• Use the Classifying Quadrilaterals practice worksheet (M-3-4-2_Classifying Quadrilaterals and KEY.docx) to determine how well students understand classifying quadrilaterals into subcategories based on attributes.
• Use the Terms and Definitions worksheet (M-3-4-2_Terms and Definitions and KEY.docx) to assess student mastery of lesson vocabulary.
• The Lesson 2 Exit Ticket (M-3-4-2_Lesson 2 Exit Ticket and KEY.docx) may be used to quickly evaluate student comprehension level of classifying quadrilaterals using attributes.

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.

Quadrilaterals and Nonquadrilaterals

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.

“Now we are going to sort the polygons we made into two groups: quadrilaterals and nonquadrilaterals.”

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.

Ask the rest of the class to study the polygons in the two different groups. Ask the seated students if they know which group they should join. “Look at your geoboard. Can you decide which group your shape belongs to?” After students have a chance to think about this, ask for volunteers. “Will someone please volunteer to take their geoboard and join a group? Can you explain why you think you belong in this group?” If the student does not belong in the group you are sorting, say, “You don’t belong in this group. Can you tell me why?” Once you get a few volunteers who can put their shapes in the right group and state the correct sorting rule, say, “Please join the group you think your shape belongs in.” Ask all students in the class, both the students seated and standing, to decide if the volunteer chose the correct group. Continue asking for volunteers until all students have joined the appropriate group, quadrilaterals or nonquadrilaterals, and the class agrees with the sorting of the geoboards. Be sure to ask students to explain their thinking, as it is important to help students focus on sorting polygons based on attributes.

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.

Parallelograms, Trapezoids, and Other Quadrilaterals

“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:

Ask the rest of the students in the class to study the quadrilaterals in the three different groups. Ask the seated students if they know which group they should join. “Look at your geoboard. Can you decide which group the shape you created belongs to?” After students have a chance to think about this, ask all students to join the appropriate groups. “Please take your geoboard and join the group you think your quadrilateral belongs in. Work with the other students to be sure you all agree that you belong in the same group.” When all students have chosen a group, ask the entire class to look at the group of parallelograms, without naming the group. Now ask, “Look at the group on the left. Do all of those quadrilaterals belong together? If not, which quadrilaterals belong in a different group and why?” Continue this questioning for the group in the center (with no parallel sides) and the group on the right (trapezoids).

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:

Ask the rest of the class to study the quadrilaterals in the two different groups. Ask the seated students if they know which group they should join. “Look at your geoboard. Can you decide which group your quadrilateral belongs to?” After the students have a chance to think about this, ask all students to join the appropriate groups. “Please take your geoboard and join the group you think your quadrilateral belongs in. Work with the other students to be sure you all agree that you belong in the same group.” When all students have chosen a group, ask the whole class to look at the rectangles group, without naming the group. Now ask, “Look at the group on the left. Do all of those quadrilaterals belong together? If not, which quadrilaterals belong in a different group and why?” Continue this same questioning for the group on the right (nonrectangles).

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:

Ask the rest of the class to study the quadrilaterals in the two different groups. Ask the seated students if they know which group they should join. “Look at your geoboard. Can you decide which group your quadrilateral belongs to?” After the students have a chance to think about this, ask all students to join the appropriate groups. “Please take your geoboard and join the group you think your quadrilateral belongs in. Work with the other students to be sure you all agree that you belong in the same group.” When all students have chosen a group, ask the whole class to look at the rhombi group, without naming the group. Now ask, “Look at the group on the left. Do all of those quadrilaterals belong together? If not, which quadrilaterals belong in a different group and why?” Continue this same questioning for the group on the right (nonrhombi).

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).

Classifying Quadrilaterals

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.