Imagine a world in which every whole number had its own symbol: a symbol for 1, 2, 3..., along with a symbol for 87, 135, and 62 million (to name a few). We would hardly be able to count, let alone perform arithmetic or any of the mathematics that we have today.
As far back as we know, humans have always recognized that numbers need some kind of pattern to make sense of them and do math with them. However, for many centuries, those patterns still used new symbols for 1, 10, 100, 1000, and increasing powers of 10, along with whatever symbols they used for the individual numbers 1, 2, 3, 4, and so on. Different number systems represented these numbers in different ways, but the real breakthrough was the invention of the number zero, which allows us to represent all whole numbers with just a few symbols. Today, we can represent every number with only 10 digits in total: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For example, we can represent the large number 61,432 with the symbols 6, 1, 4, 3, and 2 because of the place-value system.
What Is Place Value in Math?
Place value is the basis of our entire number system. This is the system in which the position of a digit in a number determines its value. The number 42,316 is different from 61,432 because the digits are in different positions. In the standard system, called the base ten number system (or decimal system), 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.). In Grades 2 and up, your students will be focusing on mastering place value for ones, tens, and hundreds. In this article, we provide two lessons for introducing and developing the concept of the base ten number system.
In standard form, the number modeled above is 233.
What Is a Place-Value Chart?
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 create a place-value model using actual objects (for example, place-value blocks, bundles of craft sticks, or—if necessary—digital manipulatives), write the digits in the chart, and then write the number in the usual, or standard form.
An understanding of the place value of numbers is vitally important to learning operations. It is how we can compare numbers; line up numbers vertically; make sense of addition, subtraction, multiplication, and division with larger numbers; and is the foundation for regrouping ("borrowing" and "carrying").
Introducing the Concept: Place Value to 1,000
Before second grade, children have usually 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 easier. Take time for practice grouping and interpreting 2-digit numbers, using the language of place value.
Materials: at least 84 snap cubes, tens and ones mat for each child or pair of children, paper to write on; if teaching remotely, use digital versions
Key Standard: CCSS.MATH.CONTENT.1.NBT.B.2Understand that the two digits of a two-digit number represent amounts of tens and ones. (CCSS 1.NBT.B.2)
Preparation: Prepare 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. Display it so children can refer to it. If teaching remotely, use digital versions of the mat and chart.
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 may use varying strategies to count the cubes. Encourage students to compare the different ways they arrived at their totals. Show that if you count the cubes by ones, there are 26 total.
- Say: Put away those cubes. I'm going to give you some more cubes.
Unless teaching remotely, have children put their cubes off to the side on a different surface or in a container. Note publicly that there were 26 cubes. Give children 26 more cubes.
Ask: Make as many groups of ten as you can. 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) How many cubes did you count before? (26) Are there the same number of cubes in both groups? (Yes)Children should understand 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 one by one or group them and count them by tens and ones.
- Continue counting cubes and making groups of tens and ones. Lead children to see the relationship of the number words to the groups 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.
- 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 seem confident in converting their groups of cubes to written 2-digit numbers.
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
Key Standard: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones. (CCSS 2.NBT.A.1)
Preparation: Create a hundreds place-value mat for each child. If teaching remotely, use digital versions of the mat.
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 place-value 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. Only use numbers that do not have any zeros in their digits.
- 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. If children have different configurations, encourage them to discuss the way they made the number with each other. Encourage them to explain what they did without calling it right or wrong.
- 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. For children who are ready, consider extending to more puzzling descriptions such as "I have 2 tens, 3 tens, 6 hundreds, and another ten" or "I have 2 tens and the same amount of hundreds and ones."
Wrap-Up and Assessment HintsPlace 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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