In this activity, students sort a set of multi-digit numbers and describe the place-value relationships they notice in the sorted numbers. They analyze numbers that have the same digits and write the numbers in expanded form, highlighting the value of each digit. Students then describe relationships they see between the digits in each number.

For example, students may note that the value of the 2 in 46,200 is 200, in 462,000 it is 2,000, and that 2,000 is ten times as much as 200. In the Activity Synthesis, they learn that the observed relationship can be expressed with multiplication and division equations, such as , , or other equivalent equations.

When students sort the cards, they look for how the numbers are the same and different, including their overall value or the digits that make up the numbers (MP7).

• Invite previously selected students to share their expressions in expanded form and what they noticed about the value of the 4.
• “What do you notice about the value of the 6 in each number? The value of the 2?” (The value of the 6 is different in each number. It is first 600, then 6,000, then 60,000.)
• Students may talk about the number of zeros in each number. Shift their focus to the place value of the 6— hundreds, thousands, ten-thousands.
• “How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?” (The value of the 2 in 462,000 is 2,000 and the same digit in 46,200 has a value of 200. 2,000 is ten times the value 200.)
• “What multiplication equation could we write to represent the relationship between the 2 in 46,200 and 462,000?” ( )
• “We can also write this equation using division: .”

In this activity, students read, write, and analyze multi-digit numbers and use expanded form to describe the relationship between the digits. The numbers in the activity are designed to highlight common errors in reading and writing large numbers. Students encounter numbers with the digit zero in the ten-thousands place and think about how to represent this in expanded and word forms.

• “How did you decide which two numbers to choose to make the statement true?”
• “How can you use the value of the digit 3 in these numbers to check if the statement is true? How can you use place value to find two numbers that make the statement true?”

• See Lesson Synthesis.