## Well Flow Rate – “Real World” Math

Keeping with the easy definition of “real world” being “math someone needed to do at their job” – I actually prefer the term “industry math”, but regardless here’s the question that needed to be answered:

My wife’s working on a project where they are building a house on a property without a well or access to city water.  So they dug a 425ft well that was 8 inch in diameter.  When they finished the dig at 3:00 pm it was completely dry.  The next day, at 7:00 am, the well had filled with water up to 45ft below the surface.

How fast is the well filling up in gallons per minute?

They wanted 1.25 gallons per minute.  Do they need to dig another well?  Wells are about \$50 a foot – so yeah, they would rather not dig another one.

(digging the well)

(well finished! That’s the cap they left on it)

They used a falling rock to determine how much of the well had filled with water.  Well (not the noun), they also used a cell phone connected to a string but isn’t that just too obvious?

Here’s the video of the rock drop:

Well Rock Drop

Cheers!

## “Real-Life” Annulus Problem!

I just can’t type “real-life” without quotes because I’m yet to resolve what “real-life” means in a math class.  But for this post it means – “Math someone needed to do at their job”.  And in this post that someone is my wife – who much to her distain and my joy – does a lot of geometry as a project manager at a construction firm.

So here’s what we need to know:  How much cement is needed to make that border?  We need the answer in cubic yards because you buy cement in cubic yards.

The image below provides some of the context for the problem – The cement border is being used to circle an existing tree.

I was actually surprised how open the middle was on this problem.  Yet students used two main strategies.  The first was the standard subtract the areas and multiply by the height, or they subtracted the volumes of both cylinders directly.  The second was the find the perimeter, think of the wall as a rectangle (dimensions of 2*pi*r X 1′), find the area of that rectangle and then multiply by the height.

One student using the second strategy used a radius of 14ft and got a solution of 16.2 cubic yards.  He told me he knew the answer would be a little too small because of only using the inner radius.  Another students used 14.5 and got the same solution as the area subtracting syndicate.

Converting from cubic feet to cubic yards is a great time to practice your perplexed I wonder why you divide by 27? face.

By the way the wall ended up costing around \$35000.  I can’t believe how long I would have to work to put a wall around a tree 🙁

Cheers!

– B

## Quick, Fun, Artistic Geometry Review

Here you go.  It’s a review of the fundamental vocabulary of geometry.  Basically students need to use all of them to draw a picture.  It is a simple idea and it works very well.  When I described it as “quick” in the title I was alluding to the fact that students are quick to understand what they are suppose to do, and they just go for it.  You don’t have to do much from a teacher standpoint – just get out of the way and see what their creative energies do.   It was made by my department chair – who along with everyone else in my department is an amazing, blogless (is that even a word?) teacher.  I suppose that’s where I come in.

The Goods:

Review Drawing Assignment

## Adding Value – A Starbucks Lesson

This lesson could be better – but it was my first shot at it. I still definitely recommend it because of the great math conversations we had around convenience and adding value. Here’s the driving question I used:

What is Convenience Worth?

But it didn’t work as well as I wanted it to because we were really only looking at the convenience of the k-cup vs. ground coffee vs. going to a coffee shop.  It’s a cool driving question, but not necessarily the focus of this lesson.  This lesson is based more on “adding value” – where value is defined as what people are willing to pay for.  Starbucks added value to it’s coffee grounds by putting them into K-cups, and that is why they are able to charge more for it.  So how much value did they add?

So here’s the lesson:

Angle 1:  K-cups vs. Ground Coffee

Starbucks makes K-cups and ground coffee, and both sell for \$10!  [excellent] So are they the same price?  The K-cups are a package of 10, 12.5 gram cups.  And the ground coffee is in a 340g bag.  Starbucks is able to elevate it’s price for K-cups because they are more convenient than traditional brewing methods.  So what’s the value of that convenience?

Angle 2:  Coffee Shop vs. Brewing At Home

It is more expensive to buy a cup of coffee at Starbucks than it is to brew it at home.  Yet people still do it.  What does that cost difference tell us about the value of convenience?  Here’s one of Starbucks menus:

There was definitely a period at the beginning where students did not know how to begin – or perhaps what they were suppose to do.  It took some time for them, and to extent me as well. to define the question.  I told them that it is natural in any problem solving situation to spend time defining the question, framing assumptions, setting up your analysis.

My teacher moves were as follows:

– Write about a time when you paid extra for convenience.  Talk it out.

– Ok so the value of convenience depends on how convenient the thing is versus the alternative.  [Future project could be to quantify convenience itself, rather than only focusing on Starbucks pricing].   That means we should standardize the convenience we are analyzing.  So let’s look at convenience through the lens of Starbucks pricing.

– What does the pricing of Starbucks coffee tell us about the value of convenience?  How could we use Starbucks to quantify its convenience.  What information would we need?

– Give various pricing data

– Ask students to come up with a value for the convenience of the k-cup vs. ground coffee, or the value of going to Starbucks versus brewing at home.

– Helping individually.   My most effective question was “For \$10 you get 340g of ground coffee.  How much would it cost to get 340g of coffee in K-cups?”

It is also an interesting conversation that ground coffee and whole bean are the exact same price.  So Starbucks is saying that the convenience of having the beans ground for you adds no value to the product.

After the activity we read an article from Proffer Brainchild titled “ADDING VALUE:  A Lesson From Starbucks”

Here are some examples of student work.  The first example was the highest quality paper I received.

Lesson Learned:

It’s clear that not all students understood what I was asking – and so I concede that “What’s the value of Convenience?” was not a good driving question for this particular activity. I do see it that prompt as a great math modeling problem where students try to quantify an inherently qualitative thing. But that’s not what they were doing here.

Next year I am going to try and focus first on the k-cup vs. ground coffee prices and stick the economics concept of “adding value”. How much value is added when Starbucks puts their grounds in K-cups vs. ground coffee? And this is not just about the calculations, it’s about the argument.

## Grazing Goat Problems (Sector Area / Trig)

I used to give this worksheet but now I draw the picture on the whiteboard and give the question orally.  Then I provide them a couple of minutes of silent time to work alone in their notebooks.  From there I group them and they work the problem at vertical erasable surfaces.  Here’s the basic question:

A goat is tethered to a 25 foot rope attached to a rectangular barn at point A.  What is the total grazing area available for the goat?

I leave them room to experiment – especially as to how to handle the situation when the goat starts to walk around the barn.  Last year I took them outside, gave a student a rope and they played the goat, with a picnic bench as the barn, and modeled the situation.  I didn’t do that this year for reasons I hope aren’t my own laziness.

But my goal is not just sector area, it’s also trig functions and thus onto to the triangular barn!  I mean – most barns are triangles aren’t they?  Here’s the next problem:

Some student work examples

The different iterations on this problem are pretty much endless:

I did this lesson the day after the students learned how to find the area of a sector.  I followed it up with another grazing goat problem the next day where they were working on paper by themselves.   It’s not like they knew exactly how to do the problem the next day  – but it was that they all persisted, asked questions, and got to a final conclusion.  So I’m happy.

I like this question because although it is not real world – it is about real things that students understand.  I also think it finds itself into the flow of a classroom where students find it appropriately challenging – they believe they can do it but yet the solution doesn’t seem immediately obvious.

Oh, and I teach in a community with a lot of agriculture and the students told me that you really don’t tether goats.  Lucky for me that didn’t stop anyone from solving it.

## Letting (making?) Them Figure It Out

I introduced exponent properties this year by writing this on the whiteboard and asking them to make observations about it.

Then I got more direct about the kind of observations I was after

There was some back and forth as we discussed what it means to simplify and how to simplify.  I was short on answers and kept the discussion geared towards observations.

I lined up another example where there was a negative exponent

From here I gave them a little half sheet with six expressions on it – grouped in them in groups of two – and off they went simplifying (do I need to add that I had them working at VNPS???).

I spent less time this year than normal covering simplifying expressions and I didn’t notice that students were any worse at it – so I’d say this approach was a success.  The next step I suppose is to apply it to other procedural concepts.

How many of the mathematical practices can we get at when we introduce concepts based in procedural fluency?  That question is swimming through my head as I begin to teach factoring.

## Geometry – First Semester Review Worksheet

I know there are some people out there who could use this.  Especially as finals near.  It covers parallel lines cut by a transversal, vertical angles, complementary / supplementary, rhombus, special right triangles, polygon interior angles, and transformations.

This was created by my colleague Dave Casey – who is definitely at the top of my “I wish they had a blog, were on Twitter, and were presenting at conferences” list.      This is just one of his worksheets – you should see his activities, amazing stuff.

The Goods (aka: the pdf)

1st sem review chart

## Standard Form Equations With An Open Middle

#CMCN14 was lights out good this year.  Amongst the many things I learned new – were a ton of reminders of things that I used to think about but had let slip.  One of those things was the importance of an open middle, where students have a defined beginning and ending, but how they get there is largely up to them.   During Dan Meyer’s talk he challenged us to find an open middle in the routine, procedural fluency building exercises students get.  Most of the great problems have it – but it is a nice tool for tipping the scale for our procedural problems towards a deeper understanding.

Here’s the typical – pretty much closed middle – version of a problem about standard form:

Find the slope, y-intercept, and x-intercept of the following equation in standard form:  3x – 4y = 20

Here’s my one up

Write the equation of a line in standard form where the both intercepts are integers, and the slope is a fraction.

We could really be here all day playing with these

Write the equation of a line in standard form where the x-intercept is a fraction, the y-intercept is 7, and the slope is a negative fraction.

We can even get at MP3

Explain why it is not possible for the slope and x-intercept of a line to be an integer, but the y-intercept a fraction.

Lastly – the Asilomar conference grounds are so amazingly beautiful.  Each tree, slightly beaten from the ocean breeze, stand in stillness as perfect landmarks to perseverance.  And as the sun begins to set, and that air begins to cool, and those stars begin to show – it’s hard to believe that it’s all just the backdrop to a professional development experience.  It’s humbling to be there – I mean you’re walking from presentation to presentation with a program booklet offering the intellects and energies of 200 amazing educators.   But you only get to pick 5…  good luck with that.

## Rotations Worksheet

Some days you just need a worksheet.  So here’s one.  I took all the problems from math-drills and just spaced them out better.  I love using matrix notation so always have my students do it.

The Goods: (aka:  The pdf’s)

RotationsWS

## VNPS & VRG Tango!

So Kristen and I are taking dance classes – so any three step process feels like a tango to me.  This activity has nothing to do with dancing (sorry).  This is one way that I use Vertical Non-Permanent Surfaces (VNPS) and Visible Random Groupings (VRG) for problems where we are working on our procedural fluency – whether it being solving equations with fractions, or factoring polynomials.

One down side of VNPS is that students believe they are finished when the correct answer reaches the whiteboard, rather than when everyone in the group understands how to do the problem.  The one student who knows how to solve it just goes up to the board and solves it.  There are a couple methods to help give students less places to hide in a group – here’s one of those methods:

Pick three problems that are similar.  Tell students that there are going to be three rounds.  In the 1st round they will be in groups of 3 (randomly assigned of course).  Then in the 2nd round they will be in groups of 2 (randomly assigned again of course), and lastly they will be back in their seats working alone on paper.  That’s it.  Throw whatever standard you want at it.  I believe it works because the thought of eventually being alone gives them the extra motivation to learn from their group.

Here are the three problems I started with in my geometry class, where my goal was to give them practice multiplying polynomials, solving by factoring, and using pythagorean theorem:

From what I’ve seen – the knowledge that they are eventually going to be alone makes those students who usually look for a place to hide in a group more apt to contribute and learn from the first two rounds.

I love to pose the question by waiting until they are in their groups and standing at whiteboards, then I pose the question on my whiteboard in the center of the room.   It’s a nice math coach moment versus math teacher moment.

Here’s a smattering of whiteboard activity from this day:

.