Discovering Patterns Within the Base 10 Number System

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: 15 x 4

During the first task, I at first demonstrated student's strategies on the board: Multiple Strategies for 15 x 4. Then, I asked students to come up with other strategies. Some students decomposed the 15 or the 4 while others decomposed both the 15 and 4 to solve the expression.

Task 2: 153 x 3

During the next task, we discussed 153 x 3. Here, a student shows how she worked on Finding Many Strategies for 153x3. This sort of activity is easy to differentiate. Higher level students develop more and more strategies. Typically, each strategy increases with complexity. Students who struggle with math are provided with the time and support to successfully model one strategy.  

Task 3: 2,153 x 3

During the next task, one student showed three different Array Models for 2153 x 3. Here's another student's board showing Multiple Strategies for 2153x3. I have loved watching student perseverance (Math Practice 1) increase through our Number Talks. 

To begin the lesson, we reviewed the Place Value Chart from yesterday's lesson, Bank Teller & Representing Numbers. The goal was to help students construct new knowledge by building upon previously learned place value concepts. Yesterday, when we began completing the equations at the bottom of the chart, I noticed some students struggled. Today, I wanted to make sure students understood that one place to the left is 10 times greater. Looking specifically at the following equations on the place value chart, the students and I discussed observed patterns (Math Practice 8: Look for and express regularity in repeated reasoning).

Equation 1: 1 ten = 10 ones

Equation 2: 1 hundred = 10 tens = 100 ones

Equation 3: 1 thousand = 10 hundreds = 100 tens = 1000 ones

Equation 4: 1 ten thousand = 10 thousands = 100 hundreds = 1000 tens

Equation 1

We reviewed with the idea that 1 ten = 10 ones. Then, I asked: What did we calculate by to get from the 1 to the 10? Students responded, "We multiplied by 10!" I labeled the chart by drawing an arrow from the 1 to the 10 and writing x 10.

Equation 2

We proceeded to the next equation, 1 hundred = 10 tens = 100 ones. I wonder if this works every time you move left one space on the place value chart. Let's look at 1 hundred. What did we calculate by to get from the 1 to the 10? Students responded, "We multiplied by 10!" Is anyone starting to see a pattern here? Students responded, "Yeah! Each time, you multiply by 10!"

We moved on to the number of ones equal to 10 tens: Well, let's see if this pattern continues! What did we calculate by to get from the 10 to the 100? Again, students responded, "We multiplied by 10!" 

Equation 3

We continued this questioning process to analyze 1 thousand = 10 hundreds = 100 tens = 1000 ones. Students caught on quickly. Soon, concepts that were confusing yesterday became more clear to my students! 

Equation 4

On the final equation, 1 ten thousand = 10 thousands = 100 hundreds = 1000 tens, I asked: If we continue this pattern...

• How many thousands are in 1 ten thousand? Students responded: "10 thousands!"
• How many hundreds are in 10 thousands? Students responded: "100 hundreds!" 
• How many tens are in 100 hundreds? Students responded: "1000 tens!" 
• Even though we didn't have room to display it on our chart, I asked: How many ones are in 1000 tens? Students responded: "10,000 ones!"

This was the perfect opportunity to increase the rigor of the task. Instead of multiplying by ten, I wanted students to see the inverse relationship between multiplication and division so I asked:

• When looking at the number of hundreds equal to 1000 tens, what could we calculate by to get from the 1000 to the 100? Turn and talk! After some time and discussion, students responded, "Divide by 10!" 
• When looking at the number of thousands equal to 100 hundreds, what could we calculate by to get from the 100 to the 10? Students responded, "Divide by 10!" 
• When looking at the number of ten thousands equal to 10 thousands, what could we calculate by to get from the 10 to the 1? Students responded, "Divide by 10!" 

Now, I wanted students to have the opportunity to model these abstract ideas using a concrete model!  

I quickly assigned math partners by asking students who were sitting next to each other to be partners. I already have student desks strategically placed based on behaviors and skills so that grouping students is quick and easy. I invited students to gather together on the carpet with a Bag of Money, student journals, and partners.

I then asked students to create a 4 column chart (Process Grid) in their math journals and to label the columns with the following headings: Question, Money Model, Expression, Answer. I wanted to use this process grid activity to carefully guide students to truly understand  that the value of a digit in one place represents 10 times what it represents in the place to its right. To aid the learning process, I color-coded each column to support the learning and categorization of each concept. Also, I only showed three of the process grid rows at a time to help break down the task. 

Question:

Prior to the lesson, I wrote the same question format in first column of each row: How many _____ does it take to get to ______? Once students were ready, I began by filling in the blanks in the first question: How many ones does it take to get to ten?

Money Model:

I asked students to use their money to find the answer to this question by counting the number of ones to get to ten: Counting Money. We documented our findings by pasting ten ones in the money column of the process grid.

Expression:

Then I asked students to help me write an expression to represent the money model. Altogether, we came up with 1 x $10.

Answer:

Finally, we completed the fourth column by rereading the question and recording the answer: 10 ones. 

We continued by answering each of the following questions by going through the same process (Question, Money Model, Expression, and Answer). 

• How many tens does it take to get 100? 
• How many hundreds does it take to get 1,000? 
• How many thousands does it take to get 10,000? 
• How many ten thousands does it take to get 100,000? 
• How many hundred thousands does it take to get 1,000,000? 

With each question, I gradually expected students to take on more responsibility for their learning. For example, I began asking students to work independently by completing a row with their partners and then, by completing a row on their own. I also provided students with the opportunity to explain their thinking to the class by completing the process grid on the board.

10 ten thousands

By the third row, students were eagerly Discovering the Pattern!. Even though students could use the pattern to discover the answer each time, I still wanted students to continue counting and using the money to help them contextualize place value concepts. In this video, 10,000x10 =100,000, a student struggles with explaining the number of ten thousands in 100,000. To help support this student, I encourage her to use the money model to justify her reasoning (Math Practice 3). 

In order to find the number of hundred thousands in one million, students counted again, only by 100,000 this time: 100,000 x 10 = 1,000,000. 

Here's an example of a completed Student Journal during this activity. Once finished, I asked students to look for patterns (Math Practice 8). Here, Analyzing the Process Grid, a student explains how each row gets 10 times bigger. Another student explains how she discovered the Adding and Taking Away Zeros I was surprised and proud when she correctly answered the question: When you take away a zero, what are you doing? (Dividing by 10.)

To bring closure to this lesson, I used the following place value magnets to create a place value sequence on the board: Analyzing the Value of the Digit 2. With each number, I rewrote the digit 2 one place over to the left. Students then explained that we are "multiplying each time we move one place to the left." I asked other students to share their conjectures. One student explained how each time, you are Moving the 2 Over to Create Another Number. Another student explained how you Just Keep Adding a Zero!. 

While I normally introduce the goal at the beginning of the lesson, I didn't want to "give away" the pattern. I wanted students to discover what happens each time we "move over one place to the left" on a place value chart. So, as a final task, I wrote the Goal on the board: I know a digit in one place represents (stands for) 10 times what it represents in the place to its right. I left the underlined words out and asked students to help fill in the "missing words." I wanted them to truly think about the goal/pattern they had "discovered" during today's lesson.