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
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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)


Standard options: 
A. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
B. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
C. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)


Your response:
C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_lesson_009431

1
options: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Optional activity
This activity is optional.  This activity is similar to the previous (optional) one from a function point of view, but now students investigate the volume of a cylinder instead of a rectangular prism. Students continue working with functions to investigate what happens to the volume of a cylinder when you halve the height. The exploration and representations resemble what was done in the previous activity, and students continue to identify the effect of the changing dimension on the graph and the equation of this function. As groups work on the task, identify those who make the connection between the graph and equation representations, and encourage groups to look for similarities and differences between what they see in this activity and what they saw in the previous activity. Invite these groups to share during the whole-class discussion.
Digital
Launch
Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.
Students using the digital activity can graph their equations using the applet.
Student Facing
There are many cylinders with radius 5 units. Let
\(h\)
represent the height and
\(V\)
represent the volume of these cylinders.
Write an equation that represents the relationship between
\(V\)
and
\(h\)
. Use 3.14 as an approximation of
\(\pi\)
.
Graph this equation.
What happens to the volume if you halve the height,
\(h\)
? Where can you see this in the graph? How can you see it algebraically?
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
Keep students in the same groups. Tell students that this activity is similar to the previous one, but they will work with a cylinder instead of a rectangular prism. Give students 2–3 minutes of quiet work time followed by time to discuss the last question with their partner. Follow with a whole-class discussion.
Students using the digital activity can graph their equations using the applet.
Student Facing
There are many cylinders with radius 5 units. Let
\(h\)
represent the height and
\(V\)
represent the volume of these cylinders.
Write an equation that represents the relationship between
\(V\)
and
\(h\)
. Use 3.14 as an approximation of
\(\pi\)
.
Graph this equation and label the axes.
[IMG]
What happens to the volume if you halve the height,
\(h\)
? Where can you see this in the graph? How can you see it algebraically?
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?
Suppose we have a rectangular prism with dimensions 2 units by 3 units by 6 units, and we would like to make a rectangular prism of volume 216 cubic units by stretching
one
of the three dimensions.
What are the three ways of doing this? Of these, which gives the prism with the smallest surface area?
Repeat this process for a starting rectangular prism with dimensions 2 units by 6 units by 6 units.
Can you give some general tips to someone who wants to make a box with a certain volume, but wants to save cost on material by having as small a surface area as possible?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask previously identified groups to share their graphs and equations.
If students completed the optional activity, display student graphs from both activities and ask:
“Compare the graph in this activity to the graph in the last activity. How are they alike? How are they different?”
“Compare the equation in this activity to the equation in the last activity. How are they alike? How are they different?”
“How can you tell that this is a linear relationship?”
For the last question, make sure students understand that “looks like a line” is insufficient evidence for saying a relationship is linear since the scale of the axes can make non-linear graphs look linear. It is important students connect that the equation is linear (of the form y=mx+b) or relate back to the relationship between the height and volume: that the volume is
\(25\pi\)
, or about 78.5, times the height.
If students did not complete the optional activity, ask:
“Which of your variables is the independent? The dependent?” (The height,
\(h\)
, is the independent variable, and volume, V, is dependent variable.)
“Which variable is a function of which?” (Volume is a function of height.)
“How can you tell that this is a linear relationship?”
See notes above on the last question in this list. If it has not been brought up in students’ explanations, ask what the volume equation looks like when we double the height
\(h\)
. Display volume equation
\(V=78.5(2h)\)
for all to see. Ask how we can write this equation to show that the volume doubled when
\(h\)
doubled (using algebra, we can rewrite
\(V=78.5(2h)\)
as
\(V=2(78.5h)\)
. Since the volume for
\(h\)
was
\(78.5h\)
, this shows that the volume for
\(2h\)
is twice the volume for
\(h\)
).
Representing, Listening, Speaking: MLR7 Compare and Connect.
Invite students to create a visual representation that shows what is the same and what is different between this activity and the previous one. If students did not complete the optional activity, students can create multiple representations that illustrate why volume is a function of height. Give students time to do a gallery walk of the displays. Look for opportunities to highlight approaches that compare the multiple representations of the function of volume. Lead a whole-class discussion that connects features on the displays, such as comparing the equations and graphs. This will help students communicate about the multiple representations of volume functions while using mathematical language.
Design Principle(s): Optimize output; Maximize meta-awareness


correct_option_index: 1