Curriculum / Math / 8th Grade / Unit 7: Pythagorean Theorem and Volume / Lesson 7
Math
Unit 7
8th Grade
Lesson 7 of 16
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Understand a proof of the Pythagorean Theorem.
The core standards covered in this lesson
8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
In the picture on the left, a right triangle has been drawn on a square grid. In the picture on the right, quadrilaterals have been built on each side of the triangle.
Below is a second right triangle, with squares built on the three sides of the triangle.
Sizing Up Squares, accessed on April 2, 2018, 10:18 a.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
The two squares below, shown as Figure 1 and Figure 2, are congruent and have side lengths $$a+b$$.
a. Describe what you notice about each figure. What is similar and different about the two figures?
b. Figure 1 is composed of 4 triangles and 1 square. Find the area of each shape then write a simplified expression to represent their sum. Do the same for the 4 triangles and 2 squares that make up Figure 2.
c. As congruent squares, the area of Figure 1 is equal to the area of Figure 2. Using your answer from part b, write an equation, in simplest terms, to demonstrate these equal areas. Explain how this relationship matches what you see in the figures.
Use the Pythagorean Theorem to find the measure of the hypotenuse, $$c$$, in the triangles below.
a.
b.
A set of suggested resources or problem types that teachers can turn into a problem set
Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
For each statement below, indicate if it is always true, sometimes true, or never true.
a. The Pythagorean Theorem shows the relationship between the sides of a triangle.
b. The Pythagorean Theorem shows the relationship between the sides of a right triangle.
c. The Pythagorean Theorem says that $${c^2=b^2+a^2,}$$ where $$c$$ is the hypotenuse.
d. The Pythagorean Theorem says that $$a^2=b^2+c^2$$, where $$c$$ is the hypotenuse.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 6
Lesson 8
Topic A: Irrational Numbers and Square Roots
Define, evaluate, and estimate square roots.
8.EE.A.2
Understand that some numbers, including $${\sqrt{2}}$$, are irrational. Approximate the value of irrational numbers.
8.NS.A.1 8.NS.A.2
Locate irrational values approximately on a number line. Compare values of irrational numbers.
8.NS.A.2
Represent rational numbers as decimal expansions.
8.NS.A.1
Represent decimal expansions as rational numbers in fraction form.
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Topic B: Understanding and Applying the Pythagorean Theorem
Understand the Pythagorean Theorem as a relationship between the side lengths in a right triangle.
8.G.B.6
Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.
Find missing side lengths involving right triangles and apply to area and perimeter problems.
8.G.B.7
Solve real-world and mathematical problems using the Pythagorean Theorem (Part I).
Solve real-world and mathematical problems using the Pythagorean Theorem (Part II).
Find the distance between points in the coordinate plane using the Pythagorean Theorem.
8.G.B.8
Topic C: Volume and Cube Roots
Define and evaluate cube roots. Solve equations in the form $${x^2=p}$$ and $${x^3=p}$$.
8.EE.A.2 8.NS.A.2
Solve real-world and mathematical problems involving the volume of cylinders and cones.
8.G.C.9
Solve real-world and mathematical problems involving the volume of spheres.
Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.
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