Lesson 2

Describing Patterns

These materials, when encountered before Algebra 1, Unit 6, Lesson 2 support success in that lesson.

2.1: Continue the Pattern (5 minutes)

Warm-up

In earlier units, students worked with linear and exponential functions. In this unit, they encounter quadratic functions, which are neither linear nor exponential but change in a predictable way. This warm-up helps students recall what makes a pattern linear or exponential. This will be useful later in this lesson when students need to identify what type of pattern they see, and explain why.

This activity is a good opportunity to practice some mental math involving fractions. The number in the list is given in fraction form so that it’s easier to compute the next values in the list mentally. If desired, this could be changed to 2.5 and students could use a calculator to find answers in the form of a decimal.

Student Facing

Consider a list that starts \(1, \frac52, \dots\) What would be the next three numbers in the list, if it followed a pattern that grew:

  1. exponentially?
  2. linearly?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Reiterate that in a pattern that grows exponentially, successive terms change by a common factor. In a pattern that grows linearly, successive terms change by a common difference. Some questions for discussion:

  • “What strategies did you use to find the next three numbers when the list grew exponentially?”
  • “What strategies did you use to find the next three numbers when the list grew linearly?”

2.2: Patterns of Sticks (20 minutes)

Activity

In this activity, students have an opportunity to encounter patterns as they do in the associated Algebra 1 lesson. There are three linear patterns, and one that is neither linear nor exponential. This work is more scaffolded than the work in the Algebra 1 lesson, giving students an opportunity to step through the same type of work.

When students express the \(n\)th term of a pattern using a variable after figuring out several numerical values in the pattern, they are expressing regularity in repeated reasoning (MP8).

Launch

Consider arranging students into groups of 2. Present the image for all to see, and give students quiet think time to describe how they see the pattern changing and generate the next two steps. Have students discuss their observations with a partner before working on the rest of this task.

Before moving on to the second pattern, you may need to clarify for the whole class how each table represents a different aspect of how the figures are growing.

Student Facing

  1. Here’s a pattern.

    Three images. Step 0, equilateral triangle. Step 1, 2 triangles, one rotated by 180 degrees, together as a parallelogram. Step 2, 3 triangles, second rotated 180 degrees, together as a trapezoid.

     

    1. How do you see the pattern changing?
    2. Extend the pattern to show your prediction of the next two steps.
  2. Here are tables that represent the pattern.

    step 0 1 2 3 6 11 \(n\)
      3 5 7        
    step 0 1 2 3 6 11 \(n\)
      3 4 5   9    
    1. In each pattern, what quantity is represented in the second row?

    2. Complete each table.

    3. Describe each pattern as linear, exponential, or neither. Be prepared to explain how you know.

  3. Here is another pattern.

    Step 0, 1 stick. Step 1, 2 sticks. Step 2, 4 sticks.
    1. Lin says that step 3 will have 8 segments. Andre says that step 3 will have 7 segments. How does each student see the pattern growing?
    2. Complete the tables to show the relationship between step number and number of segments, as Lin and Andre would see it.
    3. Describe each pattern as linear, exponential, or neither. Be prepared to explain how you know.

Lin

step 0 1 2 3 6 9 \(n\)
number of segments 1 2 4        

Andre 

step 0 1 2 3 6 9
number of segments 1 2 4      

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ensure students understand why Andre’s pattern is neither growing linearly nor exponentially. (There is no need to talk about second differences or introduce the word quadratic at this time.)

  • “How did you complete the table for Andre’s pattern?” (The number of sticks increased by 1 and then by 2. To continue the pattern, I added 3 sticks for step 4 and 4 sticks for step 5.)
  • “How did you recognize linear patterns?” (Linear patterns have a constant rate of change so you add (or subtract) the same amount each time the step increases by 1 to find the next number.)
  • “How do you recognize exponential patterns?” (Exponential patterns have a constant growth factor so you multiply by the same number each time to find the next number in the pattern, or, when you divide sequential numbers in the pattern, you always get the same quotient.)

2.3: Patterns of Dots (20 minutes)

Activity

This activity provides a scaffolded opportunity to describe the way two different patterns change.

Student Facing

  1. Here is a pattern of dots. 
    Series of 4 images. Step 0, 3 dots in a row. Step 1, 2 rows of 3 dots. Step 2, 4 rows of 3 dots. Step 3, 8 rows of 3 dots.

     

    1. Describe how you see the pattern growing.
    2. Draw the next step.
    3. Complete the table to continue the pattern. 
      step 0 1 2 3 4 6 \(n\)
      number of dots 3 6          
    4. Is the relationship between step number and number of dots linear, exponential, or neither? Explain how you know.
  2. Here is another pattern of dots. 
    Step 1= 5 dots in pentagon shape. Step 2= step 1 plus 1 dot outside of both upper left dot and upper right dot. Step 3 = step 2 plus one additional dot outside of both upper left and upper right dots.

     

    1. Describe how you see the pattern growing.
    2. Draw the next step.
    3. Complete the table to continue the pattern. 
      step 0 1 2 3 4 6 \(n\)
      number of dots 5 7          
    4. Is the relationship between step number and number of dots linear, exponential, or neither? Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of this discussion is for students to share strategies they used to complete the last few rows of each table and to clarify how they knew the relationship was linear, exponential, or neither. If time permits, highlight connections between the visual patterns and how the total number of dots is increasing in the tables.

Display a completed table for both patterns and refer to it when discussing the following questions:

  • “How did you extend the pattern to figure out how many dots were in step _____?”
  • “How could you predict the number of dots in step 10? Or step 20?”
  • “What advice would you give a classmate to help them recognize a linear relationship? An exponential relationship?”
  • “Did anyone see the pattern in question 1 growing in a different way?”