Lesson 14

The purpose of this warm-up is to elicit the idea that while there are multiple ways to represent 2 groups of 12, some ways are more useful than others. While students may notice and wonder many things about the images, how 2 images show the groups of 12 have been organized using place value and how this type of decomposition can be helpful in finding the total are the important discussion points.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “The image on the left is a drawing of equal groups. The other images are base-ten diagrams. What is the same and different about these representations?” (They all show 12. They all show 2 groups of the same size. In the base-ten diagrams you can see the tens easier. It’s harder to see the tens in the first drawing.)

The purpose of this activity is for students to see how, when multiplying a number larger than ten, the distributive property can be used to decompose the factor into tens and ones, creating two smaller products. Base-ten blocks are used to help students visualize what is happening when a factor is decomposed to make two more easily known products. Factors slightly larger than ten can be naturally decomposed into a ten and some ones using place value. This will be useful in subsequent lessons as students progress towards fluent multiplication and division within 100.

When students see that you can decompose a teen number into tens and ones and use this to multiply teen numbers, they look for and make use of structure (MP7).

• Groups of 2
• Give students base-ten blocks.

• “Take a few minutes to look at Tyler’s strategy and decide if you agree or disagree with it.”
• 2–3 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who connect the expressions \(7 \times 10\) and \(7 \times 3\) to the tens and ones portion of the place value diagram and \(7 \times 13\) to the entire diagram.
• Have students share why they agreed or disagreed with Tyler’s strategy with a focus on using the place value diagram as a justification.
• Consider asking:
  • “Where do we see the \(7 \times 13\) in the place value diagram?”
  • “Where do the 10 and the 3 come from?”
  • “How could you use the place value diagram to figure out how to find the value of \(7 \times 13\)?”
• “Now use Tyler’s method on your own to find the value of \(3 \times 14\).”
• 2–3 minutes: independent work time
• “Share your solution and your reasoning with your partner.”
• 2–3 minutes: partner discussion

• “Where do you see \(7\times13\) in the diagram?”
• “If we separate the tens and ones, what expression could we use to describe the tens? The ones?”

• Display base-ten blocks or place value diagrams that students used to solve. As students explain their work, write multiplication expressions to represent them.
• “How does this diagram (or the base-ten blocks) show how Tyler’s method could be used to multiply \(3 \times 14\)?” (We can see there are 3 tens which is 30. We can see that there are 3 groups of 4 ones which is 12. \(30 + 12\) is 42. The whole diagram represents \(3 \times 14\).)
• If there is time, ask students to find the value of \(4 \times 12\) and \(5 \times 16\) using the base-ten blocks and Tyler’s strategy.

The purpose of this activity is for students to make sense of different ways of representing multiplication of a teen number. Students analyze a gridded area diagram, base-ten blocks, and an area diagram labeled with side lengths. When they discuss how the different diagrams represent the same product, students reason abstractly and quantitatively (MP2).

• Groups of 2
• “We’re going to look at three different ways students showed the same expression. What do you notice? What do you wonder?” (Students may notice: You can see all the squares in the first 2 diagrams, but not in the last one. The middle diagram looks like base-ten blocks. Students may wonder: Why would you choose to use one of these diagrams? What numbers were they multiplying?)
• 1 minute: quiet think time
• Share responses.

• “Work with your partner to tell how you see the factors in each diagram and how you see the product in each diagram.”
• 5–7 minutes: partner work time

• “How are these ways of representing \(3 \times 15\) the same?” (They all represent 3 times 15. They all show the 15 being decomposed into 10 and 5. They are all shaped like a rectangle.)
• “How are these ways of representing \(3 \times 15\) different?” (Clare used base-ten blocks, but Andre and Diego used rectangles. Diego didn’t show the squares in his rectangle, but Clare and Andre did.)
• “How could we represent the strategy shown in all the diagrams with expressions?” (\(3 \times 10\) and \(3 \times 5\) or \(10 \times 3\) and \(5 \times 3\).)