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Activity 1

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Activity Narrative

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).

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement: “I added all the numbers together to find the answer” as “I added all the partial products to calculate the final product.”
Advances: Speaking, Representing

Launch

Groups of 2
Display the diagram in the activity.
“What do you notice? What do you wonder?”
1 minute: partner discussion
Share and record responses.

Activity

4 minutes: independent work time on the first problem about Noah’s diagram
4 minutes: partner discussion
5 minutes: group work time on the rest of the activity
Monitor for students who include the place value of each digit in 124 in explaining what is happening in the algorithm.

Noah draws a diagram and writes expressions to multiply 2 numbers.

Diagram, rectangle partitioned vertically into 3 rectangles. Left rectangle, vertical side, 7, horizontal side, 1 hundred, area 7 hundred, colored red. Middle rectangle, horizontal side, 20, area 1 hundred 40, colored yellow. Right rectangle, horizontal side, 4, area 28, colored green. Beneath the diagram, 7 hundred plus 1 hundred 40 plus 28 equals 8 hundred sixty 8.

How does each expression represent Noah’s diagram?
Noah learns another way to record the multiplication.

Step 1

Step 2
multiply. 1 hundred 24 times 7. 4 rows. First row: 1 hundred 24, the 2 is highlighted yellow. Second row: multiplication symbol, 7, highlighted yellow. Horizontal line. Third row: 28, side note 7 times 4. Fourth row: 1 hundred 40, side note 7 times 20.

Step 3
multiply. 1 hundred 24 times 7. 6 rows. First row: 1 hundred 24, the 1 is highlighted red. Second row: multiplication symbol, 7, highlighted red. Horizontal line. Third row: 28, side note 7 times 4. Fourth row: 1 hundred 40, side note 7 times 20. Fifth row: plus 7 hundred, side note 7 times 1 hundred. Horizontal line. Sixth row: 8 hundred sixty 8.

Make sense of each step of the calculations and record your thoughts.
Complete the diagram to find the value of . Use Noah’s recording method to check your work.

Diagram, rectangle partitioned vertically into 3 rectangles. Left rectangle, vertical side, 5, horizontal side, 3 hundred. Middle rectangle, horizontal side, 80. Right rectangle, horizontal side, 4.

Student Response

to access Student Response.

Advancing Student Thinking

Activity Synthesis

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.)

Activity 2

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

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).

Action and Expression: Internalize Executive Functions. Provide access to colored pencils and sticky notes. To support organization, invite students to color code each partial product as seen in the previous activity. To support working memory, invite students to use the sticky notes to record the expanded form of the multi-digit number they are multiplying by.
Supports accessibility for: Memory, Organization

Launch

Groups of 2
“What strategy would you use to find the value of ?”
1 minute: partner discussion
Share responses.
Display Noah’s and Mai’s computations.
“Take a quiet moment to make sense of Noah and Mai’s work. How do you think they arrived at the last four numbers?”
1 minute: quiet think time
1 minute: partner discussion
“Which representation is more like how you thought about it?”

Activity

“Work with your partner on the first two questions.”
5 minutes: partner work time on the first two problems
Pause for a brief discussion.
“How are Mai’s and Noah’s notation the same and how are they different?” (Both of these methods calculate partial products, but Noah’s goes in order from multiplying the ones, tens, hundreds, then the thousands. Mai’s goes in order the other way from the thousands, hundreds, tens, and then the ones.)
“Are both methods correct? How do you know?” (Yes, because they both result in the same product when you calculate it.)
“Work on the last set of problems independently before discussing with your partner.”
5 minutes: independent work time on the last set of problems

Noah and Mai want to find the value of . They record their steps in different ways.

Multiply. 2 thousand 9 hundred forty 7 times 6. 6 rows. — Noah

How are these notations alike? How are they different?
Use a diagram to show what each of the partial products 42, 240, 5,400 and 12,000 represent. Then find the value of .
Find the value of each expression. Use the algorithm that Noah used for at least one expression. Show your thinking using diagrams, symbols, or other representations.

Student Response

to access Student Response.

Advancing Student Thinking

Activity Synthesis

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

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Lesson 9

Recording Partial Products: One-Digit and Three- or Four-Digit Factors

Warm-up

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Advancing Student Thinking

Activity Synthesis

Activity 1

Standards Alignment

Building On

Addressing

4.NBT.B.5

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Advancing Student Thinking

Activity Synthesis

Activity 2

Standards Alignment

Building On

Addressing

4.NBT.B.5

Building Toward

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Response

Advancing Student Thinking

Activity Synthesis

Lesson Synthesis

Standards Alignment

Building On

Addressing

4.NBT.B.5

Building Toward