Category Archives: geometry

Systems of Equations (part 2)

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None of what I’m sharing is new…but its me reflecting on the week…so I can reference back and make adjustments in building a better unit  of learning experiences for next time around.

To address some student questions, here are examples used in class to follow-up.

On white boards:

  • y=x+1
  • Pick a value for x.  Find y.  (Ex. (3, 4)
  • Now, let’s double our equation.  What?!?  Yep, double it. 2(y=x+1)
  • Okay. 2y=2x+2
  • Use your same value for x from above and find y. (3, 4)
  • What do you notice?
  • Let’s multiply our first equation by 5. 5(y=x+1)
  • 5y=5x+5.
  • Use your same x value from above, find y.  (3,4).
  • Did that happen for everyone?  Turn and talk…
  • What if we took half of our equation?  .5y=.5x+.5
  • for the same value of x, it works again (3,4)
  • Then we go to Desmos to see the graph of our equation along with ALL of our versions of the equation.

Its a big idea that I don’t tell them.  They have observed why we can use this “magical” math thing is actually just a different version of the same equation…as one student put it “its the same equation, in disguise!”

But I also feel there is value in diverting from my original plan here to address the student’s struggle to figure out WHY? we do this in elimination, otherwise, it is literally, a “magical math thing” that just happens.

I need to do a better job of this – equivalent expressions / equations – earlier in the year, when we are looking at equations of lines…but also, how can I connect it with scale factors and similarity?  It all comes back to proportionality, but what strategies and tasks can I use to help my students make the connection and really develop a deep understanding?

Next on the list, we graphed our systems we’ve solved in Desmos.  Noticed and wonder…comparing our graphs to the work we’ve done algebraically.  Ohhhhh.  We found the intersection point!  Again, not me telling them, but they see it on their own.  I love that Desmos allows us to graph an equation in standard form.

Finally, I asked students to solve these equations and discuss their results in their groups:

  •   4x -6y   = 12                 and             7x – 4y = -11
  • -2x + 3y = -6                                     14x – 8y = 16

When does 0 = 0? ALWAYS!                   When does 0 = -6?  NEVER!

Again, we looked at the graphs in Desmos…

 

Several quickly stated the first set was only a multiple of the first equation, so it would be the SAME line!  (yes. secret happy dance!)

And the parallel lines never intersect…the equations were multiples on one side, but NOT on the other, a student noticed.  Its a translation, just moving one line up or down – another student stated.  So, how can I use their intuitive thoughts to build a better lesson?

racing dots

I found Racing Dots on teacher.desmos.com  –  based on an activity, Great Collide by Jon Orr – to bridge between special situations, graphing solutions, substitution and algebraic solutions – will share more on this task later!

 

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Posting Learning Targets yay or nay

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Thanks to @JustinAion,  I got thinking…

It depends… on my class and the students and the activity…to determine if I actually post it.

However, when I do, I refer to it at the beginning, throughout the task – to remind students of the end goal, and again as a wrap up – whether reflection, exit ticket of discussion/summary to end class.  And I like to refer to it the following day as we begin the next lesson, just as a quick review.

I, personally, would prefer to have an overarching Essential Question for each lesson to use rather than a specifically stated target.  However, I sometimes struggle a lot with Writing EQs, would love a colleague to collaborate on these.

Here’s a section of the unit organizers I’ve used this past year (thanks @lisabej_manitou). 

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And a link to this file.
Unit Organizer
Functions Overview

I give them to students toward beginning of unit, we complete the words worth knowing for vocabulary (thanks @mathequalslove). Then read through actual targets.  When quizzes are given back or practice problems checked, students have a place to reflect/record thwir level of learning as well.  Because students have this in their INBs, I can quickly refer to them if not posted on the board on any given day. 

Week 2 Sunday Summary #MTBoSchallenge & #made4math

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Week 2 is complete.  I am still trying to find my groove with having 1st hour prep.  I am a morning person, so I am ready to interact with students as soon as we arrive.  Sitting down for plan time, I lose my momentum.   Paired with having to be out of our building by 3:00 due to renovations, I have no time to sit and process the day’s events.
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3 Things That Happened This Week
I finally got my anchor chart board with sentence starters and questions completed.  I am very pleased with it and have been trying to model/give students opportunities to practice in class discussions.  Here is a link to a file of the starters.

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I giggled when I saw Sarah saying she “totally stole” from me…that’s what #MTBoS is all about. Sharing and making our classrooms better for our students!

I am using visualpatterns.org as one of my daily tasks to begin class.  I wanted students to have a page in their INBS to record these…

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Here is the file.  Print 2 up and front/back for a booklet for your INBs. 

I shared Thursday how I was a bit hesitant to allow my students to go with their process of locating the midpoint given coordinates of endpoints.  I know.  There are those that say just tell them the midpoint formula.  I could but this is the method they are owning.  Basically, they are finding the distance between the coordinates, then “moving” half the distance will put them at the midpoint. 

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But then I got to thinking about the actual standard:

CCSS.MATH.CONTENT.HSG.GPE.B.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Midpoint is the most common and yes, we’ll use it in proofs later.  But if I go in Monday and ask them to find 1/3 point which would be a 1:2 ratio, or a 2/5, 2:3? Will their method actually prove more efficient because it is actually the same process for both?

2 Things on My To-do List
I have 3 tubs that still need to be unpacked from our renovation move.  I have my shoe boxes on the shelves, but I need to get those labeled correctly.

Finish an Intro to Matrices Unit, I hope will work as  flipped/blended learning unit.

1 Good Book to Read
Thanks to @mathymeg07 for sharing Wonder by RJ Palacio. 

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Megan said it is a book everyone from 9-99 should read!  Right now, the Kindle version is on sale for $2.50.  I am making posters of Mr. Browne’s Precepts for my classroom, such great lessons to live by.

Card Tossing & Spiraling Curriculum #tmc14

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Awesome session Mary and Alex!  Thank you. Thank you. Thank you.

The session focused on their experiences with Grade 10 Applied students ( Canada).  The entire course is activity based which allows students to not miss out on big ideas as they would in a traditional unit by unit aligned course.
Students have repeated opportunities to experience big ideas. The tasks are rich  with multiple entry points and different approaches to solving.  It’s a collaborative environment with accountable talk.  There are fewer disciplinary issues with increased engagement.

Each 6 weeks a mini – exam over entire course up to that point takes place.  Questions are in context and tied to activities they have completed.

We began with beads and pennies on our desks and this task… Cole has 2 smarties and 3 juju bed for $.18 while Noah has 4 smarties and 2 juju be for $.20.  They shared that systems are presented this way – no algebraic forms- for the first several weeks of class.  I, personally, can see how effective this strategy could be.

The next activity shared was Sum of Squares (he doesn’t refer to it as Pythagoras Theorem, yet – or did he say ever?)

Students are asked to cut all squares from side length 1 to side length 26.  Each square is labeled with side length, perimeter, area.  Then they build with them.

Basically students explore and eventually they focus on triangles formed with question, are there 3 you cannot make a triangle with?   Which combinations form different types of triangles. Begin looking at 3-4-5 triangle families, similar triangles (Kate suggested dilations here), discuss opposite side and adjacent sides, then give them a TRIG table and allow them to figure it out.

Compare side lengths with perimeter, or side length with areas.  The possibilities of math concepts are endless.
We ended the day with Card Tossing by collecting data, then using rates to make some predictions.

Video of Alex & Nathan picture below is only a screenshot.

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@AlexOverwijk downed by @nathankraft 75 to 72

Each person in the room completed several trials of tossing our cards for 20 seconds.  We found our average rate of success, then determined who we thought might beat King Card Tosser.

Alex asked us to predict how long they needed to toss if he gave Nathan a 35 (?) card advantage so it would be super close and exciting.  Our prediction 38 seconds about 75 cards. Many ways of making the predictions were possible. Not to shabby, huh?

This task was fun, exciting, engaging.  Definitely on the to-do list.

This approach is definitely something I would like to consider, if administration will allow it!

Adventures in Geocaching

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This week my nephews are in town.  They love playing outdoors, have crazy fun imaginations and are loads of laughter.  My 10 year counts down the days each year until they arrive at gramma’s!  Its more exciting than end of school countdown.

On Tuesday I was brainstorming with mom and looked into geocaching.  I had never been, only heard it mentioned a few times and needed to learn how it worked.  So I went to geocaching.com registered an account, downloaded the app and yesterday, we set off on our first adventure.

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Though most were the grab and go – the kids enjoyed it and were super excited when we found a cache filled with kids party favor swag.  Only problem, we had to scavenger the car to find some trade items…a creative venture in itself. 

Each find has a log where you date and sign to prove you were there.  There are hints, clues on the app if you have trouble locating it.  For example, one clue was Australia.  We looked around trying to connect something with that and finally “down-under.”  Sure enough, the cache was underneath the landmark.

We found 10 on our first day and learned some great tips.  We even saw a robin’s egg that had fallen from the nest, a double rainbow as we drove from place to place, played on a playground and visited the gravesite of their great grandparents as one cache was near the cemetary.

Some very clever hiding techniques!  One cache was so tiny, we thought it was only a magnet that had lost its container somehow!  It took a while to roll the log back small enough to place in it again!

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Some where along the way, someone asked if I could spell supercalifragilisticexpealidocious (sp?).  My oldest nephew said, well, it shouldn’t be that difficult if you just break it apart.  True.  My response, most words are the same way.  For example, geocache.  Geo is earth, cache, a hiding place.  Which led me to geometry…geo, earth; metry, to measure.  I went on to explain how people wanted to measure distances, areas of land, patterns they noticed in the earth, so they created geometry. 

I explained how people noticed patterns with their shadows at different times of day.  I asked, when is your shadow longest, when is it shortest?  Without a hesitation, someone popped up, when the sun is on the horizon, your shadow is longest, when the sun is above you, your shadow is shortest.  I asked how do you know?  One said well, we learned that in science, but if you think about it, it just makes sense.

Again, I am convinced we force too many ideas on our students rather than just letting them think about it and develop their own sense of reasoning.

On our earlier finds, I allowed the kids to look at the GPS readings on the app.  They had to decide which direction to go -are we too close or too far north, and how far to walk.  We needed to moved west once and they could use the compass corectly, but I asked, what if we didnt have it pointing out the directions for us?  I explained how the sun sat to the west and each time after that, they immediately gauged direction based on the sun.

When I returned home after dark, I tweeted of our adventures.  In a convo,

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So, now I am curious…how can/are teachers using geocaching as a context for learning?

Honey, I Shrunk the Kids #MTBoS30 Post 15

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Many years ago, when I first began teaching, students compiled portfolios in their mathematics classes.  In the beginning it was a waste of time.  But as they adjusted them, I felt they could be useful learning tools.  About the time they got them just right, (students were making claims and supporting with reasoning /evidence, not just a bulleted list of steps to solve an equation)…well, they did away with them.

A colleague shared a task during my last year of portfolios.  I ran across it a couple of years ago in a file some where.  Once again, I forgot about it until I found this DVD:

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The task was simply for students to devise a plan to confirm or dispute Disney’s claim that the kids were shrunk to 1/4 inch tall.  Most would collect some measurements from the movie screen and support their conclusions with proportional reasoning.

Kind of interesting to determine if they held the same ratios throughout all of the scenes or if some seemed more to scale than others.

What other movies could be offered in a similar task?

Mindset #MTBoS30 Post 6

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A bit sad when I read Fawn’s post here but true.

Yes our students may come to us broken down and have given up on learning math.  But that’s when we have an opportunity to give them a chance…

By using open tasks, anyone can play with the math…not just “the smart kids” who have memorized all the steps and procedures. 

The thing I appreciated most about the staircases task was there were no rights/wrongs as we began.  Only a what do you think?  And why?  Students ordered the staircases, discussed with classmates, supported their claims with reasoning or critiqued the thinking of others. 

Then, they had to devise a measure that confirmed their claim.  In 2 years, this is one of my favorite lessons.

I will continue to search for rich, thoughtful tasks that allow ALL of my students participate and move forward. 

They may arrive at a zero love for math, but when they leave, they will know they are quite able…

Triangle Centers #MTBoS30 Day 3

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For triangle centers, I like to let students construct them on geogebra if the lab is open, then notice and wonder about them…and their properties.  Ideally, they would explore and investigate their questions and prove/dispute their claims. 

A few questions that arose this morning…

Do the 6 triangles created by medians have equal areas?

I wonder if you dilated the incenter for the inscribed circle, would it become the  circumcenter of the circumscribed circle? 

Another student stated, not always, unless your angle bisectors became the perpendicular bisectors.  (When would that happen?)  Without that happening, it’s a dilation and translation.

Comparing the areas of the circles and corresponding triangles, a student asked,

Is the area of the circumscribed circle twice the area of the triangle?

Is the area of the inscribed circle…one half the area of the triangle?

Now to explore the questions…

These were all follow-ups to discussing how these constructions would aid in solving various real-life contextual problems presented at the beginning of the lesson.

Gallery Walk #ppschat Challenge

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A common theme in many chapters of Powerful Problem Solving is Gallery Walks. Several techniques are offered throughout the book, but the common goal is to allow students to view their classmates’ approaches to problems.

One of my faults with online book chats is lack of follow-through. I can sometimes use an extra nudge of accountability. There are often so many great ideas and strategies in the books we are chatting that I get overwhelmed and not sure where to begin. Advice: pick 1 thing. Try it. Reflect. Revise. Try it again.

So here is my attempt at a gallery walk. I simply cut apart a pre-assessment for a Formative Assessment Lesson and each pair of students taped it to a large sticky note, discussed and responded. I was confident in many of the questions, but my goal was to identify the few some students were still struggling to understand completely, mostly questions involving transformations.

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1. The large majority are fine with creating a possible equation, given the x-intercepts.

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2. Initially these students tried -6, -4 and 2 as their intercepts. I asked them to graph their equation then reread the instructions. Oh. They had read write an equation, looked at the graph for possible intercepts and failed to read the y-intercept of (0, -6). One quickly stated the connection between y-intercept and factored terms and was able to adjust their response with ease. I believe it happens often to see a graph skim question and think we know what we’re supposed to do, only to realize skimming sometimes results in miseed information.

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3. Within the lesson, many students quickly realized when a factor was squared it resulted in a “double root” and the graph would not actually pass through the x-axis at that point.

The 4 transformations seemed to causes the most disagreements. These were the ones we discussed folowing our gallery walk. However, it was during the gallery walk most students were able to adjust their thinking.

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4.i. Listening to students as they were at the poster helped me realize there was not a solid understanding of the reflection across x-axis and maybe we needed to revisit. Possibly, they are confusing with across y-axis?

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ii. A few students disagreed initially, but the convo I overheard was addressing that changing the x-intercepts was not sufficient, they looked at the graphs, then said, the functions needs to be decreasing at the begiining, that’s why you have negative coefficient.

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iii. Horizontal translations always seem to trick students up. One disagreement actually stated ‘they subtracted and did not add.” Of course, we definitely followed up with this one.

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iv. This pair of students argued over which one was right. The expaned version or factored form. Simple, graph the new equations and compare to see which one translates the original up 3 units.

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A1 & A2 I believe they’ve got this one.

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B1 & B2 some confusion here due to the extra vertical line in the graphic. This student was also interchanging graph & equation in their statement.

I thought the gallery walk was a good task to overview some common misconceotions. It was not intimidating, students were able to communicate their ideas, compare their own thinking to others. I truly tried to stand back and listen. They were on task, checking each other’s work. Each station allowed them to focus on one idea at a time. They were talking math. Most misconceptions were addressed through their discussions or written comments.

Having a moment to debrief the following day highlighted the big ideas students had addressed the previously and reinforced the corrections they had made. This was so much more valuable than me standing in front of the room telling them which mistakes to watch for. Their quick reflection writes revealed majority have a better ability to transform the functions, which was my initial goal for the gallery walk. A few still have minor misgivings that can be handled on an individual basis.

No jumping in, silent/listening, no repeating & my win for the day #ppschat

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Our #ppschats the past few weeks have brought some a-has and good reminders for me.  Here are a few adjustments I am trying to mindful of:

1. When students are working in small groups, I have often jumped in to their conversation when I heard them going in the wrong direction.  Of course, my intentions may have been to redirect them. 
 
Needed adjustment:  Be silent and listen.  Giving them space to muddle through their own thinking without jumping in and telling them what I think they should think.  A key for me may be to keep my tablet, clipboard, post-its to jot down notes of their conversation.  Key points to reference back to later.  Jot down questions I might like to ask.  Not sure when to ask /share, maybe as a quick revisit before the end of class?

2. In my efforts to “value” what a student shares, I often find myself repeating. Afterall, those softspoken students need to be heard, so I repeat it so classmates across the room “hear” them.  Oh, no.

Needed adjustment: Ask student to speak up so others can hear.  If they are intimidated, offer an encouraging word, let them know you like it, find it interesting or you want others to hear it.  When I repeat, I am causing others to not listen because they know I’m going to repeat.  Oh, my.  Guilty.  Who knew?  What are ways you create a listening community of students? 

3.  I may ask for volunteers and the same 8 people are sharing.  Spread the love.

Needed adjustment:  I tried to be very purposeful in sharing this week.  In geometry, I picked a problem several seemed to have trouble with.  I structured the task with Know: what information is given, Notice: what do I notice about the diagram? Other information I can use to move me further?  Wonder: What other measures can I determine? How can I justify my reasoning? 

Students took couple of minutes individually, to jot down a couple of things in each bullet, then in their groups of 3 to discuss.  I asked each group to pick 1 thing they felt was important to share.  Yep.  Good ol’ Think-Pair-Share.  As I went around to groups, I arrived at one who said…they already shared ours, so I used the suggestion from PPS to +1 on the board.  It seemed that others were really listening to what was being said.

A win in class today as I gave students a diagram with no questions.  They noticed/wondered and it was a statement from a student that 2 chords were congruent.  When I asked them to convinece me…their statement was quite fuzzy ending with “it just seems like they would be.” I challenged the others to prove or dispute the statement… 

“Oh yea” high fives, “we got it!” & “you’re genius!”  Students celebrating something they hadn’t seen before. 

What was even better, the way they justified their reasoning…all different, not one that I had seen myself.  And that is why I don’t care for the answer key as the answer key. Students sharing their strategy and each confirming the others.  Hearing them say ‘that’s cool’ to another student’s strategy.  Engaged while looking at a different approach.  I truly feel the take aways from a single problem approached this way is valuable.  It was a productive day.