few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Mathematical concept/skill options: 
A. Reason with shapes and their attributes.
B. Understand similarity in terms of similarity transformations.
C. Use coordinates to prove simple geometric theorems algebraically.
D. Translate between the geometric description and the equation for a conic section.
E. Understand congruence and similarity using physical models, transparencies, or geometry software.
F. Analyze, compare, create, and compose shapes.
G. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
H. Understand congruence in terms of rigid motions.
I. Classify two-dimensional figures into categories based on their properties.
J. Draw, construct, and describe geometrical figures and describe the relationships between them.
K. Explain volume formulas and use them to solve problems.
L. Prove theorems involving similarity.
M. Prove geometric theorems.
N. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Apply trigonometry to general triangles.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
S. Make geometric constructions.
T. Apply geometric concepts in modeling situations.
U. Solve real-world and mathematical problems involving area, surface area, and volume.
V. Graph points on the coordinate plane to solve real-world and mathematical problems.
W. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
X. Understand and apply theorems about circles.
Y. Understand and apply the Pythagorean Theorem.
Z. Experiment with transformations in the plane.
AA. Find arc lengths and areas of sectors of circles.


Your response:
A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_lesson_009431

1
options: Extend the domain of trigonometric functions using the unit circle.

Build new functions from existing functions.

Build a function that models a relationship between two quantities.

Model periodic phenomena with trigonometric functions.

Use functions to model relationships between quantities.

Analyze functions using different representations.

Understand the concept of a function and use function notation.

Interpret expressions for functions in terms of the situation they model.

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.

Define, evaluate, and compare functions.

Interpret functions that arise in applications in terms of the context.
problem_activity: Activity
The goal of this activity is to write an expression that models the volume of a box as a function of the side length
\(x\)
of the square cutouts. Students then graph the function, use the graph to estimate the value of
\(x\)
that will produce the box with the largest volume, and establish a reasonable domain for the function by considering the context.
During the discussion, students interpret their expression in context (MP2). Of particular interest, values of
\(x\)
greater than 5.5 inches result in positive volumes even though cutouts of this size do not make sense when constructing a box from an 8.5 inch by 11 inch sheet of paper. There is no need for students to rewrite the expression for
\(V(x)\)
at this time in standard form. In later lessons, students will practice rewriting equations for polynomials in different forms.
Launch
Arrange students in groups of 2. Tell students there are many possible answers for the first question. After quiet work time, ask students to compare their response to their partner’s and decide which plan they like best. Select 2–3 groups to share their plans. Provide access to devices that can run Desmos or other graphing technology.
Representation: Internalize Comprehension.
Use annotations to highlight connections between representations in a problem. For example, prior to developing an equation for volume, have students label the edges of their box with the calculations they used to determine the length, width, and height.
Supports accessibility for: Visual-spatial processing
Student Facing
[IMG]
The volume
\(V(x)\)
in cubic inches of the open-top box is a function of the side length
\(x\)
in inches of the square cutouts. Make a plan to figure out how to construct the box with the largest volume.
Pause here so your teacher can review your plan.
Write an expression for
\(V(x)\)
.
Use graphing technology to create a graph representing
\(V(x)\)
. Approximate the value of
\(x\)
that would allow you to construct an open-top box with the largest volume possible from one piece of paper.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
The surface area
\(A(x)\)
in square inches of the open-top box is also a function of the side length
\(x\)
in inches of the square cutouts.
Find one expression for
\(A(x)\)
by summing the area of the five faces of our open-top box.
Find another expression for
\(A(x)\)
by subtracting the area of the cutouts from the area of the paper.
Show algebraically that these two expressions are equivalent.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may forget that the width and length of the paper are based on removing length
\(x\)
twice due to cutting out squares at each corner. Encourage these students to revisit their work from the previous activity and think about how they can calculate length and width without measuring directly.
If students have trouble finding a general expression for the volume of the box, encourage them to revisit their results from the previous activity and write the volumes of the boxes without simplifying their answers (e.g., 6.5 in
\(\times\)
9 in
\(\times\)
1 in instead of 58.5 in
3
) in order to see the pattern more easily.
Activity Synthesis
The purpose of this discussion is for students to think critically about the equation they’ve written to model the volume of the box. Here are some questions for discussion:
“What is a value of
\(x\)
that wouldn’t make sense?” (
\(x=6\)
would not make sense, because the sheet of paper is only 8.5 inches long on one side, so cutting out 6 inch by 6 inch squares from each corner is not possible.)
“What is the smallest value that would work for
\(x\)
? The largest?” (The smallest value is some number greater than 0 inches that you could still use as a side length to cut out the squares and fold up the box sides to make a very short box. The largest value is some number less than half of 8.5 inches that you could still use as a side length to cut out the squares and fold up the box sides to make a very tall box.)
“What is the range of
\(x\)
values where the volume is increasing? Decreasing?” (The volume increases as the cutouts get larger, until
\(x\)
is about 1.6 inches. Then the volume decreases until
\(x\)
is 4.25 inches, which creates a box with a volume of 0. The volume also starts increasing again when
\(x\)
is about 4.9 inches, but these values of
\(x\)
don’t make sense because we can’t make cutouts that large.)
If no students bring up the word
domain
during the discussion, remind students of the meaning of the word and select students to recommend a reasonable domain for the function
\(V\)
.
Conclude the discussion by telling students that
\(V\)
is a type of
polynomial
function and that they will learn about other situations polynomial functions can model along with what graphs of polynomial functions can look like in future lessons.
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to interpret the equation they’ve written to model the volume of the box. Call students’ attention to language such as “range of values,” “groups of parentheses,” or other key parts of the equation representing the length, width, or height. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Use this opportunity to define a “polynomial function.” Remind students to borrow language from the display as needed. This will help students use mathematical language to interpret the equation for the volume of the box.
Design Principle(s): Maximize meta-awareness


correct_option_index: 11