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Interior Angles

I. Section Objectives

  • Identify the interior angles of convex polygons.
  • Find the sums of interior angles in convex polygons.
  • Identify the special properties of interior angles in convex quadrilaterals.

II. Multiple Intelligences

  • Divide this lesson into two parts. The first part is going to focus on finding the sum of interior angles of polygons. The second part is going to focus on finding the sum of the interior angles of a quadrilateral.
  • Define convex.
  • Define polygon.
  • Define interior angles.
  • Note that the number of interior angles matches the number of sides of the polygon.
  • Show how to use the Triangle Sum Theorem to divide the polygon into triangles.
  • Demonstrate using an equation to solve for the sum of the measure of the interior angles.
  • Have students do this with two new polygons in their notebooks- a pentagon and a decagon.
  • Allow time for students to share their work when finished.
  • Be sure student answers were found using the equation.
  • Part 2 – move onto quadrilaterals
  • Test out the equation with a rectangle and a trapezoid.
  • Intelligences- logical- mathematical, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Define all terms on the board. Request that students copy these notes into their notebooks.
  • Here are the steps to solving these problems.
  • 1. Divide the figure into triangles.
  • 2. Use an equation to find the sum of the angles.
  • 3. With a quadrilateral- check does the sum equal 360^\circ?

IV. Alternative Assessment

  • Check student work. Be sure that the students are writing an equation when solving for the sum of the interior angles of both polygons and quadrilaterals.
  • Provide correction when necessary.

Exterior Angles

I. Section Objectives

  • Identify the exterior angles of convex polygons.
  • Find the sums of exterior angles in convex polygons.

II. Multiple Intelligences

  • Each of the examples in this lesson can be used as an extension to include multiple intelligences as you teach this lesson.
  • I recommend not using the text with the students but teaching this lesson as an exploration.
  • Have students work with rulers, protractors, paper and pencils at their desks in small groups.
  • Have students work in pairs and/or groups of three.
  • Then, present each section of the lesson.
  • Have the students explore each example in their seats.
  • For example, you draw the diagram on the board or overhead. Present a leading question, and then ask the students to work with the figure to solve the dilemma.
  • When finished, allow time for class discussion.
  • After going through each example in the text. Provide students with the notes for the lesson. They will have an experiential understanding of the information through working with each example.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Write all vocabulary and terms on the board/overhead.
  • Define exterior angles.
  • 1. Formed by extending the side of a polygon.
  • 2. Two possible exterior angles for any given vertex.
  • Define supplementary angles.
  • Define vertical angles.
  • Exterior Angle Sum

IV. Alternative Assessment

  • Alternative Assessment is done through observation of student group work.
  • Use an observation checklist to be sure that the groups are working to discover the big ideas of this lesson.

Classifying Quadrilaterals

I. Section Objectives

  • Identify and classify a parallelogram.
  • Identify and classify a rhombus.
  • Identify and classify a rectangle.
  • Identify and classify a square.
  • Identify and classify a kite.
  • Identify and classify a trapezoid.
  • Identify and classify an isosceles trapezoid.
  • Collect the classifications in a Venn diagram.
  • Identify how to classify shapes on a coordinate grid.

II. Multiple Intelligences

  • To differentiate this lesson, I recommend beginning by having students design a Venn diagram to classify the quadrilaterals. Do this BEFORE teaching the lesson.
  • Begin by going through a brief explanation of a Venn diagram. Most students will be familiar with them.
  • Then tell students that they are going to work with the text in small groups and design a Venn diagram to classify the following quadrilaterals.
  • List the quadrilaterals on the board and provide students with chart paper and colored pencils.
  • Students have been working with these figures for a long time. This exercise allows you an opportunity to walk around and assess student understanding about these figures.
  • Allow students to devise their own classifications system to engage higher level thinking.
  • When finished, ask students to explain how and why they chose to classify the figures the way that they did.
  • Then move to the text.
  • Go through the material in the text with the students having a deeper understanding of different quadrilaterals.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Define each type of quadrilateral on the board.
  • Help students to create a chart of each with a drawing and a brief description.
  • Define Opposite Sides of Parallelogram Theorem
  • Define Opposite Angles of Parallelogram Theorem
  • Review distance formula
  • Review slope

IV. Alternative Assessment

  • Use the Venn diagram exercise to assess student learning and understanding.
  • Look at how the students have chosen to classify the different figures- does this make sense? What adjustments are needed?
  • Did the students classify according to sides and angles?
  • Provide feedback to expand student understanding.

Using Parallelograms

I. Section Objectives

  • Describe the relationships between opposite sides in a parallelogram.
  • Describe the relationship between opposite angles in a parallelogram.
  • Describe the relationship between consecutive angles in a parallelogram.
  • Describe the relationship between the two diagonals in a parallelogram.

II. Multiple Intelligences

  • Begin by handing out pieces of string to each student. Some of the strings can be the same length and some can be different lengths.
  • Have the students use these strings to work through the examples at their seats.
  • This is a hands- on way to demonstrate the power of congruent side lengths.
  • Then have the students design a quadrilateral on the coordinate grid. They can each decide whether it is a parallelogram or not.
  • Exchange papers with a peer. Each peer needs to use the distance formula to test out whether the figure that they have been given is a parallelogram or not.
  • Allow time for sharing.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review the meaning of congruent.
  • List out the following description of a parallelogram. Request that the students copy this information down in their notebooks.
  • Parallelogram
  • 1. Quadrilateral with 2 pairs of parallel sides.
  • 2. Opposite sides are congruent.
  • 3. Opposite angles are congruent.
  • 4. Consecutive angles are supplementary.
  • 5. Diagonals bisect each other.
  • Walk through each step of filling in the proofs.
  • Provide a review of each “Reason” as it is presented.

IV. Alternative Assessment

  • Listen to the student sharing and assess whether students understand what makes a parallelogram a parallelogram.
  • Allow time for questions.

Proving Quadrilaterals are Parallelograms

I. Section Objectives

  • Prove a quadrilateral is a parallelogram given congruent opposite sides.
  • Prove a quadrilateral is a parallelogram given congruent opposite angles.
  • Prove a quadrilateral is a parallelogram given that the diagonals bisect each other.
  • Prove a quadrilateral is a parallelogram if one pair of sides is both congruent and parallel.

II. Multiple Intelligences

  • To differentiate this lesson, begin by going through the material in the lesson and stop when you get to the diagram in Example 4.
  • Divide students into five groups.
  • Each group is going to use this diagram to PROVE that the figure is a parallelogram.
  • Each group is assigned a characteristic of a parallelogram to prove. Explain that they will also need to prove the converse of each statement.
  • Group 1- quadrilateral with two pairs of parallel sides.
  • Group 2- opposite sides are congruent
  • Group 3- opposite angles are congruent
  • Group 4- consecutive angles are supplementary
  • Group 5- Diagonals bisect each other.
  • When finished, have students present their work.
  • As a class decide if the group was successful in proving their statement.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • List characteristics of a parallelogram on the board.
  • 1. Quadrilateral with two pairs of parallel sides.
  • 2. Opposite sides are congruent.
  • 3. Opposite angles are congruent.
  • 4. Consecutive angles are supplementary.
  • 5. Diagonals bisect each other.
  • Be sure that students understand how to find each characteristic in a diagram.
  • Walk through each proof.
  • Explain each “Reason” as it is covered. This will require students to review previously learned information.

IV. Alternative Assessment

  • Students will provide the assessment in this lesson when they decide whether each group has successfully proven their statement.

Rhombi, Rectangles, and Squares

I. Section Objectives

  • Identify the relationship between the diagonals in a rectangle.
  • Identify the relationship between the diagonals in a rhombus.
  • Identify the relationship between the diagonals and opposite angles in a rhombus.
  • Identify and explain biconditional statements.

II. Multiple Intelligences

  • Break down the information in this lesson to provide students with the following notes on rectangles and rhombi.
  • Rectangle
  • 1. Demonstrate diagonals are congruent using the distance formula
  • Provide students with an example on the overhead that they can they figure out on grid paper using the distance formula. You can even divide the class in half. Ask one half of the class to work on one diagram and the other half of the class to work on another diagram.
  • Rhombi
  • 1. Diagonals are perpendicular bisectors of each other.
  • 2. Diagonals bisect the interior angles.
  • Define a biconditional statement as a conditional statement that also has a true converse. “if and only if”
  • In pairs, have students write a biconditional statement for a rectangle and a biconditional statement for a rhombus.
  • Allow time for students to share their statements.
  • The class decides whether it is true biconditional statement or not.
  • If not, provide counterexamples.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal

III. Special Needs/Modifications

  • Walk through each proof in the lesson.
  • Provide a brief explanation of each “Reason” as it is presented. Do not assume students remember the definitions of each.
  • Review the following:
  • Distance formula
  • Definition of Perpendicular bisector
  • Review finding the slope of a line.
  • Review conditional statements.

IV. Alternative Assessment

  • Create a checklist of what would be acceptable correct biconditional statements. Use this checklist to assist students in evaluating the biconditional statements written by each pair.

Trapezoids

I. Section Objectives

  • Understand and prove that the base angles of isosceles trapezoids are congruent.
  • Understand and prove that if base angles in a trapezoid are congruent, it is an isosceles trapezoid.
  • Understand and prove that the diagonals in an isosceles trapezoid are congruent.
  • Understand and prove that if the diagonals in a trapezoid are congruent, the trapezoid is isosceles.
  • Identify the median of a trapezoid and use its properties.

II. Multiple Intelligences

  • Break down the information in this lesson into sections to assist student understanding.
  • Define a trapezoid.
  • 1. One pair of parallel sides
  • 2. NOT parallelograms
  • Define an Isosceles trapezoid.
  • 1. One pair of non- parallel sides that are the same length.
  • 2. Base angles are congruent.
  • 3. Diagonals are congruent.
  • Activity- have students draw two trapezoids and two isosceles trapezoids.
  • With the isosceles trapezoids request that they do the following.
  • 1. Label angles, sides and diagonals to show that it is an isosceles trapezoid.
  • 2. Write a statement and its converse for each label to explain it.
  • Notes of Trapezoid Medians
  • 1. Connects the medians of the non- parallel sides in a trapezoid.
  • 2. Located half-way between the bases in a trapezoid.
  • Theorem- \frac{\mathrm{sum\ of\ base\ lengths}}{2}
  • Use this equation with the example in the text. Request that students practice this as well.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Write all notes on the board/overhead. Request that students copy this information in their notebooks.
  • Define symmetry.
  • Use an example so that students understand symmetry in connection with trapezoids.

IV. Alternative Assessment

  • Collect student diagrams.
  • Assess understanding based on labels and statements.

Kites

I. Section Objectives

  • Identify the relationship between diagonals in kites.
  • Identify the relationship between opposite angles in kites.

II. Multiple Intelligences

  • Write out the following notes on kites.
  • 1. No parallel sides.
  • 2. Two pairs of congruent sides adjacent to each other.
  • 3. Two vertex angles
  • 4. Two non- vertex angles that are congruent
  • 5. Diagonals are perpendicular
  • Have students design a kite on a coordinate grid.
  • They must prove it is a kite by providing proof that the diagonals are perpendicular, and that the non- vertex angles are congruent.
  • They can do this through statements or by writing a proof with reasons.
  • Finally, once their diagram has been approved, they can use it to design and decorate their own kite. Students can use chart paper and colored pencils to do this.
  • Hang all work up in the class.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, bodily- kinesthetic, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Define adjacent.
  • Define vertex angles.
  • Define non- vertex angles.
  • Walk through each proof.
  • Explain each “Reason” in the proof.

IV. Alternative Assessment

  • Approve all students kite diagrams before students design their final kite.
  • Be sure that the points have been successfully proven.

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