targeted content standard(s): student friendly learning

Grade: _Math 1___ Lesson Title: ___Proving Congruence_____________________________ Unit: __2 (Lesson 4 or 4)_ Time Frame: __135 minutes_______ Essential Question:

What are the roles of hypothesis and conclusion in a proof?

What criteria are necessary in proving a theorem?

Targeted Content Standard(s): Student Friendly Learning Targets 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.

I can…

I can prove that opposite sides are congruent in a parallelogram.

I can prove opposite angles are congruent in a parallelogram.

I can prove that the diagonals of a parallelogram bisect each other.

I can prove that rectangles diagonals are congruent.

Targeted Mathematical Practice(s): 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively 3 Construct viable arguments and critique the reasoning of others 4 Model with mathematics 5 Use appropriate tools strategically 6 Attend to precision 7 Look for and make use of structure 8 Look for an express regularity in repeated reasoning

Supporting Content Standard(s): (optional) G.CO.1 G.CO.2 G.CO.4 G.CO.5

Explanation of Rigor: (Fill in those that are appropriate.) Conceptual: G.CO.11 In the Introductory Straw Activity, students manipulate straws to create parallelograms. They develop a conceptual understanding of the characteristics and qualifications to create a parallelogram.

Procedural: G.CO.11 In the #1 Spot the Parallelogram assignment, students identify which figures are parallelograms based on limited information.

Application: G.CO.11 In the #2 Prove It activity, students are given a real world example of railroad tracks. They use their knowledge of parallel lines to prove that railroad tracks are made up of a series of parallelograms. Students then ponder the affects if tracks were not made up of parallelograms.

Vocabulary: Transformation Rotation Rigid Motion Reflection

Congruence Translation Hypothesis

Dilation Conclusion Point

Proof Line Postulate

Angle Theorem Corresponding

Evidence of Learning (Assessment): Pre-Assessment: 4 Corners Activity, Introductory Straw Activity (Segment 1) Formative Assessment(s): #1 Spot the Parallelogram activity, #2 Prove It! Activity Summative Assessment: #3 G.C0.9-11 End of Unit Assessment




Self-Assessment: 4 Corners Activity




Lesson Procedures: Segment 1

Approximate Time Frame:

45 minutes (one class period)

Lesson Format: Whole Group Small Group Independent

Modeled Guided Collaborative Assessment

Resources:

4 Corners Activity

Introductory Straw Activity

#1 Spot the Parallelogram Assignment

Focus:

Students discover the qualifications and properties of parallelograms as they manipulate straws to create an assortment of examples.

Modalities Represented: Concrete/Manipulative Picture/Graph Table/Chart Symbolic Oral/Written Language Real-Life Situation

Math Practice Look For(s):

MP#2 Reason abstractly and quantitatively. Students will create abstract examples of parallelograms to reason and discover the hidden properties.

MP#3 Construct viable arguments and critique the reasoning of others. Through class discussion, students will critique the reasoning of others and make conjectures based on their discoveries of parallelograms.

MP#7 Look for and make use of structure. Students will make conjectures based off of their examples of parallelograms. They will then examine the examples created by the rest of the class to test their conjectures.

Differentiation for Remediation:

Students may create parallelograms using computer programs, graphing paper, geoboards, etc. if they are having difficulty visualizing with straws.

Differentiation for English Language Learners: Share and/or collaborate with partners. Have visual definitions of parallel lines.

Differentiation for Enrichment: Students can divide their parallelograms into triangles and develop a proof illustrating why the opposite sides, opposite angles, and diagonals of parallelograms are congruent.

Potential Pitfall(s):

Students trying to classify hexagons or other polygons with opposite sides parallel as parallelograms.

Students having difficulties using protractors to measure the angles.

Independent Practice (Homework): #1 Spot the Parallelogram assignment

Steps: 1) As a class, do the 4 Corners Activity to access their

prior knowledge.

2) To introduce the lesson, have students work with a

partner.

Teacher Notes/Reflections:

By working with the straws, students should discover that in order to have a closed, parallel figure the opposite straws must be the same measure in order to connect.




3) Give students the instruction to create at least 3

different parallelograms using their straws. Give

them as little information as possible. They may

cut and/or use as many straws as they want.

a. Helpful Scaffolding: Is it possible to make

a parallelogram with 5 straws? 5 sides?

b. Does the figure need to be a closed

polygon?

c. What sizes of straws do you need? Do all

of your straws have to be the same?

d. Are certain polygons parallelograms?

(rectangles, squares, rhombi)

2) Students draw a diagram on graph paper of each

parallelogram that was created.

3) Have students make observations about the

characteristics of their parallelograms.

a. Make a conjecture about the sides and

angles of all parallelograms.

4) Students measure and label the sides and angles

of their diagrams. Partners make conjectures

about their examples.

5) Have each group draw their labeled diagrams on

the board for class observations.

6) As a class, identify the characteristics of

parallelograms and test their conjectures.

7) Have partners answer the following based off of all

of the created examples:

a. What is true about the opposite sides of a

parallelogram?

b. What is true about the angles of a

parallelogram?

c. What polygons are considered

parallelograms?

8) For homework, assign the #1 Spot the

Parallelogram Activity.






Segment 2


15 minutes



Resources:

#2 Prove It! Activity

Focus:

Students visualize parallelograms in the real world.

Students prove the properties of parallelograms given a real world example.


Math Practice Look For(s): MP#1 Make sense of problems and persevere in solving them MP#2 Reason abstractly and quantitatively MP#3 Construct viable arguments and critique the reasoning of others MP#4 Model with mathematics MP#6 Attend to precision MP#7 Look for and make use of structure

Differentiation for Remediation: Help students create a diagram highlighting the parallel lines in one color and transversals in another. Label all congruent angles and sides.

If students need help constructing a proof, have them explain their reasoning to you verbally. This will help the student translate and communicate their verbal explanation to their paper.

Differentiation for English Language Learners:

Share and/or collaborate with partners.

Have visual definitions of parallel lines and parallelograms.

Differentiation for Enrichment:

Have students take one of the real world examples identified at the beginning of class. Have them create their own problem that utilizes the real-word application.

Potential Pitfall(s):

Students having difficulty visualizing the parallelograms in

the rail road tracks.

Students needing a push to begin their proofs.

Students communicating their reasoning through proof.

Independent Practice (Homework):

Steps:

1. As a class, discuss where you may see parallelograms in the real world. Discuss their purpose.

2. Have students work in pairs on the #2 Prove It! Activity.

3. After the students have had time to familiarize themselves with the assignment, take a section of the railroad track and draw it as a diagram of parallel lines cut by transversals on the board. Have students make


Be aware that this is the first time students have worked with formal proofs. They may need help initiating their proof as well as concluding it.




connections between parallel lines, angles, and parallelograms.

4. Class can identify all angles and sides that are congruent.

5. Students return to their small groups to develop a proof.

6. Towards the end of the hour, students present their method of proof to the class.






Segment 3




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Differentiation for Enrichment: Potential Pitfall(s):


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Segment 9




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Segment 10




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targeted content standard(s): student friendly learning

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