few_shots: 
Example:
Concept/skill: 
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Problem/activity:
Task
Cruz and Erica were both getting ready for soccer.
Cruz ran 1 lap around the school.
Erica ran 3 laps around the playground.
Erica said,
I ran more laps, so I ran farther.
Cruz said,
4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.
Who is right? Draw a picture to help you explain your answer.


Answer: 
Yes. This task asks students to compare two fractions that are equal. The students need to explain why the two fractions are equal by drawing a picture, which involves using visual fraction models.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: im_task_000212_pos_0

4
standard_description: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
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
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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