December 19, 2014

Lemonade Stand

I came across this activity way back when - during student teaching. I've modified it a few times and while this year was an improvement I'm still not 100% satisfied. I had to do a ton of hand holding today. So this post is a stream of consciousness as I work through what happened and how the activity should adjust to reflect that. Help me out?

Your class needs to raise $100 to go on a field trip.  You decide to sell cups of iced tea and lemonade after school.  At the stand, iced tea costs $0.50 per cup and lemonade costs $0.80 per cup.

We started noticing and wondering with just the situation, no prompts. This was a particularly engaging context for some of my students since they had run a business in their math class last year. I clarified a few things based on their commentary (ex: assume all the supplies were donated so you have 100% profit) and told them that I wondered "How many cups of lemonade and iced tea will we need to sell to get $100?"

Then I handed out the photocopy and told them to find 3 possibilities. Then check in with me and I'd give them graph paper. Everyone was able to start, they grabbed a calculator and tried something. A majority of them could figure out 200 cups of iced tea was a solution. Not everyone recognized it as (200 ice tea, 0 lemonade). Of those, some went directly to lemonade, others I gave minimal prompting got them to (0 ice tea, 125 lemonade). Everyone who had started with one or the other needed a lot of guidance to get a third point. I'm thinking that two points might be enough, and I'll just let them assume the situation is linear because we're studying lines. They can pull some other points off the graph later. So the pressing issue in (after?) #1 is getting students to see the solutions as ordered pairs. Many students who found another solution, say, (40 ice tea, 100 lemonade) recorded the final answer as 140 cups.

give them a table (that's more structure than I want in the first question, so perhaps #2 becomes a table)
give them some sentence frames (I was pushing to get them to write "40 cups of ice tea is $20, 100 cups of lemonade is $80") so that they clarify their thinking
What about a 5 column table? Cups ice tea | Profit from ice tea | Cups lemonade | Profit from lemonade | Total Profit
That's too much structure to have anywhere on the paper while they're pondering - but I could leave the situation and the question on the board, then after students have done some work in their notebook they could have the handout which would start with that five column table... I might like that best because that will also assist with the equation writing process.

OMG, making a coordinate plane on blank graph paper was such an ordeal for several kids! I told them to count by 10's so the graph would fit on the page and a few of them put 10 after one box, then skipped a line and put 20 after two boxes. This is a conversation worth having (If the first box is 10 units, all the boxes are 10 units on that axis.) so no changes here. Just a reminder to myself to watch for this error and make sure to watch when the kid restarts so they don't have to do it three times (sorry kid I walked away from!).

More than one kid remarked on the fact that the line was decreasing as soon as they drew it. If they're noticing it, let's capitalize on this. Ask if it's increasing or decreasing and what the slope is. Then ask what the slope means. For my fundamentals kids I think a fill in the blank sentence is the way to go here. I want them to say "As they sell more cups of ice tea, they sell less cups of lemonade to maintain a constant profit." So we'll start with "As x increases, y _____." Then put it in context and explain why this makes sense?

Now they are well equipped to find a variety of other points. They need to check that each point actually works to give a profit of $100 (especially since the scale is so... big? small? uh... especially since each box is worth 10). So let's go back to that table from before and fill in some more rows.

By now they've done enough repeated calculation that they're ready to write an equation. In class today I had to show them how to write out their one point that wasn't an intercept in a single equation, and then they were able to substitute the variables in to generalize. Actually, let's make the last row of the table x | ___ | y | ___ | ___. Then they can pull the equation almost directly from there. Nice.

Then let's capitalize on the fact that they're bound to have found a point that requires selling a fraction of a cup. I think the color coding questions are okay. I like that there are a variety of answers that don't make sense (fractions, negatives).

The last part is to rewrite in y=mx+b form and recognize the parts. I am not sure if this is worthwhile. It is a nice aha moment when they solve the equation and then see that b matches the y-intercept and m matches the slope. So maybe it's helpful for making connections to one of the ways we've written equations in the past?

Okay, I put some of those thoughts into a word doc. Here is the newest draft, eagerly awaiting your ideas. Remember, they will notice and wonder, then work on finding at least two solutions before they get the handout. Oh, and if you're interested, here's the version I used 5 years ago when I last taught Algebra.

December 6, 2014

Useful Tech

Over the summer I found out someone had written a grant for iPads and an Apple TV, but then got a district position so I would be receiving the tech. I spent some time researching, then learned I wouldn't be getting anything until October. I wanted to make a plan for the year where I wouldn't be "making do" until the tech arrived (good thing since we're still sorting out issues in December!).

I realized I had a document camera that I never used because it misaligned the smart board every time I switched between them, so I wanted to create an alternative to the document camera that didn't depend on the Apple TV. 

My current flow, which works great:
I have a google drive account specifically for school. One of the folders is titled classroom photos. That folder exists on my phone and my computer. Whenever I want to project something, I take a photo on my phone, wait two seconds for it to sync with the computer and then drag the image from the folder onto the slide I'm projecting. 

For this to work well you need: 
Good service or wifi in your classroom.
Drive installed on your computer (otherwise you would have to open drive in your browser, download the file, then insert it. Doable, but several extra clicks).

Ways I've used this flow:
I do out the PreCalc homework, take a photo of my solutions and have them in the slides before class starts. 

Projecting problems from a textbook. I use GeniusScan+ (Apple, Android) to scan and send it to the drive folder. It does really well with typed material (but it's not a significant improvement with handwritten material). 

Yesterday I was expecting to have each algebra student put up a homework problem on one of the boards (simultaneously), but then most of the class hadn't done the assignment so it made more sense to do a few as a class (one at a time). Rather than me having to draw all those balance problems ( I took a photo of the workbook page and projected it. 

When I'm grading I take photos of good student work to show the class what a clearly explained solution looks like. 

Every student in PreCalc made a triangle with a hypotenuse of 4 inches and I arranged them to build a unit circle. Putting the photo up meant everyone could see and I could annotate the image with their observations. 

What I would like to do:
Take more photos of student work during class and post them immediately for students to discuss. I think there are two reasons why I don't do this. First, my classes are small and kids work at different paces so I tend to frontload the discussions (notice and wonder), then talk to students/groups individually without a wrap up discussion. Second, if it is something worthy of a class discussion I don't want to put the whole solution up at once. I remember being really excited about comparing student work at PCMI, but I somehow haven't fit it into my classroom routine. Suggestions?

To note:
This tech is useful because it lets me teach. It's not the focus of my lesson but it helps me speed up the boring parts (students don't have to watch me draw a diagram, I don't have to recreate something that exists) and gives everyone access (work is visible to the whole class and saved for reference later). I'm not sure I'll use the Apple TV other than to occasionally have kids share a Desmos graph because it's not my job to make something work in my classroom just because I have it (in fact I gave away my document camera so I'd stop feeling guilty about not using it). It's my job to teach my students math, using appropriate tools strategically.

December 1, 2014

TMC15 Speaker Proposals

Have you heard of Twitter Math Camp? It's the best weekend of professional development and enthusiasm replenishment around. Don't end up in the jealousy camp this summer! Sign up to present and you'll get early access to registration:

We are starting our gear up for TMC15, which will be at Harvey Mudd College in Claremont, CA (outside of LA – map is here) from July 23-26, 2015. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc ( It’s an open GDoc for people to list their interests and someone who might be good to present that topic. If multiple people were interested in a session idea, he/she added a “+1” after it. The doc is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, so we really, honestly and truly need you to submit/present! What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing.

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is January 19, 2015 at 11:59 pm Eastern time. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Kemlage, Jami Packer, Max Ray, Glenn Waddell, and Darryl Yong

November 24, 2014

Boston TweetUp

Wish that TMC existed as a year round social/rejuvenation option? Global Math Department inspired tweetups are happening worldwide December 12-13! Okay, it's just NYC and Boston so far. Don't live in either of those cities? Organize a tweetup in your own city and join in on the fun!

When: Saturday, December 13, 2014 at 4 pm

Cornwall’s Pub, 654 Beacon Street, Boston MA
What: Connect or reconnect with math tweeps, share practices/experiences/tips, play pool/board games, MATH!
RSVP here!
Questions? Post them here in the comments! Or contact @heather_kohn or @crstn85 on Twitter
Check out the Global Math Department – we sponsor weekly virtual professional development (hosted at Big Marker) and have a weekly newsletter with blog reviews.

November 22, 2014

Age vs. Birth Year (#tmwyk)

Last night I was talking to my 13 year old foster daughter (she's been here for over 3 months! hard to believe!) and the conversation wound around to my college 80's themed dodgeball competition, which somehow transitioned to J asking what year her friend Katie was born if Katie is 30 now.

You can figure that out.
J: I don't know
Well, how old was she in 2004?
J looks at me like this is an equally difficult question and she has no idea why I asked it.
It's 2014 now, how old was she in 2004?
J: Oh. She was... 20.
Okay, so how old was she in 1994?
J: She was 10... So she was born in 1984!

Conversations about the 80's continue. Then J asks what year I was born (I'm 29), then answers her own question - 1983! I shake my head, "I'm one year younger than Katie." J realizes that I was born in 1985, and proceeds to share her thought process (which I don't remember word for word but I love that she already knows we're going to have this conversation and wants to share). We discuss how it seems like one year younger should mean subtract one from the birth year, but it actually means I was born one year more recently.

Then J turns the conversation to how old she will be in the future.

J: How old will I be in 2025?
You can figure that out.
J (pulls out a chair and sits): Let me think about this... In 2015 I'll be 14.
(Mentally I'm super excited that she's about to use the count by decades strategy I walked her through earlier)
J: In 2016 I'll be 15.
(Mentally I'm sad she didn't use the strategy but interested to see if she'll count all the way there. And keeping my mouth shut with a neutral/interested expression on my face.)
J: In 2017 I'll be 16.
(Her face lights up and I realize the alternate strategy at the same time she does.)
J: So in 2025 I'll be 24!
What did you just realize?
J: Since I was born in 2001 I can find my age by subtracting one from the year! So if I forget how old I am I can always ask someone what year it is.
People might think you have a concussion if you don't know what year it is.
J: Well if I forget what year it is I can always ask myself, "How old are you, J?" (we laugh because she asks this very expressively, sometimes 13 is a really fun age) ... In class today they were asking us about what life would be like in 2050. So I was wondering how old I would be and I figured out that subtract one thing.

So then she wanted to know what year it would be when she turned 99. My first thought was year 3000, but as I was thinking that she was saying 99+1=100 so I realized that it would be 2100, not 3000. While I was realizing how much more sense that made, J was saying how it wouldn't really be 100, it would be 3000. I should have asked her how she got that, but I was doing too much thinking of my own so instead I went with, "You were born in 2001, how long from then is 3000?" She realized her mistake and then I shared that I'd done the same thing!

Conversation turns to getting old and how long she wants to live and me telling her that 70 is not old enough to plan on being done living. I told her that the average lifespan is in the 70's and that average means middle - so lots of people live longer than that. That factoid didn't lead to her doing any more math, which was just fine with me.

Things that make me happy about this entire interaction:
J asked all the questions.
When I took her down a path where I modeled a strategy, she figured it out and continued the strategy on her own.
She was thinking about math and patterns outside of her math class (during the morning, plus this conversation).
She wanted time to think, and told me as much.
She didn't just tell me the pattern (year - 1=age) she also told me why (I was born in 2001).