# Category Archives: algebra

## 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.

## 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.

## Visual Patterns and VNPS & VRG!

I feel like I’ve been preaching the gospel of vertical non-permanent surfaces and visible random groups everywhere I go these days.  The norm is set in my room – I pose the problem, give them a couple minutes of silent thought, put them in groups, and away they go.

Below is a pattern I made up quickly one morning.  It doesn’t look exciting – but guess what?  It’s doesn’t have to be.  It was close enough to full class engagement for me, which was due to a nice combination of:

1.  They believed they could do it.

2.  Vertical non-permanent surfaces and visible random groupings.

3.  Probably some other things I can’t quite pin down yet.

I’ve settled on these as my go-to questions for visual patterns.  I know I got the sketch the 10th idea from Fawn’s blog.  I never used to have them do that but when I started requiring it I was impressed with how helpful it was for a lot of my students when they ultimately wrote the equation.

1.  Sketch the 10th

(helps them immensely when writing the equation)

(sketches aren’t exact drawings.  I tell them I should be able to have them sketch the 1,000,000th)

2. How many blocks are in the 49th?

(too big for a table!  For students struggling to write an equation, having them sketch the 49th usually gets them to get it)

3. How many blocks are in the nth?

(I start the year asking it this way:  “Write an equation that relates the step number to the number of blocks in that step.  (another way to ask this question is:  How many blocks are in the nth step?)”

I would literally have that parenthesis in each problem, until I finally got to drop it.)

4. What is the largest step I could build with 1000 blocks?

The first extension.  My true goal here is the equation in #3.

5. How much of the sequence could I build with 1000 blocks?

A second extension.  It’s quadratic and I haven’t directly covered quadratics, so it will challenge those kids.  We have talked about Gauss addition so it is not completely out of their range.

On the whiteboards below you will see graphs because in this particular case I also asked them to graph the number of blocks per step, and the total number of blocks needed to build the entire sequence per step. I wanted them to have to graph something non-linear.  I think it helps further highlight what makes things linear when they work with things that aren’t.

They don’t go directly to the whiteboards.  I first give them about 5 minutes to develop their own thoughts in quiet.  Then I group them and they do their thing.

After class I always look at every whiteboard and judge how much of the conclusions are in their writing vs my writing.  I’m not sure what I gain from that but it is a research point for me right now.  There is a little bit of my writing on boards 7 and 5, but they are supplementary thoughts and not the main thinking that I wanted to the students to do.  Here are some of the whiteboards after the activity:

Lastly, after it was finished I had them go back to that paper with their initial thoughts and complete the problem on paper. I give them graph paper and rulers and have them make nice graphs to turn into me.  In some sense, one could think of the paper as the assignment as the whiteboard as a giant scaffolding.  But in another sense the whiteboards could be the assignment, and the paper is something that goes in the notes.  Or in another sense…

## Monomial Partners

“What’s your name, what’s your monomial?”

This is a great activity that was inspired by Matt Vaudrey’s Equation Speed Dating.  In this lesson each student gets to create their own monomial – which I constrained to having to be even and with a variable.  Then they break up their paper into three columns:  Partner / Our Binomial / Our Rectangle.  The students pick a partner and join each others monomials together to create “Our Binomial”.  Then they factor their binomial and represent it as a rectangle by labeling it’s dimensions and indicating the area.  I circulate the room and once it appears every group is finished, I have everyone get up and find a new partner.  I’m demanding here that all students get up out of their seats and move somewhere new.

After a couple rounds I started having them draw their monomial and their partners monomial as separate rectangles, and then draw them together.

I have been focusing on a geometric approach to factoring, so the rectangle column was a great addition to previous times when I have done this activity but only asked for the solution.

The column “Our Binomial” does a nice job reinforcing that a binomial is the combination of two monomials.

The Advice

Don’t require them to say “what’s your monomial?”, “do you agree that our binomials is….  “, but inspire them to say it by modeling it.  A lot of my students were saying it because I was giving them messages that anytime they get the chance to say “monomial” or “binomial” they need to take it.

Tell the students not to move onto a new partner when they are finished.  They need to wait until you tell them to switch partners.

Remind them that you are really counting on the partners to catch any errors! Because you can’t do the problems on the board since every pair is working a different problem.  “And yes, you are the partner I am counting on for someone else.”

“What’s your name, what’s your monomial?”   No that’s not a pickup line for Speed Dating…  or is it?

## Explaining “Explain”

Here is a released question from Smarter Balanced (I even answered it!!!):

Ok I lied.   That was an edited version of a Smarter Balanced question – here’s the original:

Now all of a sudden my answer doesn’t seem sufficient anymore 🙁   Here’s my best guess at a popular student answer:

This word “explain” is keeping me up at night lately.  In this problem I’m not sure adding the word explain to the end gains us enough to warrant it.  To achieve Common Core we can’t just throw the word “explain” after every problem we did last year and call it a day.  By the way I’m not saying that’s what the Smarter Balanced Consortium did on this particular problem.  But this use of the word “explain” does bring two things to mind:

1.  It’s hard to explain your mathematical reasoning without access to drawing diagrams.

2.  If we ask students to explain something – it should be something worth explaining.

With respect to #1 – my focus this year has been on explanations through multiple representations.  Basically I have students make connections between diagrams, tables, graphs, mathematical symbols, and written descriptions.  I feel underwhelmed asking students to explain with just a typed explanation.  I want explanations to look like this:

In the student work above – image if it was only the conclusion.  Look at how much would be lost.

There are certainly better answers to the rectangle problem from Smarter Balanced than I offered up here.  I actually really like the problem itself, I just do not think having them explain it gains us much versus just solving it.

It’s hard to explain the word explain.  It’s a word that only makes sense to me until I try to explain it.

## T-Block Visual Pattern

I created this visual pattern as a followup to the I Rule! exercise from MVP.  It is intended to be more difficult than I Rule!.  When I gave this to my students, I included a linear T-Block just like MVP does for I Rule!

I asked them two questions:

1.  How many squares are in the 10th sequence

2.  How many squares are in the nth sequence

Only a couple of my students actually got to the right answer, but the effort was tremendous.  I had students coming to me during lunch and saying they had asked all their friends and they couldn’t figure it out.  Students were telling me they worked with their parents and couldn’t get it.  I had a student (who failed first semester mind you) tell me that her and her two math tutors stayed 45 minutes after their session working on it and couldn’t figure it out.  She had two pages of work.  I have a couple students who get 100% on everything they touch, and they didn’t figure it out.  So yay me!  I challenged them 🙂

Here’s the I Rule! pattern:

Check out many more visual patterns at visualpatterns.org – a site created and curated by my conference buddy Fawn Nguyen (@fawnpnguyen)

The Goods:

Here is the worksheet I used, not sure if I will include the linear T next year.

TBlockWithLinear

## The Handshake Problem

The Overview:

I had a lot of fun with the Handshake Problem this year, so I figured I would write about it.  My goal was to use less structure (meaning no worksheet – especially one with pre-staged t-tables).

I revealed the question in three parts, each time raising the number of people shaking hands:

– 5 people go to a party and shake every one’s hand once.  How many handshakes are there?

– If everyone in the class shakes everyone else’s hand, how many handshakes would that be?

– If everyone in the school shakes everyone else’s hand, how many handshakes would that be?

Lastly, in a rather last second “I want them to process this more” moment,  I had them create solution guides for it.

The Description:

I began with the following two questions as warmup problems:

1.  5 people go to a party and shake every one’s hand once.  How many handshakes are there?

2.  If a 6th person shows up to the party, how many handshakes will they give?

After the warmup I raised the bar a little bit by increasing the amount of people who shake hands:

“If everyone in the class shakes everyone else’s hand, how many hand shakes would that be?”

The class problem raises the bar a little, but still leaves the door open for the students to add up each individual scenario.  For example, they noticed that the 31st student would shake 30 hands, and the the 30th student in the room would shake 29 hands and so forth.  So the final answer would be 30 + 29 + … + 2 + 1.  No equation needed, no additional math tools needed.  But I still took this moment to show them a new math tool that would have made there job easier – the summation!  I pulled up the Wolfram Alpha Summation calculator and let them know that a math symbol will do all that addition for you.

Then I asked them – How many handshakes would there be if everyone in the school shook eachothers hand?  Now they are dealing with a numbers that is far too big for them to simply do the summation on their own.  But luckily they had this new math tool I just gave them.  I kept that summation calculator up on my computer for them to use.  Pattern found – now execute.  And some of them literally ran to my computer to find the new sum.  Now I can ask them whatever the hell I want – If everyone on earth shook each others hand?  The size of the number is irrelevant now!  They got this.

Finding the above pattern was the most popular solution method.  But other students created a t-table and looked for a pattern to model with an equation.  They pretty quickly noticed that the equation that describes this situation had to be quadratic because we’ve looked at quadratic patterns before.  From there they made things fit and discovered the equation:  (x^2-x)/2.  I told the students that they just had to make the numbers fit.  Which was fine for the students who are good at creating equations like that.  They know what they want the function to equal, they know it’s quadratic – go to work.  The rest of the class was not amused.  And that’s when a student walked up to the whiteboard and amazed all of us.

Jose came up to the board and said, “we know that it is quadratic from the t-table.  So let’s assume there are 3 people at the party.  3 squared would be 9 handshakes, which accounts each person shaking the other two peoples hands, and their own hand.”

“But they can’t shake their own hands, so we have to get rid of those three handshakes”

“Here we are still double counting each handshake.  So then we must divide by 2 in order to only count each handshake once.”

Oh my God Oh my God Oh my God!!!  That was soo excellent!  I had never thought about the problem like that!

And with this equation in hand, when I raised the bar to how many handshakes there would be if every student in the school shook hands, they saw how quickly they could answer it by plugging in the school’s population.

I had students just work in their notebooks, but afterwards I had them formalize their work and create a solution guide.  I will probably write about these solutions guides next – but for the time being, here are some nice ones:

The Reflection:

I am most excited about the idea of these solution guides.  It was kind of a last minute idea I threw together, but turned out kind of gold for me.  Ended up doing a gallery walk and have great classroom talks about what THEY liked and disliked about each others guides.  I”m looking forward to see the quality of the next round of them!  Bring on the “Guess What I Heard?” problem!!!

## Real World Math: Project Manager

When I first started this blog the idea was to categorize assignments based on a series of twitter style hashtags, which would ultimately allow a teacher to quantify how differentiated their lessons had been – in a macro sense at least.  I have not really stuck to that idea, but one of the original hashtags I had was #industry.  The purpose of #industry was pretty simple – someone had to do this in their job.

I was hoping that #industry would end up as a collection of problems that come directly from people’s  work experiences.  These experiences would be served to student’s unedited from the workplace to the classroom.  Inherent in this hashtag would be the answer to the question “why?” because presumably anybody doing something for their job would have a clear reason as to why they were doing it. (presumably?)

So here’s the problem:  My fiance is a project manager and she had one site that was 3/4 an acre and a price from that site for \$54,000 for some work. Then she had another site that was 2.5 acres and needed to know how much that same work would be for the larger site.  That was the first thing she needed to calculate, but she ended up just wanting to know how much 1 acre was worth, so she could scale it to all her other jobs.

Here’s an error analysis angle to this question –  To scale the cost for 1 acre she had initially multiplied 54,000 by 1.25 and she was genuinely curious about why that did not work.  Hhhhmmmmm…

## Stacking Cups Assessment

When three of your favorite bloggers all write about the same lesson (Dan, Andrew, Fawn) it is a pretty safe bet that you should do the lesson.  I used Andrew’s 3Act video because my students can be pretty green and I might not hear the end of it if I couldn’t find an additional use for all these cups I was bringing into the class.

I don’t have anything to add to what was already said by Dan, Andrew, and Fawn, so I will just share a problem I created that you can put on your midterm that is a slight twist to the presentation of the original problem:

1.  How many cups would stack in a 250 cm door?

2.  What are the dimensions of the cup?  Draw it and label it with the dimensions.

I suppose you could ask for the y-intercept and slope and all that stuff too if you wanted.

Moving on from test questions – The actual lesson went great for me and I am definitely looking forward to doing it again next year.  When I did this problem in algebra I had the students make a Stacking Cups comic that was supposed to describe how to solve the stacking cups problem.

I like the comic concept because I think this is a very visual problem, and since I didn’t provide them with actual cups they needed to create their own visuals.  I have been trying to get students to give me a visual for every word problem they do this year.  My stated reasoning for that has been that visuals help you give a clearer and more convincing justification for your solution.

In order for students to learn how to construct a viable argument and critique the reasoning of others (Let’s hear it for MP.3!!!), we are going to have to have an iterative process on a couple problems where they essentially hand in drafts, and we keep having them make improvements.  I think this is a great problem to do for that since it has a couple nice extensions for system of equations (different sized cups) and geometry (here).

## My Attempts At CCSS Word Problems

I’m trying to prepare for CCSS, so this year I have been looking at word problems with a specific goal of improving student literacy by connecting each problem to graphs and having students explain their solutions.  Then afterwards coming back to their solutions and analyzing them and improving them.  It’s been successful thus far based on my last assessment so well the hell – figured I share.

I started on day 1 with my Las Vegas Problem.  Then on day 2 I played this video of myself graphing the equations we wrote on day 1 for the two airports.

[youtube=http://www.youtube.com/watch?v=Ir1EHue9qIE&w=420&h=315]

Then I passed out this worksheet and have the students try to figure out why Wolfram Alpha was calling 14 the “solution”.

I like the worksheet because it has the students try to explain which graph is for which airport, and this is before we have learned to graph or talked about things like y-intercepts and slopes.  It is just them connecting the story to the picture of the story.  The last problem on the worksheet was meant to highlight that context drives the graph, and that this particular graph should not have any negatives because you can’t have negative days.  But we had a good discussion there.

Since this is day 2 and these students aren’t used to having to “explain” their reasoning, I got a lot of papers that gave an answer without any explanation.  So I went back and did a Math Hospital and had the class analyze how to explain their solution.

During the Math Hospital I introduced them to one of the English languages most powerful and poetic words: “because”.  I showed that all they have to do is put that word after their solution and it will literally force you to state the reason for the answer.  CCSS literacy for me isn’t about having the students explain their thought process, rather it is about having the students explain why they made the decision they did.

I did a couple worksheets that were styled like the Vegas one and then I put one of them on Ch1 test (The Internets was on the test).  Every single student explained their answer on the exam.  After the test the class and I did the “Math Gym” where we take their healthy answers and make them healthier!  Basically teaching students that it’s great to choose internet company A because it is cheaper, but we can’t just say it’s cheaper, we need to also explain why it’s cheaper.

Attached are several other worksheets similar to Vegas.  Each of the problems was taken directly from our textbook.  I just provided the graphs and asked the questions in a similar manner to the Vegas trip.  One of the things I really like about these worksheets is that they provide students with a graph of the situation and ask them to make some connection between the graphs and the situation they represent.

The Goods:

RockClimbingGym

TheInternets

VegasGraph