Understanding Decay

5 min

Narrative

The purpose of this Warm-up is to recognize linear and exponential patterns that involve fractions, which will be useful when students write expressions to represent these patterns in a later activity. While students may notice and wonder many things about these tables, the linear and exponential patterns are the important discussion points.

Launch

Arrange students in groups of 2. Display the two tables for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about the tables. 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

What do you notice? What do you wonder?

Table A

xx yy
0 2
1 3123\frac12
2 5
3 6126\frac{1}{2}
4 8

Table B

xx yy
0 2
1 3
2 92\frac92
3 274\frac{27}{4}
4 818\frac{81}{8}

Sample Response

Students may notice:

  • Both tables start with x=0x=0 and y=2y=2.
  • The successive yy-values in Table A all differ by 32\frac{3}{2}.
  • The successive rows in Table B do not differ by the same amount.
  • In Table B, the yy-value of each row is 32\frac32 the value in the row above it. The pattern has a common factor.
  • The yy-values for Table A were initially greater than those in B, but later they became less than those in Table B.

Students may wonder:

  • Is there a simple pattern for Table B?
  • Will the yy-values in the two tables ever be the same again, as they were when xx was 0?
  • Will the yy-values in the two tables stay pretty close to each other?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display their responses, without editing or commentary, for all to see. 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 to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the way that the values in the tables change from row to row does not come up during the conversation, ask students to discuss this idea.

If no students wonder about whether the values in the two tables will ever agree again, ask, “Will a linear growth and an exponential growth that start with the same value have the same value again at some later point?” Consider posting the question for students to think about throughout the unit.

Standards
Building Toward
  • F-BF.A·Build a function that models a relationship between two quantities
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • F-LE.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
  • HSF-BF.A·Build a function that models a relationship between two quantities.
  • HSF-LE.A.2·Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

15 min

15 min