objects together, and practice with those. Display 4 bags and 3 leftover beans. How many are there? The four bags means... forty, and besides those there are three. Total, forty-three.

## A counting game for place value

You need beans or other counters and small bags. The main rule is that you are ONLY allowed to use the words from one to ten when you count! In other words, you are NOT allowedto use words like eleven, thirteen, twenty, etc.

In the game, each player adds one more counting object to the common pile on the table, and says the amount of total objects in a broken-down form. For example, eleven is said as "ten and one", twelve is said as "ten and two", twenty is "two tens", twenty-five is "two tens and five", and so on.

Whenever a whole ten is fulfilled, those ten beans are bundled together into a bag.

You can modify the game so that on their turn, each player adds two beans to the pile instead of one. Another variation is to name the number both in the usual way and in the broken down form.

Next would come the representation of this idea on paper, with numbers. The crucial point in place value is that a certain position or place represents a certain size group. The digit in that place tells you how many of those certain size groups you have.

(For that matter, we could start a different system of writing numbers where font size tells you the place value: for example 782 would be 7 tens, 8 hundreds, and 2 ones = 872.  Or, where the color of the digit tells you what group it represents, for example if black is ones, blue is tens, and red is hundreds, then 74 would be 470.  You could use these to further illustrate how the way we write numbers is just a convention; other cultures have done it differently.)

## A more abstract idea

Here's a different idea to teach place value and the concept of representing certain size groups by something else (from the article The Concept and Teaching of Place Value). It does use something quite abstract, so may not help all students.

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.

When we use same-size objects, children are forced to picture the color chip in their mind as representing a group, instead of actually being that group. This works just like in writing numbers, when the symbol or digit '5' can represent fifty if written in a certain place, but it is not in reality fifty.

Now, do not use the word "represent" unless you are sure it won't confuse.  Instead, you can talk about trading, exchanging, and making: "You can trade ten white ones into one blue one."  Show the students 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?

This would be three hundred four. Here you see fifty-two.

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.

You can 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 regrouping ten ones as 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 are on your way to explaining regoruping a ten into 10 ones for the purpose of subtracting.

After doing math with the color chips, it should be easier for the children to grasp the idea that when numbers are written, the columns or places 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.

## Resources for place value

Example exercises for learning tens and ones

Online games for teaching place value

Using a 100-bead abacus in elementary math

I have a student that is adding from left to right. He is also regrouping in the same manner. I need a creative way of teaching him that you must begin in the ones place when adding the numbers.
Keith Myers

If your pupil's way of adding works, would he really need to change it? It is certainly possible to add from left to right - or in other words, starting from the biggest place value. See these simple examples:

 tens first ones first ```67 + 58 11  15 125``` ```67 + 58 15 11 125```
 hundreds first ones first ```857 + 979 17 12  16 1836``` ```857 + 979 16 12 17 1836```

In fact, the way many people add mentally is adding from left to right:

 35 +  47 = (30 + 40) + (5 + 7) | adding tens first = 70 + 12 = 82. or 123 + 457 = (100 + 400) + 23 + 57 | adding hundreds first = 500 + (20 + 50) + 3 + 7 | then tens, and ones = 500 + 70 + 10 = 580

There is more to learn about adding left to right, of course, for example how to carry when adding in columns. This book from Amazon explains about adding from left to right (and other mental math tricks):
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