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, in json format:
{
  "explanation": "This activity asks students to reason how shapes can combine to make other shapes.",
  "answer": "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: Model periodic phenomena with trigonometric functions.

Interpret functions that arise in applications in terms of the context.

Define, evaluate, and compare functions.

Build a function that models a relationship between two quantities.

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.

Analyze functions using different representations.

Understand the concept of a function and use function notation.

Build new functions from existing functions.

Extend the domain of trigonometric functions using the unit circle.

Use functions to model relationships between quantities.
problem_activity: Activity
In this activity, students again apply what they learned about quadratic expressions and their graphs. This time, they solve a contextual problem. Students consider how to change the parameters of a quadratic expression that models the movement of a digitally animated object—a peanut—such that it travels in a parabolic path and meets certain restrictions (MP4).
Providing time to notice and wonder will elicit the idea that this equation shows the horizontal and vertical movement of the peanut, which will be useful when students adjust the equation in a later activity. While students may notice and wonder many things about the graphs and equation, connecting the vertex of the graph to the parameters in the equation are the important discussion points. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).
For the second question, there are many possible values of
\(h\)
and
\(k\)
that students could use. As students work, notice those who choose to:
Shift the starting position of the peanut closer to the wall (translating the graph horizontally to the right).
Stay in the same starting position but shift the vertex to a point closer to the wall.
Shift both the peanut and the vertex of the parabolic path.
If students choose to use graphing technology to help them experiment with expressions or verify their predictions, they practice choosing tools strategically (MP5).
Digital
Launch
You may wish to use the applet during the synthesis to demonstrate student equations. There is a copy of the applet with an option to show the path of the peanut.
Representing, Conversing: MLR7 Compare and Connect.
As students share their values of
\(h\)
and
\(k\)
with the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the
\(h\)
and
\(k\)
represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall.
Mai uses the equation
\(y = \text-0.1(x-h)^2+k\)
to represent the path of the jump.
\(y\)
represents the height of the peanut as a function of the horizontal distance it travels
\(x\)
. On the screen, the base of the wall is located at
\((22, 0)\)
, with the top of the wall at
\((22,4.5)\)
.
The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.
[IMG]
What are the values of
\(k\)
and
\(h\)
in this equation?
This applet is programmed to use Mai’s equation. Experiment with the applet to find an equation to get the peanut over the wall, and keep it on the screen. Be prepared to explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Ask students to read the first couple of paragraphs in the statement. Then, display this graph for all to see. Tell students that when Mai used the equation
\(y = \text -0.1(x-16)^2+7\)
, this is the path of the peanut’s jump.
[IMG]
Ask students what they notice and wonder about the equation and the graph. Make sure students see the connection between the parameters in the equation and the vertex of the graph before they proceed with the activity.
Provide continued access to graphing technology, in case students choose to use it.
Representing, Conversing: MLR7 Compare and Connect.
As students share their values of
\(h\)
and
\(k\)
with the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the
\(h\)
and
\(k\)
represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to model the path of the main character, an animated peanut, jumping over a wall.
[IMG]
Mai uses the equation
\(y = \text-0.1(x-h)^2+k\)
to represent the path of the jump.
\(y\)
represents the height of the peanut as a function of the horizontal distance it travels,
\(x\)
.
On the screen, the base of the wall is located at
\((22, 0)\)
, with the top of the wall at
\((22,4.5)\)
. The dashed curve in the picture shows the graph of 1 equation Mai tried, where the peanut fails to make it over the wall.
[IMG]
What are the values of
\(h\)
and
\(k\)
in this equation?
Starting with Mai’s equation, choose values for
\(h\)
and
\(k\)
that will guarantee the peanut stays on the screen but also makes it over the wall. Be prepared to explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may randomly choose values for
\(h\)
and
\(k\)
until they find a combination whose graph meets the requirements. Encourage them to reason about the problem more systematically, by considering what they learned about the vertex form.
Activity Synthesis
Select students with contrasting strategies to share their equations and explanations for the second question. To verify that students’ proposed equations meet the animation criteria, consider graphing students’ proposed equations for all to see. Highlight the three strategies mentioned in the Activity Narrative.
Point out one distinction between the graphs in this activity and previous graphs that represent projectile motions. In earlier cases, the output represented the height of an object as a function of
time
, so the graph of the function and physical trajectory of the object may be very different (for example, the object may be going straight up and then fall straight down, but the graph shows a parabola). Here, they see a model where the height of an object is a function of
horizontal distance
, so the path of the object in a projectile motion does resemble the graph representing the function.


correct_option_index: 7

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