Judge the validity of arguments and give counterexamples to disprove statements.


Clarification 1: Within the Geometry course, instruction focuses on the connection to proofs within the course.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Logic and Discrete Theory
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

Terms from the K-12 Glossary


Vertical Alignment

Previous Benchmarks

Next Benchmarks

Purpose and Instructional Strategies

In grades 7 and 8, students explored the reasons why some geometric statements concerning angles and polygons are true or false. In Geometry, students learn and construct proofs for many of the geometric facts that they encounter and they learn to use counterexamples to check the validity of arguments and statements. The content of this benchmark is to be used throughout this course. In later courses, students continue to learn to judge the validity of many different arguments and statements. 
  • Instruction includes the understanding that a valid argument can be a statement or sequence of statements supported by valid reasons, a part of a proof or an entire proof. 
  • One way to show that an argument is not valid is to provide at least one counterexample to at least one statement in the argument. 
    • For example, if the argument is “All rectangles have opposite sides parallel; therefore, given a quadrilateral is not a rectangle, the quadrilateral does not have opposite sides parallel,” then a student can provide a parallelogram as a counterexample to show that concluding statement of the argument is not valid.

Common Misconceptions or Errors

  • Students may think a statement is true because they cannot think of a counter example.

Instructional Tasks

Instructional Task 1 (MTR.3.1
  • Part A. Which of the following statements are true? 
    • If a quadrilateral is a square, then it is a rectangle. 
    • All trapezoids are parallelograms. 
    • Any quadrilateral can be inscribed in a circle.
  • Part B. Provide counterexamples to prove the invalid statements from Part A are not true.

Instructional Items

Instructional Item 1 
  • Puaglo said the following statements are true. Select all the statements that are false. 
    • a. All quadrilaterals have four right angles. 
    • b. A triangle is a polygon with three sides. 
    • c. All circles are similar. 
    • d. All equiangular quadrilaterals are congruent. 
    • e. A trapezoid must have at least one obtuse angle.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912070: Access Mathematics for Liberal Arts (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 - 2023, 2023 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))
1207350: Mathematics for College Liberal Arts (Specifically in versions: 2022 and beyond (current))
1212300: Discrete Mathematics Honors (Specifically in versions: 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.LT.4.AP.10: Select the validity of an argument or give counterexamples to disprove statements.

Related Resources

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Lesson Plan

Musical Chairs with Words and a Ball:

This lesson introduces students to concepts and skills that they will use throughout the year. Students will learn that the terms point, and line are considered "undefined." Students will play musical chairs while learning to develop precise definitions of circle, angle, parallel line, and perpendicular line, using counterexamples at different classroom stations. Students will identify models, use notation, and make sketches of these terms.

Type: Lesson Plan

Student Resources

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Parent Resources

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