Lesson 5

In this warm-up, students practice estimating a reasonable answer using known information, rounding, and multiplicative reasoning strategies. For example, students may create a diagram and arrive at \(7 \times 7 \times 7 \times 7\) as an estimate of the number of fish. Then, they may approximate \((7 \times 7) \times (7 \times 7)\) with \(50 \times 50\), or estimate \((7 \times 7 \times 7) \times 7\) with \((50 \times 7) \times 7\) or \(350 \times 7\). Some students may notice that the question is vague and asks, “How many are going to the park?” rather than “How many people are going to the park?” and account only for the number of children and teachers in their estimation. If students ask for clarification, ask them to make their own assumptions and explain why they made those assumptions when they share their estimates.

• Groups of 2
• Display the description.

• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high? Too low? About right?”
• 2 minutes: quiet think time
• 1 minute: partner discussion
• Record responses.

• Consider asking:
  • “¿Alguien hizo una estimación menor que 500?, ¿mayor que 5,000?” // “Is anyone’s estimate less than 500? Greater than 5,000?”
  • “¿Alguien incluyó el número de profesores y de estudiantes en su estimación?” // “Did anyone include the number of teachers and students in their estimate?”
• Invite students to share their estimation strategies. After each explanation, ask if others reasoned the same way.
• Consider revealing the actual value of 2,457 teachers, students, and fish.

In this activity, students revisit two algorithms for multiplying numbers. They recall that, in the standard algorithm, the digit in one factor is multiplied by each digit in the other factor, but the partial products are not recorded on separate lines. Rather, the standard algorithm condenses multiple partial products into a single product. 

• Groups of 2
• Give students access to grid paper, if needed for aligning the digits in a multiplication algorithm.

• 2 minutes: independent work time
• Pause to discuss the first set of questions. Display the two algorithms in the first question. Ask students to share responses.
• “¿En qué se parecen los dos algoritmos? ¿En qué son diferentes?” // “How are the two algorithms alike? How are they different?”
• Highlight student responses to emphasize:
  • In method A, each partial product is listed separately before being added at the end.
  • In method B, only one digit is recorded at a time. The values for any place value unit are added and only one digit is recorded. Any new units are recorded in the next highest place.
• 6–10 minutes: independent work time
• 2–4 minutes: partner discussion

• Select students to share their responses to the second set of questions.
• If not mentioned in students’ explanations, highlight that:
  • The result of \(3 \times 5\) is 15, a two-digit number, so the 1 ten should be carried over to the tens place and added to the 3 tens that result from \(3 \times 10\).
  • One ten and 3 tens make 4 tens.
• Poll the class on whether their preferred method is A, B, or is dependent on the problem. Select a student from each camp to explain their reason.

Earlier, students compared and made connections between two algorithms for multiplying a multi-digit number and a single-digit number. In this activity, students compare an algorithm that uses partial products with the standard algorithm for multiplying 2 two-digit numbers. As students analyze and critique each method, they practice looking for and making use of base-ten structure of whole numbers (MP7). 

• Groups of 2
• Access to grid paper, in case needed to align digits when multiplying

• 3–4 minutes: independent work time on the first two problems
• 1–2 minutes: partner discussion
• Monitor for students who can explain the numbers in the standard algorithm.
• 5 minutes: independent work time on the remaining questions

• Invite students to share how they used each method to show that ​\(44 \times 12 = 528\)​ and ​\(63 \times 21 = 1,\!323\)​