standard_description: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
The mathematical work in this activity centers on interpreting graphical and symbolic representations of a piecewise-defined function.
First, students interpret a graph of a piecewise function and make sense of the rules in terms of a situation. In particular, students consider the meaning of the open and solid circles on the graph and the input values that appear to "break" the graph.
Next, students interpret a couple of equations in cases notation that could define the function, making connections between the symbols and numbers in the notation and features of the graph.
Launch
Display the graph in the activity statement for all to see. Give students a minute to notice and wonder about something on or about the graph. Then, invite students to share what they noticed and wondered. If no students wondered whether the graph represents a function, ask students about it.
Arrange students in groups of 2. Give students a moment of quiet time to think about the first two questions, and then time to discuss their responses with their partner. Some students may not be familiar with the fact that mailing rates may depend on the weight of the items being mailed. Give students a brief introduction, as needed. Ask partners to pause for a whole-class discussion before continuing to the last question.
Ask students how they knew what to pay for mailing a letter that weighs 1 ounce, given that there are two output values that correspond to the input value of 1.
Students are likely to remember that open and closed circles are used to mark the boundary points of graphs of inequalities in one variable, and transfer that understanding to this graph. If not mentioned in students' explanations, clarify that:
An open circle at
\((0,0.5)\)
, for instance, means that, when 0 is the input, 0.5 is
not
the output.
A closed circle at
\((1, 0.5)\)
, for instance, means that, when 1 is the input, 0.5 is the output.
Next, ask students to analyze the two sets of rules in the last question.
Conversing: MLR2 Collect and Display.
As students work in small-groups, listen for and collect language and  gestures students use to describe a piecewise graph and equation. Capture the language that reflects the various ways students describe the notation in the equation, the visual appearance of the graph, the inequality sign, and open or closed circles. Write the students’ words on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students connect the notation with the features on the piecewise graph.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
The relationship between the postage rate and the weight of a letter can be defined by a
piecewise function
.
The graph shows the 2018 postage rates for using regular service to mail a letter.
[IMG]
What is the price of a letter that has the following weight?
1 ounce
1.1 ounces
0.9 ounce
A letter costs $0.92 to mail. How much did the letter weigh?
Kiran and Mai wrote some rules to represent the postage function, but each of them made some errors.
\(\displaystyle K(w) = \begin{cases} 0.50, & 0\leq w\leq 1 \\ 0.71, & 1\leq w\leq 2 \\ 0.92, & 2\leq w\leq 3\\ 1.13, & 3\leq w\leq 3.5\\ \end{cases} \)
\(\displaystyle M(w) = \begin{cases} 0.50, & 0< w< 1 \\ 0.71, & 1< w< 2 \\ 0.92, & 2< w< 3\\ 1.13, & 3< w< 3.5\\ \end{cases}\)
Identify the error in each person’s work and write a corrected set of rules.
Student Response
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Student Facing
Are you ready for more?
Here is an image showing how the postal service specifies the different mailing rates.
Notice that it uses the language "weight not over (oz.)" to describe the different rates.
Explain or use a sketch to show how the graph would change if the postal service uses "under (oz.)" instead?
[IMG]
Student Response
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Activity Synthesis
Invite students to share their analyses of Mai and Kiran's work and identify the errors each person made. Highlight explanations that point out that:
In Kiran's rules, a letter that weighs 1, 2, or 3 ounces each has two possible rates. For example, for 1 ounce, it could be
$
0.50 or
$
0.71. For 2 ounces, it could be
$
0.71 or
$
0.92.
Mai's rules, on the other hand, excludes letters that weigh exactly 1, 2, or 3 ounces each. No rates are specified for those weights.
In both cases, the problem is with the inequality symbols used.
Discuss what the rules should be, making sure to connect the notation with the features on the graph. Ask students:
"How do we know whether to use
\(<\)
or
\(\le\)
by looking at the graph?" (A point with an open circle means the coordinate values of that point is not included, but values that are greater or are less, depending on the graph, are included. So an open circle corresponds to
\(<\)
or
\(>\)
. A point with a solid or closed circle means the coordinate values of that point are included, so it corresponds to
\(\le\)
or
\(\geq\)
.)
"By looking at the graph, how can you tell how many cases should be included in the equation?" (Each piece of the graph is a case.)
If time permits, ask students about the domain and range for this function. Assuming that mail item heavier than 3.5 ounces no longer qualifies as a letter, the domain would include weights that are greater than 0 but no more than 3.5 ounces (
\(0<x \le 3.5\)
). The corresponding range includes 0.50, 0.71, 0.92, 1.13, in dollars.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: piecewise function, point with an open circle, point with a solid or closed circle.
Supports accessibility for: Conceptual processing; Language


aligns: no