Lesson 12

This warm-up prompts students to carefully analyze and compare features of 4 multi-digit numbers. Although students used place value terminology in the previous section, this activity lets the teacher hear the terminologies students use as they describe and compare each number. The observations and reasoning here prepare students to reason about the relative size of numbers later in this lesson.

• Groups of 2
• Display image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “These numbers have mostly the same digits—0, 1, 3, 5, and 9. Are they all the same size?” (No)
• “Which of these is the greatest? How do you know?” (30,195, because it has five digits and is in the ten-thousands. The others have fewer digits and are in the thousands.)
• “The other numbers are all four-digit numbers. How might we compare them?” (Compare the digits in the thousands place and see which one is greater.)

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 that the digits in certain places matter more than others for making comparisons (MP7).

• Groups of 3–4
• Give each group a set of cards 0–10. Ask students to remove the cards that show 10.
• “What three-digit numbers can we make with 5, 7, and 3?” (357, 375, 537, 573, 735, 753)

• “Work with your group to make pairs of numbers using the digits on the cards. First use 3 cards, then 4, then 5, and lastly 6. Compare each pair of numbers.”
• “Think about how you go about comparing each pair of numbers.”
• 5 minutes: group work time
• Monitor for students who:
  • notice that not all digits need to be compared in order to tell which number in a pair is greater
  • use place-value language in describing the numbers
  • make two numbers with the same first digit or the same first and second digits

When answering the question about how they compared the numbers, students may say that they “just know” which one was greater. Encourage them to think about the features of the numbers that gave them an immediate clue about the size of the numbers, or the features that they didn’t find as useful.

• Select students to display their number statements and read them. Ask if the class agrees with their comparison.
• “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.
  • “Did you pay attention only to some digits but not others?”
  • “Which ones did you prioritize? Were there any you tended to ignore?”

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 here 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).

• Groups of 2
• “Let’s look at some other pairs of multi-digit numbers, but this time the numbers are missing a digit. Can they still be compared?”

• “Take a few quiet minutes to think about the first problem. Then, share your thinking with your partner.”
• 1 minute: independent work time
• 1–2 minutes: partner discussion
• Pause for a whole-class discussion. Invite students to share their responses, hearing first from those who agree with Han, and then those who agree with Clare.
• “If the missing digit is not the same digit, can we compare the two numbers?” (No) “Why not?” (Because we don’t know which number has the greater missing digit, so there’s no way to compare.)
• If not mentioned by students, point out the features of the pair of numbers being compared: Both are three-digit numbers, both are missing the first digit, and 62 is greater than 17.
• “Let’s compare some other numbers that have a missing digit.”
• 5 minutes: partner work time
• Monitor for students who use place-value language to explain their thinking.

When the missing digit in a pair of numbers is in different places  (such as 27,__95 and 2__,745), students may generalize about how the numbers compare after trying one possibility for the missing digit, not realizing that a different possibility may change the relative size. Ask them to check their conclusion with other numbers for the missing digits.

• 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 \(4 \underline{\phantom{5}},\!300\) to be less than \(3 \underline{\phantom{5}},\!400\), or for \( \underline{\phantom{5}}4,\!300\) to be less than \( \underline{\phantom{5}}3,\!400\).

• Each group of 2 needs a set of cards from the previous activity.

• Groups of 2
• Give each group a card with a digit between 0 and 9 for each group

• “Use the digit on the card to complete the comparison statements in the first problem and decide if they are true.”
• 4–5 minutes: group work time
• Pause to collect responses from all the groups.
• “Is the first comparison statement true for all digits?” (Yes, because if the digits are the same, we can just compare 999 to 500.)
• Repeat the question with all statements. Record responses in a chart such as shown:

statement	true	false
\( \underline{\phantom{00}} \ , 999 > \underline{\phantom{00}} \ , 500\)	all digits
\(15,2 \underline{\phantom{00}}0 > 15, \underline{\phantom{00}}02\)	0, 1, 2	3, 4, 5, 6, 7, 8, 9
\(4 \underline{\phantom{00}},700 < 7 \underline{\phantom{00}} ,400 \)	all digits
\(1 \underline{\phantom{00}}5,000 > 5 \underline{\phantom{00}}1,000\)		all digits

 

• “Why are the statements in parts a and c true no matter what digit is used?” (Sample response:
  • In part a, both numbers are missing the first digit, in the thousands place. Since the missing digit is the same, we’re comparing the hundreds, and 999 is always greater than 500.
  • In part c, the missing digit is in the thousands place of both numbers, but the digits in the ten-thousands place need to be compared first, and 4 ten-thousand is always less than 7 ten-thousand.)
• “Why is the statement in part d false no matter what digit is used?” (Ten-thousand is never greater than 50 thousand.)
• “Take a few quiet minutes to work on the last two problems independently.”
• 5 minutes: independent work time

• Select students to share their responses to the last two problems.
• Highlight explanations that make it clear that:
  • \(4 \underline{\phantom{5}},\!300\) will always be greater than \(3 \underline{\phantom{5}},\!400\) because 4 ten-thousand is always greater than 3 ten-thousands.
  • \( \underline{\phantom{5}}4,\!300\) will always be greater than \( \underline{\phantom{5}}3,\!400\) because, given the first digit is the same, the digits to compare are the thousands, and 4 thousands is always greater than 3 thousands.