EngageNY
The Volume Formula of a Sphere
What is the relationship between a hemisphere, a cone, and a cylinder? Using Cavalieri's Principle, the class determines that the sum of the volume of a hemisphere and a cone with the same radius and height equals the volume of a...
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The Volume Formula of a Pyramid and Cone
Our teacher told us the formula had one-third, but why? Using manipulatives, classmates try to explain the volume formula for a pyramid. After constructing a cube with six congruent pyramids, pupils use scaling principles from previous...
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Scaling Principle for Volumes
Review the principles of scaling areas and draws a comparison to scaling volumes with a third dimensional measurement. The exercises continue with what happens to the volume if the dimensions are not multiplied by the same constant.
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Definition and Properties of Volume
Lead a discussion on the similarities between the properties of area and the properties of volume. Using upper and lower approximations, pupils arrive at the formula for the volume of a general cylinder.
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General Pyramids and Cones and Their Cross-Sections
Are pyramids and cones similar in definition to prisms and cylinders? By examining the definitions, pupils determine that pyramids and cones are subsets of general cones. Working in groups, they continue to investigate the relationships...
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Three-Dimensional Space
How do 2-D properties relate in 3-D? Lead the class in a discussion on how to draw and see relationships of lines and planes in three dimensions. The ability to see these relationships is critical to the further study of volume and other...
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Proving the Area of a Disk
Using a similar process from the first lesson in the series of finding area approximations, a measurement resource develops the proof of the area of a circle. The problem set contains a derivation of the proof of the circumference formula.
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Properties of Area
What properties does area possess? Solidify the area properties that pupils learned in previous years. Groups investigate the five properties using four problems, which then provide the basis for a class discussion.
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What Is Area?
What if I can no longer justify area by counting squares? Lead a class discussion to find the area of a rectangular region with irrational side lengths. The class continues on with the idea of lower approximations and upper...
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End-of-Module Assessment Task - Geometry (module 2)
Increase the level of assessment rigor with the test of performance tasks. Topics include similar triangles, trigonometric ratios, Law of Sines, Law of Cosines, and trigonometric problem solving.
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Mid-Module Assessment Task - Geometry (Module 2)
Challenge: create an assessment that features higher level thinking from beginning to end. A ready-made test assesses knowledge of dilations using performance tasks. Every question requires a developed written response.
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How Do 3D Printers Work?
If we stack up all the cross sections of a figure, does it create the figure? Pupils make the connection between the complete set of cross sections and the solid. They then view videos in order to see how 3D printers use Cavalerie's...
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The Volume of Prisms and Cylinders and Cavalieri’s Principle
Young mathematicians examine area of different figures with the same cross-sectional lengths and work up to volumes of 3D figures with the same cross-sectional areas. The instruction and the exercises stress that the two figures do not...
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General Prisms and Cylinders and Their Cross-Sections
So a cylinder does not have to look like a can? By expanding upon the precise definition of a rectangular prism, the instructional activity develops the definition of a general cylinder. Scholars continue on to develop a graphical...
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The Scaling Principle for Area
As they investigate scaling figures and calculate the resulting areas, groups determine the area of similar figures. They continue to investigate the results when the vertical and horizontal scales are not equal.
West Contra Costa Unified School District
Key Features of Graphs
The key is ... After a day of instruction on key features of graphs, groups create a poster and presentation on the key features of their given function graph. The resource provides an extension activity of "telephone" using graphs.
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End-of-Module Assessment Task - Algebra 1 (Module 5)
This unit assessment covers the modeling process with linear, quadratic, exponential, and absolute value functions. The modeling is represented as verbal descriptions, tables, graphs, and algebraic expressions.
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Modeling a Context from a Verbal Description (part 2)
I got a different answer, are they both correct? While working through modeling problems interpreting graphs, the question of precision is brought into the discussion. Problems are presented in which a precise answer is needed and others...
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Modeling a Context from a Verbal Description (part 1)
When complicated algebraic expressions are involved, it is sometimes easier to use a table or graph to model a context. The exercises in this lesson are designed for business applications and require complex algebraic expressions.
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Modeling a Context from Data (part 2)
Forgive me, I regress. Building upon previous modeling activities, the class examines models using the regression function on a graphing calculator. They use the modeling process to interpret the context and to make predictions based...
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Modeling from a Sequence
Building upon previous knowledge of sequences, collaborative pairs analyze sequences to determine the type and to make predictions of future terms. The exercises build through arithmetic and geometric sequences before introducing...
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Analyzing a Data Set
Through discussions and journaling, classmates determine methods to associate types of functions with data presented in a table. Small groups then work with examples and exercises to refine their methods and find functions that work to...
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Analyzing a Verbal Description
What function will describe the insect population growth? Pairs or small groups work together to determine which type of function and specific function will model given scenarios. The scenarios differentiate between linear, exponential...
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Analyzing a Graph
Collaborative groups utilize their knowledge of parent functions and transformations to determine the equations associated with graphs. The graph is then related to the scenario it represents.