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From Visual Patterns to Numerical Patterns

Grade 4: Unit 6: Lesson 3

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Lesson Purpose

The purpose of this lesson is for students to analyze, describe, and extend numerical patterns that

follow a rule.

Lesson Narrative

Previously, students explored growing and repeating patterns and reasoned about the patterns

using words, numbers, and operations. In this lesson, students investigate patterns in a geometric

context and explore how the side lengths, area, perimeter, and other features of a rectangle change

when the rectangle changes by a rule. In doing so, students practice looking for and making use of

structure (MP7).

Students also practice reasoning quantitatively and abstractly (MP2) as they interpret the values in

number sequences that represent geometric features of rectangles, and vice versa. (For example, 6,

8, 10, 12, ... may represent the area, in square centimeters, of a series of rectangles whose width is 2

centimeters and whose length grows by 1 centimeter each time.)

The second activity in this lesson is optional as it allows students more time to work with the ideas

from the first activity.

Learning Goals

• Analyze patterns represented visually and numerically.

• Use numbers, words, and the idea of factors and multiples to describe and extend patterns in

the features of rectangles.

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Let’s look at numerical patterns we can write to describe patterns in rectangles.

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Number Talk: Patterns in Multiplication

Warm-Up

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Find the value of each expression mentally.

20 x 3

b. 21 x 3

c. 40 x 3

d. 42 x 3

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Let’s Talk About It!

How can one of the previous expressions help us find the value of 42 x 3 ?

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Growing Rectangles

Activity 1

MLR 3: Clarify, Critique Connect

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Here is a pattern of rectangle

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.”

a. Can you think of possible reasons that all of them could be correct even though they describe the patterns differently?

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b. Revise the statement made by each student so that what they mean is clearer and more precise.

c. 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.

d. Predict what number Priya, Noah, and Lin will write for step 20 if the pattern of rectangles continue. Explain or show your reasoning.

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Let’s Talk About It!

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More Growing Rectangles

Activity 2

MLR 1: Stronger and Clearer Each Time

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Here is another pattern of rectangles that also follows a rule.

a. The number list 1, 2, 3, ____, ____,_____ represents the number of vertical columns in the first six steps of the pattern. Complete the number list.

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b. 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: ______, ______ , _____ , _____ , _____ ,______

c. 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__________

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Let’s Talk About It!

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.

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No Grid This Time

Activity 3

MLR 2: Collect and Display

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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.

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

a. What are you representing? :_________

Numerical pattern: ______, ______, _____ ,_____ ,_____ ,____

b. What are you representing? :_________

Numerical pattern: _____ , ____ , ____ , _____ , ____ ,_____

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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:

a. Could 82 inches be a side length of a rectangle?

b. Could 300 square inches be the area of a rectangle in the pattern?

c. Could 100 inches be the perimeter of a rectangle in the pattern?

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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.

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Another Set of Rectangles

Cool-Down

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Cool-Down: Another Set of Rectangles

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