# Midpoint – on a different day than Distance

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In years past, I’ve usually taught Midpoint and Distance on the same day or at least on consecutive days.  After a reminder of some brain research last fall – how our brains store information by similarities but retrieves information by differences – I decided to try things in split them up this semester – hoping to lessen the confusion students often face (do I add or subtract with midpoint/distance formulas?).  Again, this confusion stems from teaching a procedure without paying close attention to in-depth student understanding.

I chose to introduce “Midpoint with Coordinates” the same day we were working with segments, bisectors, midpoints of segments.  No bells/whistles here – just the basics

I gave students a grid index card and the points A(2, 1) B(8, 11) and C(8, 1) to attach to their INB RPH.  Simply starting with locating the midpoint of the AC and BC.  But also asking them to compare/contrast the coordinates of ACE and BCF each time.

Finally, asking them to locate G, the midpoint of AB.  Walking around the room, it was quite fun watching the various strategies.  The great thing was asking students to share their different strategies.  One used rise/run, several “counted diagonals” from A and B until they got to the middle, one used the midpoints of AC and BC and traced up from E / over from F until he found where G was located.  After discussing methods using the graph, a student stated “I just added my x’s then divide by 2 and added my y’s then divide by 2.”  When discussing how the coordinates were alike/different, a student asked “Isn’t that, what C____ did? Just averaging the x’s and averaging the y’s?”

So, I never actually gave them the “Midpoint Formula.”  Awesome.  Of course, we went on to practice the skill a few times.  I also chose 8 questions from Key Curriculum’s Discovering Geometry (did I mention, I *LOVE* this book?!?!? And have since the mid-90’s!)  – that required a  bit more thinking beyond skill/drill.  Two questions that led to some great discussion today was:

Find two points on segment AB that divide the segment into three congruent parts.  A(0,0) and B(9,6).  Explain your method.

Describe a way to find points that divide a segment into fourths.

But in class, I offered another – what about if I need to divide it into fifths?  Students worked individually, pair-share – then class discussion.  Quite different approaches.  I loved it.

What was even better, a student asked, “But  the examples we’ve used all have an end point at the origin.  Will it still work if the endpoint is not at the origin?”  Aaahhhhhhhhhhhh! That’s music to my ears!  Wow. Wow.  I love it.  I love it.  I love it.

This is a nice little open question to share with your students.  It definitely allowed me to see student understanding of the task by their work / responses / discussion.

# #MyFavFriday – Kagan Geometry

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Last week I stood glancing at a shelf of books left behind by my colleague.  I’m not sure why I didn’t notice it before – but there on the middle shelf was a Kagan Geometry book.

Two days this week I’ve smiled at the end of the day and it felt great.  Becky Bride has compiled simple to implement, engaging activities.  I’ve read snippets about the Kagan books – but never really sat down to read/do any of the activities.

## Boss-Secretary

One of the activities this week was using a strategy called Boss – Secretary.  Students work in pairs.  The boss tells the secretary what to write, explaining their reasoning for the steps/work.  IF the secretary sees the a mistake, he/she respectfully points out the mistake to the boss and praises her/him when they corrects their work.  If they work through it correctly, the secretary is asked to praise the boss, vice-versa.  After completing a problem, they switch roles.
The students have been funny with this simple, yet VERY effective activity.  Speaking of resumes, tough bosses, etc.  One asked today – do I really have to praise them when they do it correctly?  I’m really not a praise-y kind of person…  I said a high 5 would suffice.
Here is what I love about this – Students are talking/explaining their work so the secretary can do it.  Secretaries are listening, following directions, hopefully picking up on any mistakes.  I’ve heard multiple times – student exclaim – oh, now I get it.  They’ve all said they like this activity – its helped them really figure out “their thinking” – having to say what they’re doing – is difficult, and sometimes what they say/tell the secretary to write it not exactly what they meant.
This is a great formative assessment activity to observe / listen to students.  I’ve learned a lot about their thinking this week and I believe they have as well.  When students, notice plural, ask to do an activity again because it really helped them, well – isn’t that what we’re here to do?

## INB LHP assignment

As a left-hand page assignment in the INBs, I asked them to pick one problem they completed as a secretary – and they had to write out the boss’s diaglogue to solve the problem.  (midsegments or isosceles triangles this week).
Another activity in the Kagan book was something I have completely taken for granted… Processing altitudes.  Students draw one of each type triangle, and are asked to draw an altitude. Pass their paper to the next person, who then draws another altitude, etc.   Even after a couple of examples / illustrated definition for reference…they still struggled with “drawing” it.  What?  If they cannot draw an altitude, how can they actually know what one is in order to use it to solve problems?

## Applying Some Brain Research

Its been many, many years since I taught geometry – but I always remember students confusing medians, altitudes, perpendiuclar bisectors and angle bisectors of triangles.  I remember attending a David Sousa How the Brain Learns training several years back.  An example was shared how students often confuse concepts that are closely related because they are often times taught on the same day.  Concepts are stored by similarities, but are retreived by differences.  When we teach similar things on the same day, they are stored together, at the same time – when students are asked to retrieve that information, there’s not enough distinction between the two – therefore, they are often mixed-up, confused.  Hmmm.
So do I choose to teach each of these similar concepts (special lines/segments in triangles) on separate days – but is that even enough space between?  Should I skip a day between them?  Anyone with experience pacing it out this way – please share successes / need to make adjustments!  I really think this is an opportunity, by using Bride’s processing lessons – to make a difference, giving students the chance to build concrete understanding of these other-wise intertwined concepts.
If you’re not familar with the Kagan series – I think its definitely worth checking out.  There is very little prep time – other than working through the lessons yourself.  All blackline masters needed are included in the book!
I am soooooo excited about using more of these strategies in the weeks to come!  🙂