Tag Archives: midpoint

Writing Equations of Lines – with Some Novetly

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So I picked up some packs of foam number cubes at Mighty Dollar last week.  I sat wondering – what could I do with these?  Finally, during supper one night – an idea came across.  I’d use them to generate coordinates of points and students could write equations of lines.  Hmm.

2013-02-03 14.52.59

But as is, all points would fall in Quadrant I.  On half of the dice, I added a negative to 1, 2, 3 and the other half, onto 4, 5, 6.  So, have students roll the dice…thanks to ROY G BIV, we’ll know what order to place them for some consistency.

2013-02-03 14.56.50  Students record the coordinates of 3 points.  2013-02-03 14.57.33

Directions will be:

  1. i. Find the slope between RO & YG.
  2. Write an equation of a line that passes through RO & YG in slope-intercept form.
  3. Write an equation of a line parallel to ROYG and through the 3rd point IV.
  4. Write an equation of a line perpendicular to ROYG and through the point IV.
  5. Find the midpoint coordinates.
  6. Calculate the distance between RO & YG.

Yes, skill and drill – but with a bit of novelty, hopefully to engage the students a bit more than a black & white worksheet.

I’ve read several posts about activities similar to this – they are not easily assessed.  Students in the group – hold each other accountable.  I prefer same ability grouping – this allows students who are able to move along – while I can spend time with a student who has been absent/struggling to catch them up.  I purposefully walk around the room and spot-check each group to ensure they are on the right track.  If students are recording their coordinates/work/equations – its very easy to take up their work and spot check 2 or 3 sets to ensure correctness.

stoplightcups

Sometimes when working in small groups  such as this – I like to have the stop light cups out – If students are okay, the green cup is showing, if they have a question – but can keep on going, the yellow cup and finally, if they need my help – the red cup showing.  I can easily glance around the room for a quick look to see how everyone is doing.

 

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

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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.

Geo-board Investigations

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I was clearing out some files this weekend and ran across this packet from a presentation at KCTM in 2002.  I had just completed my initial National Board Certification earlier that spring (still didn’t know if I had certified yet) and thought these lessons were worth sharing.

I’m not sure if you’ll be able to read the first two pages – orginal files are long gone and just by happenstance I rance across this packet.  Reading through it – its almost like I was “blogging” 10 years ago – but it reminds how important reflection on your lesson will always be – how much you can learn about teaching by pausing to think about student thinking/responses.  Whether you use actual geo-boards, paper/pencil or modify to www.geogebra.org – maybe they will give you some ideas for your classroom.

Geo-board Investigations

  • Parallel & Perpendicular Investigation – use rectangle properties to find relationship with slopes
  • Amusement Park – distance between 2 points (I hate using distance formula and often allow students to find slope triangle, then apply Pythagorean Theorem)
  • Midpoint Investigation
  • Midsegment Investigation
 
 

*I used the reinforcement tabs for students to write coordinates/label points on geo-boards.  BUT don’t let them peel and stick…just leave on paper and drop over the geo-board tab.