ALEX | Alabama Learning Exchange

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

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This lesson provided by:

Author:	Kenneth Webb
System:	Tuscaloosa City
School:	Tuscaloosa City Board Of Education
The event this resource created for:	CCRS

General Lesson Information

Lesson Plan ID:	33163
Title:	PER - PIN - TIC - YOU - LER!(Part 2 to Take the Stairs Lesson)
Overview/Annotation:	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.

Associated Standards and Objectives

Content Standard(s):

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis

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

Example: Find the equation of a line parallel or perpendicular to a given line that passes through a given point.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a line,

Create lines parallel to the given line and compare the slopes of parallel lines by examining the rise/run ratio of each line.
Create lines perpendicular to the given line by rotating the line 90 degrees and compare the slopes by examining the rise/run ratio of each line.
Use understandings of similar triangles and logical reasoning to prove that parallel lines have equal slopes and the slopes of perpendicular lines are negative reciprocals.Given a geometric problem involving parallel or perpendicular lines.
Apply the appropriate slope criteria to solve the problem and justify the solution including finding equations of lines parallel or perpendicular to a given line.

Teacher Vocabulary:

Parallel lines
Perpendicular lines
Slope
Slope triangle

Knowledge:

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.

Skills:

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.

Understanding:

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.

Diverse Learning Needs:

Essential Skills:

Learning Objectives:
GEO.33.1: Define slope, point slope formula, slope-intercept formula, standard form of a line, parallel lines, and perpendicular lines.
GEO.33.2: Demonstrate and explain algebraically how perpendicular lines have only one common point.
GEO.33.3: Demonstrate and explain algebraically how parallel lines have no common points.
GEO.33.4: Write and solve equations of parallel and perpendicular lines.
GEO.33.5: Illustrate graphically how perpendicular lines have only one common point.
GEO.33.6: Illustrate graphically how parallel lines have no common points.
GEO.33.7: Write an equation of a line in slope intercept form.
GEO.33.8: Find the slope of a given line.

Prior Knowledge Skills:

Define slope, intercept, linear, equation, and bivariate.
Recall how to determine the rate of change (slope) from a graph.
Identify the parts of the slope-intercept form of an equation.
Recognize how to read a graph.
Recall how to write an equation in slope-intercept form.
Apply the identification of the slope and the y-intercept to a real-world situation.
Create a graph to model a real-word situation.
Define proportional relationships, unit rate, and slope.
Demonstrate how to graph on a Cartesian plane.
Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
Define linear functions, nonlinear functions, slope, and y-intercept.
Recognize linear equations.
Identify ordered pairs.
Recognize ordered pairs.
Generate the slope of a line using given ordered pairs.
Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
Graph a function given the slope-intercept form of an equation.
Recognize that two sets of points with the same slope may have different y-intercepts.
Graph a linear equation given the slope-intercept form of an equation.

Alabama Alternate Achievement Standards

AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Local/National Standards:

Standard 1: Make sense of problems and persevere in solving them

Standard 2: Reason abstractly and quantitatively

Standard 3: Construct viable arguments and critique the reasoning of others

Standard 4: Model with mathematics

Standard 5: Use appropriate tools strategically

Standard 6: Attend to precision

Standard 7: Look for and make use of structure

Standard 8: Look for and express regularity in repeated reasoning

Primary Learning Objective(s):

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.

Preparation Information

Total Duration:	Greater than 120 Minutes
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.
Background/Preparation:	Teacher: Have rectangle/square tiles or shaped objects available for students. Have graphing calculator ready to be assigned. Students: 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.

Procedures/Activities:

ENGAGE

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.

EXPLORE

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 : http://www.youtube.com/watch?v=VVvErpLG2EQ (if your school system does not allow youtube access, you may download the video using http://www.keepvid.com/)

Refer to https://www.khanacademy.org/math/algebra/two-var-linear-equations/slope/v/slope-of-a-line.

EXPLAIN

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

Assessment Strategies

EVALUATE

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.

Acceleration:

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: https://www.khanacademy.org/math/geometry/congruent-triangles/cong_triangle/v/two-column-proof-showing-segments-are-perpendicular

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.

Intervention:

For extra practice or for students who were absent, have them visit: https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-geometry-topic/cc-4th-classifying-shapes/v/classifying-shapes-examples

View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.

ALEX Lesson Plan

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