Warm-up: Number Talk: Partial Product (10 minutes)
- Display one problem.
- “Give me a signal when you have an answer and can explain how you got it.”
- 1 minute: quiet think time
- Record answers and strategy.
- Keep problems and work displayed.
- Repeat with each problem.
Find the value of each product mentally.
- \(3 \times 3\)
- \(3 \times 20\)
- \(3 \times 600\)
- \(3 \times 623\)
- “How is the last product related to the first three?” (It is the sum of the first three.)
- “Did the first three calculations help you find the last product?” (Yes, I was able to add them together to find \(3 \times 623\).)
Activity 1: Compose with the Standard Algorithm (20 minutes)
The goal of this activity is for students to understand how to record newly composed units when using the standard algorithm for multiplication. Students compare the familiar partial products algorithm to the standard algorithm. Students may draw on their experience with the standard algorithm for addition to make sense of the new units being composed.
When students discuss their interpretation of Elena's calculation and improve their explanations they construct viable arguments and critique the reasoning of others (MP3).
- Groups of 2
- Display the problems.
- “Take a moment to look at how Elena and Han calculated \(318 \times 3\). Explain to your partner what each student did.”
- 2 minutes: quiet think time
- 2 minutes: partner discussion
- 5–6 minutes: independent work time
- Monitor for students who:
- explain that the 2 represents the 20 in 24, which is the product of \(3 \times 8\).
- explain that the 5 represents 5 tens because 3 times 1 ten is 3 tens, plus 2 more tens is 5 tens.
- “Share your response with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
- 3–5 minutes: structured partner discussion
- Repeat with 2–3 different partners.
- “Revise your initial draft based on the feedback you got from your partners.”
- 2–3 minutes: independent work time
Here is how Han calculated \(318 \times 3\) using partial products.
Here is how Elena calculated \(318 \times 3\) using the standard algorithm.
- What does the 2 in Elena’s calculation represent? Explain or show your reasoning.
- What does the 5 in Elena’s solution represent? Explain or show your reasoning.
Advancing Student Thinking
If students do not explain what the 2 or 5 represents, ask, “How can we use partial products to figure out what the 2 or 5 represents?”
- Invite previously selected students to share their revised explanations.
- “Elena used the standard algorithm for multiplication to find the product. When we compose a new unit in the standard algorithm, we record the new number of new units over the place value to the left of the digit we are multiplying.”
Activity 2: Use the Standard Algorithm (15 minutes)
The purpose of this activity is for students to use the standard algorithm to multiply a multi-digit number by a one-digit number. Students find products that involve one or more compositions of a new unit.
Supports accessibility for: Memory, Organization
- Groups of 2
- “You are going to practice Elena’s multiplication strategy, the standard algorithm.”
- 5-6 minutes: independent work time
- 2-3 minutes: partner discussion time
Calculate each product using Elena’s strategy.
- \(3,\!615 \times 4\)
- \(16,\!023 \times 3\)
- \(27,\!326 \times 3\)
- \(10,\!215 \times 6\)
- Invite students to share their solutions and reasoning for the first and second problems.
- “Which new units did you compose in the first problem?” (New tens because \(4 \times 5 = 20\) and ten-thousands because \(3,\!000 \times 4 = 12,\!000\)).
- “Which new units did you compose in the second problem?” (New ten-thousands because \(3 \times 6,\!000 = 18,\!000\).)
- Invite students to share their solutions for the third and fourth problems.
“Today we learned the standard algorithm to multiply whole numbers.”
Display student work for \(27,\!326 \times 3\).
“Which new units were composed here? How do you know?” (They composed new tens and new ten-thousands. I see a 1 above the 2 tens and a 2 above the 2 ten thousands.)
“How did they keep track of the new units?” (They wrote them above the first factor in the correct place value.)