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.

• Invite several students to share diagrams they drew to represent .
• As each student shares, consider asking the class:  
  • “Where do you see the parts of the factor that were 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).

• Select 1–2 students to share their solutions and reasoning for finding the value of .

• Consider asking students to write expressions that represent the way they found each product. For example: