Curriculum / Math / 8th Grade / Unit 7: Pythagorean Theorem and Volume / Lesson 16
Math
Unit 7
8th Grade
Lesson 16 of 16
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Lesson Notes
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Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.
The core standards covered in this lesson
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.
The foundational standards covered in this lesson
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.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Look at the four figures shown below.
a. What do you notice about each figure? What do you wonder about each figure?
b. What measurements would you need in each figure to determine its volume?
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Option #1: Glasses
The diagram shows three glasses (not drawn to scale). The measurements are all in centimeters.
Glasses from the Summative Tasks is made available through the Mathematics Assessment Project under the CC BY-NC-ND 3.0 license. Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham. Accessed March 20, 2017, 3:12 p.m..
Option #2: Shipping Rolled Oats
Rolled oats (dry oatmeal) come in cylindrical containers with a diameter of $$5$$ inches and a height of $${9 {1\over2}}$$ inches. These containers are shipped to grocery stores in boxes. Each shipping box contains six rolled oats containers. The shipping company is trying to figure out the dimensions of the box for shipping the rolled oats containers that will use the least amount of cardboard. They are only considering boxes that are rectangular prisms so that they are easy to stack.
a. What is the surface area of the box needed to ship these containers to the grocery store that uses the least amount of cardboard?
b. What is the volume of this box?
Shipping Rolled Oats, accessed on June 8, 2017, 2:01 p.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.
Option #3: Comparing Snow Cones
Pablo's Icy Treat Stand sells homemade frozen juice treats as well as snow cones. Originally, Pablo used paper cone cups with a diameter of 3.5 inches and a height of 4 inches.
His supply store stopped carrying these paper cones, so he had to start using more standard paper cups. These are truncated cones (cones with the "pointy end" sliced off) with a top diameter of 3.5 inches, a bottom diameter of 2.5 inches, and a height of 4 inches.
Because some customers said they missed the old cones, Pablo put a sign up saying, "The new cups hold 50% more!" His daughter Letitia wonders if her father's sign is correct. Help her find out.
Comparing Snow Cones, accessed on June 2, 2017, 9:38 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.
Option #4: Flower Vases
My sister’s birthday is in a few weeks and I would like to buy her a new vase to keep fresh flowers in her house. She often forgets to water her flowers and needs a vase that holds a lot of water. In a catalog, there are three vases available and I want to purchase the one that holds the most water. The first vase is a cylinder with diameter 10cm and height 40cm. The second vase is a cone with base diameter 16cm and height 45cm. The third vase is a sphere with diameter 18cm.
Flower Vases, accessed on April 7, 2018, 9:16 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.
Option #5: Tennis & Golf Balls
Tennis balls are packaged into cylindrical cans to ship and sell. Three tennis balls fit into one cylinder so that the height of the three balls is equal to the height of the cylinder, and the diameter of one ball is equal to the diameter of the cylinder.
Golf balls are sometimes packaged into square prisms to ship and sell. Three golf balls fit into one square prism, similar to how the tennis balls fit into the cylinder.
Choose a problem that you did not solve already and review the work of your peers on their poster.
a. Briefly describe their approach to the problem.
b. Would you have taken a similar approach? Why or why not?
c. Do you agree with their solution?
A set of suggested resources or problem types that teachers can turn into a problem set
15-20 minutes
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
5-10 minutes
At the beginning of the summer, Alessandra bought a new piece of chalk, shown below. She used it for several weeks until it was almost gone. What was left of the chalk is also shown below. All measurements are given in inches, and the figures are not drawn to scale.
How much chalk did Alessandra use? Give your answer in cubic inches.
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.
Topic A: Irrational Numbers and Square Roots
Define, evaluate, and estimate square roots.
Standards
8.EE.A.2
Understand that some numbers, including $${\sqrt{2}}$$, are irrational. Approximate the value of irrational numbers.
8.NS.A.18.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.28.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.
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