The purpose of this activity is to analyze an algorithm that uses partial products. Students are not required to use a specific notation, but analyzing each algorithm deepens their understanding of the structure of place value in multiplication.

When students interpret and make sense of Noah's work, they construct viable arguments and critique the reasoning of others (MP3).

• Select students to share their interpretations of the steps in the second problem.
• Highlight these key ideas along the way:
  • Multiply the 7 by the 4 ones in 124 and write the product below the line.
  • Multiply the 7 by 2 tens in 124 and record this product below the first product.
  • Multiply the 7 by the 1 hundred and record this product below the previous product.
  • Add all the partial products to calculate the final product.

    

• If needed, review these steps for .
• Reiterate that we typically begin by multiplying the single-digit factor by the digit in the ones place, and then repeat with the digit in the tens place, and so on.
• “The recording strategy that Noah learned is an algorithm that uses partial products. We used an algorithm to add and subtract large numbers in our last unit.”
• “How do we know if we have finished finding all the partial products?” (We keep track of all the partial products or smaller parts of the whole product as we go. We went in order from the smallest place value to the largest place value.)

In this activity, students continue to analyze an algorithm that uses partial products and learn that there are different ways to write the partial products. While students are not required to use a specific algorithm for multiplication, analyzing variations in the partial products notation deepens their understanding of how to use base-ten structure to multiply efficiently (MP7). 

• “In the Lesson Synthesis, let’s analyze different ways to keep track of partial products.”