9.3.2A Logical Arguments
Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.
Overview
Standard 9.3.2 Essential Understandings
Examples suggest, whispering, "It might be true." One counterexample, however, thunders, "It is false." (Attributed to Polya.)
Using logic to construct and refute arguments is the essence of proof and distinguishes formal high school geometry from previous geometric learning. Students should continue to explore proposed conjectures using dynamic geometry software, but then use formal arguments to move from proposed conjectures to proved theorems. The structure of geometric proof helps the students to organize their thinking and make connections between ideas. "Students should see the power of deductive proof in establishing the validity of general results from given conditions. The focus should be on producing logical arguments and presenting them effectively with careful explanation of the reasoning, rather than on the form of proof used (e.g., paragraph proof or two-column proof)." (NCTM, PSSM, p. 310) Students need to center on creating, evaluating and communicating accurate mathematical arguments in a valid logical progression.
All Standard Benchmarks
9.3.2.1 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.
9.3.2.2 Accurately interpret and use words and phrases such as "if...then," "if and only if," "all," and "not." Recognize the logical relationships between an "if...then" statement and its inverse, converse and contrapositive.
9.3.2.3 Assess the validity of a logical argument and give counterexamples to disprove a statement.
9.3.2.4 Construct logical arguments and write proofs of theorems and other results in geometry, including proofs by contradiction. Express proofs in a form that clearly justifies the reasoning, such as two-column proofs, paragraph proofs, flow charts or illustrations.
9.3.2.5 Use technology tools to examine theorems, make and test conjectures, perform constructions and develop mathematical reasoning skills in multi-step problems. The tools may include compass and straight edge, dynamic geometry software, design software or Internet applets.
Benchmark Group A
9.3.2.1 Understand the roles of axioms, definitions, undefined terms and theorems in logical arguments.
What students should know and be able to do [at a mastery level] related to this benchmark:
- Use formal definitions to classify figures and their attributes.
- Explain the difference between an axiom and a theorem.
- Know that axioms and definitions are used to prove theorems.
- Determine if an axiom, definition or theorem applies to a given situation.
Work from previous grades that supports this new learning:
- Students have used definitions to classify shapes and look for patterns.
- Students have made conjectures and developed informal arguments related to shape and space.
- Students have used theorems and properties to solve problems.
NCTM Standards: Geometry
Instructional programs from pre-kindergarten through grade 12 should enable all students to:
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
In grades 9-12 all students should-
● analyze properties and determine attributes of two- and three-dimensional objects;
● explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
● establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
Use visualization, spatial reasoning, and geometric modeling to solve problems
In grades 9-12 all students should-
● draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;
● visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections;
● use geometric models to gain insights into, and answer questions in, other areas of mathematics;
● use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Common Core State Standards (CCSS)
HS.G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
HS.G-CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
HS.G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
HS.G-CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
HS.G-CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
HS.G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
HS.G-SRT.6 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
HS.G-C.3 Construct the inscribed and circumscribed circles of a triangle, and proveproperties of angles for a quadrilateral inscribed in a circle.
Misconceptions
Student Misconceptions and Common Errors
- Students believe that every theorem is biconditional.
- Students believe definitions need to be proved.
- Students confuse properties of shapes with their formal definitions.
Vignette
In the Classroom
Ms. Xiong's Geometry class is starting a unit on triangles. She knows that one major struggle in geometry is learning all of the associated vocabulary. She decides to begin the unit using vocabulary strategies with familiar terms as a foundation for future work.
Teacher: We are going to start by using the KHSN method to mark a list of words. Who can remind us about KHSN?
Student: It's how well we already know vocabulary words. We mark up a list of words to show our level of understanding.
T: Yes, the K represents "Know" for a term that you are confident about, H represents "Hunch" for a term that you think you know but aren't certain, S represents "Seen" for a term that you have read or heard before, and N represents "New" for a completely unfamiliar term.
Ms. Xiong gives each student a copy of the following list of terms: acute triangle, equiangular triangle, equilateral triangle, isosceles triangle, obtuse triangle, right triangle, scalene triangle.
T: Take the next 2 minutes and use KHSN to mark these terms.
Ms. Xiong circulates through the room as students mark their lists. She notices that all the terms except equiangular and equilateral are being marked as K or H by nearly everyone. After two minutes, she asks the students to discuss the words with their partners and continues to circulate. After five minutes, she brings the whole group back together.
T: What did you notice about these words?
S: They are all about the sides and angles of triangles.
T: Yes. Which were the most familiar to you?
S: Right, acute, and obtuse are all about how big the angles are. Equilateral, isosceles and scalene are about how many of the side lengths are repeated. In equilateral, all the sides are the same, in isosceles two of the sides are the same and in scalene none of the sides are the same.
T: Does isosceles have to have exactly two equal sides?
S: I'm not sure.
T: OK, that's something we'll look at later. I noticed that none of you mentioned equiangular. Let's see if we can determine what it means. (Ms. Xiong writes both equiangular and equilateral on the board.) What do you notice about both of these words?
S: They both start with "equi-."
T: Is this a prefix that you are familiar with?
S: Yes, it means equal.
T: What about the rest of each word?
S: I suppose that "-angular" means angles. A lateral pass in football is off to the side, so "-lateral" means sides. Equilateral is all equal sides, so equiangular must mean all equal angles.
T: Nice use of your knowledge of root words. Now go to page 217 and compare your definitions with the ones in the book.
S: Our book says that isosceles have at least two equal sides.
T: This may be different than what you used before. The most descriptive term for a triangle with three equal sides is equilateral, so that is the best term to use when you know all three sides are equal. We call a triangle isosceles when we know for sure that two sides are equal, but this still allows the possibility that all three sides could be equal. It's important to check the precise definitions with your book to make sure that everyone is using the same words to mean the same thing. Now that we have previewed these terms, it's time to make our four-square cards.
Ms. Xiong has been using four-square cards to create vocabulary study cards. The plan is for every student to have his or her own set of index cards to review terms during the year. She varies the exact instructions for each vocabulary set to best fit the terms. For today's terms she has the students put each term in the center oval and then labels the four boxes as: sides, angles, drawing, and real world example.
Sample Vocabulary Card:
T: For each of our terms, create a four-square card. Write something that is true about its sides and something that is true about its angles in your own words and label which is the definition of the term. Remember that your drawing needs to be accurate, so use a protractor and a ruler. Work with your partner to think of examples. Your book homework tonight uses all of these vocabulary terms.
Ms. Xiong circulates through the classroom and checks in with each pair to ensure that everyone understands the directions and that misconceptions have been resolved.
Resources
Teacher Notes
- Teachers may model the difference between "if...then" and "if and only if" by finding counterexamples for the converses of "if...then" statements.
- Teachers should model that definitions are equivalence statements and use vocabulary strategies to develop them.
- Teachers need to make the distinction between definitions and theorems clear. The definition is the starting point determining what our term identifies; a theorem is a logical conclusion based on this definition. Teachers also need to distinguish between formal definitions and naive understandings about shapes.
Cartoon depictions of the differences between undefined terms, axioms, and theorems
Additional Instructional Resources
For more information on Marzano's vocabulary strategies
Dan Meyer's blogsite includes his version of a complete Geometry course. Many of the activities can be incorporated for use with any textbook.
conjecture: an unproven statement that is based on observations
theorem: a generalization that can be proved
axiom: a statement that is accepted as true without proof (aka postulate)
Reflection - Critical questions regarding teaching and learning of these benchmarks:
- What other instructional strategies can I use to engage my students with the roles of axioms, definitions, undefined terms and theorems in logical arguments?
- How do I scaffold my instruction for my students?
- What additional scaffolding do I need to provide ELL students?
- Do the tasks I've designed connect to underlying concepts or focus on memorization?
- How can I tell if students have reached this learning goal?
- How do I differentiate this lesson?
Kimberling, C. (2003). Geometry in action. Emeryville, CA: Key College Publishing.
Marzano, R.J. & Pickering, D.J. (2005). Building academic vocabulary. Alexandria, Va: ASCD.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, Va: NCTM.
Assessment
Assessment
Teacher Note: This benchmark is best assessed within the context of proof-writing (see 9.3.2.2 and 9.3.2.4). One way to assess understanding of individual definitions and theorems is via sketches and constructions. Students may be asked to sketch a figure that fits multiple definitions, fits one definition but not another related term, or a figure that is a counterexample to the converse of a theorem.
Sketch (or construct) a figure that fits the following conditions:
1. A pair or supplementary angles that is not a linear pair
2. A congruent linear pair
3. A pair of corresponding angles that are not congruent
4. A pair of consecutive interior angles (same-side interior) that are congruent
5. Non-congruent triangles each with an angle of 30°, one side of length 7, and one side of length 4.
Answers:
1.
2.
3.
4.
5.
Differentiation
Provide students with a pre-printed blank four-square cards containing each term and space to fill in their own words and examples.
Provide students with a packet of axioms and theorems to use during tests and quizzes.
English Language Learners
Talk to the Language Arts department about the vocabulary strategies developed in English classes. Using the same strategies in multiple classes helps ELL students make vocabulary connections.
Compare and contrast Euclidean geometry with taxi-cab and/or spherical geometry. How do we define line segments, angles, triangles, circles, etc. with a different world view? Students can determine how to measure length, angles and areas using one of these systems. How do our familiar theorems change? For example, taxi-cab circles resemble squares and spherical triangles may have more than one right angle.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: (descriptive list) | Teachers are: (descriptive list) |
working with their partner to create four-square cards | guiding each pair to complete their four-square cards |
making sketches of figures that do and do not fit the descriptions | checking student sketches for accuracy and giving suggestions when needed |
using rulers and protractors to create accurate diagrams | ensuring that students are using their geometry tools correctly |
discussing their work with their partners | checking in with pairs that include ell students to ensure that the terms are clear |
Parent Resources
- Sophia and Khan Academy are websites with uploaded lessons teaching multiple topics. Most of these lessons were developed by teachers and reviewed.
- Teacher Tube and You Tube include multiple uploaded lessons on most school topics.
- Many textbook publishers have websites with additional resources and tutorials. Check your child's textbook for a weblink.