In this activity, students compare pairs of numbers with the same number of digits and the same set of digits (for example, 278 and 872, or 1,356 and 3,156). The goal is to elicit the idea that both the placement and the size of the digits matter in determining the value of a number, and when comparing two numbers with different digits in the largest place. We do not need to compare the other digits in the number. The place value system only allows up to 9 in each place value and this will always be less than 1 of the largest place value units.

• Select students to display their number statements and read them. Ask if the class agrees with their comparison.
• “¿Cómo decidieron cuál número es mayor? ¿Compararon todos los dígitos?” // “How did you decide which number is greater? Did you compare every digit?”
• Select students who wrote numbers with the same first digit (or the same first two digits) to share their number statements. Ask them to explain how they compared the numbers.
• If no students mentioned that the digits in some places matter more than those in others, ask them about it.
  • “¿Tuvieron en cuenta solo algunos dígitos, y otros no?” // “Did you pay attention only to some digits but not others?”
  • “¿A cuáles les dieron prioridad? ¿Tendieron a ignorar a algunos?” // “Which ones did you prioritize? Were there any you tended to ignore?”
  • “¿Por qué podemos comparar sin mirar esos dígitos?” // “Why are we able to compare without looking at these digits?”

In the previous activity, students noticed that the digits in certain places within numbers matter more than others when comparing numbers. In this activity, students deepen that understanding by comparing pairs of numbers with a missing digit. The missing digit is the same for each pair but may not be in the same place in the two numbers. The reasoning prompts students to pay closer attention to place value. It also reinforces the idea that the digit with the greatest place value affects the size of the number the most, followed by the digits with the second greatest place value, and so on (MP7).

• Invite selected students to share their explanations.

This optional activity gives students additional opportunities to compare multi-digit numbers by reasoning about the value of the digits in different places. It also prompts students to generalize the relative size of two numbers based on their understanding of place value. Students practice constructing logical arguments (MP3) as they explain whether it is possible for to be less than , or for to be less than .

• Select students to share their responses to the last two problems.
• Highlight explanations that make it clear that:
  • will always be greater than because 4 ten-thousand is always greater than 3 ten-thousands.
  • will always be greater than because, given the first digit is the same, the digits to compare are the thousands, and 4 thousands is always greater than 3 thousands.