Partial Products in Algorithms

10 min

Narrative

This Warm-up prompts students to compare four representations of multiplication. Students compare diagrams and equations that represent multi-digit multiplication. It gives students a reason to use language precisely (MP6) and prepares them for the work of the lesson, in which they compare different ways to represent products as sums of partial products. 

Launch

  • Groups of 2
  • Display the image.
  • “Pick 3 representations that go together. Be ready to share why they go together.”
  • 1 minute: quiet think time
Teacher Instructions
  • “Discuss your thinking with your partner.”
  • 2–3 minutes: partner discussion
  • Share and record responses.

Student Task

Which 3 go together?

A
Diagram, rectangle partitioned vertically into 4 rectangles.
Diagram, rectangle partitioned vertically into 4 rectangles. Left rectangle, vertical side, 4, horizontal side, five thousand. Area, twenty thousand. Colored blue. Second rectangle, horizontal side, three hundred. Area, question mark. Colored red. Third rectangle, horizontal side, 40. Area, one hundred sixty. Colored yellow. Right rectangle, horizontal side, 2. Area, 8. Colored green

B
math expression. 4 times 5 thousand, plus, 4 times three hundred, plus, 4 times 40, plus, 4 times 2

C
Diagram, rectangle partitioned vertically into 3 rectangles.
Diagram, rectangle partitioned vertically into 3 rectangles. Left rectangle, vertical side, 4, horizontal side, five thousand. Area, twenty thousand. Middle rectangle, horizontal side, three hundred. Area, one thousand two hundred. Right rectangle, horizontal side, 42. Area, one hundred sixty eight. 

D
Diagram, rectangle partitioned vertically into 4 rectangles.
Diagram, rectangle partitioned vertically into 4 rectangles. Left rectangle, vertical side, 5, horizontal side, five thousand. Area, twenty five thousand. Colored blue. Second rectangle, horizontal side, three hundred. Area, one thousand five hundred. Colored red. Third rectangle, horizontal side, 40. Area, two hundred. Colored yellow. Right rectangle, horizontal side, 2. Area, 10. Colored green.

Sample Response

Sample responses:

A, B, and C go together because:

  • They represent the product 4× 5,3424 \times 5,342.

A, B, and D go together because:

  • They are broken into 4 partial products by place value.

A, C, and D go together because:

  • They are diagrams.
  • They show values of the partial products.

B, C, and D go together because:

  • They show partial products or none.
Activity Synthesis (Teacher Notes)
  • “Does the value of Expression B match the value represented in any of the diagrams?" (Yes, Diagrams A and C both represent the product 4×5,3424 \times 5,342, and that's the same as B.)
Standards
Building Toward
  • 5.NBT.5·Fluently multiply multi-digit whole numbers using the standard algorithm.
  • 5.NBT.B.5·Fluently multiply multi-digit whole numbers using the standard algorithm.

20 min

15 min