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# Kendall Hunt Geometry Answer Key

## Unit : Conditional Probability

Kendall Hunt, Illustrative Mathematics K-5 Curriculum, Beta Pilot Overview

In grade 7, students learned about probability by conducting chance experiments. Along with looking at experimental data, students created and analyzed sample spaces for situations. In this unit, students extend that knowledge by considering situations with two events, for example: roll a number cube and flip a coin. Students find probabilities when events are combined in various ways including both occurring, at least one occurring, and one event happening under the condition that the other happens as well.

The unit begins with students creating different models for understanding sample spaces and probability. The models include tables, trees, lists, and Venn diagrams. Venn diagrams allow students to visualize various subsets of the sample space such as A and B, A or B, or not A. Tables help students determine the probability of those subsets occurring, and support students understanding of the Addition Rule, \ = P + P – P\).

Conditional probability is discussed and applied using several games and connections to everyday situations. In particular, the Multiplication Rule \ = P \boldcdot P\) is used to determine conditional probabilities. Conditional probability leads to the definition of independence of events. Students describe independence using everyday language and use the equation \ = P\) when events A and B are independent.

## Unit : Constructions And Rigid Transformations

In grade 8, students determine the angle-preserving and length-preserving properties of rigid transformations experimentally, mostly with the help of a coordinate grid. Students have previously studied angle properties, including the Triangle Angle Sum Theorem, but no formal proofs have been required. In this unit, students create;rigid motions using construction tools with no coordinate grid. This;leads to more rigorous definitions of rotations, reflections, and translations. Students begin to explain and prove angle relationships like the Triangle Angle Sum Theorem using these rigorous definitions and a few assertions.

In previous courses, students developed their understanding of the concept of functions. In this unit, the concept of a transformation is made somewhat more formal using the language of functions. While students do not use function notation, they do move away from describing transformations as moves that act on figures and towards describing them as taking points in the plane as inputs and producing points in the plane as outputs.

A blank reference chart is provided for students, and a completed reference chart for teachers.;The purpose of the reference chart is to be a resource for students to reference as they make formal arguments. Students will continue adding to it throughout the course. Refer to About These Materials;in the Geometry course for more information.

## Geo1 Constructions And Rigid Transformations

In this unit, students first informally explore;geometric properties using straightedge and compass constructions. This allows them to build conjectures and observations before formally defining rotations, reflections, and translations. In middle school, students studied;transformations of figures in the coordinate plane. In this unit, they transition to more formal definitions;that don’t rely on the coordinate plane, and the focus shifts from transforming whole figures towards a more point-by-point analysis. Students then begin to use the rigorous definitions they developed to prove statements involving angles and distances, preparing them for congruence proofs in the next unit.

A blank reference chart is provided for students, and a completed reference chart for teachers.;The purpose of the reference chart is to be a resource for students to reference as they make formal arguments. Students will continue adding to it throughout the course. Refer to About These Materials;in the Geometry course for more information.

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## Unit : Solid Geometry

In previous grades, students solved problems involving area, surface area, and volume for various solids. In grade 6, students worked with areas of triangles and quadrilaterals, as well as surface areas and volumes of right rectangular prisms including those with fractional edge lengths. In grade 7, students found areas of circles, solved problems involving the volume and surface area of right prisms, and described plane sections of three-dimensional figures. In grade 8, students solved problems involving volumes of spheres, cones, and cylinders using given volume formulas.

In this unit, students practice spatial visualization in three dimensions, study the effect of dilation on area and volume, derive volume formulas using dissection arguments and Cavalieris Principle, and apply volume formulas to solve problems involving surface area to volume ratios, density, cube roots, and square roots.

Then, students extend their study of scaling to solids. They conclude that dilating a solid by a scale factor of \ multiplies all lengths by \, surface areas by \, and volumes by \. They work backwards from a scaled volume or surface area to find the scale factor involved, requiring the introduction of cube roots. Students create a graph representing \ and use it to answer questions about how changes in volume affect changes in the corresponding scale factor.

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Lesson 45 uses another transformation reflection to examine the square root family with parent function y x. In middle school students studied transformations of figures. Proving Geometric Theorems Algebraically.

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## Kendall Hunt Beyond Words

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## Proving Geometric Theorems Algebraically

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## Unit : Coordinate Geometry

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Prior to beginning this unit, students will have spent most of the course studying geometric figures not described by coordinates. However, students have seen figures on the grid as well as lines and curves on the coordinate plane . This unit brings together students experience from previous years with their new understanding from this course for an in-depth study of coordinate geometry.

The first few lessons examine transformations in the plane. Students encounter a new coordinate transformation notation which connects transformations to functions. Students transform figures using rules such as \ \rightarrow \);and connect the geometric definitions of reflections and dilations to coordinate rules that produce them. They prove objects similar or congruent using reasoning, including distance , angle , and definitions of transformations.

The next set of lessons focuses on building equations from definitions. Students examine circles and parabolas through the lens of distance. A circle is the set of points the same distance from a given center, and a parabola is the set of points equidistant from a given point and line . Based on these definitions, students develop a general equation for a circle, and they write equations that represent specific parabolas.

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## Unit : Right Triangle Trigonometry

Prior to beginning this unit students will have considerable familiarity with right triangles. They learned to identify right triangles in grade 4. Students studied the Pythagorean Theorem in grade 8, and used similar right triangles to build the idea of slope. This unit builds on this extensive experience and grounds trigonometric ratios in familiar contexts.

The first few lessons of this unit examine some special cases of similar right triangles to solidify the idea that any right triangles with a single congruent acute angle are similar. Two of these three lessons are optional. While the standards do not specifically call for special right triangles they are an opportunity to practice, build on important ideas, and are frequently included on college entrance exams. From there students generate data for the side length;ratios of many sets of right triangles. This data is organized into a table which students apply to problems. Taking the time to both build and use the table helps students build a solid foundation before they learn the names of trigonometric ratios.;Once students have practiced estimating both side lengths and angle measures using the table, they learn the names cosine, sine, and tangent.

Students will continue to use and add to their reference charts. The completed reference chart for this unit is provided for teacher reference.