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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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)


Concept/skill: 
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.”)

Answer: 
Yes. This activity asks students to compose three-dimensional shapes, using geoblocks, to create composite shapes.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_lesson_009431_pos_0

1
standard_description: Solve quadratic equations with real coefficients that have complex solutions.
problem_activity: Optional activity
This activity is optional because it goes beyond the depth of understanding required by the standards.
The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to tell how many solutions an equation has, and whether those solutions are real or non-real. Students have already solved two of the equations in the previous activity, and the equation they haven’t solved involves the perfect square that connects all three of the equations. Graphically,
\(y=f(x)\)
intersects the
\(x\)
-axis twice, which corresponds to the two real solutions to
\(x^2 - 8x + 13 = 0\)
. The graph
\(y=g(x)\)
intersects the
\(x\)
-axis once, which corresponds to the single real solution to
\(x^2 - 8x + 16 = 0\)
. However, the graph
\(y=h(x)\)
doesn’t intersect the
\(x\)
-axis, which reflects the fact that the complex solutions to
\(x^2 - 8x + 19 = 0\)
can’t be described with a real number line.
Launch
Provide access to graphing technology.
Student Facing
How many real solutions does each equation have? How many non-real solutions?
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 16 = 0\)
\(x^2 - 8x + 19 = 0\)
How do the graphs of these functions help us answer the previous question?
\(f(x) = x^2 - 8x + 13\)
\(g(x) = x^2 - 8x + 16\)
\(h(x) = x^2 - 8x + 19\)
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 attempt to plot the complex solutions in the coordinate plane as if it were the complex plane. Remind these students that points in the coordinate plane are pairs of real numbers
\((x,y)\)
, whereas points in the complex plane are single complex numbers. In other words, the point
\((2,4)\)
in the coordinate plane is a different kind of object than the point
\(2+4i\)
in the complex plane.
Activity Synthesis
Select students to share their solutions and discuss the connections between the graphs and the equations. Discuss the idea that if a graph of a quadratic function doesn’t cross a given horizontal line (in this case,
\(x=0\)
) this doesn’t indicate that the related equation has no solutions, but rather that it has no
real
solutions that can be graphed with real number lines.
Speaking: MLR8 Discussion Supports.
To help students produce statements about the connection between graphs of quadratic functions and solutions to related quadratic equations, provide sentence frames such as: “If _____ then _____ because…” and  “On the graph, I see _____, so I know….” Show concepts multi-modally by displaying the graphs of
\(f(x)\)
,
\(g(x)\)
, and
\(h(x)\)
.
Design Principle(s): Support sense-making


aligns: yes