Page 48 - IM_FL_Geometry_Print Sample

The Rrst few lessons examine transformations in the plane. Students encounter a new coordinate transformation notation which connects transformations to functions. Students transform Rgures using rules such as . They prove objects similar or congruent using reasoning including distance (via the Pythagorean Theorem), angle (calculated using trigonometry), and deRnitions of transformations.
The next set of lessons focuses on building equations from deRnitions. Students examine circles, perpendicular bisectors, parallel lines, and perpendicular lines to identify the properties necessary to describe the set of points in an equation. In each case students prove the equation they derived will always form that object. Then students are asked to apply these ideas to other proofs such as categorizing quadrilaterals.
At the end of the unit students use weighted averages to partition segments, scale Rgures, and locate the intersection point of the medians of a triangle. In the Rnal lesson all students locate the intersection points of the altitudes of a triangle. Then there are several optional activities which oUer diverging paths towards the Euler line or practicing equations of lines through constructing and describing tessellations.
G7 Circles
In grade 7, students used formulas for circumference and area of circles to solve problems. Earlier in this course, students used straightedges and compasses for constructions, studied similarity and proportional reasoning, and proved theorems about various angle relationships. This unit builds on all of these skills and concepts to investigate the geometry of circles more closely.
In the Rrst set of lessons, students rigorously deRne radian measure and apply proportional reasoning to solve problems involving circular arc length and sector area. Students revisit the proof that all circles are similar and use proportional reasoning to observe several ratios in circles that are invariant under dilation. Students connect angles and circles by observing that the ratio of arc length to radius of a central angle doesn't depend on the size of the circle, and the ratio increases as the angle takes up a larger portion of the circle. This gives a new way to measure an angle: draw a circle of any size at the vertex of the angle and measure the ratio of arc length to radius. This ratio is known as the radian measure of the angle. Observing ratios that are invariant under dilation is precisely the same line of reasoning that led to deRning trigonometric ratios in an earlier unit. Students then solve problems involving arc length and sector area to develop Suency with radian measure. This is important for the transition towards Algebra 2, where they will work with radian measure to deRne trigonometric functions in terms of the unit circle. However, students still have ample opportunity to maintain their Suency with degree measure.
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