Trash the Lesson

Literally. We trashed a lesson today.  It was a lesson I’ve used for a long, long time.  Triangle Midsegment Investigation.  In recent years, I’ve used geogebra to test/ explore student findings.

It lead a small group of students through realizing midsegment is parallel and 1/2 the length of the corresponding base.

Well.

For whatever reason, it was not sinking in. 

I was getting frustrated.  On the edge of sarcasm.  I remembered a comment from my colleague last Friday, to remember to have fun with them.

The lesson wasn’t clicking with them. I wanted to trash it.

So, I picked up a student’s paper, wadded it up and instructed the entire class to follow my lead.

Noone was allowed to leave class until they threw a paper wad at someone.

They laughed. I LAUGHED.  It was totally worth it.

Then I grabbed some poster graph paper and they opened their INBs…we got the big ideas and went on to use them in a few problems…successfully.

A couple of times students made comments how all we had been doing was coming together in the new problems. They were seeing the connections.

Yep. Sometimes we just need to trash the lesson and move on.

Painting a Bridge

In my Algebra I, we are looking at parent functions. Students said this week was quite easy, they felt they were doing 3rd grade work.  But I assured them
recognizing the parent equation and making connections to the parent graphs may seem easy, but it’s a lead-in to more intense math!

We’ve done several data collections throughout the semester, mostly linear, a few quadratic and exponential.   But today we took a look at rational with Painting the Bridge, which is embedded in a MARS lesson. 

Students are asked to sketch the relationship x:# workers and y: # hours each works to complete the given job.

Those are a good overview of what we saw.  I allowed students to ask questions about things they wondered about others’ graphs.  At first glance, a couple of the graphs may look odd, but given the chance to share thwir thimking, student reasoning made perfect sense in the real world.

Though I didn’t have an actual student create this graph, I included it on the board.

I followed the suggested questioning in the MARS lesson, which led most students to some A-ha moments.  What does point Q mean? Points S? Does it make more sense for the graph be solid or dotted? Why?

As a data collection to follow up this discussion, we picked up erasers. One student held a cup in their dominant hand and picked up one eraser at a time and placed it in the cup, we timed.  Then another student helped.  Continued adding workers and it eventually became too crowded, they were dropping erasers and slowed them down.

We compared the shape of our scatter plot and decided maybe exponential or quadrant 1 of a rational (inverse) function.

The calculator power regression resulted in
y =76x^-1.  Which gave us a chance to discuss that -1 exponent.  How it meant the inverse of multiplying by x, which was to divide by x.  So we graphed y=76/x. Nice. They were seeing the connection to our Painting the bridge discussion. 

Oh wait, how many erasers were we picking up? 78. Not bad, huh?

My goal is to give them a concrete data collection for which they can access and connect back to the math.

To end the day, they asked if they could draw a graph on the board and everyone guess the parent function name.  Sure.   They were on task and engaged so I was fine with it.
They began graphing the endpoints of their graphs,  so their classmates were finishing the graph and naming the function. It was humorous. But again, they were engaged.

I love these kids.  They were my favorites today.  It’s been a tough semester at times, but I want to end these last weeks strong. I want them to leave our classroom having grown in confidence and changed their attitude toward math.  That’s my goal. 

Systems Linear Programming

As an intro to this lesson, I shared this scenario…

You are bidding a contract for Company ABC.  The order is for 12,000 dozen of a product and needs to be completed within 3 months.

First student question, why would anyone need 12,000 dozen of anything?  They felt this amount was a ludicrous number.  (After many summers working at Fruit of the Loom, I knew this was within reason, but a nice discussion anyhow.)

Well, is it?
According to Apple Press Info, if it’s as popular as  iPhone6…no. 

First Weekend iPhone Sales Top 10 Million, Set New Record

We figured if we had equal distribution among all 50 states, this was quite doable.

Do we have the man hours to fulfill this order in 3 months?

There are…21 (bc there were 21 students in class today) workers in this particular unit…who work 8 hours per day, each of you can complete 10 products in 1 hour.  Yes, I just made these up, but that’s what we worked with.

After a few minutes, we started sharing processes, quickly a bit of an argument – why did you do it this way? Should you have….?  Others arrived at the same solution, but with varying approaches.

I could kick myself for not taking a picture of their suggestions.  Some nice verifying one another going on.  However, they were not sure what those values represented…they could get the “right” values but lost when I asked for a label.

Watching students grapple with the numbers, made me realize how far out of reality we’ve taken students math skills.  I just want to do a better job of letting them make sense of problems themselves.

We determined it would cut it close, but we could likely finish this job, maybe requiring a bit of overtime to meet the deadline.

Now, as we make an offer for the contract, what are costs west consider? This leading to an idea of our linear programming. 
Wages, materials, utilities, insurance, packaging,  shipping, etc.  One student even said, there’s a lot to consider. Me, knodding, yes.

Is this a great example intro. Nah. But I feel it’s a nice way to show students there are many options a company must consider prior to the contract, production, sale.

Now, to the hard part.  A variety of students, some with adequate graphing skills, others struggling to find the line x> 3.