Using Hands-On Inquiry in High School Geometry

The main reason that High School Geometry is my favorite course to teach is that it lends itself so well to inquiry-based learning.

Before introducing each property, let your class work in pairs to discover the rule for themselves. 

There are so many advantages to this.  The students will remember the concept and be able to reproduce the rule at any time because they observed it themselves and know why it works.

Teach your students the SPORT structure for inquiry at the beginning of the year.  After a few lessons, they'll be comfortable with this style.

There are two main ways to get your class working hands-on with Geometry concepts.  Try manipulatives, software, or a combination of both.

Check out Geogebra for a great tech-approach to inquiry learning.  To use the software in a lesson, have students create the diagram (or use a template).  Be sure all measures are displayed.  Then, as students manipulate the points, lines, and angles, they can observe how the measures change in relation to one another. 

For example, to discover properties of angles along a transversal, your students can quickly sketch a pair of parallel lines on the screen.  Then, after drawing the transversal, they can label all angles and have the software display the measures.  They will notice congruent pairs right away.  Then, as they drag the transversal and change the diagram, the angle measures will keep updating.  The students will see that certain pairs of angles are always congruent and certain pairs are always supplementary.

Have students write these observations in complete sentences.  I often have my classes write their rules in an "if___, then ___" format.  I also have them give their own examples.

If you do not have access to technology, or prefer to follow it up (or mix it up) with a hands-on activity, the same properties can be discovered by hand.  Have your students trace an angle, then slide it down the transversal to overlap perfectly with its corresponding angle.  They can discover Corresponding Angles Postulate, and then move on to make observations about Alternate Interior & Exterior Angles.

The key is really just to avoid GIVING a theorem or property any time that you can.  When students discover it for themselves, they can remember it, understand it more deeply, and apply it more smoothly in the future.

You can use patty paper for this, but I usually just cut up tissue paper or tracing paper. 

You can also have your class measure the angles with a protractor.  They can draw a few diagrams and record angle measures and observations when the lines are parallel and compare these to similar diagrams where the lines are not parallel. 

Be sure that each student records observations in complete sentences and then develops a property also written as a sentence.

I do a similar setup for teaching vertical angles.  Using a small piece of tracing paper, the kids draw a pair of intersecting lines.   By folding different ways, they can see pairs of congruent "overlapping" angles.

I use an inquiry-based introduction for Triangle Theorems as well.  For Triangle Sum Theorem, there are plenty of options:

1.  Use Geometry software to sketch a triangle and display its measures.  Find the sum of the interior angle measures, then drag one vertex to create a new triangle.  Find the sum again. (Repeat)

2.  Use a protractor.  Draw a few different triangles with different classifications (right, obtuse, etc.)  Measure the angles and find the sum for each triangle.  (There will be some error with this method, so I have students do plenty of examples and notice that their sums are all approximately 180 degrees.)

3.  Use cut-up paper triangles and have students line up the vertex angles to create a straight angle.

I do Exterior Angles Theorem in a similar way. 

A few tips:

• Have all materials ready when the kids enter the room.  Either have Geogebra set up on the computer, or have the protractors, straightedge, tracing paper, etc. on the tables.  Start the lesson by having them jump right in.

• Allow students to work in pairs.  Encourage discussion.  Often, the students will have trouble writing out in words what they've discovered.  Saying it out loud first helps them to make that transition into writing.

• Stay out of the way.  Listen and walk around, but avoid giving hints until you are absolutely sure a group needs you.  They will be tempted to use you as a crutch.  If you are tough about this early on, your students will get used to investigating and will build their own skills and confidence.

• After the activity, come together as a whole class to compare discoveries.  Clear up any misconceptions.  Check that the properties always work.  Make sure each team tested all possible cases.  This is where I insert the "notes" for the lesson.  Have students formally write up the theorems, then do a few examples that apply the new knowledge.
  

• Use a standard format for students to record observations from all inquiry activities.  I like to have a space for a diagram, the "math language" explanation of the theorem or property, the rule in their own words, and sometimes spaces for student-provided examples and non-examples.
  

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Looking for more detail or more examples??:

Here are links to some of my Geometry specific inquiry posts to get you started -

1 - How to actually structure an inquiry-based lesson plan
2 - The specificbenefits of an inquiry approach
3 - Questioning strategies for inquiry learning
4 - Discovering Congruent Triangles
5 - Discovering Impossible Triangles
6 - Discovering Surface Area (middle school)
7 - Discovering Segment Addition Postulate

Or, click any of the images above to purchase worksheet packs and materials to accompany your lesson.

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