Would adults use algebra? What if we give them Oreos?

In my recent post, "Teaching Algebra without algebra", I tried to build algebraic understanding in my students by comparing their intuitive solutions to algebraic ones. I began to wonder how the colleagues in my school would try to solve a similar problem. Would they simply reason their way through it, or would they actually use a system of equations?

So I proposed this problem to them which is completely stolen from Christopher Danielson. Anyone in the mathtwitterblogosphere is probably already familiar with it. Chris is well known within that community. And he made this video of his son which I'll never forget.

Here's the scenario:

My co-teacher, Mrs. Runkle, and I both love to eat Oreo cookies. She especially likes the ones called Double Stuf because they have twice the filling. She loves the filling....but she doesn't like the cookie wafers. They get in her teeth and she looks funny when she smiles. I'm the opposite. I can't get enough of the wafers, but the filling is disgusting. It looks like some kind of heavily processed glop that could double as an adhesive...maybe that's why it's called "Double Stuf". We came up with a solution. We agreed to buy a package of Double Stuf Oreos and share them. She would eat the filling off of each cookie, and then hand the remaining wafers to me to eat.

Note: I actually re-enacted this in a class with her. She scraped off the cream with her teeth, then I took the two cookies and ate them. She didn't know I was going to do that. The kids thought it was gross.

Then I tell the kids that Mrs. Runkle and I had an argument. We can't decide who is consuming more calories. I say the cream is made of stuff that is bad for you because it's mostly sugar and fat and who knows what, and therefore, has more calories. She thinks the two wafers are bigger than the filling and therefore, they have more calories. We look at the nutrition facts to see who is right, but it only says that two double stuff cookies have 140 calories. This didn't seem to help. Luckily, we had a package of regular Oreos, but that didn't help much either. All it told us was that three regular Oreos have 160 calories.

So...who's getting more calories? What can we do to figure this out?



I sent this out as an email to all of my co-workers for them to figure out. I told them that if they came up with a unique solution, they would get one Double Stuf Oreo. (Yeah, just one. I'm cheap.)

Here are some of their responses:

Solution 1 (5 people came up with this method):

    160 calories divided by 3 cookies = 53.3 calories per regular cookie
    140 calories divided by 2 cookies = 70 calories per Double stuff cookie……

    70 – 53.3 calories = 16.7 more calories in a double stuff

    16.7 x 2 (double stuff) = 33.4 calories in the double stuff stuffing alone

    70 calories total – 33.4 calories in the double stuff stuffing (total) = 36.6 calories in the cookies.

    The person who only eats the cookies consumes more calories. 3.2 calories


Solution 2 (1 person came up with this...believe it or not, this was not a math teacher):

    3 Regular Oreos (R ) = 160Cal
    2 Double Stuff Oreos (D) = 140Cal

    R = 160/3 
    R = 53 1/3Cal
    D = 140/2 
    D = 70Cal

    If it is accepted that the components of a regular Oreo are cookies (C ) and filling (F)
    AND
    If it is accepted that the components of a Double Stuff Oreo are cookies (C ) and Double 
    filling (2F)
    Then

    R = C + F
    D = C + 2F

    Therefore the difference between a regular Oreo and a Double Stuff Oreo is a single serving 
    of filling (F):

    D-R = (C+F)-(C+2F)
    D-R = C + F – C – 2F = C + F – C – 2F 
    D-R=F

    If D = 70 and R = 53 1/3:
    AND
    D-R = F
    Then
    F = 70 – 53 1/3
    F = 16 2/3 calories

    Since a Double Stuff Oreo contains 2F, the friend who eats only the cookies consumes:
    2F = 2X(16 2/3) 
    2F = 33 2/3Calories per cookie

    To find the calorie content of a Double Stuff Oreo Cookie, we need only to go back to the 
    equation (D = C+2F) and solve for C:

    70 = C + 33 2/3
    C = 70 – 33 2/3 
    C = 36 2/3 Calories

    The cookie-eating friend consumes 36 2/3 calories per cookie.
    The filling-eating friend consumes 33 1/3 calories per cookie.

    Answer:
    The friend who eats only the cookie part of the Double Stuff Oreo will consume 3 1/3 MORE 
    calories per cookie than the friend who eats only the filling part of the same cookie.

Solution 3 (again, not a math teacher - also, this person solved it using a system of equations and two different methods, elimination on the left and substitution on the right), 2 non-math and 1 math teacher used this method:

 

So about half of the responses used algebra...half didn't. While I think it's great that so many were eager to figure this out, I wonder why some would prefer to use an algebraic method. Did they do that because that's the most natural way for them to solve this problem? Was there something about the problem that tipped them off to think about using a system of equations? Or did they assume that the solution had to be an algebraic one because I teach Algebra?

And finally, can we really say if one solution is better than another? (And what do we mean by "better"?) I can see the benefits of doing it either way. For solution 1, you really have to be thinking about what's going on at each step. In solution 3, once you set it up, your hours and hours of practicing solving systems of equations take over and you don't need to worry about context anymore. You're on auto-pilot. Is that a good thing? And which way was more efficient? Solution one appears to take five calculations. Solution three has about 12. As an Algebra teacher, this concerns me.


Other thoughts:

1. A big thank you to Marshall Thompson for inspiring me to look into this! He also expressed interest in how people solve these problems differently.

He told me that someone once used this diagram as a solution. I love how simple this makes everything. It is basically using the same method as solution 1, but it practically screams a system of equations.



2. "Double Stuf"? Where the heck is the second "f"?

3. Here is an article about Stuf from The Onion.

4. The amazing Fawn Nguyen has made her own contribution. She actually saw this method first. She is nothing short of genius...or bizarre.

Torturing Kids with Toothpicks - Part 1

Last week, I had my 7th grade Pre-Algebra students work on "Toothpicks" by Dan Meyer.

Students had to figure out how many layers of toothpicks would be in the entire structure once Dan ran out of toothpicks.

This might have been a bit too easy. One student simply drew a diagram with 250 little lines, each one representing a toothpick. I was a little worried at first that many students would do this. I wanted them to think about and use patterns. So I came up with a sequel where drawing would be the last thing you'd want to do....more on that later.

Later I decided that I was really glad that a student tried this method. Good problems are supposed to have multiple solution paths and it really falls on me to show how these different solutions are similar/different.

Other students did something similar, but first divided the number of toothpicks in the container by the number of toothpicks in a triangle. They then drew 83 triangles.

And finally, some looked for patterns: there is 1 triangle in the first layer, 3 triangles in the first two layers, 6 triangles in the first three layers, 10 triangles in the first four layers, and so on. They actually saw this pattern a week ago when they tried to figure out how many cups were in my son's cup pyramid. Sadly, no one made the connection.

Knowing that this problem would be so easy, I came up with a sequel that I knew would keep them busy. How many toothpicks would it take to make the largest triangle I could fit in the classroom?

To be continued.

Space Elephants

This was my warm-up for today. I put this on 101qs a while ago and I used this to help my students better understand a sequel I'm doing for Dan Meyer's toothpicks (I know that seems like a stretch...more on that later). The poster is from Fake Science and I love it because it reminds me of all those goofy facts about bottles being wrapped around the Earth fifty times.

Most of my seventh grade students solved this fairly quickly. It was interesting to see a very common wrong answer of 6,358,563. Can you guess what they did wrong?

We did discuss whether or not all of the elephants would die. This is probably more intriguing as a science question than as a math question. How many would die from a lack of oxygen? How many would die from being crushed by other elephants? Would they explode from the lack of air pressure? How many would float for a bit, slowly be brought down by Earth's gravity, and burn up in the atmosphere? I'm not sure I know the answers to these questions, but it was interesting to see the kids hypothesizing.

By the way, one of my students thought that there would be eleven elephants. She said she just counted them on the picture. I told her that if she sees an elephant that big, she better hope that it doesn't have a bowel movement on top of her. (There's my sequel. How big would that elephant's bowel movement be?)

Using Desmos for Scatter Plots

My students have always struggled with scatter plots and lines of best fit. I blame myself. I don't use a textbook, but I've been teaching this particular topic in the same fashion that my book presents it. In fact, I've used many of their examples to help teach scatter plots including this one:

There are couple problems I have with this example:

• Students have difficulty visualizing a trend line given a scatter plot, especially with the data above. This topic would be better introduced with scatter plots that have a strong positive or negative correlation.
• It relies on using point-slope form. I think slope-intercept form is much more accessible to my students. By the time they see these scatter plots, they should have a firm grasp of the meaning of slope and y-intercept.

This year I tried something different. I gave my students some data points.

The had to plot the points on graph paper, draw a trend line, find the slope and y-intercept, and write their equations in slope-intercept form. Because this data has such strong correlation, it was very easy to see where the trend line should be. After all students made an attempt at creating a trend line, I sampled a few of them (strategically) and compared them on the Smartboard using Desmos's online graphing calculator. Three of the students' trend lines are shown below.

As we added each trend line, students were asked to comment on how each could be improved: "It's not going through the points." "It's slanted the wrong way." "The slope is too big." "There are more points below the line than above the line." Naturally this became a little competitive as some students thought their trend lines were better than others'. I welcome competition...as long as it doesn't get too ugly. Oddly enough, it did get a little ugly.

To settle any disputes, we compared the students' equations and graphs to one I calculated using Microsoft Excel. It's pretty cool when a student's trend line (in red) almost perfectly matches the line of best fit (in purple).

Note #1: If you're feeling risky, do the exercise with the kids and submit your own equation into the mix. I have done this and I have been beaten by a student. They loved it. And of course, I pretended to be devastated by it.

Note #2: Prior to doing this activity, we looked at the McDonald's menu and compared calories and grams of fat on a scatter plot. Without drawing a trend line, I asked students to predict how many grams of fat would be in McDonald's new burger, The McHeart Attack, which boasted a whopping 1000 calories. Without ever mentioning what a trend line is, students were seeing and drawing trend lines on the Smartboard to predict the grams of fat.

Texting Algebraic Expressions

I try to read as many other blogs as I can. Sometimes I'll find a really cool idea and it somehow gets lodged into my brain somewhere, waiting to be used at just the right moment.

One of the posts/podcasts that I read/heard was from Dan Meyer and his lesson on scientific notation (I think I got it here). In it he explains how writing numbers in scientific notation can be related to texting. When texting, we often abbreviate or use acronyms to speed up the process (ttyl = talk to you later). Or maybe we do this because we're lazy. Or maybe we do this because it's the cool thing to do. Or maybe there is an annoying limit on how many characters you can use.

Yesterday I was teaching how to write algebraic expressions and this idea popped back into my head. We were looking at the expression "seven more than a number". This has twenty-four characters, and being the ancient 34-year-old man that I am, would take a while for me to text. (And it doesn't help that I only just got a cell phone a couple of months ago.) I then demonstrated how long it would take for me to text this expression to another teacher. I sent her the text which was a little odd for her since I didn't explain why I was sending it. She was nice enough to send back a response:

Not bad for an English teacher who curls up into a fetal position every time I talk about math. And look! She used a variable! That saved a lot of time. And only five characters were needed! Awesome! What a convenient way to write that expression! Now kids, here's 200 to do on your own. Good luck.

Good Day or Bad Day? (Adding Integers)

I love this clip from Seinfeld where Kramer describes what happens at the dinner table when you're married. It would seem that one of the worst parts of marriage is the time when you talk about your day. Was it a good day or a bad day? Eventually this became the inspiration for how I chose to teach integer operations.



I used to teach adding integers with gang violence. There were two gangs (positives and negatives) who would fight and kill each other. The only problem with this metaphor was that it was only really good for adding integers. It didn't help me teach some of the other operations (subtracting a negative, negative times a negative).

I also had the problem of offering students too many ways to think about adding integers. I used gang violence, money, number lines, drawn positive/negative signs, and the boring rules. But it was way too much information. I was trying to offer my students choice, but many were becoming confused.

So I'd like to say that I have one way to explain integers, and it all has to do with having a good day or a bad day. I start the lesson off by showing this slide:

I ask my students how they know if they've had a good day or a bad day. What kind of things could help you move to either side of the spectrum? I then use specific examples, always in pairs (see below). I use little arrows to indicate which way we are moving on the happy face (soon to be number) line.

The first two examples are pretty obvious. If two good things happen, then it's a good day. If two bad things happen, then it's a bad day. 

This third scenario (the lost cell phone) had some debate. Maybe the loss of a cell phone was a good thing...perhaps this would convince your parents to buy a new/better phone. But the point was to show that when a bad thing happens, and then an equally good thing happens, you end up back where you started. It was neither good nor bad.

The kids laughed at this one. Not sure why hamster death is so funny. Perhaps it was the juxtaposition of the two scenarios that made the second seem so ridiculous. But this is exactly what I wanted. Sure, it was nice to find the dollar. But your hamster is dead. Finding the dollar does not make up for the fact that your hamster died, so overall, it was a bad day.

And finally, the lost/found money. This last scenario transitioned into the use of a number line (at which point my students thought, "Oh, so this is a math lesson").

Where do I go next? Basically negatives are bad things that happen and positives are good things that happen. If more bad things happen, it's a bad day. Or if there are more negatives, then the answer is negative.

I can easily see how I can relate this now to subtracting and multiplying. Taking away bad things helps make your day better (subtracting a negative). When bad things happen over and over again, it is a bad day (positive times negative). Taking away bad things over and over again makes it a good day (negative times negative).

Nathan Kraft

Exploiting My Son for Math

At some point I'm going to have to apologize to my four-year-old son, Emmett. Over the last year I've been using him for all sorts of math lessons - many times under the guise that I'm spending quality time with him. It all started with this video:


I posted this on facebook and lot of people thought it was cute. But many also wanted to know the answer to the question. I used the same video in class and one of my students went home and worked on the problem with her dad. She was so proud when she came in the next day with the right answer.

I quickly learned that I could use his "cuteness" to teach math. He was helping me teach probability:


He was helping me teach problem-solving:


And sometimes he'll even inspire lessons. The other night I was sitting in the kitchen doing some work. Emmett should have been in bed sleeping, but instead he was in the living room stacking cups.


He was so proud of his design and he wanted to make an even bigger pyramid. So we went out the next day and bought a bunch of cups, and hence, a new math lesson was created.


So again, I apologize to Emmett for using him this way. But I think he's having some fun in the process. Besides, how many kids have had the chance to do this?


I only hope his cuteness doesn't run out any time soon.

Nathan Kraft