A Different Kind of Change

5 min

Narrative

The purpose of this Warm-up is to introduce students to a new pattern of change, which will be useful when students examine quadratic functions in a later activity. While students may notice and wonder many things about these tables, recognizing the linear and exponential patterns, as well as evoking curiosity about the new pattern, are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). They should recognize the structure of linear and exponential growth from prior units and notice that the third pattern does not fit either of those patterns.

Launch

Arrange students in groups of 2. Display the tables for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things they notice and wonder about.

Student Task

Look at the patterns in the three tables. What do you notice? What do you wonder?

xx yy
1 0
2 5
3 10
4 15
5 20
xx yy
1 3
2 6
3 12
4 24
5 48
xx yy
1 8
2 11
3 10
4 5
5 -4

Sample Response

Students may notice:

  • The xx values are 1, 2, 3, 4, 5 in all three tables.
  • In the first two tables the yy values increase, while in the third table they increase and then decrease.
  • The yy values in the first table are all multiples of 5 and they grow linearly. In the second table, the yy values grow by a factor of 2 each time xx increases by 1. In the third table, there isn’t an obvious pattern in how the yy values change.

Students may wonder:

  • Is there a rule for the relationship in the third table?
  • Will the yy values in the third table continue to decrease, or will they increase again at some point?
  • What would the third relationship look like if graphed?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display, for all to see, their responses without editing or commentary. If possible, record the relevant reasoning on or near the tables. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If curiosity about a rule for the relationship in the third table or how the outputs are changing does not come up during the conversation, ask students to discuss this idea.

Standards
Building On
  • F-LE.1·Distinguish between situations that can be modeled with linear functions and with exponential functions.
  • F-LE.1·Distinguish between situations that can be modeled with linear functions and with exponential functions.
  • F-LE.1·Distinguish between situations that can be modeled with linear functions and with exponential functions.
  • HSF-LE.A.1·Distinguish between situations that can be modeled with linear functions and with exponential functions.

20 min

10 min