Lesson 7

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. Listen for the different ways students use their understanding of place value to estimate the area and explain why an estimate is too low or too high.

• Groups of 2
• Display the image.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “Is your estimate greater or less than the actual product?” (greater)
• “How could we find the actual product?” (\(6 \times 400 - 6 \times 5\), or subtract \(6 \times 5\) from 2,400)

In this activity, students use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Though students may decompose the multi-digit factor in different ways, the activity is designed to encourage them to decompose it by place value—into hundreds, tens, and ones.

• Groups of 2

• “Work with your partner on the first problem. Then, try the rest of the activity independently.”
• 2 minutes: group work time on the first problem
• 5–7 minutes: independent work time on the rest of the activity
• 3 minutes: partner discussion
• Monitor for students who:
  • write expressions that show the multi-digit factor decomposed by place value
  • draw diagrams that partition the multi-digit factor by place value

• Invite several students to share diagrams they drew to represent \(6\times252\).
• As each student shares, consider asking the class:  
  • “Where do you see the parts of the factor that was decomposed?”
  • “What expressions are represented in the diagram?”
• Record expressions as students share.

This activity extends students' work with multiplication to include a factor with up to four digits. Students begin to generalize that they could decompose any number into parts and multiply the parts. In this activity, students analyze a common error when multiplying. The work they look at does not apply place value understanding and therefore represents a product that is unreasonable for the given expression. When students analyze Jada's work, find her errors and explain their reasoning, they critique the reasoning of others (MP3).

Students see partial products as a way to describe the sub-products when a factor is decomposed and multiplied using the distributive property. Continue to refer to partial products in students’ diagrams and calculations by their mathematical name to build students’ intuition for their meaning (though students are not expected to use them in their reasoning).

• Groups of 2

• “Take a few quiet minutes to analyze Jada’s errors. Then, share your thinking with your partner.”
• 2 minutes: independent work time on the first problem about Jada’s diagram
• 2 minutes: partner discussion
• Pause for a discussion.
• “What did Jada do correctly?” (She multiplied the non-zero digits correctly. She decomposed the 6,489 by place value, which is helpful.)
• “What did Jada miss?” (She multiplied only the non-zero digits. She didn’t account for the place value of the digits being multiplied. For example: The partial product of \(3 \times 6,\!000\) is 18,000, not 18.)
• Display and correct Jada’s diagram as a class.
• “Now complete the remaining problems independently.”
• 5 minutes: independent work time on the last two problems
• Monitor for students who:
  • draw diagrams that partition the multi-digit factor by place value
  • write expressions that show the multi-digit factor decomposed by place value

• Select 1–2 students to share their solutions and reasoning for finding the value of \(8\times4,\!973\).
• Consider asking students to write expressions that represent the way they found each product. For example:
  \(\displaystyle 8 \times 4,\!973\\ 8 \times (4,\!000 + 900 + 70 + 3)\\ (8 \times 4,\!000) + 8 \times 900 + 8 \times 70 + 8 \times 3\\ 32,\!000 + 7,\!200 + 560 + 24\\ 39,\!784\)