standard_description: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
problem_activity: Activity
Up to this point in the unit, students have written and interpreted equations representing exponential change which were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a context. The independent variable is time,
\(t\)
, so negative values of
\(t\)
refer to times
before
\(t\)
is 0.
For the last question, to find the time when the coral had the given volume, students may:
Calculate the values of
\(y\)
when
\(t\)
is -3, -4, -5, etc. (i.e., extend the table of values, if they previously created one) until they reach 37.5 or a value close to it.
Repeatedly divide by 2 (or multiply by
\(\frac12\)
) the value of
\(y\)
when
\(t\)
is -2 until it approaches or hits 37.5.
Use the equation
\(1,\!200 \boldcdot 2^t = 37.5\)
to find the value of
\(2^t\)
and then reason about
\(t\)
from there.
Look for students using these or other strategies so they can share later.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Encourage them to think quietly about the questions before discussing with their partner. Creating a table or spreadsheet may help students organize the work in the second question. If needed, encourage students to do so.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students make sense of the language of exponential situations, start by displaying the task statement “A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.” Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the activity’s questions. Listen for and amplify any questions referring to the value of
\(t\)
, especially those wondering about time in the past. This will build student understanding and interpretation of the language used to represent negative exponents.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Encourage students to create a timeline next to their tables, selecting a starting point year for zero and then labeling the years correlated with the positive exponents prior to labeling years associated with negative exponents.
Supports accessibility for: Conceptual processing; Memory
Student Facing
A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.
[IMG]
Write an equation of the form
\(y=a \boldcdot b^t\)
representing the relationship, where
\(t\)
is time in years since the coral was measured and
\(y\)
is volume of coral in cubic centimeters. (You need to figure out what numbers
\(a\)
and
\(b\)
are in this situation.)
Find the volume of the coral when
\(t\)
is 5, 1, 0, -1, and -2.
What does it mean, in this situation, when
\(t\)
is -2?
In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Students may struggle to think of how to start finding the values for
\(a\)
and
\(b\)
. Suggest that they make a table of corresponding values of
\(t\)
and
\(y\)
, and think about starting with
\(t\)
and calculating
\(y\)
.
Activity Synthesis
Review how students found an equation,
\(y = 1,200 \boldcdot 2^t\)
, to represent the volume of coral. Make sure that they recall that the 1,200 represents the 1,200 cubic centimeters of coral when it was first measured and the 2 indicates the doubling of its volume each year.
Select students to share their interpretations of negative values of
\(t\)
and how they found the value of
\(t\)
when given a volume. Ask questions such as:
“In this situation, what does it mean to say that when
\(t\)
is -3,
\(y\)
is
\(1,200 \boldcdot 2^{\text-3}\)
or 150?” (3 years before the coral was measured to be 1,200 cubic centimeters, its volume was 150 cubic centimeters).
“How did you estimate or find when the volume of the coral was 37.5 cubic centimeters?”
It may be helpful to display a graph of
\(y = 1,200 \boldcdot 2^t\)
, such as the one embedded in the digital version of these materials. Display the graph (either with or without the points from the table plotted) and ask students to identify the volume of coral when
\(t\)
is -3 and the time when the volume of coral is 37.5 cubic centimeters.
If using the digital applet, consider initially turning off the function and the point, and hiding the expression list. Ask students to gesture at points on the graph corresponding to questions in the activity. Then, turn on the function and the point for demonstration. Remember that if you click on a plotted point, its coordinates are revealed.


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