Building Quadratic Functions to Describe Situations (Part 1)

5 min

Narrative

The purpose of this Warm-up is to elicit the idea that the values in the table have a predictable pattern, which will be useful when students consider the context of a falling object in a later activity. While students may notice and wonder many things about this table, the patterns are the important discussion points, rather than trying to find a rule for the function. Because the rule is not easy to uncover, studying the numbers ahead of time should prove helpful as students analyze the function later.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that all the yy values are multiples of 16 and perfect squares. Some may notice that the pattern is not linear and wonder whether it is quadratic.

Launch

Arrange students in groups of 2. Display the table 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 that they notice and wonder about.

Student Task

Study the table. What do you notice? What do you wonder?

xx 0 1 2 3 4 5
yy 0 16 64 144 256 400

Sample Response

Students may notice:

  • The xx-values increase by 1 as you move across the table.
  • The yy-values increase.
  • All yy-values are even and divisible by 4, 8, and 16.
  • The yy-values are all perfect squares.
  • The successive differences are growing.

Students may wonder:

  • Is there a rule for these numbers?
  • Do the numbers represent the growth of a geometric pattern?
  • Do the numbers represent a quadratic relationship?
  • What is the next number in the table?
Activity Synthesis (Teacher Notes)

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

If the patterns for the values do not come up during the conversation, ask students to discuss this idea.

Standards
Building Toward
  • F-BF.1·Write a function that describes a relationship between two quantities
  • F-BF.1·Write a function that describes a relationship between two quantities
  • F-BF.1·Write a function that describes a relationship between two quantities
  • F-BF.1·Write a function that describes a relationship between two quantities
  • F-BF.1·Write a function that describes a relationship between two quantities
  • F-BF.1·Write a function that describes a relationship between two quantities
  • HSF-BF.A.1·Write a function that describes a relationship between two quantities.

15 min

15 min