PER - PIN - TIC - YOU - LER!(Part 2 to Take the Stairs Lesson)

Learning Resource Type

Lesson Plan

Subject Area



9, 10, 11, 12


This lesson allows students to investigate the slope criteria and characteristics of perpendicular lines using graphing calculators and rectangle/square tiles.  Students will also use equations and graphs.  Students will work cooperatively to develop and justify ideas/conjectures.

This is a College- and Career-Ready Standards showcase lesson plan.

Mathematics (2019) Grade(s): 09-12 - Geometry with Data Analysis


Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.



  • Parallel lines
  • Perpendicular lines
  • Slope
  • Slope triangle


Students know:
  • Techniques to find the slope of a line.
  • Key features needed to solve geometric problems.
  • Techniques for presenting a proof of geometric theorems.


Students are able to:
  • Explain and justify conclusions reached regarding the slopes of parallel and perpendicular lines.
  • Apply slope criteria for parallel and perpendicular lines to accurately find the solutions of geometric problems and justify the solutions.
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.


Students understand that:
  • Relationships exist between the slope of a line and any line parallel or perpendicular to that line.
  • Slope criteria for parallel and perpendicular lines may be useful in solving geometric problems.

Primary Learning Objectives

Students will discover and use the properties of perpendicular lines.  Students will find equations of perpendicular lines given other equations of lines passing through other points.   Students will use slope definitions with rise over run and slope/distance formulas to solve problems and justify reasoning for developed discoveries.

Additional Learning Objective(s)

Students will work cooperatively to discover patterns in the graph table of values while problem solving. Students will verify conclusions by finding slope/distance and understanding properties of perpendicular lines.  Students will measure and document lengths of the L shape of a rectangle/square traced on a coordinate grid or a triangle verified with pythagorean triples verified and compare values to determine perpendicular slopes.



As students enter the classroom, they should receive a group that tells where to sit with all needed materials available.  Students should be informed that their mission today is to determine whether a pair of equations of lines is perpendicular. 


1.  Student groups should trace the rectangular/square tile/object on a coordinate plane.  Each group can decide to use a L shape from the graph to receive coordinate pairs to input into graphing calculator to generate equations of lines and graphs. Each group can also decide to use a triangle from the traced shape, use distance formula to calculate length values and then verify values as a Pythagorean triple.

2.  The students should key their information into their graphing calculators (using the list key on the graphing calculator) to determine if the L shape of the traced object makes perpendicular lines and have perpendicular slopes. They should look for patterns, develop equations based on data and list slope conclusions based on data.  The groups must agree on findings and be able to justify. 

3.  If students do not understand perpendicular lines and slopes, provide necessary feedback to move them forward with their thinking.  Refer to : (if your school system does not allow youtube access, you may download the video using

Refer to


1.  The teacher should be sure that students are using slope/distance or pythagorean theorem correctly to verify perpendicular lines. 

2.  Groups will discuss properties discovered and begin justifying their ideas; along with creating equations and graphs to match their tables in the calculator.

3.  Allow students time to prepare attractive posters to share with class to begin discussions about perpendicular, along with having the groups share their findings using the Smartview function of the calculator.


Assessment Strategies


Use the rubric provided to assess student work individually and as a group. Students should justify all reasoning stated on poster.  The Assessment Process is on-going. The teacher should visit each group and ask questions to make sure students apply the properties of perpendicular line definitions.  Post the posters in the classroom and have students present conclusions.  As students present posters and graphing calculator finds, they should compare posters for other information not listed on their group's poster.


Extensions can be to include formal proofs in geometry or have students justify the properties of perpendicular lines using more coordinate  and two column proofs.

  Refer to:   Groups can be random or you can select groups to allow struggling students to find slopes using formulas or inputing information into calculator for graphs.

Total Duration

Greater than 120 Minutes



  • Have rectangle/square tiles or shaped objects available for students.
  • Have graphing calculator ready to be assigned.


  • Geometry students should be able to graph coordinates on Cartesian plane and measure the lengths of models with ruler. 
  • Students should be able to find slope algebraically and graphically, use slope intercept form to solve problems, know/understand the pythgorean theroem/distance formula and have some knowledge of using the graphing calculators.

Materials and Resources

Poster Paper or Poster with grid lines, sticky notes of different types, straight edge, markers, rectangle/square tiles or rectangular/square objects, student may use their own devices or teacher issued graphing calculators.

Technology Resources Needed

Computer with Internet access, LCD Projector and Document camera to share student work (helpful but not necessary.) Smartview with calculator is needed.