Lesson 3

This Number Talk encourages students to rely on what they know about place value, multiples of 3 and 4, and properties of operations to find the value of products mentally. The reasoning elicited here will be helpful as students use multiplication to find the area and perimeter of rectangles and look for patterns in these measurements. 

• Display one expression.
• “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.

• “How can one of the previous expressions help us find the value of \(42\times3\)?” (We can take \(21\times3\) and double it to get \(42\times3\). We can take \(40\times3\) and add 2 threes.)

In this activity, students examine a pattern of rectangles and consider different numerical patterns that could represent the rectangles. Students begin by analyzing claims about how the rectangles are growing and work to make the claims clearer and more precise (MP6). In doing so, they notice that the numerical patterns that represent side lengths, perimeter, or area of the rectangles each display a different rule. Students use the rules they observe to predict the value in later terms in the sequence.

• Groups of 2
• Display the image of the rectangles.
• “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion
• Provide access to graph paper, in case requested.

• Read aloud Priya, Noah, and Lin’s claims about the rectangles and the first question.
• “Let’s look at Priya’s statement together.”

MLR3 Clarify, Critique, Correct

• Display and read aloud Priya’s claim: “Each step increases by 1.”
• “What do you think Priya is trying to say? Is anything unclear?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• “With your partner, work together to write a revised statement that Priya could say so that her intention is clearer.”
• Display and review the following criteria:
  • Write in a complete sentence.
  • Include mathematical vocabulary when possible.
  • Include an example, if possible.
• 2–3 minutes: partner work time
• Select 1–2 groups to share their revised explanation with the class. Record responses as students share.
• “What is the same and different about the revisions to Priya’s claim?”
• “Take a few quiet minutes to analyze Noah and Lin’s claims and revise them so that their intentions are clearer. Then work with your group to complete the activity.”
• 3–4 minutes: independent work time
• 5–6 minutes: group work time
• Monitor for students who attend to precision and clarity as they revise Noah’s and Lin’s claims. Select them to share during the synthesis.

Students may count individual squares within rectangles to find the area. Consider asking:

• “How might you find out the number of squares in the rectangle you just drew without counting?”
• “How might you find out the size of the next rectangle without drawing it?”

• Invite previously selected students to share their explanations and revisions of Noah and Lin’s claims. Record the revised claims for all to see.
• Select other students to share the numerical patterns they wrote to represent the rectangles, and how they made predictions for the 20th number in each pattern. Record their responses for all to see.

• Groups of 2
• Display the image of the rectangles.
• “How many vertical columns do you see in the rectangles?” (One in step 1, two in step 2, and three in step 3.)
• “How is the number of columns changing?” (It grows by 1 each time.)
• “Besides vertical columns, what other features of the rectangles could we count or measure?” (Sample responses: Number of rows, area or number of square units, side lengths, perimeter)
• “Let’s see what other patterns you can find in this set of rectangles.”

• “Work on the activity independently for a few minutes.”
• 5 minutes: independent work time

MLR1 Stronger and Clearer Each Time

• “Find a partner who wrote a different numerical pattern.”
• “Take turns being the speaker and the listener. If you are the speaker, share your numerical pattern and your explanation for the last problem. If you are the listener, ask questions and give feedback to help your partner improve their explanation for the last problem.”
• 3–5 minutes: structured partner discussion.
• Repeat with 1–2 other partners who chose a different feature than you did.
• “Revise your initial response to the last question based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

Students continue to analyze and describe patterns related to rectangles. In this activity, the rectangles show no grid. To reason about possible patterns in the features of the rectangles, students rely on what they know about the relationship between side lengths of a rectangle and its perimeter and area.

• Groups of 2
• Read the opening paragraph and the first question as a class.
• “Take a quiet minute to sketch the missing rectangles. Label the sides.”
• 1 minute: independent work time
• Share responses and display the completed sequence of four rectangles.
• Display this sequence of numbers: 3, 3, 3, 3
• Ask students: “How does this numerical pattern represent the rectangles?” (The length of the shorter side of the rectangle, which doesn’t change.)
• “Let’s think about other numerical patterns that can represent the rectangles.”

• 8–10 minutes: independent work time
• 3 minutes: partner discussion
• Monitor for the different ways students reason about the last set of problems.

• See lesson synthesis.