Posted by: degarcia | November 24, 2008

Place Value

Background Knowledge

Place value helps us make decisions that are used in our daily lives ex) costs, weight, distances, time etc. Our number system is based on a Base Ten system.

Base ten means our number system has a base of ten. We group our numbers by clusters of ten. Example: 12= one group of ten and two ones. 46= four groups of ten and six ones.

Place Value

Place Value means that each position of digit in a number represents its value based on powers of ten.

Place value

Let’s compare the numbers 4321 and 1234.

4321 = (4 x 10^3) + (3 x 10^2) + (2 x 10) + 1

In this number we have 4 groups of 1,000, 3 groups of 100, 2 groups of 10 and 1 group of 1.

1234 = (1 x 10^3) + (2 x 10^2) + (3 x 10) + 4

In this number we have 1 group of 1,000, 2 groups of 100, 3 groups of 10, and 4 groups of 1.

We know that there are four groups of one thousand in the number 4321 thus it making it greater then 1234 since that number only has one group of one thousand. With these two numbers its easy to see how the digits can remain the same but once their positions change the value of the number changes drastically.


Lesson Plans

Grades: K-2

For a good daily routine count the days of the school year as they progress making sure to group some sort of counter in sets of tens and one hundreds.

1st Grade

1. Groovy Grouping: 1st Grade Standard 1 Obj. 1

Summary: These activities are designed to help students see how grouping objects into tens — when possible — does bring organization to amounts of objects to be counted. These activities are also meant to introduce students to the various ways a number may be written (e.g. standard form and expanded form).

Utah Lesson Plans. (2003). Place Value. Retrieved: December 16, 2008. Utah Education Network.


2nd Grade

1. Place Value as Easy as Pie: 2nd Grade Standard 1 Obj. 1

Summary: Place Value is a difficult concept for students to understand. This activity provides a basic hands-on introduction to place value.

Utah Lesson Plans. (2006). Place Value. Retrieved: December 16, 2008. Utah Education Network.


2. Value that Number: 2nd Grade Standard 1 Obj. 1

Summary: Students will complete a variety of activities to gain an understanding of the place values for ones, tens, and hundreds.

Utah Lesson Plans. (2004). Place Value. Retrieved: December 16, 2008. Utah Education Network.



Background and Definition:

Our number system is based on a base ten place value. The place of a digit tells how many ones, tens, hundreds, and so forth. Earlier number systems used different bases for counting. For example, the Babylonians counted in base 60. “They used finger segments of one hand to total 12, and the other hand was used for double counting, or keeping track of the numbers of 12s they counted” (Bahr and de Garcia, 2010, p.103). The Chinese and Egyptians used non-proportional symbols to represent place value, like colored coins or counters. This is similar to our money system we have today.

The Hindu-Arabic number system was the first to use base ten. “In our system, ten symbols represent the quantities 1-9 and zero” (Bahr and de Garcia, p. 104). It’s characterized by base ten, place value, multiplicative principle, additive principle, and zero as the place holder (Bahr and de Garcia, p. 104). This system is easier than previous systems because we can refer to our fingers to count on base ten.

The previous post stated, “Place Value means that each position of digit in a number represents its value based on powers of ten.” In any one place value, there are 1-9 groups of numbers. As soon as the group adds up to ten, that ten is clustered and moved up to the next place value. The multiplicative principle of the base ten system states that “Each place value to the left of another is 10 times greater than the one to the right” (Charles, 2005, 13). Our base ten system also uses the additive principle which states, “You can add the value of the digits together to get the value of the number” (Charles, 2005, 13). Our place value is placed on proportional representation. For example, the two in the number 27 represents two tens and the seven represents seven ones. An example of nonproportional representation is our money system. The size or color of the coin or bill is not proportional to the amount of money it represents. For example, a dime, which is smaller than a penny, is worthy ten times the amount of a penny.

How to teach place Value:

Building Numbers: The concept of building numbers helps students make visual and kinesthetic connections to the numbers. Students should build numbers with a partner. Using a place-value mat and base-10 blocks, student A builds a number and student B records the number on a piece of cash register tape (long and skinny for several recordings) or regular paper. Student B should record the number one as “01” because there are no cubes in the “tens place.” The students will continue to add more numbers to their place-value mat. When they get to “10,” make sure the students understand that the tenth cube indicates a change in place value. Instead of placing ten ones in the ones column, the students place one ten in the tens column, or place ten ones in a cup and place the cup in the tens column. As recording continues, students will see a pattern develop amongst the numbers. There should be 9 numbers starting with 0 (or ten if the students started with “00”) (e.g. “01, 02, 03…”), ten numbers starting with the number “1” (e.g. “10, 11, 12, 13…”), ten numbers starting with the number “2,” etc. The students will see the tens value increase after ten entries each time. They will also see the value in the ones column increase from 0-9 for each place value in the tens column. The students can come together after building numbers up to 100 to discuss patterns they saw while building (Bahr and de Garcia, pp. 108-109).

Using a Place-Value Mat: It is really helpful to use a place-value mat to teach place value. It can be beneficial when working individually with children to “help them see the relationship between what they are building and how they read the number” (Bahr & de Garcia, 2010, p. 110). “One activity is to put an amount of cubes and cups on the place-value mat and have each child write the numbers of cups underneath the cups and the numbers of singles underneath the singles” (Bahr & de Garcia, p. 110). Then have the student read the number aloud and then count to confirm the quantity.

Race to 100 is a great game to reinforce the notion of trading up and Race to Zero is a game that reinforces trading back. These can be used whether the students are using cups to hold the cubes or if they are now using the base 10 rod. “These games can be played to any desired quantity and with different-sized dice, depending on the number size the student is working with. Two players play against each other using their own place-value mat and set of cubes or beans” (Bahr & de Garcia, p. 110). In Race to 100, “one player rolls the dice and puts that many cubes in her ones spot. She then says how many more she needs to make 10 and whether she thinks she can get it on her next roll” (Bahr & de Garcia, p. 110) Player 2 then takes a turn. Whenever a player adds more cubes, that player needs to “say the new total and whether there are enough ones to make a 10. If so, the player says how many leftovers there are. If not, the player says how many more are needed” (Bahr & de Garcia, p. 111).

While playing Race to Zero, children struggle identifying what to do when they roll their first amount and have started with a 100 flat. “It takes a long time for them to realize that they have to break the flat into 10 tens and break a 10 into 10 ones so that they can take some away” (Bahr & de Garcia, p. 112).

Websites that emphasize Place Value:




By Hayley Hicks and Heather Wilkinson


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