Warm-up: True or False: Decomposed Numbers (10 minutes)
The purpose of this True or False is to elicit the insights students have about the composition of multi-digit numbers in terms of place value. It also reinforces the idea that the same digit has different values depending on its place in a number—that is, digits cannot be viewed in isolation of their positions. The reasoning students do here will be helpful later when students compare and order numbers within 1,000,000.
- Display one statement.
- “Give me a signal when you know whether the statement is true and can explain how you know.”
- 1 minute: quiet think time
- Share and record answers and strategy.
- Repeat with each statement.
Decide if each statement is true or false. Be prepared to explain your reasoning.
- \(1,\!923 = 1 + 90 + 200 + 3,\!000\)
- \(1,\!923 = 1,\!000 + 90 + 20 + 3\)
- \(19,\!203 = 10,\!000 + 9,\!000 + 200 + 3\)
- \(190,\!023 = 10,\!000 + 90,\!000 + 20 + 3\)
- “How would you correct the false statements so that they become true?”
- \(3,\!291 = 1 + 90 + 200 + 3,\!000\)
- \(190,\!023 = 100,\!000 + 90,\!000 + 20 +3\)
Activity 1: Ways to Compare (25 minutes)
This activity prompts students to examine more closely how multi-digit numbers can be compared, and to use their insights to order several numbers. Students solidify their awareness that looking only at the first digit is not a definitive way of comparing numbers. They also practice constructing a logical argument and critiquing the reasoning of others (MP3) when they explain why the strategy of analyzing only one digit is not reliable. When students refine Tyler's statement about comparing numbers to include making sure to compare digits with the same place value, they attend to precision in the language they use (MP6).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing
- Groups of 2
- Read the first problem as a class.
- Ask 1–2 students to restate Tyler’s claim in their own words.
- “Take a few quiet minutes to work on the first two problems about Tyler’s strategy. Then, share your responses with your partner.”
- 4–5 minutes: independent work time
- 2–3 minutes: partner discussion
- Monitor for students who use place-value understanding to explain why Tyler’s strategy is not reliable.
- “Work on the last two problems independently.”
- 4–5 minutes: independent work time
Tyler compares large numbers by looking at the first digit from the left.
He says, “The greater the first digit, the greater the number. If the first digit is the same, then we compare the second digit.”
In each of these pairs of numbers, is the number with the greater first digit also the greater number?
985,248 and 320,097
72,050 and 64,830
320,097 and 58,978
54,000 and 587,000
58,978 and 547,612
146,001 and 1,483
Does Tyler's strategy work for comparing any pair of numbers? Explain your reasoning.
- How would you compare large numbers? Describe your strategy for comparing 54,000 and 587,000.
Use your strategy to order these numbers from least to greatest.
Advancing Student Thinking
- “Share your strategy for comparing multi-digit numbers with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their explanation.”
- 3–4 minutes: structured partner discussion
- Repeat with 1–2 other partners.
- “Revise your initial description based on the feedback you got from your partners.”
- 2–3 minutes: independent work time
- Invite students to briefly share their ordered sets of numbers from the last problem and their reasoning. Record and display their responses.
- If not mentioned in students’ explanations, point out that in the last set of numbers, the third digit (in the thousands place) in each 630,951 and 631,051 is what tells us how the two numbers compare. The third digit (in the hundreds place) also tells us how 63,591 and 63,951 compare.
Activity 2: Video Game Scores (10 minutes)
In this activity, students apply their understanding of place value to order multi-digit whole numbers and solve problems in context. They also reason about the range of numbers whose values are between two given numbers.
Supports accessibility for: Memory
- “Who enjoys playing video games? What games do you enjoy playing?”
- “Who has played a game where the scores of the players get accumulated or added up over multiple rounds?”
- “Let’s use what we know about big numbers to compare some video game scores and rank some players.”
- “Take a few quiet minutes to work on the activity. Then, discuss your responses with your partner.”
- 6–7 minutes: independent work time
Mai and her friends had a video game tournament one weekend.
Here are the scores at the end of the tournament:
Rank the scores from highest to lowest. Who is in first place?
Andre’s score was accidentally deleted but everyone agreed that he is in second place. Could Andre’s score be a six-digit number?
Describe what Andre’s score could be and give a couple of examples.
- Invite students to share the ranking of the players and their reasoning. Record their responses.
- Invite students to share examples of what Andre's score could be.
- If not mentioned in students’ explanations, point out that 101,012 (the highest score) is 1,012 greater than the first six-digit number, 100,000. This means there are many six-digit numbers that could be the second-highest score.
“Today we compared and ordered numbers within 1,000,000.”
“Is it true that whole numbers with more digits are always greater than those with fewer digits? Why or why not? Can you give an example?” (Yes. More digits means greater place values. A three-digit number has hundreds for its largest place value. A four-digit number has thousands.)
“Write down two large numbers that show that it is possible to tell which number is greater by comparing the first or leftmost digits. Then, share the numbers with your partner.” (Sample response: 6,315 and 4,315)
“Write down two other numbers that show that we can’t rely on the first or leftmost digits to tell us which number is greater. Share them with your partner.” (6,315 and 42,315)