Place value is the basis of our entire number system. A place value system is one in which the position of a digit in a number determines its value. In the standard system, called base ten, each place represents ten times the value of the place to its right. You can think of this as making groups of ten of the smaller unit and combining them to make a new unit.

Ten ones make up one of the next larger unit, tens. Ten of those units make up one of the next larger unit, hundreds. This pattern continues for greater values (ten hundreds = one thousand, ten thousands = one ten thousand, etc.), and lesser, decimal values (ten tenths = one one, ten hundredths = one tenth, etc.). At this level, however, your students will be focusing on mastering place value for ones, tens, and hundreds.

In standard form, the number modeled above is 233.

A place-value chart is a way to make sure digits are in the correct places. A great way to see the place-value relationships in a number is to model the number with actual objects (place-value blocks, bundles of craft sticks, etc.), write the digits in the chart, and then write the number in the usual, or standard form.

Place value is vitally important to all later mathematics. Without it, keeping track of greater numbers rapidly becomes impossible. (Can you imagine trying to write 999 with only ones?) A thorough mastery of place value is essential to learning the operations with greater numbers. It is the foundation for regrouping ("borrowing" and "carrying") in addition, subtraction, multiplication, and division.

## Introducing the Concept: Place Value to 1,000

Last year the children worked with place value through 99. Before beginning place value to 1,000, review place value through 99. If the children spend some time reviewing, the transfer of knowledge to 1,000 will be much easier. Take time for practice of many groupings for 2-digit numbers, using the language of place value.

**Materials: **84 snap cubes, tens and ones mat for each child or pair of children

**Preparation: **Copy a tens and ones mat for each child. Prepare a chart of number words. One column should include the number words one to nine, a second column ten to ninety by tens, and a third column, eleven to nineteen. Place it so children can refer to it.

**Prerequisite Skills and Concepts: **Children should know how to count to 100. They should be familiar with number words and what they look like when written.

Give each child (or pair of children) 26 cubes.

**Ask**:*How many cubes do you have?*

Children should count and say that there are 26 cubes.**Say**:*Put away those cubes. I'm going to give you some more cubes.*Give children 26 more cubes.

**Say**:*First make as many groups of ten as you can.***Ask**:*How many groups of ten can you make?*

Children should say they have 2 groups of tens, or 2 tens.**Ask**:*How many tens and ones do you have?*(2 tens and 6 ones)*How many cubes are there?*(26)*Do you remember how many cubes you counted before?*(26)*Are there the same number of cubes in both groups?*

Children should say that the two groups have the same number of cubes.**Ask**:*Does it change the number when you group the tens?*

Children should say that the number is the same whether you count them or group them and count them by tens and ones.- Continue counting cubes and making groupings of tens and ones. Lead children to see the relationship of the number words to the groupings of tens and ones.
- Give each child a tens and ones mat and a sheet of paper to write on. Then give each child 32 cubes. Have them group the cubes by tens and ones on their mat. You may want to have children make trains of ten at this point.
**Ask**:*How many tens do you have?*(3)*How many ones?*(2)**Say**:*Now let's write that number.***Ask**:*What number did you write?*(32)*How did you know that was the number?*

Children should say that the 3 shows how many tens and the 2 shows how many ones.- Repeat with other 2-digit numbers until children have a good understanding of place value to 99.

## Developing the Concept: Place Value to 1,000

Once children show a good understanding of place value with tens and ones, introduce place value with hundreds, tens, and ones.

**Materials: **place value blocks, hundreds place value mat

**Preparation: **Copy a hundreds mat for each child.

**Prerequisite Skills and Concepts: **Children should know place value with tens and ones.

**Say**:*We have been working on place value with 2-digit numbers. Today we are going to work with 3-digit numbers. You will be using a new model to show your numbers.*- Introduce the ones, tens, and hundreds place value blocks. Allow time for children to line up and compare ten ones to one ten, and ten tens to a hundred block. Have children place the blocks in the correct positions on their place value mats.
**Say**:*I have ten ones. I want to trade them for another block that has the same value. Who will trade with me?*

Have a volunteer show how you can trade ten ones for one ten. Repeat using ten tens and a hundred block. Then demonstrate trading 20 ones for 2 tens or 20 tens for 2 hundreds.**Ask**:*If you have the number 162, how many hundreds, tens, and ones will you place on your mat?*

Allow time for children to place their blocks, repeating the number as necessary. Children should place 1 hundred block, 6 tens blocks, and 2 ones blocks in the appropriate sections of their mats.- Repeat the activity until the children place the blocks correctly and with ease.
**Say**:*Now we are going to try another number. It is 205.***Ask**:*How many hundreds are there?*(2)*How many tens?*(0)*How many ones?*(5)**Ask**:*Are there blocks in every section of your mat?*

Children should say that there are no blocks in the tens section.**Ask**:*How will we write the number?*

Lead children to express that they must write a zero when there are no blocks.- Continue by challenging children to show numbers on their mats when you give the digits out of order. For example, say, I have 2 tens, 3 hundreds, and 6 ones. What number is that? I have no hundreds, 7 ones, and 3 tens. What number is that? This will make children pay careful attention to the place value words.

**Wrap-Up and Assessment Hints**

Place value needs lots of practice. Reinforce the vocabulary. Remind children that it is very important to listen and write a number carefully; that the numbers should be in order and that the numbers should be in the correct position. As you assess each child, check the placement of the written numbers.

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