Lesson 3
From Visual Patterns to Numerical Patterns
Warmup: Number Talk: Patterns in Multiplication (10 minutes)
Narrative
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.
Launch
 Display one expression.
 “Give me a signal when you have an answer and can explain how you got it.”
 1 minute: quiet think time
Activity
 Record answers and strategy.
 Keep expressions and work displayed.
 Repeat with each expression.
Student Facing
Find the value of each expression mentally.
 \(20 \times 3\)
 \(21 \times 3\)
 \(40 \times 3\)
 \(42 \times 3\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 “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.)
Activity 1: Growing Rectangles (20 minutes)
Narrative
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.
Supports accessibility for: Conceptual Processing, Language
Required Materials
Materials to Gather
Launch
 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.
Activity
 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.
Student Facing
Here is a pattern of rectangles that follows a rule.
 Priya says, “Each step increases by 1.”
 Noah says, “Each step increases by 4.”
 Lin says, “Each step increases by 2.”
 Can you think of possible reasons that all of them could be correct even though they describe the patterns differently?
 Revise the statement made by each student so that what they mean is clearer and more precise.

Priya writes the number list 1, 2, 3, 4, 5, 6 to represent the first six steps of the pattern she sees. Write a list of numbers to represent the first six steps of the pattern that Noah and Lin see.
 Predict what number Priya, Noah, and Lin will write for step 20 if the pattern of rectangles continue. Explain or show your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
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?”
Activity Synthesis
 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.
Activity 2: More Growing Rectangles [OPTIONAL] (20 minutes)
Narrative
Launch
 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.”
Activity
 “Work on the activity independently for a few minutes.”
 5 minutes: independent work time
Student Facing
Here is another pattern of rectangles that also follows a rule.
 The number list 1, 2, 3, _____, _____, _____ represents the number of vertical columns in the first six steps of the pattern. Complete the number list.

Find another feature of the rectangles that can be represented with a number list and would show a pattern. Write at least one list of numbers for the first six steps of that feature.
Feature: ________________________________________
Number list: _____, _____, _____, _____, _____, _____

Without writing out all the numbers, predict the 30th number in your list. Explain your reasoning by completing this sentence frame:
I know that the 30th number is _____ because . . .
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
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
Activity 3: No Grid This Time! (15 minutes)
Narrative
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.
Advances: Conversing, Reading
Launch
 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.”
Activity
 8–10 minutes: independent work time
 3 minutes: partner discussion
 Monitor for the different ways students reason about the last set of problems.
Student Facing
Here are steps 1 and 4 in a pattern of rectangles. One side length of the rectangle increases by 5 units each time.
 Sketch the missing rectangles in steps 2 and 3. Label the sides with their lengths.

Write two numerical patterns that each represent the rectangles, from step 1 to step 6.

What are you representing? : ________________________________________
Numerical pattern: _____, _____, _____, _____, _____, _____

What are you representing? : ________________________________________
Numerical pattern: _____, _____, _____, _____, _____, _____


For each of the following questions, if you answer yes, show how you know and state the step number. If you answer no, explain or show why not.
If the pattern continues:
 Could 82 inches be a side length of a rectangle?
 Could 300 square inches be the area of a rectangle in the pattern?
 Could 100 inches be the perimeter of a rectangle in the pattern?
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 See lesson synthesis.
Lesson Synthesis
Lesson Synthesis
“Today we looked at a pattern of rectangles that follow a rule. We saw that we could write different numerical patterns to represent the rectangles.”
Display the numerical patterns that represent the side length, area, and perimeter of the rectangles in the last activity.
Side length: 5, 10, 15, 20, 25, 30
Area: 15, 30, 45, 60, 75, 90
Perimeter: 16, 26, 36, 46, 56, 66
Invite students to share their responses and reasoning to the last set of questions in the last activity (on whether a number in the pattern for the longer side length, area, and perimeter of the rectangles in the sequence could have a certain value). Record their reasoning for all to see.
Highlight reasoning that reinforces the idea of multiples (“82 is not a multiple of 5”), multiplicative reasoning (“20 times 15 is 300”), additive reasoning (“the perimeter increases by 10 each time”), and place value (“the perimeter always has 6 in the ones place”).
Cooldown: Another Set of Rectangles (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.