Category Archives: slope

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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Setting Personal Social-emotional Goals pt. 2 #julychallenge Post #17

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This morning as I responded to a commented from @bpagirls on my post about an Essential Questions Board, a thought hit me, so I typed it in my reply so I wouldn’t forget…

… I have just realized as I type, why not add a spot for personal-social goal-setting on my organizer for each student to set, write and reflect.

It stems back to this post and one of the 14 ways to think about good teaching post, 3. Include social-emotional learning goals as well as academic goals.

I got that I needed to do this, but I was not quite sure how to set and record these goals.  My plans are to include a place on the back of our unit organizer students receive at the beginning of each unit.  These are formatted in a booklet style to fit our INBs.  Students can set a personal/social goal to focus on for the duration of the unit. Ideally, following the SMART goal format.  Commit to it by writing it on their organizer.  I will ask to see it, but they may choose whether to share with a peer.  Wouldn’t it be great to have accountability partners for the unit? 

Throughout the unit or even at beginning of class, ask them to read it to themselves.  Maybe even allow someone to share their progress.  Revisit them as we end the unit and write a brief reflection:  How did I do?  Did I meet my goal?  If not, did I at least move toward it? What do I need to modify?  Follow the format: 2 stars and a wish for their quick-write reflection.  Celebrate their progress, maybe through our Shout-Out Board (more on that later).

I realize this type of goal setting may be tough for students… I am hoping after completing this task, it will allow for students to generate ideas.

Initially, I think goals can range from:
Improved / good attendance
Be to class on time
Being prepared for class
Completion of assignments
Asking for help
Asking questions or participating in class discussions.
Attend tutoring if needed
Work in a group with people I don’t know.
Share my ideas in class
Share my assessments and progress with parents/guardian
Choose better practice/study options
Listen to others ideas
Evaluate how my choices are impacting my learning.

Here is a sample of the back of my unit organizer.  I plan to insert personal goals below the unit reflection.  Here is an updated version of a complete unit organizer and student assessment tracker. Feel free to modify for use in your personal classroom. Thanks to Crazy Math Teacher Lady and Math = Love for inspiring through their posts?

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My next task is to locate a fill-in the blank for a SMART to include on the first unit.  Kind of a madlibs style to get us started.

If you have a system in place or use LIM or AVID in your school, I welcome input and suggestions.

Equations of Lines FAL

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So this is from a Formative Assessment Lesson from MARS site a couple of weeks ago. 

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As I think back, the pre-assessments were very lacking, some even left blank or only minimal scribbles.  Their post-assessments were much better.  They were more confident in manipulating equations to a similar form so they could more easily compare, picking those that were parallel and those perpendicular.

However, a handful really struggled with the given graph in the lesson.  It had 3 lines without the x- & y-axes. 

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Part of the task asked them to place & label the axes on the graph.  Some actually drew the graph and all lines forming the rectangle outside the given graph, then transferred their work to the graph.  Interesting.  It seemed easier for them to graph the entire thing than to simply add the missing information.  I wonder why?

Several a-ha’s were noted throughout the lesson.  Students thinking opposite slopes would be perpendicular, how to find the x-intercept, in the beginning naming equations like y+4x=3 and y= 4x+5 as parallel.  It was definitely a task where I had to bite my tongue, let them struggle a little, then ask questions without telling them how I did it.

As I look over the first sort, I recall several having trouble getting started simply because the equations were in different forms.  Once they realized putting them in similar forms would allow for easier comparisons.  I gave them the categories for the sort, but I wonder how they would have sorted them had I chosen an open sort?  One reason I chose to use the lesson’s headings was because a couple served as quick reviews of checking to see if a point was on the line and how to find the x-intercept.

Would a better assessment be to create equations (not in slope-intercept form) to fit it given categories?

Just Tell Us the Answer! No.

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I was curious how my Geometry students would handle Fawn’s Staircases and Steepness Task.  I will be honest, since it was high school, and slope is no longer an introductory concept, I was afraid it would be too simple…

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I was wrong. 

The discussion and sharing were so worthwhile!  This is a keeper task.  I shared it with other geometry teachers, fingers crossed they’ll give it a try!

About 1/3 of students used protractors.  When asked why they chose to measure angles…some replied it just made sense (was that intuitiveness coming through?), one student stated it took the math out of it? Huh? He explained if he had chosen to measure height and base, there was more of a chance of making a mistake…twice because there are 2 measures.

Was it because it was open, choose your own measure tool/strategy, that allowed them to think without it being so ‘mathy’???

1/3 of students measured the segment (hypotenuse) length.  But when asked if it confirmed their rankings, several realized they needed to adapt their plan.

The remaining students used a classic slope height/base.  Some wondered if measuring each step would result in the same value as the entire staircase.

Here are measures shared by 3 students.  The top angle measures confirmed her ranking, but a classmate wondered “if its least steep to steepest, it would make more sense to me for the steeper to have the larger angle measure.”  So the discussion led, where did she get her measures and how are they related to student CWs? 

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There are some errors in measurements shown.  But what made this task so great, they wanted to know who was right.  And were furious when I would not tell them.  I told them I didn’t have a key, that they needed to revisit their measures and be ready to defend their rankings with measures that confirmed.  Could they critique their classmates reasoning?

With a bit more sharing, they all agreed there was a relationship between the rise/run, step height/step base and the angle measures.  I asked if they had ever heard of Trig Ratios.  Some said yes, in 8th grade, so hard! Others stated it sounded difficult.

What is trigonometry anyway? Lets break it down.  Tri-gono-metry.  They recognized tri as 3 and metry as a measure, but gono is from gonia which is angle…3 angle measures…hello! That has to do with triangle measures!  Connecting it back to a sketch a student shared and another said there’s mini triangles in each step of the staircase. 

Anyway, I am rambling, but I shared the idea of tangent and explained that it is simply the ratios they used to measure their steepness.  We did a few examples, connecting to angle measures- using 45º as a reference. Thinking of our angle as a hinge on a door and looking at different ratios for different angles. 

  I hope to pull some of their examples and share more in a later post.  But when a student tells you thank you because you made it simple for them to “see it”…that makes it worthwhile.  In reality, I didn’t make it simple.  It was already simple.  I only provided a task (thanks, Fawn!),  that helped them see the connection for themselves.

Linear Equations Card Match #made4math

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Let me first say – I did NOT create this set of cards.  I received them in a session at KCM about 3 years ago.  Kudos to whomever they belong. 

 I was looking for resources to use during my RTI and ran across a box I had used in the past. 

LinearEquationsMatch – the file of the cards.

You can do several different sorts with them.  POINTS-SLOPE, POINTS-EQUATION, GRAPHS-EQUATIONS, etc.

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I have each complete set on different colors of cardstock, so I can have several sets out at once, but none of them get shuffled.

INBs – A New Adventure – “Flip 4 Answers”

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INBs – A New Adventure

All summer long I searched for ways to improve my literacy in math class – I learned so much chatting with my tweeps during our #lit4math book study.  It helped me redefine what literacy is / should be in math class – not just about reading.  And writing.  Not about creating something completely new – but improving what I already do to emphasize communication – discussing – giving students opportunities to make connections.

As I ran across various posts on the Interactive Notebooks – I knew this was something I wanted to do.  At first I had the wrong perception – thinking the interaction was between student and teacher – I struggled, wondering how in the world would I find the time to “grade” and evaluate that many notebooks efficicently and effectively and keep them in students’ hands for continuous learning???

After reading – mostly from @mgolding – I realized I had it all wrong.   The interaction was between the students and their own notebooks – to provide them with opportunities to engage with the information I gave them.  I was overly excited when I saw @mgolding would be presenting at #TMC12 – then crushed to find out my session was at the same time. boo. and I would miss out.

Listening to conversations that came out of her session and reading more once I returned home only confirmed my decision to move forward with INBs.  My science colleague had decided to pursue this learning tool as well – so grateful to have an in-person to collaborate/share ideas with too!

During the first Global Math Department meeting, she brought calm to me – answering so many of my questions in her session that night – thank you, thank you, thank you @mgolding!!!

I have begun my venture with INBs.  I feel a bit stronger in one class than the other – but I have been upfront with my students – this is a learning experience for me as well.

Some things I’ve quickly learned:

  1. I MUST keep my TOC up to date – its easy to get off track if I don’t!
  2. I MUST do the INB along with students – having completed the pages myself – knowing exactly what I want to go on them;
  3. I MUST practice any foldables / graphic organizers to make sure they’ll fit/work.  I may have a great idea in my head – in theory anyway- but I have to put it to the paper to see if it will acutally do what I need it to do!
  4. I MUST think about what I want to accomplish with the LHP assignments.  This is the one I tend to struggle with some…thankfully I have lunch with my colleague and bounce ideas to get feedback.

Flip 4 Answers

I plan to blog my list of RHP ideas later, but for today, I want to share an idea that came from my students.  Its similar to something I saw @mgolding share at #TMC12.  She had used post-its to cover hints/work/solution to an assignment she left with a substittue teacher.

When asked to create a practice quiz, one of my students used an index card to cover their work – thus “Flip for answer.”  When  I shared the student’s example, I never dreamed others would follow.  Yesterday during our first cumulative test – I oberved several others started playing off the concept.

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As I look at the sample below, a CWP (color with purpose) would be VERY easy to assign…I think on Monday – that may be a good warm-up – turn to page 12 and CWP…  positive or negative or zero or undefined, even identifying which letters model parallel & perpendicular.

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Another idea I think I’ll lean toward using for my LHP assignments – is the use of “Open Questions” – an idea I got during a book chat last fall from More Good Questions Small/Lin.  The second part of RHP 12 was an example of this…give coordinates of two points: with zero slope, undefined slope, positive slope, perpendicular to the slope in part c.

I believe the INBs require me to be more organized in my example choices.  It helps students be more focused / organized as well.  Looking through INBs yesterday – those who were having some trouble with their INBs / not completing their LHP assignments – seem to be the same (few) students who were struggling to make the connections I need them to.  This confirms to me the choices I am making – since most are finding great success.

Distance & Midpoint on a Map

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Been playing with this idea for a couple of days – here’s a rough sketch.  Rather than having students work a gazillon problems – I’ve decided to use a school map.  I ran a copy of grid paper on a transparency and overlayed on a map of school – copied, added a rough set of axes.  Placed points throughout.

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Questions range from:

  • Calculate the distance between Room 137 & Room 114.
  • Find the coordinates between Room 137  & Room 114.  What room are you closet to at this point?
  • Connect Ag, Kitchen, Cafeteria & Workshop.  What type of quadrilateral have your formed?  How do you know?   Prove it.  We have not covered types quads – but they can use their BYOD to find this information if needed, right?
  • Connect Library, Room 128, Room 116 and Room 114.  Is it a rectangle?  Or a square?  (LOL) How do you know.  This always comes up in discussion – I must say I love the “disagreements”.
  • Connect Room 145, Room 142 and Band.  What type of Triangle have you created?  How do you know?  Prove it.
  • Connect the Gym, Library and Tan Hall – Find the perimeter & area of this triangle.
  • What about having them “map” out their schedule and calculate “as the crow flies” distances between their destinations.

Anyone else have better ideas?  Other uses for this?

I’m thinking it was @k8nowak who did a scavenger with other geometric concepts.

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.

Open Questions: My First Attempt

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I’ve been reading More Good Questions and am so excited about this book!  #sbarbook study Monday nights 9:30 est. on twitter  – the big ideas so far, have been defining open questions and parallel tasks and how easy it is to create them. 

@druinok that’s what I’m enjoying too – very low stress, but HUGE dividends! #sbarbook

@druinok @jmalpass totally agree!!! I think what I’ve gotten out of it the most is rich questions don’t have to be hard for the teacher to do 🙂

Such simple, quick changes – yet great opportunity for thinking at ALL levels!

Today, on a target quiz on slope/rate of change – I made my first planned attempt to use an open question. The last question (no discussion on the first 4 ?s) took us into a great in-depth discussion.  The question was this:

Slope is 4/5. 

Give me two points on the line.

A student asked…does it have a y-intercept?  My response, Does (voice inflection) it have a y-intercept?  When I asked students for responses – I called on this student because I wanted to talk more about his question.  Student stated – its not vertical, so it has to have a y-intercept – even if its (0, 0) – the y-intercept is zero.  Good point.

While students were working – I observed their various stratgies for getting their coordinates- THIS is the part I *LOVED*!!!!  There were graphs, tables of values, slope formulas, and other strange strategies I would have never been aware of – if I hadn’t given this question!  I attempted to call on students with different strategies for getting their solutions.  Even calling on a few I knew had incorrect answers to allow for discussion.  I didn’t have to correct them – other students were able to ask questions.** 

One student looked confused as she asked, “How can we have so many points, but the same slope?”  My answer, “How can we have so many points, but the same slope?”  Another replied – “the lines are different but they have the same slope – so it makes them parallel.”  A concept not included in the objective – but I think it will stick.  On the board graph – students were able to quickly identify points that were not giving the correct slope and able to explain – usually inverted coordinates.

About 1/3 of the class struggled with where to start on this question – but my feeling is as the year goes on and they are given more open questions, I’ll see a higher number successfully attempting it.  One student made a comment on her way out – how ‘seeing other ways really helped (her) to better understand slope and how it works’ –

How difficult was it to come up with the question?  Not at all.  Level of cognitive demand – much higher that #’s 1 – 4; Level of discussion – much more in-depth! 

This idea may be something most of you use in your classroom.  I consider myself a good teacher – but my thinking has shifted – when I’m looking at examples / assessments – my thought is, how can I make this an Open Question???  I will continue to share my experiences initiated through this book!

** When a student sees a mistake another student has made – I encourage them to  question by asking, “What question can you ask them about their work/answer,” rather than tell them what they did wrong.  It gives both students a chance to reflect.