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What is reasoning and why is it such an important part of mathematical problem solving?
What is reasoning and why is it such an important part of mathematical problem solving?
Mathematics is learned through questions that arise while solving well-constructed problems. Our students begin with problems, they use strategies to solve the problems, and they learn the necessary mathematics along the way. Many classroom investigations are designed so that students will collaboratively or individually discover the mathematical properties. The properties are then discussed in class, summarized, and become part of the students’ mathematical knowledge to be applied to future problems.
The discovery of the mathematics is an essential part of the development of each student’s confidence as a mathematician. Knowledge that is gained through inquiry is more likely to be remembered for the long term. Teachers and parents work together to promote this discovery of math through investigation, problem solving, and reasoning. Students develop the conceptual understandings embedded within the mathematics. Our goal is for students to be problem solvers and to understand that mathematics makes sense.
Course Outline
Course Outline
Throughout all the units, students will deepen their understanding of algebraic principles and how they help to solve geometric problems.
Throughout all the units, students will deepen their understanding of algebraic principles and how they help to solve geometric problems.
I. Communicating and Reasoning in Geometry
I. Communicating and Reasoning in Geometry
Conceptual Lens: Constructing Viable Arguments
What is the body of mathematical knowledge called Geometry and how do we learn it? The importance of communicating clearly and precisely through diagrams, words, and symbols is explored. Students are introduced to how mathematicians learn and make conjectures (inductive reasoning) and how they prove and communicate thinking (deductive reasoning). Through the course of this unit students develop the understanding that Mathematicians use universally understood language, symbols, and pictures to communicate clearly and use logical reasoning.
Common Core State Standards: HSG-CO.A.1, HSG-CO.D.12, HSG-GPE.B.6, HSG-CO.C.9
II. Lines and Angles
II. Lines and Angles
Conceptual Lens: Parallel Lines
How are lines and angles used to define different quadrilaterals? Building upon the learning from the previous unit, students continue to reason logically to discover more geometry. The relationships between lines and angles, especially those formed by parallel lines, are discovered and then proven using deductive reasoning. Finally, students use algebraic reasoning to determine the types of quadrilateral or parallelograms when they are placed on the coordinate plane. All of these concepts and skills flow logically so that students develop the understanding that the relationships between lines and angles logically leads to new learning about more complex figures such as triangles and parallelograms.
Common Core State Standards: HSG-CO.A.1, HSG-CO.C.9, HSG-CO.D.12, HSG-GPE.B.5, HSG-GPE.B4, HSG-CO.C.11, HSG-CO.D.10, HSG-GPE.B.7
III. Rigid Transformations, Mapping & Congruence
III. Rigid Transformations, Mapping & Congruence
Conceptual Lens: Transformations
How does using mapping and transformations develop the understanding of congruence? Students explore transformations on the coordinate plane and the mapping rules that define them. This leads to an understanding of isometries (rigid motions) and how they can be used to prove congruence of two figures. Students will understand that mathematicians establish triangle congruence criteria based on analyses of rigid motions.
Common Core State Standards: HSG-CO.B.6, HSG-CO.B.7, HSG-CO.B.8, HSG-CO.A.2, HSG-CO.A.3, HSG-CO.A.4, HSG-CO.A.5
IV. Triangles
IV. Triangles
Conceptual Lens: Congruence
How can congruence be used to draw conclusions about triangles? Students will begin by proving that pairs of triangles are congruent, using previous definitions, postulates and theorems as well as methods developed from the understanding of previous theorems and definitions. Students will extend these proofs to include sides and angles by using the fact that corresponding parts of congruent triangles are congruent. Students will continue to extend their knowledge of congruent triangles to other proofs and diagrams and be able to apply reasoning to new constructions. Students will develop an understanding of how proving triangles congruent lead mathematicians to draw conclusions about angles and segments in triangles.
Common Core State Standards: HSG-CO.B.8, HSG-CO.C.9, HSG-CO.C.10, HSG-CO.D.12, HSG-GPE.B.4
V. Parallelograms
V. Parallelograms
Conceptual Lens: Parallelograms
How can we use previous learning to discover properties of quadrilaterals? This unit will begin by exploring the properties of parallelograms. Students will analyze several theorems related to angles, sides and diagonals of parallelograms and apply them in different contexts. Students will deepen their understanding of proof involving triangles and make connections to the theorems of parallelograms. Students will explore other quadrilateral properties to determine if they can be classified as a parallelogram. Students will realize that parallelograms are quadrilaterals with special properties proved through triangle congruence.
Common Core State Standards: HSG-CO.C.11, HSG-GPE.B.7, HSG-GPE.B.4, HSG-SRT.B.5
VI. Non-Rigid Transformations and Similarity
VI. Non-Rigid Transformations and Similarity
Conceptual Lens: Similarity
How does using mapping and transformations develop the understanding of similarity? Building upon the previous unit, students explore other non-rigid transformations. The concept of scale factor (from grade 7) develops into a deeper understanding of dilation and similar shapes. Properties of similar shapes as well as proportional reasoning (also a well-developed concept from grade 7) are used to solve problems. Throughout this unit students will develop the understanding that mathematicians apply proportional reasoning in a geometric setting.
Common Core State Standards: HSG-CO.A.2, HSG-SRT.A.1, HSG-SRT.A.1a, HSG-SRT.A.1b, HSG-SRT.A.2, HSG-SRT.A.3, HSG-SRT.B.4, HSG-SRT.B.5, HSG-GPE.B.6
VII. Right Triangles
VII. Right Triangles
Conceptual Lens: Trigonometry
How do mathematicians use constant relationships and similarity to solve measurement problems? Right triangles were explored, then Pythagorean Theorem was discovered and first used in grade 8 mathematics. This unit builds upon concepts to use triangles to solve more complicated problems. The properties of special right triangles are discovered and also used to solve problems. All in all, the connections between proportional reasoning and similarity build a deeper knowledge base and repertoire of skills and understandings to solve problems. Students will understand that mathematicians can use known relationships between sides and angles of triangles to calculate missing measurements.
Common Core State Standards: HSG-SRT.C.6, HSG-SRT.C.7, HSG-SRT.C.8
VIII. Circles
VIII. Circles
Conceptual Lens: Circles
In what ways is the similarity in circles special? Students begin this unit by discovering the precise definition of a circle by extending their knowledge on what happens to polygons as the number of sides increases. As they develop a deep understanding of the locus of points in a plane equidistant from a given point, students reason logically to discover other relationships between circles, angles and lines of circles. Throughout the unit students will develop the understanding that the similarity of all circles creates consistent proportional relationships.
Common Core State Standards: HSG-CO.A.1, HSG-CO.D.13, HSG-GPE.A.1, HSG-GPE.B.4, HSG-C.A.1, HSG-C.A.2,
HSG-C.A.3, HSG-C.A.4(+), HSG-C.B.5, HSG-GMD.A.1
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