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Students study the eUect of dilation on area, surface area, and volume. They establish that dilating by a scale factor of multiplies all lengths by , all areas by , and all volumes
by . They do this by observing the eUect of dilation on rectangles and rectangular prisms, and then generalizing to all areas and volumes by covering arbitrary areas with rectangles and Rlling arbitrary volumes with rectangular prisms. Students then solve problems that start with a given area or volume and ask what scale factor was used to produce it. This brings forth the need to introduce square roots, cube roots, and their graphs.
This unit then builds on students' prior knowledge about volumes of prisms, and introduces a technique of comparing prisms by analyzing their corresponding cross sections that are formed when they are intersected by a plane parallel to their bases. This establishes that the volume of a prism with base area square units and height units
is cubic units, regardless of the particular shape of the base and regardless of whether the prism is oblique.
Students derive the volume of a square pyramid with base area square units and height units as cubic units by Rrst noting that a unit cube can be divided into three
congruent square pyramids and then using informal arguments involving cross sections and limits. This formula is then extended using Cavalieri's Principle to all pyramids with the same base area and height, regardless of the particular shape of the base and regardless of whether the pyramid is oblique. This derivation relies on the fact that any cross section of a pyramid that is parallel to its base is the image of the base when dilated from the apex. Cavalieri's Principle applies because corresponding cross sections of any two pyramids with the same base area and height each have the same area, which is the base area multiplied by the square of the scale factor used to produce each cross section.
In the Rnal few lessons, students apply what they know about volume to investigate problems in context. These problems include considerations of density, surface area to volume ratio, and graphing cube root and square root functions.
G6 Coordinate Geometry
Prior to beginning this unit students will have spent most of the course studying geometric Rgures which are not described by coordinates. However, students have seen Rgures on the grid (notably transformations in grade 8) as well as lines and curves on the coordinate plane (in previous courses). This unit brings together students’ experience from previous years with their new understanding from this course for an in-depth study of coordinate geometry.
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Course Guide