Lesson 9 Recording Partial Products: One-digit and Three- or Four-digit Factors

- Let’s analyze and try an algorithm that uses partial products.

Warm-up Which One Doesn’t Belong: Expressions Galore

Which one doesn’t belong?

$7 \times 50$
$(3 \times 50) + (4 \times 50)$
$(5 \times 10) \times 7$
$50 + 50 + 50 + 50 + 50 + 50 + 50$

Activity 1 An Algorithm for Noah

Noah drew a diagram and wrote expressions to show his thinking as he multiplied two numbers.
$7 \times 124$
$7 \times (100 + 20 + 4)$
$(7 \times 100) + (7 \times 20) + (7 \times 4)$
$700 + 140 + 28$
How does each expression represent Noah’s diagram? Be prepared to share your thinking with a partner.
Later, Noah learned another way to record the multiplication, as shown here.

Step 1

Step 2

Step 3

Make sense of each step of the calculations and record your thoughts. Be prepared to explain Noah’s steps to a partner.
Complete the diagram to find the value of $217 \times 8$ . Use Noah’s recording method to check your work.

Activity 2 Try an Algorithm with Partial Products

Problem 1

Noah and Mai want to find the value of $8 \times 3, 419$ . They recorded their steps in different ways, as shown.

multiply. 3 thousand 4 hundred 19 times 8. 6 rows. — Noah

multiply. 3 thousand 4 hundred 19 times 8. 6 rows. — Mai

How are Mai’s and Noah’s notation alike? How are they different?
Use a diagram to show what each of the partial products 72, 80, 3,200 and 24,000 represent. Then, find the value of $8 \times 3, 419$ .

Problem 2

Find the value of each expression. For at least one expression, use the algorithm that Noah used. Show your reasoning.

$4 \times 5, 342$
$7 \times 983$

Practice Problem

Problem 1

The diagram and calculations show two ways for finding the value of $2, 518 \times 6$ .

Diagram, rectangle partitioned vertically into 4 rectangles.

multiply. 2 thousand 5 hundred 18 times 6.

How does each part of the vertical calculation relate to the diagram?
Find the value of $3, 172 \times 5$ using a method of your choice.