Math / 8th Grade / Unit 7: Pythagorean Theorem and Volume
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.
Math
Unit 7
8th Grade
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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)
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The following assessments accompany Unit 7.
Have students complete the Pre-Unit Assessment and Pre-Unit Student Self-Assessment before starting the unit. Use the Pre-Unit Assessment Analysis Guide to identify gaps in foundational understanding and map out a plan for learning acceleration throughout the unit.
Pre-Unit Student Self-Assessment
Have students complete the Mid-Unit Assessment after lesson 11.
Use the resources below to assess student mastery of the unit content and action plan for future units.
Post-Unit Assessment
Post-Unit Assessment Answer Key
Post-Unit Student Self-Assessment
Use student data to drive your planning with an expanded suite of unit assessments to help gauge students’ facility with foundational skills and concepts, as well as their progress with unit content.
Suggestions for how to prepare to teach this unit
Unit Launch
Prepare to teach this unit by immersing yourself in the standards, big ideas, and connections to prior and future content. Unit Launches include a series of short videos, targeted readings, and opportunities for action planning.
The central mathematical concepts that students will come to understand in this unit
Terms and notation that students learn or use in the unit
converse statement
cone
cube root
cylinder
hypotenuse
irrational number
legs
perfect square
pythagorean triplet
pythagorean theorem
rational number
sphere
square root
To see all the vocabulary for Unit 7, view our 8th Grade Vocabulary Glossary.
The materials, representations, and tools teachers and students will need for this unit
To see all the materials needed for this course, view our 8th Grade Course Material Overview.
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
Understand a proof of the Pythagorean Theorem.
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.
Key
Major Cluster
Supporting Cluster
Additional Cluster
The content standards covered in this unit
8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.
8.G.B.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.B.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.C.9 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
8.NS.A.2 — Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Standards covered in previous units or grades that are important background for the current unit
6.EE.B.5 — Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
6.G.A.1 — Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
6.G.A.3 — Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
7.G.B.6 — Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
6.NS.C.6 — Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
7.NS.A.2.D — Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.A.3 — Solve real-world and mathematical problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
Standards in future grades or units that connect to the content in this unit
G.GPE.B.7 — Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
G.GMD.A.1 — Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
G.GMD.A.2 — Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
G.GMD.A.3 — Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G.SRT.C.6 — Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.C.7 — Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.C.8 — Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.
Unit 6
Systems of Linear Equations
Unit 8
Bivariate Data