Structuring Guided Inquiry in Secondary Math

I am completely passionate about using guided inquiry in math classes!  However, I know that it takes time to re-work the typical lessons that you have been using into a more discovery-friendly format.  I've gotten questions about how to actually set up an inquiry lesson.  Do not fall into the trap of believing that inquiry has to be completely aimless and open-ended.  There is a middle ground where you can implement structured, organized framework for consistent guided inquiry.

Personally, I do not like unstructured chaos, so this modified inquiry style is what works well for me.

Many people find it easy to allow young students to explore, but up in middle school and high school, incorporating inquiry-based learning can be pretty challenging as far as lesson planning.  However, it is just as important as ever that at this age, students discover properties for themselves.  As the teacher, it's not as hard as you may think to direct this exploration.

I developed a "SPORT" method that I like to follow.  Here is a specific example of how a class period flows using the steps outlined above (feel free to print the main graphic at the top for reference, or pin it to come back to).

Sample Lesson: Triangle Inequality Theorem (Grades 7-10)

Intro:  Start by saying "Ok, we are working with triangles again today, so quickly before we start, I'm going to need the desks in a triangle arrangement.  Please move the desks quickly - Let's do 14 desks in a straight line across the back of the room, facing the board, and then just slide the rest in front to make the rest of the triangle."

Students will begin moving the desks around. (I used 14 as the number for the straight row, but you should use a number that is slightly more than half of the number of desks in your classroom for this to work.)  They will start out, then quickly realize that it is impossible to create this triangle.

Try to sit back and watch this unfold.  You will likely hear some students start trying to convince others that it is impossible.  Listen carefully to their explanations.  Be ready for them to engage you in the discussion and complain to you about this task once they realize that you are "tricking" them.

Once the class has discovered that the desks cannot possibly form a triangle this way, make a quick statement about the fact that it seems that some triangles (with certain side lengths) cannot be constructed.  Then lead into the discovery portion of the lesson by distributing manipulatives.

Manipulatives:  I have seen similar activities done with toothpicks, but I prefer markers.  If your students have the kind with the caps that snap onto the end, these are perfect.  They will attach to make rigid sides for the triangles.  Each student or pair will need about 12.

Once your students have been trained to follow the "SPORT" structure, they will have no problem getting started.  Direct them to record their findings from each triangle or non-triangle in a chart, then look for patterns after they have plenty of examples.


S -  Students come up with ten examples of triangles and record the side lengths (from least to greatest).  Then, they do the same for ten non-triangles.  They can record these in a chart.


P - After reviewing the examples, some students will already have an idea of what's going on.  Some may need to analyze the chart more closely to look for patterns.  Depending on the grade level, sometimes it is nice to have hint cards to hand out for those who are stuck.  Encourage the students to find the patterns and not give up.  They sometimes need to be trained not to ask for your help (and you may need to train yourself to step back).


O - Always require students to write their observations out in complete sentences.  They will get used to this.  For this particular lesson, sentences may look like this:

•     The two short sides have to add up to at least the long side.
•     It can't be a triangle if two of the sides aren't as long as the third.

Note that these sentences are not perfect, but this is what the kids usually come up with.  They are noticing the reasoning behind the property or theorem.  As you walk around, encourage kids to improve their sentences before writing their official rule.  In this case, a really great observation sentence would look like this:

• In the triangles, the sum of the two shorter sides is greater than the length of the longest side.  In the non-triangles, the sum is less than or equal to the length of the longest side.

R - Students will next develop a rule that should come close the official definition for the Theorem.  When groups get stuck here, I encourage them to try writing in an "if ___, then ___" format.  Sometimes using a conditional structure helps them to get their rule into writing.


T - Once students or teams have a rule written, they must test and revise it.  For example, if a student wrote a rule stating that the length of the longest side of a triangle must not be greater than the sum of the lengths of the two short sides, then they would have to adapt it.  They would need to include in their rule that the sum of the lengths of the two short sides also cannot be EQUAL to the length of the third side.

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Follow up the discovery portion of the lesson with practice.  Have students apply and use the theorem.  They will understand it much more deeply after working in a hands-on way and observing the properties. 

Take some time to look at a few other key lessons, and see if you can start by adjusting just one lesson per week to follow a guided inquiry format.  Take the time with your classes to go over the expectations.  They will slowly get used to it and start depending on you less.  It will take a few days before they will accept this.  Don't give up!  The kids are so used to asking things like "What are we supposed to notice?" and "Is this right?" and it can be so hard to feel like you are ignoring them.  I had to resist the temptation to give in after a few "You're supposed to help us!" complaints.  But once I overcame the guilt and let them struggle through a few lessons, I learned how to teach effectively this way, and the kids learned how to LEARN effectively this way.  It's amazing what this style did for my classes.

Once you train your students to follow this pattern, they will take off and be able to discover plenty of theorems and properties on their own.  You will just have to supply the tasks and materials to kick off each lesson, and they will know the expectations and be able to take over from there.  Check out some other specific examples of teaching with inquiry using the links below:

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