Lego Toys and Optimization

So playing off Fawn’s Funky Furniture…students decided today they would make Lego Airplanes and Lego Unicorns. 

Here’s part of slide from our discussion.

Dummy me, erased other equations and inequalities they had shared when I asked for a function to describe our data. 
In this situation, we were ONLY looking at the number of large Legos used and how our total airplanes constructed would affect our total unicorns.

Good ideas came out in our discussion and almost every single student was on task.  They noticed several things in the data.  One student wanted to use our “vertex form” strategy (thanks mr. Waddell!)  to write the equation.  But they were not sure how to handle the repeated numbers 8 and 6 to find their rate of change.

Here are some of the suggestions from them…

What was great about this lesson, I didn’t correct them.  Their classmates were confirming or disagreeing.  This is huge because many of the students in this particular class entered the year not confident in their math or not liking it at all.  They can reason very well and have so much ability when I let them think on their own. 

Here was their data graphed…and one of the equations for the boundary line.

We were unable to finish the lesson due to a College & Career Counselor visit, but tomorrow, I look forward to seeing their work with considering only the small Lego pieces.  One student even brought up…but you can’t make… and I replied, “exactly…that is where we are heading, you are just a step ahead!”  What will they notice when we graph all data together?!?!?

My hope is to give them a chance to really understand the equations, make those connections for themselves without me saying Look at this! Or do it this way. They are such amazing, hard working kiddos…they will find success in life…

Reflection on Previous Post -Equations of Lines

As I thought back on the lesson, I wondered why so many students had trouble with placing the axes on the 3 graphed lines.  As I just read @wmukluk’s comments in the previous post, I realized the lesson focus was not on graphing.

I wondered, did I miss that part somehow?  One of the targets was to Identify and use intercepts.  However, nowhere, except the assessment task, were they asked to do anything with a graph.  They were asked to identify in the sort, but never asked to use the intercepts.  In the materials section, it stated, Graph paper should be kept in reserve, and only used when requested.

Students had graphs on their whiteboards as they worked, but only a hand full used them in their discussions and even fewer offered them as a strategy to confirm.  Now, as I look back, I find this interesting because more often, students will choose a graphical method over algebraic. Hmm.?.?.

In our wrap up discussion, we actually graphed possible equations for a given equation to show a rectangle was formed. 
I wonder if I add the element “Verify your card sorts by showing graphically.”   Though, I am still not sure this would help the fact that they had trouble placing the axes.

I wonder what type of responses I would receive if they were given multiple copies of the same sloped line, but asked to place the axes with the origin in different locations…how does this affect/change the equation of the line.  Which could lead to great discussion on function tranformations.

Equations of Lines FAL

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

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. 

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?

Purple Circle Card Sort

Last spring, I placed equations of circles after distance between 2 points.  The idea came from a mini-investigation in my Discovering Geometry book (formerly Key Curriculum, now Kendall Hunt). 

Earlier in the semester a new colleague shared the success her students experienced with the Formative Assessment Lesson Equations of Circles 1.  I decided to use this lesson…

In my early geometry class yesterday, we literally stared at circles.  It felt like a wasted class.  No matter what example I referred back to, or what question I asked, it just didn’t work.  Thankfully, I had planning immediately following and I was able to reflect very quickly.  For my last geometry class of the day, I adjusted my sequence of leading examples.  Reviewing our previous work from last week. 

The remainder of the lesson went smoothly.  A quick white-board quiz at the beginning of class today allowed me to address some small errors.  Once again, I had them create their own notes/examples in their INBs.  Yes, a few are still lacking, but the majority are very thorough in what they are including.  Asking questions about specific what-ifs, like one student brought up none of our examples today had a center located at the origin, so I asked the class if they could remind her.  Several went on to include a similar example on their page.

The lesson continued with a collaborative pair.  They were given 12 equations to sort by center and radius.  There were 4 blank blocks in their grid that required them to create their own equations.  At the beginning, some were “cheating” so I stopped them to remind them 1 person picked a card, explained why they were placing it, the other person had to agree and understand before taking their turn.  They are getting better at disagreeing and telling why when their partner is making a mistake.

Their assignment was to create an artistic picture incorporating 5 different circles and listing their equations on the back. Short, sweet, simple.  Can’t wait to see them.

Dice #tlapmath

Well, @druinok shared this amazing find from Dollar Tree!

First of all, let me just say, JEALOUS!  But soon after, she shared an idea from #tlapmath Walk the Plank ideas to make lessons Pirate-Worthy.  So, being on the road home from a visit to my brother’s near St Louis, I had plenty of time to think. Hmmmm.

Here are a couple of  thoughts. 

Roll the dice, generate 3 sets of coordinates.  Prove what type of triangle they form.  Find the perimeter. 

OR roll 6 sets of coordinates.  Which triangle is “closest to being equilateral”?

Create coordinates for a quadilateral.  Prove what type of quadrilateral.

In a group, compare your quadrilaterals.  Who has the one closest to being a square? Rectangle? Or other polygon. Why?

Use your 8 coorindates and can you arrange (x,y) pairs to create quadilateral closest to _____?

Pythagoras, His Formula and a Teacher Who Didn’t Teach

So a simple lesson today. 

A segment on a grid and asked students to find the length of it. Yep, most sketched in their right triangles and pulled out the Pythagorean Theorem.  But what if we don’t have a grid? How can you find the distance between the two points without graph paper? Or if one of your points is (543, 97)? 

After their sharing, while practicing, some wondered, “Is it okay to use the slope if its in lowest terms?”

Good question.  Does it matter?  What could you do to determine if it matters?

And their suggestion:

…with their finding.  Makes me smile when they answer their own questions.

Best part of lesson today?  Their INB notes.  A post by @justinaion made me wonder how I could be more purposeful in student notes.  Today’s notes…after completing the lesson, students put their whiteboards away, and created their own notes.  Some had step by step instructions.  Others had pictures drawn, paragraphs with a couple of examples.  But in the end, they wrote what was important to them. 

I loved the question a student asked while walking out the room.  “How am I supposed to find the distance with three coordinates in space?”

My response, How are you supposed to find the distance with three coordinates in space?  A smile with an a-ha look on her face…you just…yes, child, you knew how all along.

A day when I didn’t teach a thing but my students left knowing something new (well, except for the kid who sulled up because I wouldn’t TELL them ‘the formula’, use your device) …its been a good day.

Even better, a FB post from a former student-

Just Tell Us the Answer! No.

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…

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? 

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.