Lesson Plan

Objectives

In this lesson, students will study the different types of quadrilaterals, focusing on their commonalities and distinctions. Students will:

• identify and describe the attributes of quadrilaterals.
• classify quadrilaterals based on characteristics into subcategories of quadrilaterals, including parallelograms, rectangles, rhombi, squares, and trapezoids.

Essential Questions

• How can patterns be used to describe relationships in mathematical situations?
• How can recognizing repetition or regularity assist in solving problems more efficiently?
• How can the application of the attributes of geometric shapes support mathematical reasoning and problem solving?
• How can geometric properties and theorems be used to describe, model, and analyze situations?
• How are spatial relationships, including shape and dimension, used to draw, construct, model, and represent real situations or solve problems?

Vocabulary

• Pentagon: A polygon with exactly five sides.
• Polygon: A closed plane figure bounded by three or more line segments that only meet at their endpoints.
• Quadrilateral: A polygon with exactly four sides.
• Rhombus: A quadrilateral with sides of equal length.

60–90 minutes

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

• geoboards and rubber bands

(a virtual geoboard can be found at http://www.mathplayground.com/geoboard.html)

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

• geoboards and rubber bands

(a virtual geoboard can be found at http://www.mathplayground.com/geoboard.html)

• View

Instructional Procedures

• View

The lesson focuses on using attributes to define subcategories of quadrilaterals, including parallelograms, rectangles, rhombi, squares, and trapezoids. The lesson also focuses on helping students understand that some quadrilaterals belong to many subcategories. For example, a rectangle is also a parallelogram and a square also belongs to the subcategories rhombi, rectangles, and parallelograms.

Distribute a geoboard and rubber bands to each student. “Please use a rubber band to construct a polygon on your geoboard.”

Students will likely create a variety of polygons. Observe students as they are creating the polygons.

Facilitate the sorting of the polygons into two categories, quadrilaterals and nonquadrilaterals. To do so, ask 3 to 4 students who constructed quadrilaterals to stand in the front of the room on the left side. These students should bring their geoboards and hold them so the rest of the class can see the quadrilaterals they constructed. Then, ask 3 to 4 students who constructed nonquadrilateral polygons to stand in the front of the room on the right side. Be sure there is space between the groups.

Distribute a Terms and Definitions worksheet (M-3-4-2_Terms and Definitions and KEY.docx) to all students. Help students define the term quadrilateral on the Terms and Definitions worksheet. Be sure to provide the correct spelling. Also, ask the class to help write a definition. Be sure all students draw at least two examples of quadrilaterals. Drawing at least two helps students remember to be flexible in their thinking of what the term represents. Students can draw some of the shapes created on the geoboards as examples.

Remove the rubber band. Now, use the rubber band to construct a quadrilateral on your geoboard.”

Students will likely create a variety of quadrilaterals. Observe students as they create the quadrilaterals.

Facilitate the sorting of these quadrilaterals into three categories: parallelograms, trapezoids, and others. To do so, ask 2 to 3 students who constructed parallelograms to stand in the front of the room on the left side. These students should bring their geoboards and hold them so the rest of the class can see the parallelograms they constructed. Ask 2 to 3 students who constructed trapezoids to stand in the front of the room on the right side. Be sure there is a lot of space between the groups. Now, ask 2 to 3 students who constructed quadrilaterals with no pairs of parallel sides to stand in the front of the room in the center.

For example:

Be sure to ask students to explain their thinking, as it is important to continue to help students focus on sorting polygons based on attributes.

Help students define the terms parallelogram and trapezoid on the Terms and Definitions worksheet (M-3-4-2_Terms and Definitions and KEY.docx). Be sure to provide the correct spelling for each term. Also, ask the class to help write a definition. Be sure all students draw at least two examples of each shape. Drawing at least two helps students remember to be flexible in their thinking of what the term represents. Students can draw some of the shapes created on geoboards as examples.

Rectangles and Nonrectangles

“Remove the rubber band. Now, use the rubber band to construct another quadrilateral on your geoboard.”

Students will likely create a variety of quadrilaterals. Observe students as they create quadrilaterals.

Facilitate the sorting of these quadrilaterals into two categories, rectangles and nonrectangles. To do so, ask 2 to 3 students who constructed rectangles to stand in the front of the room on the left side. Be sure at least one of these students constructed a square, which is a rectangle with four equal or congruent sides. These students should bring their geoboards and hold them so the rest of the class can see the rectangles they constructed. Ask 2 to 3 students who constructed nonrectangles to stand in the front of the room on the right side. Be sure there is a lot of space between the groups.

For example:

Be sure to ask students to explain their thinking, as it is important to continue to help students focus on sorting polygons based on attributes.

Help students define the terms rectangle and square on the Terms and Definitions worksheet (M-3-4-2_Terms and Definitions and KEY.docx). Be sure to provide the correct spelling for each term. Also, ask the class to help write the definitions. Be sure all students draw at least two examples of each shape. Drawing at least two helps students remember to be flexible in their thinking of what the term represents. Students can draw some of the shapes created on geoboards as examples.

*Be prepared: Some students may think that squares and rectangles do not belong in the same group. Help students define both terms, as stated above. Now, write the following sentence on the board, and read it aloud.

• “All squares are rectangles.”

Ask students to show thumbs up if they believe this is true, thumbs down if they believe this is false, and thumbs to the side if they are not sure. Ask a few students from each group to explain their thinking. For example, “Can two people volunteer to tell me why they think this sentence is true?” and so forth.

To help support all students in understanding that this statement is true, ask students to construct squares on their geoboards and lay them on their desks. Now, challenge students to find at least one square that is not a rectangle. “Look at all of the squares created on the geoboards. Can you find at least one that is not a rectangle?” Encourage students to get up and move around the room looking at all of the geoboards.

Remind students that a rectangle is merely a quadrilateral with 4 right angles. Encourage students to remember that a square is a quadrilateral with 4 right angles AND 4 equal sides, so a square is a special rectangle. All squares are rectangles, but they also have one other attribute, 4 equal sides.

[Note: Sometimes it is helpful to use nonmathematical examples to support students in their understanding. In this case, a rectangle is like a boy and a square is like a boy wearing a red shirt. The square is a rectangle with one additional attribute, likened to wearing a red shirt in this example. Another example is a rectangle is fruit and a square is a banana. The square is a particular type of rectangle, as the banana is a particular type of fruit.]

Now, write the following sentence on the board, and read it aloud.

• “All rectangles are squares.”

Ask students to show thumbs up if they believe this is true, thumbs down if they believe this is false, and thumbs to the side if they are not sure. Ask a few students from each group to explain their thinking. For example, “Can two people volunteer to tell me why they think this sentence is true?” and so forth.

To help support all students in understanding that this statement is false, ask students to construct rectangles on their geoboards and lay them on their desks. Now, challenge students to find at least one rectangle that is not a square. “Look at all of the rectangles created on the geoboards. Can you find at least one that is not a square?” Encourage them to get up and move around the room looking at all of the geoboards.

You must remind students that a rectangle is merely a quadrilateral with 4 right angles. Encourage students to remember that a square is a quadrilateral with 4 right angles AND 4 equal sides. This statement is false because a square must also have an additional attribute beyond the 4 right angles in a rectangle. All rectangles are not squares, as all rectangles do not have 4 equal sides. Instead, only some rectangles are squares.

[Note: Again, you can use nonmathematical examples to support students in understanding the concept. Using the examples stated above, in this case all boys are not boys wearing red shirts. The rectangle, or the boys, is general, and the square, or the boys wearing red shirts, is a specific case. Similarly, all fruit are not bananas. The fruit is general and includes oranges, bananas, blueberries, and many others. The bananas are a specific type of fruit.]

Rhombi and Nonrhombi

“Remove the rubber band. Now, use the rubber band to construct another quadrilateral on your geoboard.”

Students will likely create a variety of quadrilaterals. Observe students as they create the quadrilaterals.

Facilitate the sorting of these quadrilaterals into two categories, rhombi (plural of rhombus) and nonrhombi. To do so, ask 2 to 3 students who constructed rhombi to stand in the front of the room on the left side. Be sure at least one of these students constructed a square, which is a rhombus with four right angles. These students should bring their geoboards and hold them so the rest of the class can see the rhombi they constructed. Ask 2 to 3 students who constructed nonrhombi to stand in the front of the room on the right side. Be sure there is a lot of space between the groups.

For example:

Be sure to ask students to explain their thinking, as it is important to continue to help students focus on sorting polygons based on attributes.

Help students define the term rhombus on the Terms and Definitions worksheet (M-3-4-2_Terms and Definitions and KEY.docx). Be sure to provide the correct spelling for the term. Also, ask the class to help write the definition. Be sure all students draw at least two examples of a rhombus. Drawing at least two helps students remember to be flexible in their thinking of what the term represents. Students can draw some of the shapes created on geoboards as examples.

*Be prepared: Some students may think that squares and rhombi do not belong in the same group. Help students understand the definitions of both terms, as stated above. Now, write the following sentence on the board, and read it aloud.

• “All squares are rhombi.”

Ask students to show thumbs up if they believe this is true, thumbs down if they believe this is false, and thumbs to the side if they are not sure. Ask a few students from each group to explain their thinking. For example, “Can two people volunteer to tell me why they think this sentence is true?” and so forth.

To help support all students in understanding that this statement is true, ask students to construct squares on their geoboards and lay them on their desks. Now, challenge students to find at least one square that is not a rhombus. “Look at all of the squares created on the geoboards. Can you find at least one that is not a rhombus?” Encourage them to get up and move around the room looking at all of the geoboards.

You must remind students that a rhombus is merely a quadrilateral with 4 equal sides. Encourage students to remember that a square is a quadrilateral with 4 equal sides AND 4 right angles, so a square is merely a special rhombus. All squares are rhombi, but they also have one other attribute, 4 right angles.

Now, write the following sentence on the board, and read it aloud.

• “All rhombi are squares.”

Ask students to show thumbs up if they believe this is true, thumbs down if they believe this is false, and thumbs to the side if they are not sure. Ask a few students from each group to explain their thinking. For example, “Can two people volunteer to tell me why they think this sentence is true?” and so forth.

To help support all students in understanding that this statement is false, ask students to construct rhombi on their geoboards and lay them on their desks. Now, challenge students to find at least one rhombus that is not a square. “Look at all of the rhombi created on the geoboards. Can you find at least one that is not a square?” Encourage them to get up and move around the room looking at all of the geoboards.

You must remind students that a rhombus is merely a quadrilateral with 4 equal sides. Encourage students to remember that a square is a quadrilateral with 4 equal sides AND 4 right angles. This statement is false because a square must also have an additional attribute beyond the 4 equal sides in a rhombus. All rhombi are not squares, as all rhombi do not have 4 right angles. Instead, only some rhombi are squares (those that do have 4 right angles).

Distribute a copy of the Classifying Quadrilaterals practice worksheet (M-3-4-2_Classifying Quadrilaterals and KEY.docx) to each student.

Ask students to work in pairs to complete this activity. Students should first cut out the quadrilaterals. To save time, suggest that the students work together to cut out one set of the quadrilaterals, instead of having both students cut out their own set. Now, each pair of students should work to trace all quadrilaterals that belong in each category. Some of the quadrilaterals may belong in multiple subcategories.

When they are finished, ask one pair of students to show which quadrilaterals they traced in the parallelogram category. Be prepared to ask why they didn’t trace other quadrilaterals in that category. (Enlarge the quadrilaterals using a photocopier, and cut out the quadrilaterals to use in the discussion.) Hold a specific quadrilateral up and ask why they did not consider it to be a parallelogram. After completing the discussion of parallelograms, continue this discussion with the remaining categories.

Extension:

This section may be used to tailor the lesson to meet the needs of the students. The Routine section provides ideas for reviewing lesson concepts throughout the school year. The Small Group section is intended for use by students who would benefit from additional practice. The Expansion section includes a challenge for students who are prepared to move beyond the requirements of the standard.

• Routine: To review the lesson concepts during the school year, ask students to describe objects in the classroom using the appropriate vocabulary, including parallelograms, quadrilaterals, rectangles, rhombi, squares, and trapezoids. For example, ask students to identify three quadrilaterals in the classroom. Be sure to ask them to give all of the names that are appropriate. For example, a square is also a parallelogram, a quadrilateral, a rectangle,
• Small Groups: Students who need additional practice may by pulled into small groups to work on the following activity.

Use a blank set of index cards. Create an index card with each term, such as parallelograms and nonparallelograms, and place these on the table apart from one another.

Ask individual students to pick an index card, draw a shape on the index card, and place it in the appropriate group. The other students must decide why they agree or disagree about the placement of the card. Be sure to ask them to explain their reasoning, using specific terms about the attributes of the shape. Continue this until all individual students understand the attributes of parallelograms.

Repeat the game using trapezoids and nontrapezoids, rhombi and nonrhombi, rectangles and nonrectangles, squares and nonsquares.

Draw a square on an index card, and ask if it is a parallelogram, a trapezoid, a rhombus, and a rectangle. A square has the attributes of many other subcategories of quadrilaterals, so it can be classified as all of these except a trapezoid.

• Expansion: Students who are prepared to move beyond the requirements of the standard should work individually or in pairs to play the polygon game found at this link: http://www.math-play.com/Polygon-Game.html

The game introduces students to additional polygon terms such as regular and equilateral. Students can also be challenged to use the game found at this link to examine properties of the diagonals of quadrilaterals and the sum of the angles in quadrilaterals: http://www.mathsisfun.com/geometry/quadrilaterals-interactive.html

To do so, click on the angles and diagonals buttons.

Related Instructional Videos

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Final 05/24/2013