# Tag Archives: modeling

## Car Jack Problem – Math Modeling

My colleague Dave Casey designed the initial lesson.

I enjoy this modeling activity.  It’s based on developing a relationship between the width and height of a car jack – which just happens to be a perfect shape to be modeled by a rhombus.  I bring a car jack, but also like to show these videos because the jack opens a little slow for the students to get a great feel of the movement.

Or check it out in double time

Check out the cool graph a scissor jack makes (height as a function of width)

Here is one of the guiding questions I use – it’s also the big thinking moment that I make sure to stay very intentional about:

Is it more difficult to raise the car in the beginning or the end?

I phrase it that way as to make students have to define what it means to be difficult.  Some may think the end is more difficult because it takes longer to lift the car, and others think the beginning is harder because it takes more force (since you are lifting the car more).  Either way, they need to resolve what it means for it to be difficult.  Here is another way I phrase the question:

Let’s say 16 of us each signed up to move the jack screw 1″.  Would you rather be the 1st person, the last person, or does it not matter?

Here is an activity handout you can use.  I didn’t use it – but it was nice to have for students who were absent that day.

I start by talking about the shape.  I have the students draw the shape and we make conjectures about the key characteristics of the shape.  The students have graph paper, so I recommend a uniform figure – I tell them to make it eight gridlines long, and +/- two gridlines up and down.

A critical point here is let the students know that the car jack is actually not a rhombus because the sides don’t actually touch.  So we are modeling the real car jack as a rhombus even though that is not actually the shape.  The rhombus is a mathematical model of an actual car jack.   Even the word “modeling” itself implies that we are not working with the real thing – rather a model of the real thing.

To improve the visual I usually draw a representation on the board of the jack close to starting position and ending position.  I add the length of the sides as 8″, and then I ask the these two questions:

1.  What is the largest width we need to consider?
2. What is the shortest width we need to consider?

I then put them at whiteboards and tell them the whiteboard number they are at is the width of the car jack they need to calculate the height for.  I also tell them to pick one other width.  I do not tell them how to do it – just what to do.  I’ve done this with classes that knew the pythagorean theorem and used it, and I’ve done it with classes that didn’t and we used this a vehicle to reteach the pythagorean theorem.

Common Mistake Alert:  A few students will anticipate a linear relationship and just quickly make a table that goes 16-0, 15-1, 14-2 etc.

Once they have the table, I have them graph it.  I show the Desmos graph, talk about how to write the function.

Once the graph is up, and the table is complete, I give them a couple minutes to consider when is it the most difficult to lift the car.  I survey the entire room.  Once each student is on record with an opinion – moving them into a talking point on the following prompt:

“It doesn’t matter.  It will be the exact same difficulty to turn the crank at all times because the car’s weight doesn’t change.”

Once the talking circle is over I am generally satisfied that they have all had sufficient opportunity to think and discuss.  I go around the room again and tally the results of the talking point and have a group discussion, calling on various students to give their reasoning.

I also like to bring up the slope of the lines, and ask them “What does slope represent in the context of this problem?”.   Between the first and second person is the largest slope – what does that represent in this context?” (By the end most of us are thinking of the scenario where 16 of each are each going to turn the screw one inch)

My goal by the end is to get the students to see that the car moves more in the beginning than in the end, and understand how that validates arguments for the beginning or end being more difficult based on how the word “difficult” has been defined.

Here is a students work – but a lot of the great student thinking that happened here today was not caught on paper.  I didn’t take any pictures of the whiteboards.

Regrets:  I never had them write their arguments for when they thought it was the most difficult to use the Car Jack.  At least not formally.  I need to do a better job with the written communication piece of modeling.

Cheers!

-B

And even my textbook thought this was a good problem:

## 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.