standard_description: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
problem_activity: Activity
This task contains two distinct parts. Depending on what students need and time constraints, you could decide to do either or both.
In the associated Algebra 1 lesson, students will need to evaluate the functions
\(6x^2\)
and
\(3^x\)
for different input values. The purpose of the first part is to prepare them to avoid some common mistakes. Analyzing and correcting incorrect work is an opportunity to critique the reasoning of others (MP3).
The purpose of the second part is to revisit ways to tell whether a function is linear, quadratic, or exponential, and to practice the skills of generating a table and sketching a graph for a given function.
Student Facing
Different students are evaluating two expressions,
\(3\boldcdot 6^x\)
and
\(5^x\)
. Analyze their work, describe any errors made, and then evaluate each expression correctly.
|                                                                                                                                     | Noah’s work | Mai’s work | corrected work |
|-------------------------------------------------------------------------------------------------------------------------------------|-------------|------------|----------------|
|       Evaluate                       \(5^x\)                      when                       \(x\)                      is 6.       |             |            |                |
| Evaluate                       \(3 \boldcdot 6^x\)                      when                       \(x\)                      is 2. |             |            |                |
Here are three functions. For each function:
Complete the table of values.
Sketch a graph.
Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know.
\(f(x)=3 \boldcdot 2^x\)
|  \(x\)   |   |   |   |   |   |
|----------|---|---|---|---|---|
| \(f(x)\) |   |   |   |   |   |
[IMG]
\(g(x)=3 \boldcdot x^2\)
|  \(x\)   |   |   |   |   |   |
|----------|---|---|---|---|---|
| \(g(x)\) |   |   |   |   |   |
[IMG]
\(h(x)=3 \boldcdot 2x\)
|  \(x\)   |   |   |   |   |   |
|----------|---|---|---|---|---|
| \(h(x)\) |   |   |   |   |   |
[IMG]
Student Response
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Activity Synthesis
For the first part, invite students to describe the errors they see, and share how to correctly evaluate each expression.
For the second part, display the tables and graphs for all to see. Ask students to share some ways they can tell a function is linear, exponential, or quadratic using the different representations. For example:
The graph of the linear function is a straight line.
The table shows the exponential function has a growth factor of 2. Each time
\(x\)
increases by 1, the
\(f(x)\)
values increase by a factor of 2.
The table shows the quadratic function does not have a constant rate of change or a constant growth factor.


aligns: no