Students learn about irrational numbers, approximating square roots of non-perfect square numbers, and investigate the well-known Pythagorean Theorem to solve for missing measures in right triangles.
In Unit 7, eighth-grade students extend their understanding of the Number System to include irrational numbers. This new understanding supports students as they study square and cube root equations and relationships between side lengths in right triangles, both concepts that fall within the major work of the grade. Students start the unit by investigating solutions to equations like $$x^2=2$$ and realize that the solution is not an exact point on the number line. They approximate square roots of non-perfect square numbers and represent rational numbers written in decimal form as fractions. The focus of the unit shifts to right triangles, and students investigate the well-known Pythagorean Theorem. They apply their understanding of square roots to solve for missing measures in right triangles and other applications. They look closely at geometric figures to identify and create right triangles, opening up the opportunity to apply the Pythagorean Theorem to find new information (MP.7). The focus shifts once more as students learn about cube roots and apply this new concept to various volume applications involving cylinders, spheres, and cones. Throughout the unit, students must attend to precision in their work, their solutions, and their communication, being careful about specifying appropriate units of measure, using the equals sign appropriately, and representing numbers accurately (MP.6).
Prior to this unit, students learned many skills and concepts that prepared them for this unit. Since elementary grades, students have been learning about and refining their understanding of area and volume. They have learned how to use composition and decomposition as tools to determine measurements, they’ve learned formulas and how to use them in problem-solving situations, and they’ve encountered various real-world situations. Standard 8.G.9 is a culminating standard in the Geometry progression in middle school, which will lay the foundation for much of the work they will do in high school geometry.
In high school, students will more formally derive the distance formula and other principles, they will expand their work with right triangles to include trigonometric ratios, and they will solve more complex problems involving volume of cylinders, pyramids, cones, and spheres.
Pacing: 20 instructional days (16 lessons (17 days), 2 flex days, 1 assessment day)
This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.
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legs
square root
cube root
perfect square
irrational number
rational number
converse statement
Pythagorean Theorem
Pythagorean triplet
hypotenuse
cylinder
cone
sphere
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8.EE.A.2
Define, evaluate, and estimate square roots.
8.NS.A.1
8.NS.A.2
Understand that some numbers, including $${\sqrt{2}}$$, are irrational. Approximate the value of irrational numbers.
8.NS.A.2
Locate irrational values approximately on a number line. Compare values of irrational numbers.
8.NS.A.1
Represent rational numbers as decimal expansions.
8.NS.A.1
Represent decimal expansions as rational numbers in fraction form.
8.G.B.6
Understand the Pythagorean Theorem as a relationship between the side lengths in a right triangle.
8.G.B.6
Understand a proof of the Pythagorean Theorem.
8.G.B.6
Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.
8.G.B.7
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).
8.G.B.7
Solve real-world and mathematical problems using the Pythagorean Theorem (Part II).
8.G.B.8
Find the distance between points in the coordinate plane using the Pythagorean Theorem.
8.NS.A.2
8.EE.A.2
Define and evaluate cube roots. Solve equations in the form $${x^2=p}$$ and $${x^3=p}$$.
8.G.C.9
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.
8.G.C.9
Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.
Key: Major Cluster Supporting Cluster Additional Cluster
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