# Lesson 11

Partial Products and the Standard Algorithm

## Warm-up: Number Talk: The Value of the Digits (10 minutes)

### Narrative

This Number Talk routine encourages students to think about decomposing factors by place value to multiply multi-digit numbers by one-digit numbers. As students look for and make use of structure, they may notice that multiplying the ones place has a result in the pattern of 5, 10, and 15. Students may use the partial products equations to multiply.

This routine helps students pay attention to the value of the digits when multiplying. This is important for setting up the conversation about the standard algorithm, in which students will find partial products mentally and use what they know about the value of the digits to condense the number of steps to multiply by multi-digit numbers.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expression and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$5 \times 101$$
• $$5 \times 102$$
• $$5 \times 203$$
• $$5 \times 404$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What connections or relationships do you see between each expression?” (Each problem involves one more group of 5 or sets or 100 more.)

## Activity 1: Two Algorithms to Multiply (20 minutes)

### Narrative

This activity introduces students to the standard algorithm for multiplication. Students make sense of it by comparing and contrasting it to an algorithm that uses partial products for multiplying three- and four-digit numbers by one-digit numbers where no regrouping is necessary. When they interpret the given student work showing the standard algorithm students construct a viable argument for what Kiran did in his calculation (MP3). They also have an opportunity to make use of the structure they notice to compute the value of other products.

MLR8 Discussion Supports. Synthesis: Create a visual display of Diego’s algorithm. As students share their strategies, annotate the display to illustrate connections. For example, next to each step, write the multiplication expression used to find each partial product.

• Groups of 2

### Activity

• “Make sense of Kiran and Diego’s algorithms. Talk to your partner about how they are alike and different, and how Kiran might have found his answer.”
• 3 minutes: partner discussion
• Pause for a discussion.
• Invite students to share their conjectures on how Kiran might have reasoned about the product.
• Alternatively, consider displaying a few scenarios and polling students on which one might be closest to their conjecture, if any. For example:
• A: Kiran found $$(3 \times 700) + (3 \times 13)$$ mentally and wrote down the result of $$2,\!100 + 39$$.
• B: Kiran used the same method as Diego but added the 9, 30, and 2,100 mentally, without writing them down.
• C: Kiran drew a diagram and did the computation on another sheet of paper and wrote the result here.
• D: Kiran multiplied the single-digit 3 with each digit in 713 and wrote each partial product in a single line.
• Explain that Kiran had multiplied 3 by each digit in 713, but instead of reasoning about $$3 \times 3$$, $$3 \times 10$$, and $$3 \times 700$$, Kiran reasoned about $$3 \times 3$$, $$3 \times 1$$, and $$3 \times 7$$, while paying attention to the place value of each digit.
• Demonstrate Kiran’s process:
• Because the 3 in 713 means 3 ones, he wrote the result of $$3 \times 3$$ or 9 in the ones place.
• Because the 1 means 1 ten, he wrote the result of $$3 \times 1$$ in the tens place.
• Because the 7 means 7 hundreds, he wrote the result of $$3 \times 7$$ in the hundreds place.
• “Try using Kiran’s algorithm to find the value of the last two products.”
• 3 minutes: independent work time
• 2 minutes: partner discussion

### Student Facing

1. Here are two algorithms for finding the value of $$3 \times 713$$.

1. How are Kiran’s algorithm and Diego’s algorithm alike? How are they different?
2. How do you think Kiran found the product 2,139?
2. Find the value of each product.

1. $$212 \times 4$$
2. $$3 \times 4,\!132$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Select students to share and explain their calculations of the last two products.
• Consider demonstrating the process of Kiran’s strategy to find $$3 \times 4,\!132$$.
• “Kiran’s strategy is called the ‘standard algorithm for multiplication.’ We’ll take a closer look at this algorithm in the next activity.”

## Activity 2: Algorithm Comparison (15 minutes)

### Narrative

The purpose of this activity is for students to compare the standard algorithm for multiplication and an algorithm that uses partial products. The focus of the synthesis is on the convention used for composing a new unit and how it connects to their work with the standard algorithm for addition.

Engagement: Internalize Self-Regulation. Synthesis: Provide students an opportunity to self-assess and reflect on their own progress. Remind students what kinds of multiplication problems they were working on before this unit, and ask them to notice how the multiplication they are working on now is different. Invite them to list the new strategies they have learned in this section.
Supports accessibility for: Conceptual Processing, Social-Emotional Functioning

• Groups of 2

### Activity

• “Work with your partner on the first problem.”
• 3–4 minutes: partner work time
• Pause for brief discussion. Invite students to share how the two methods are alike and how they are different.
• “In Kiran's algorithm, why is a 1 written in above the 2 in the tens column?” (It represents 1 ten from the number 12. It helps us remember to add it to the tens place because we are doing a lot of calculations in our heads.)
• “Where have we seen this notation before?” (When we add using the standard algorithm, we use this notation to show that we have more than 9 ones, tens or hundreds, and so on, in a given place, we add each group of 10 units to the place value to the left.)
• “Kiran's used the standard algorithm for multiplication. Try using it to find $$512 \times 3$$.”
• 3 minutes: independent work time
• 1–2 minutes: partner discussion

### Student Facing

1. Analyze the two algorithms used to find the value of $$4 \times 223$$.

1. How are Kiran and Diego's algorithms alike? How are they different?
2. Where is the 12 in Kiran's algorithm?
1. Try using Kiran's algorithm to find the value of $$512 \times 3$$.
2. Check your work using a different method.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we compared the standard algorithm for multiplication to an algorithm that uses partial-products. Let’s see how we’d find $$3 \times 512$$.”

Display:

“How would you find the value of the product?” (Sample responses:

• I know that 500 x 3 is 1500 and 12 x 3 is 36 and 1500 + 36 = 1,536.
• Three times 2 is 6, so that goes in the ones place. 3 times 1 is 3, so that goes in the tens place. 3 times 5 is 15, so that goes in the thousands and hundreds place.)

Assure students that they are not expected to use a particular method for multi-digit multiplication in grade 4. Explain that they will study this algorithm more in grade 5. Invite them to try to use it to multiply as we continue to work through lessons.

## Cool-down: Choose a Way to Multiply (5 minutes)

### Cool-Down

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