Introductory Investigation of Quadrilaterals

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Objective

Students will be able to investigate the specific features of the different quadrilaterals.

Big Idea

Students familiarize themselves with quadrilaterals by using a ruler and protractor to investigate the characteristics that define each polygon.

Ensuring Familiarity with the Concepts

10 minutes

As the students walk in the room, I hand each one a small, square colored piece of paper.  For 24 students, there are 6 different colored squares, marked with the numbers 1 through 4.  I will use these to formulate different groups throughout the today's lesson.

There is also a list on the board of diagrams that I would like them to draw in their notebooks.  The board says:

Use a ruler and protractor to draw (no need to construct!) in your notebook the following diagrams. 

  • two line segments that are congruent
  • two angles that are congruent
  • two line segments that bisect each other
  • two intersecting segments in which one is bisected and one is not
  • two parallel lines
  • two perpendicular lines
  • two line segments that are perpendicular and bisect each other
  • an angle that is bisected 
  • a quadrilateral and its diagonals

I ask that the students compare their diagrams with the diagrams of the other students in their group. I walk around the room, keeping an eye out for anyone who is struggling. I also direct questions to the students:

  • How do you know the segments are congruent?  
  • What does congruent mean?  
  • What does bisected mean? 
  • How do you know when segments are perpendicular? Parallel?  

I have found that the concept of segments that bisect each other is often difficult for some students; they don’t always make the connection between the concepts bisect and midpoint.  The diagonals of a quadrilateral can also be challenging, as well; I have had some students who connect midpoints of two sides, rather than connecting two vertices, and this certainly merits discussion (see my Deep Prior Knowledge Required! reflection for more on this issue).

When the task has been discussed and completed, I ask if everyone feels certain of these concepts and if anyone has any questions. We are now ready for the main activity of the lesson.

Close Investigation of Two Quadrilaterals

15 minutes

I give each student the handout entitled Table. I then instruct the students to divide themselves into 4 groups, and to move their desks accordingly.  Their group is determined by the number that they received on the colored piece of paper as they entered the room. (All the 1’s go together, all the 2’s, etc.) I remind my students not to throw their slips of paper out – we will use them again!

Before the lesson, I created a handout for each type of quadrilateral by dividing a sheet of graph paper into sixths and drawing six different figures of the same type of quadrilateral. For example, one page has six different sized rectangles, one page has six different sized trapezoids, etc.  

I give each group two copies of Quadrilateral_Cutouts.  I ask that that one person in each group quickly cut one of these pages apart and distribute one figure to each member of the group.  All in the group will have the same shape (a rectangle, for example) but everyone’s shape will be of slightly different dimensions (the example page is obviously trapezoids).

I explain that the group members work together to measure and investigate their quadrilateral and to develop hypotheses about their figure.  When the group comes to a consensus about a particular feature of their quadrilateral, they each record their conclusion on the table that I have provided for them.  When the group has finished with one figure, they repeat the process with their second figure.

The groups are divided as follows:

Group 1:  quadrilateral, rhombus

Group 2:  rectangle, isosceles trapezoid

Group 3:  square, trapezoid

Group 4:  parallelogram, kite

I ask that the groups investigate and discuss lengths and relationships of sides (including parallel and perpendicular), measures of angles, and characteristics of the diagonals.  The handout entitled Table is set up to record these observations, as well as any other the students might have.  (I’m hoping that someone will remember the sum of the interior angles of a quadrilateral!)

 

Share the Knowledge

19 minutes

When the students have finished investigating and recording their conclusions in their tables, I instruct the students to regroup according to the color of their slips of paper, taking their diagrams and Tables with them.  This places the students in 6 groups with 4 students each. 

I ask the students to report out to each other, one at a time, so that each member of the group teaches the other members of his group about all the features of his two quadrilaterals that his group observed.  The students record this information in their tables as they discuss it.

My experience with this jigsaw activity has been that very little is required of me during this phase of the lesson. I walk around the room and listen as the students instruct their classmates.  Each time I have done this, I have been so impressed with how seriously they take this task and with their mastery of the vocabulary as they describe the characteristics of their figures.  

Wrap-Up

1 minutes

I take the last minute to wrap things up – to state that in the next lesson we will try to the relationships between all of these quadrilaterals in perspective.  We will look at what they have in common and how they differ. We will begin to prove some of the hypotheses that the students made in this lesson.