Student Actions

Teacher Actions

• Develop Mathematical Reasoning by identifying definitions, postulates, and theorems appropriately when assessing diagrams or mathematical relationships and using these items when constructing mathematical arguments and proofs.

• Develop the ability to communicate mathematically when using appropriate geometric terminology to explain the reasoning of ideas in their own words in multiple representations (pictorially, written, and verbal) 

• Pose purposeful questions that allow students to explain their thought processes when describing definitions and proving theorems.

• Elicit and use evidence of student thinking when students present proofs of the same argument that are correct but show different methods of proving the same given statement. 

• Facilitate meaningful discourse by allowing students to critique others’ thinking and discuss that there are alternate ways to logically reach a desired conclusion of an argument. 

Key Understandings

Misconceptions 

• The undefined terms in geometry are point, line, and plane. These terms are the building blocks of other geometrical ideas. Points show location. Lines are a collection of points extending infinitely in two directions. Planes are flat, two-dimensional surfaces that extend infinitely.

• A plane contains an infinite number of lines, thus an infinite number of points.

• Proofs are written in various forms and are used to justify a mathematical argument, such as a two-column or paragraph. 

• A two-column proof lists statements in the left column and the corresponding reasons (either definitions, previously proved theorems, or postulates) in the right column. The given information is listed at the top and the statements flow in logical order. 

• A paragraph proof seems less formal but explains in logical order the statements and reasons leading to the conclusion of a mathematical argument. 

• Students believe definitions need to be proven.

• Students confuse the given statement with what they are to prove. 

• Students do not understand subtle differences in some definitions, for example, the difference between a segment and a line, or a line and a ray.

• Students think there is only one correct way to organize statements and reasons when writing a  proof.

• Students believe every proof must be two-column proofs. 

  Knowledge Connections

Prior Knowledge

Leads to 

• Justify steps in generating equivalent expressions by combining like terms and using order of operations (to include grouping symbols). Identify the properties used, including the properties of operations (associative, commutative, and distributive). (PA.A.3.2)

• Solve mathematical problems using linear equations with one variable where there could be one, infinitely many, or no solutions. Represent situations using linear equations and interpret solutions in the original context.(PA.A.4.1)

• Analyze and interpret associations between graphical representations and written scenarios. (A1.A.4.5) 

• Evaluate reports by making inferences, justifying conclusions, and determining appropriateness of data collection methods. Show how graphs and data can be distorted to support different points of view. (A2.D.2.1)

• Identify and explain misleading conclusions and graphical representations of data sets. (A2.D.2.2)