ALEX Classroom Resource

  

Grade 8 Mathematics Module 7, Topic D: Applications of Radicals and Roots

  Classroom Resource Information  

Title:

 Grade 8 Mathematics Module 7, Topic D: Applications of Radicals and Roots

URL:

 https://www.engageny.org/resource/grade-8-mathematics-module-7-topic-d-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 7, Topic D, students learn that radical expressions naturally arise in geometry, such as the height of an isosceles triangle or the lateral length of a cone. The Pythagorean Theorem is applied to three-dimensional figures in Topic D as students learn some geometric applications of radicals and roots (8.G.B.7). In order for students to determine the volume of a cone or sphere, they must first apply the Pythagorean Theorem to determine the height of the cone or the radius of the sphere. Students learn that truncated cones are solids obtained by removing the top portion above a plane parallel to the base. Students know that to find the volume of a truncated cone they must access and apply their knowledge of similar figures learned in Module 3. Their work with truncated cones is an exploration of solids that is not formally assessed. In general, students solve real-world and mathematical problems in three dimensions in Topic D (8.G.C.9). For example, now that students can compute with cube roots and understand the concept of rate of change, students compute the average rate of change in the height of the water level when water is poured into a conical container at a constant rate. Students also use what they learned about the volume of cylinders, cones, and spheres to compare volumes of composite solids.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
28. Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications
Unpacked Content
Evidence Of Student Attainment:
Students: Given real-world and mathematical problems in two and three dimensions,
  • Apply the Pythagorean Theorem in order to solve problems and justify solutions and solution paths for finding side lengths in right triangles within the problem contexts.
Teacher Vocabulary:
  • Pythagorean Theorem
Knowledge:
Students know:
  • The Pythagorean Theorem.
  • Appropriate labeling of a right triangle, leg and hypotenuse.
Skills:
Students are able to:
  • Solve equations involving one variable and square root.
  • Represent real-world and mathematical contexts involving right triangles in a variety of formats (drawings, equations).
  • Justify solutions and solution paths using conceptual understandings and vocabulary related to the Pythagorean Theorem (right angle, hypotenuse).
Understanding:
Students understand that:
  • the properties of right triangles can be used to solve problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.28.1: Discuss strategies for solving real-world and mathematical problems.
M.8.28.2: Solve problems using the Pythagorean Theorem, with or without a calculator.
M.8.28.3: Identify right triangles.
M.8.28.4: Demonstrate how to find square roots, with or without a calculator.
M.8.28.5: Solve problems with exponents, with or without a calculator.

Prior Knowledge Skills:
  • Define exponent, numerical expression, algebraic expression, variable, base, power, square of a number, and cube of a number.
  • Compute a numerical expression with exponents, with or without a calculator.
  • Restate exponential numbers as repeated multiplication.
  • Choose the correct value to replace each variable in the expression (Substitution).
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.27 Use the pythagorean theorem to find the hypotenuse when given the measures of two legs in a real-world context. Limit to Pythagorean triples.
Mathematics
MA2019 (2019)
Grade: 8
30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Understand that the application of volume formulas and the relationship between these three formulas can be used in combinations when determining solutions involving real-world cylinders, cones, and spheres.
Teacher Vocabulary:
  • Radius
  • Pi
  • Volume
  • Cylinder
  • Cone
  • Sphere
Knowledge:
Students know:
  • The volume formulas for cylinders, cones, and spheres.
  • That 3.14 is an approximation of pi commonly used in these volume formulas.
  • That composite three dimensional objects in the real-world can be created by combining cylinders, cones, and spheres in part or whole.
Skills:
Students are able to:
  • Calculate the volume of cones, cylinders, and spheres given in real-world contexts. often times approximating solutions to a specified decimal place.
  • Identify the components of a composite figure as being portions of or whole cylinders, cones, and spheres.
  • Combine the results of calculations to find volume for real-world composite figures.
Understanding:
Students understand that:
  • the application of volume formulas and the relationship between these three formulas can be used in combinations when determining solutions involving real-world cylinders, cones, and spheres.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.30.1: Define formula, volume, cone, cylinders, spheres, and height.
M.8.30.2: Discuss the measure of volume and give examples.
M.8.30.3: Solve problems with exponents, with or without a calculator.
M.8.30.4: Recall how to find circumference of a circle, with or without a calculator.
M.8.30.5: Identify parts of a circle.
M.8.30.6: Calculate the volume of three-dimensional figures.
M.8.30.7: Solve real-world problems using the volume formulas for three-dimensional figures, with or without a calculator.

Prior Knowledge Skills:
  • Define volume, surface area, triangles, quadrilaterals, polygons, cubes, and right prisms.
  • Discuss strategies for solving real-world mathematical problems.
  • Recall formulas for calculating volume and surface area.
  • Identify the attributes of triangles, quadrilaterals, polygons, cubes, and right prisms.
  • Define diameter, radius, circumference, area of a circle, and formula.
  • Identify and label parts of a circle.
  • Recognize the attributes of a circle.
  • Apply the formula of area and circumference to real-world mathematical situations.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.30 Use the formulas for perimeter, area, and volume to solve real-world and mathematical problems (where volume problems are limited to finding the volume of cylinders and rectangular prisms).

Tags: cones, cylinders, formula, Pythagorean Theorem, right triangles, spheres, threedimensions, twodimensions
License Type: Custom Permission Type
See Terms: https://www.engageny.org/terms-of-use
For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses
AccessibilityText Resources: Content is organized under headings and subheadings
Comments

There are five lessons in this topic.

This resource is free for teachers to access and use. All resources required for the lessons are available to print from the site. 

  This resource provided by:  
Author: Hannah Bradley
This site may contain links to other websites or content belonging to or originating from third parties or links to websites and features in banners or other advertising. Such external links are not investigated, monitored, or checked for accuracy, adequacy, validity, reliability, availability, or completeness by us. WE DO NOT WARRANT, ENDORSE, GUARANTEE, OR ASSUME RESPONSIBILITY FOR THE ACCURACY OR RELIABILITY OF ANY INFORMATION OFFERED BY THIRD-PARTY WEBSITES LINKED THROUGH THIS SITE.
Copyright © 2017, Alabama Learning Exchange
Alabama State Department of Education
Alabama State Department of Education