Expanded Form

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SWBAT represent numbers in expanded form.

Big Idea

Being able to understand and explain numbers will help students make sense of multi-digit computation and problem solving.


20 minutes

Today's Number Talk

For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model.

Task 1: 7 x 3

For the first task, 7x3, I modeled a few students' strategies on the board to begin with: 7x3 Models. Then students tried to find more strategies on their own. Some students were Mixing up Adding & Multiplying for 7x3. I was glad that we started today's Number Talk with a simpler problem so that clearing up this confusion would be easier and more likely. 


Task 2: 17 x 3

During the next task, most students were extremely motivated and worked hard at Finding Multiple Strategies for 17x3. I came back around to this student later on and found Four Strategies on her board! 


Task 3: 217 x 3

For 217 x 3, many students were able to correctly provide the solution to the problem prior to solving it on their boards! Here, a student finds his mistake when asked to explain his thinking: Finding Mistakes through Discourse. This goes to show the importance of Math Practice 3: Constructing viable arguments.


Task 4: 3217 x 3

Here, a student shows how 3217=2[(3x100)+200+17]. Again, through the process of defending his answer, he discovers his own mistake. I loved watching him Fixing the Mistake! in front of all his classmates.  

Teacher Demonstration

30 minutes

To begin, I asked students to get out their whiteboards. I then introduced the Goal: I can represent numbers in expanded form. Even though most students have learned about standard, word, and expanded forms in third grade, I took the time to review each of the Forms of Numbers.


Forms of Numbers:

I explained: There are three basic ways to represent numbers. The first way is called standard form. (I wrote the word and definition on the board). Standard form is when you represent a number using a base-10 numeral, such as 123. 

I moved on to explaining the meaning of "written form" on the board: Written form is when you represent a number using words. For example, I would write "one hundred twenty-three." Did anyone notice a special mark that I made between the twenty and three? This is called a hyphen. I wanted to pre-teach this concept for tomorrow's lesson.

Then I wrote the meaning of "expanded form" and explained: Finally, there's expanded form. This is when you represent a number by adding the value of its digits. For example 123 = 100 + 20 +3. 

Place Value Blocks:

Today we are going to be using place value blocks to represent numbers. This will help us truly understand the meaning of expanded form. I placed magnetic color-coordinated base 10 blocks on the board to demonstrate with: 

I continued: Let's take a closer look at place value blocks. The orange pieces are called units and each one is equal to one unit. The blue pieces are called rods. Each one is equal to 10 units. Then we have flats. These are yellow and are equal to 100 units. Finally, we have the thousand cube, which is equal to 1000 units. What else are you noticing about the place value blocks? Turn and talk to a partner!

After giving students some time to discuss their observations, one student said, "It takes 10 orange units to get to a blue rod and ten blue rods to get to a yellow flat." I drew the following representation on the board to further show the student's thinking: Place Value Blocks Relationships. Students came up to the board to explain their thinking: Analyzing Base 10 Blocks and Number of Units in Each. This was a great opportunity to solidify the idea that the value of a digit in one place represents 10 times what it represents in the place to its right. 

I wanted students to get their hands on their own math manipulatives! Excitement filled the air as I passed out the following math tools to students: Place Value Flip Charts (one per student) and Base-10 Blocks: (one set per group).


Guided Practice:

I wanted to give students the opportunity to use multiple tools (place value blocks, flip charts, and whiteboards) to represent numbers (Math Practice 5: Use appropriate tools strategically). 


1. Modeling with Place Value Blocks:

For this activity, I first used the base ten blocks on the board to model the number 17. I used one rod and seven units. I asked groups to work together to model "the number on the board" using their place value blocks. 

2. Using the Place Value Flip Charts:

Next, I asked students to use their individual place value flip charts to represent the number shown. This chart was particularly helpful to students as each place in the number was labeled (ones, tens, hundreds, etc.) and in the correct order. 

3. Expanded Form on Whiteboards:

Then, students wrote the number in expanded form on their white boards: Flip Chart & Expanded Form.  


Next, we went through the same process with larger numbers. I made sure to include numbers with zeros to clear up any misconceptions: 300 + 6.JPG400 +30 + 51000 + 301000+300+50

Student Practice

30 minutes

At this point, I knew students were ready to practice representing numbers in expanded form without using the manipulatives as a support.

I asked students to place the manipulatives aside, to get out their math journals, and to create the following t-chart on a new page in their journals: Base Ten Numeral vs. Expanded Form Chart. I began by writing 17 on one side and asked students to record the expanded form. One student said, "That's 10 + 7." Then we moved on to 207. I waited for students to record their thinking before discussing the answer as a class. I took this time to monitor student understanding by checking student responses in their journals.

For the first five numbers, I tried to use similar digits to reinforce the idea that digits have different values when located in different places within a number. For the last four tasks in the chart, I provided students with the expanded form for the last four numbers all at once and asked them to fill in the matching base ten numerals. I wanted to be sure students were provided with a variety of tasks to develop a well-rounded understanding of expanded form. 

As soon as students were finished with their charts, I asked them to check their work with a peer before moving on to the final task, placed on the corner of my desk. 


30 minutes

To ensure students were able to independently meet the goal of today's lesson, the final task was two practice pages (front to back) from the site below. 

1. Practice Page #1: Expanded Form to Numeric Form Within 1 Million

2. Practice Page #2: Numeric Form to Expanded Form Within 1 Million

During this time, I checked on each student and provided support and questioning as needed. One student in particular was more successful when he used the place value flip chart to represent the larger numbers in particular. I'll address this student in the reflection section. Here's an example of a conference with a proficient student: Expanded Form Practice.

Most students finished within 15 minutes! Again, I asked students to check their answers with peers. Peer-checking gives students the opportunity to practice Math Practice 3: Construct viable arguments and critique the reasoning of others. Here are examples of proficient students: Proficient on Practice Page #1 and Proficient on Practice Page #2. Alongside the edge of one page, a student checked off each problem as she verified her answers with a peer.