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)


Concept/skill: 
Recognize area as an attribute of plane figures and understand concepts of area measurement.

Answer: 
No. This activity does not teach students area as an attribute of plane figures. Instead, the activity focuses on solid, three-dimensional figures.

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_neg_0

1
standard_description: Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).
problem_activity: Activity
Students may be familiar with this proof from middle school. Revisiting this proof helps students connect a familiar diagram and proof to the similarity proof. It also gives students an opportunity to compare and contrast different types of proof and think about why mathematicians might bother to re-prove something that has already been proved.
Launch
Use the Notice and Wonder routine to help students make sense of the context and images they will be using. Ask students what they notice and wonder about the images. If not mentioned by students, ask where
\(c\)
shows up in these images.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for how Pythagoras used the diagrams to prove
\(a^2+b^2=c^2\)
. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do you know that the two large squares are the same size?”, and “How do you know that the area of the blue square is
\(c^2\)
?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify the proof of the Pythagorean Theorem.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Representation: Develop Language and Symbols.
Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide access to the different pieces of the figures so students can rearrange and compare them.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
[IMG]
When Pythagoras proved his theorem he used the 2 images shown here. Can you figure out how he used these diagrams to prove
\(a^2+b^2=c^2\)
in a right triangle with hypotenuse length
\(c\)
?
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?
James Garfield, the 20th president, proved the Pythagorean Theorem in a different way.
Cut out 2 congruent right triangles
Label the long sides
\(b\)
, the short sides
\(a\)
and the hypotenuses
\(c\)
.
Align the triangles on a piece of paper, with one long side and one short side in a line. Draw the line connecting the other acute angles.
How does this diagram prove the Pythagorean Theorem?
[IMG]
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
Encourage students to look for
\(a^2\)
. Students have scissors and tracing paper in their geometry toolkits, they may want to trace, cut out, and reassemble the pieces.
Activity Synthesis
Invite students to talk about the differences between this proof and the one in the previous activity, including which one made more sense to them, which one they preferred writing, or which one felt most convincing. Ask students why they think mathematicians might continue to try to come up with different ways to prove a theorem as old as the Pythagorean Theorem.


aligns: no