Standard Algorithm: Multi-digit Numbers without Composing
Warm-up: Number Talk: Partial Products (10 minutes)
The purpose of this Number Talk is to elicit strategies and understandings students have for mentally multiplying numbers that require composing a new unit. Students apply this understanding in the lesson when they compose a new unit using the standard algorithm.
- 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 expressions and work displayed.
- Repeat with each expression.
Find the value of each product mentally.
- \(20 \times 3\)
- \(24 \times 3\)
- \(120 \times 3\)
- \(140 \times 3\)
- “How do the tens in \(20 \times 3\) compare to the tens in \(24 \times 3\)?” (There is one more ten in \(24 \times 3\) that came from \(3 \times 4\).)
- “How do the hundreds in \(120 \times 3\) compare to the hundreds in \(140 \times 3\)?” (There is one more hundred in \(140 \times 3\) that came from \(40 \times 3\).)
Activity 1: Compare Two Algorithms (20 minutes)
The purpose of this activity is for students to connect an algorithm that uses partial products to the standard algorithm when multiplying a three-digit and a two-digit number. The standard algorithm shows 2 partial products while the other algorithm shows 6 partial products. While the products are recorded differently, the same 6 partial products are still part of both calculations, and this activity gives students a chance to see this common structure while also appreciating the different way the standard algorithm records the calculations.
Students use the common structure in the two algorithms (MP7) to make sense of the standard algorithm before they use it themselves in the next activity.
Advances: Listening, Speaking
- Groups of 2
- Display the algorithms.
- “We are going to learn about a new algorithm today.”
- 1–2 minutes: quiet think time
- 8–10 minutes: partner work time
- Monitor for students who notice that:
- the two algorithms show the same products in the same right to left order.
- the two algorithms record the results of the products differently.
Two algorithms for finding the value of \(413 \times 21\) are shown.
- How are the two algorithms the same? How are they different?
- Explain or show where you see each step from the first algorithm in the second algorithm.
- How do the final steps in the two algorithms compare?
Advancing Student Thinking
- Invite students to share what was alike in the two algorithms, highlighting:
- all six partial products are calculated in both.
- they are calculated in the same order.
- they both need to add up their partial products at the end.
- Invite students to share what was different in the two algorithms highlighting:
- one algorithm lists each partial product on a separate line while the standard algorithm lists some of them on the same line
- Circle the first partial product (413) in the standard algorithm.
- “What does the first partial product 413 represent?” (It's \(1 \times 413\).)
- Circle the second partial product (8,260) in the standard algorithm.
- “What does the second partial product 8,260 represent?” (It's \(20 \times 413\).)
Activity 2: Use the Standard Algorithm (15 minutes)
The purpose of this activity is for students to practice multiplying a two-digit and a three-digit number using the standard algorithm. The problems do not involve composing new units so that students can practice the procedure of multiplying each place in one factor by each place in the other factor. In the last problem, students look at incorrect work where the value of the digit in the tens place is not accounted for. This problem encourages them to use estimation to assess the reasonableness of their answers and is the focus of the lesson synthesis.
Supports accessibility for: Attention, Conceptual Processing
- Groups of 2
- 8-10 minutes: independent work time
- 2-3 minutes: partner discussion
Use the standard algorithm to find the value of each expression.
- \(202 \times 12\)
- \(122 \times 33\)
- \(321 \times 24\)
Diego found the value of \(301 \times 24\). Here is his work. Why doesn’t Diego’s answer make sense?
- Invite students to share their solution for \(122 \times 33\).
- “How is multiplying 122 by the 3 in the ones place of 33 the same as multiplying 122 by the 3 in the tens place?” (In both cases I get 366.)
- “How is it different?”(The 3 in the tens place represents 30 and so the 366 needs to shift one place to the left because it is really 366 tens or 3660.)
“Today, we used the standard algorithm to multiply a two-digit number and a three-digit number.”
Display Diego’s work from the last problem.
“Why doesn’t Diego’s answer make sense?” (The product is too small. \(300\times20\) is 6,000, so the product is greater than that.)
“What advice would you give Diego to revise his thinking?” (Remember that the 2 in 24 is 2 tens. So 2 tens times 1 should be 20, so you need to write the 2 in the tens place.)