Warm-up: Number Talk: Increasing Factors (10 minutes)
The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying multi-digit whole numbers. These understandings help students develop fluency. The products in the number talk get increasingly more cumbersome to keep track of mentally so students can identify times when products are more easily found mentally and when the algorithm, with pencil and paper, might be preferable.
- 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 expression mentally.
- \(230 \times 10\)
- \(230 \times 12\)
- \(230 \times 15\)
- \(232 \times 15\)
- “Which problem was the most challenging to solve mentally? Why do you think so?” (Answers vary, but students will likely mention one of the last three because there are different products to keep track of and add.)
Activity 1: Choose a Multiplication Strategy (20 minutes)
The purpose of this activity is for students to consider the numbers when they choose a strategy to find the value of a product. Some students might choose to use the same strategy for any multiplication problem. In this activity, the numbers were chosen to encourage students to choose different strategies in order to generate discussion about why certain multiplication problems lend themselves to different strategies.
When students choose a multiplication strategy based on the factors, using the associative and commutative properties of multiplication and the distributive property, along with known facts, they make use of structure to facilitate their calculations (MP7).
- Groups of 2
- “Sometimes we use different strategies to solve problems, depending on the numbers in the problem. Without solving, look at the numbers in each expression and think about the strategy you would use to find the value.”
- 1–2 minutes: quiet think time
- “Choose 2 problems you would solve using a different strategy and describe the strategy to your partner. Explain why you chose different strategies.”
- 1–2 minutes: partner work time
- “Find the value of each expression. If you find the value mentally, record the steps you used in your head to arrive at the product.”
- 5–10 minutes: independent work time
- Monitor for students who used different strategies for the same problem. For example, for \(14\times25\), some students might multiply \(14\times20\) and \(14\times5\) and some students might double 25 and halve 14 and find \(7\times50\).
- 8–10 minutes: partner work time
Find the value of each expression. Explain or show your reasoning.
- Invite students to share their strategies for finding the value of \(14 \times 101\).
- “Is there a mental strategy you can use to find the value of this product?” (Yes. I know \(14 \times 100\) is 1,400 and then I can add 14 to that.)
- Invite students to share their strategies for finding the value of \(14 \times 136\).
- “Which strategy did you use to find the value of \(14 \times 136\)? Why?” (I used the standard algorithm because I did not see a quick mental strategy for finding the value.)
Activity 2: Compare Strategies (15 minutes)
The purpose of this activity is for students to try new ideas from the previous activity and practice multiplying using the standard algorithm.
Supports accessibility for: Organization, Conceptual Processing, Language
- Groups of 4
- 10 minutes: independent work time
- “Compare your strategies with your small group. Where did you use the same strategy? Where did you use a different strategy?”
- 3–5 minutes: small-group discussion
Find the value of each expression.
- Ask students to share their strategies for the first 3 problems.
- “Did anyone use a strategy other than the standard algorithm on any of the problems? Why did you choose that strategy?” (For \(15 \times 199\) I knew 199 is just 1 away from 200 and I could find \(15 \times 200\) in my head. I just needed to take away 15 and I could do that in my head also.)
- “Does the standard algorithm also work for these problems?” (Yes. The standard algorithm always works.)
- “What strategy did you use to find the product \(24\times154\)? Why?” (I used the standard algorithm. The numbers are complex and I could not see a good mental strategy.)
“Today we reasoned about how the factors in a problem can influence the multiplication strategy we use. Can someone describe specific numbers that made you choose one strategy over another?” (Sometimes the numbers help me use a mental strategy. For \(100 \times 15\) I just know it’s \(1,\!500\). Or for \(99 \times 15\) that would be 15 less than \(100 \times 15\) or \(1,\!485\).)
“What are some examples of problems in which we would use the standard algorithm to find a product?” (If the numbers are complicated like \(573 \times 86\). I don’t see a mental approach so the standard algorithm would be a good method.)