A Proof of the Pythagorean Theorem
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to study a diagram in preparation for understanding a proof of the Pythagorean Theorem. While students may notice and wonder many things about these diagrams, the fact that the construction on the left has right triangles, while the construction on the right does not is the important discussion point.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that the construction on the left leads to a composite figure of a square, while the composite figure on the right is not. This is due to whether or not the construction has right triangles.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Things students may notice:
- There are two figures both made up of a square and four triangles.
- The triangles in the figure on the left are right triangles and the triangles in the figure on the right are not.
- The figure on the left is a square.
- The figure on the right is not a square.
- The smaller squares in the middle look the same size.
Things students may wonder:
- Are the squares in the middle the same size?
- Are the triangles in the left figure all the same size?
- Are the triangles in the right figure all the same size?
- Does it matter if the triangles are right triangles?
- Does it matter if they are equilateral triangles?
- What are these figures going to be used for?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
Tell students that when there is a square with congruent right triangles on each side, as shown on the left, they form a larger square (they will be able to prove this in high school). But it doesn’t work if the triangles are not right triangles. This construction will be used in the next activity.
Standards
- 8.G.6·Explain a proof of the Pythagorean Theorem and its converse.
- 8.G.B.6·Explain a proof of the Pythagorean Theorem and its converse.
20 min
10 min
Knowledge Components
All skills for this lesson
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A Proof of the Pythagorean Theorem
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to study a diagram in preparation for understanding a proof of the Pythagorean Theorem. While students may notice and wonder many things about these diagrams, the fact that the construction on the left has right triangles, while the construction on the right does not is the important discussion point.
This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that the construction on the left leads to a composite figure of a square, while the composite figure on the right is not. This is due to whether or not the construction has right triangles.
Launch
Arrange students in groups of 2. Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Things students may notice:
- There are two figures both made up of a square and four triangles.
- The triangles in the figure on the left are right triangles and the triangles in the figure on the right are not.
- The figure on the left is a square.
- The figure on the right is not a square.
- The smaller squares in the middle look the same size.
Things students may wonder:
- Are the squares in the middle the same size?
- Are the triangles in the left figure all the same size?
- Are the triangles in the right figure all the same size?
- Does it matter if the triangles are right triangles?
- Does it matter if they are equilateral triangles?
- What are these figures going to be used for?
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
Tell students that when there is a square with congruent right triangles on each side, as shown on the left, they form a larger square (they will be able to prove this in high school). But it doesn’t work if the triangles are not right triangles. This construction will be used in the next activity.
Standards
- 8.G.6·Explain a proof of the Pythagorean Theorem and its converse.
- 8.G.B.6·Explain a proof of the Pythagorean Theorem and its converse.