Lesson 3 From Visual Patterns to Numerical Patterns

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

Warm-up Number Talk: Patterns in Multiplication

Find the value of each expression mentally.

Activity 1 Growing Rectangles

Here is a pattern of rectangle that follows a rule.

pattern of gridded rectangles. Step 1, 1 row of 4 squares. Step 2, 2 rows of 4 squares. Step 3, 3 rows of 4 squares.

  • Priya says, “Each step increases by 1.”

  • Noah says, “Each step increases by 4.”

  • Lin says, “Each step increases by 2.”

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

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

  3. 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.

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

Activity 2 More Growing Rectangles

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

pattern of gridded rectangles. Step 1, 1 column of 8 squares. Step 2, 2 columns of 8 squares. Step 3, 3 columns of 8 squares.

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

  2. 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: , , , , ,

  3. 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

Activity 3 No Grid This Time!

Problem 1

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.

pattern of rectangles, all with vertical sides 3 inches.

Problem 2

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

  1. What are you representing? :

    Numerical pattern: , , , , ,

  2. What are you representing? :

    Numerical pattern: , , , , ,

Problem 3

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:

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

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

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

Practice Problem

Problem 1

Here is the first rectangle in a pattern. For each step in a pattern of rectangles, the short side stays the same and the long side grows by 2 centimeters.

Rectangle. Horizontal side, 3 centimeters. Vertical side, 1 centimeter.

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  1. Draw the next 4 steps in the rectangle pattern. Include the length and width of each rectangle.

  2. Can the perimeter of the rectangle, in centimeters, be an even number? Explain your reasoning.

  3. Can the area of the rectangle, in square centimeters, be an even number? Explain your reasoning.