There are really helpful diagrams and details about the chemicals in the brain and how they work together for each learner. 

If you have not read about the first chapter, check out Ellie's post here.

I'm going to give you a quick recap of Chapter 2, which is all about Growth Mindset vs. Fixed Mindset and how important the classroom culture and environment are for the success of differentiation.

Chapter 2 started with a really interesting sample scenario in which the teacher has different expectations of two different students.  I had thought that I never did this to students until I saw this sample in which the teacher barely reacts when one student did not complete homework, but is surprised and disappointed when another student did not complete it. 

I realized that I have actually done this!  It made me feel horrible, but we do need to think about how we react to different students.  Every student should feel that the teacher has confidence in his or her ability. 

If an under-performing student hands in a blank test, the teacher should not just shrug it off as if it's what they expect from that particular child.  The teacher needs to react and respond to the situation just the same as if it was a high-achieving student.  Students are SUPER aware of the fact that we are surprised at a behavior from one student while we accept it from another.

The book led into a discussion of mindsets.  Even the most perfect teacher has a set of assumptions and expectations regarding student behavior and performance.

My biggest takeaway from the mindset portion was that BOTH the student AND the teacher have to have a growth mindset.  If EITHER of them is in a fixed mindset, there is a decreased level of student learning.

We, as teachers must believe that all students truly want to succeed.

This chapter also discussed the safety and security of the learning environment.  There was a great diagram representing the hierarchy of response to sensory input.

Summarized, it showed that since the limbic system takes over to handle emotional responses, the brain focuses its attention first and foremost on data that affects the student's survival.  If the child does not feel safe, the brain will never move beyond that point. If the safety criteria is satisfied, the brain will then turn its attention to input that causes an emotion. 

This is SO important for us as teachers to realize and remember. 

Essentially, this means that if we cannot offer an EMOTIONALLY safe environment, then our students' brains physically cannot learn.

This is especially important to overcome in a middle or high school class.  Student brains are receiving so much input that affects them emotionally throughout the day.

It is only when the survival/safety AND the emotional needs are met that the brain can turn to focus on the data for new learning.  This sweet spot is where we must strive to keep our classroom environment.

When a student is in a healthy learning climate and feels emotionally safe and respected, the body releases chemicals that are called endorphins.  The endorphins flow through the blood and heighten the student's mood.  The frontal lobe of the brain works to help the student learn.  He or she can then classify the new learning material as important and store it in his/her memory.

However, when a student is in a stressful environment, cortisol is released into the bloodstream.  This causes an anxiety-based reaction.  The brain shuts out what it considers to be low-priority input.  Unfortunately, this includes the learning objective of the lesson!

I have not finished the book yet, but am really enjoying peeking ahead at the specific strategies that teachers can use to optimize the classroom and increase effective differentiation.  So far, its the best I've read on differentiation.  If you are interested in working on differentiation this summer, check it out (amazon affiliate link):

Ready to go on?  Check out what Brittany at The Colorado Classroom learned in Chapter 3.

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.
  

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.