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Despite my attempts to maintain a somewhat healthy diet, I still succomb often to sweets. If there are cookies or cupcakes in the teacher planning room, I’m there…and often regretting the indulgence later.

I especially enjoy chocolate cake.  My fingers tremble in anticipation just as I type those wonderful words – chocolate cake.

It’s a great day for baking, so I did an online search for a real kick-butt chocoloate cake recipe.  There were many, many great candidates, but I stumbled upon a recipe touting itself as Heavenly Chocolate Cake

Heavenly, you say?…tell me more…

Need to make sure I have all of the ingredients around, or else it is off to the grocery store:

• Eggs
• Milk
• Sugar
• Flour

Check, check, check, check….we are good so far.

• Bicarbonate of Soda – uhhhh…what?
• Instant espresso – is this really necessary?
• Powdered gelatine

{{Sigh}}….let’s look ahead. Maybe I can skip some of this stuff?  Perhaps the instructions will give me an out here:

Let chocolate mixture cool to room temperature. Whip the double cream to soft peaks and fold into the chocolate mixture.

What are soft peaks? And is folding just a fancy way of stirring, or is that whisking?

Be careful not to over cook and curdle the mixture. Pour egg-milk mixture through a strainer into the melted chocolate. Melt the gelatine and water

Do I own a strainer? And how do I know when I have reached the event horizon for curdling?

OK, I surrender. I’m probably a little over my head here.  Fortunately, there is an option for the cooking-challenged like me:

Thanks to Mrs. Crocker and her boxed wonderfulness, I can make a tasty cake in just 30 minutes!  Eggs, oil and water.   And just 3 simple steps: heat – stir – bake. These are some steps I can get behind!

And just that quickly, I am enjoying cake!  It’s the way I have always made cake, and the cake has always been quite tasty. My mom made cakes using this recipe; don’t go telling me that your cake is any better!

But in my heart, I know it’s no match for the Heavenly Chocolate Cake, which I salivate for.  I once had a cake like that which a neighbor made: such a memorable cake – I want more of that cake! So many sophisitcated flavors.  I can admire its beuaty, subtlety, its intricacies and I am aspire to be just half of the kitchen pro my neighbor is.

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It occurs to me that Betty Crocker’s cake products share a lot with the ongoing debate over the necessity of Common Core math methods: the cakes you bake are simple and satisying, but in no way are they a suitable replacement for the genuine cooking experience in both the path taken, and the finished product.

NOTE: I understand that the Common Core does not suggest a method for mathematical operations.  Many of the methods confused as Common Core methods have been around for quite a long time, and are commonplace in math programs. Ideally, it would be wonderful to discuss these methods separate from the Common Core debate.  My intent here is provide a justification for seemingly more convoluted methods, through the lens of the Standards for Mathematical Practice.

A worksheet full of correct answers doesn’t mean you are good at math, in the same way that successfully baking a Betty Crocker cake isn’t cooking.  There’s a real disconnect over what it means to do math.  And the disconnect is not just between what educators expect from students and what parents hope to see from schools.  There are also wide differences from teacher to teacher, and school to school.  Yesterday, I ran across a post concerning an “insane how-to-add guide” which represents the worst of both worlds: a frustrated parent wondering why so many steps are needed to add, and a weak addition “guide” which is overly helpful.  Math, like cooking, cannot be diluted down to simple steps without a loss in complexity and reflection. In my recent post on Common Core subtraction, I suggested that reflection and adaptation are far more important to me as a math teacher than filled worksheets.

Betty Crocker = Core Standards. Gourmet cooking = Standards for Mathematical Practice.  I’m still here enjoying my cake, and I’ll likely make Betty Crocker cakes again.  Maybe next time I’ll toss in some extra chocolate chips, but the cake won’t be much different from the mandated recipe.  A true chef can experiment with flavor profiles, adjust and develop new ideas for cakes.  They can “Make Sense of Cakes and Persevere is Baking Them” (even if a few attempts don’t taste so heavenly) and “Construct their own Recipes and Critique the Recipes of Others”.  The end result – the cake – is still the star of the show.

Basic skills matter.  Being “right” matters.  But true chefs, and mature math students, can demonstrate understanding through explanation, exploration, and tackling rich problems.

The recipe with the least number of steps ultimately leads to a less-satisfying product. I recognize that my cake is good, but not great.  I’d really like to experience the Heavenly cake, but understand that it will require time and effort.  Worksheets allow for lists of correct answers, but this is not the most-satisfying mathematics. Effective math teachers cause thinking. We can add fractions, but what next? How do we use this skill? How can we extend it to other ideas? Can we explain how to add fractions to other?  There may be some brain sweat, and many eggs to crack, before we reach our goal.  When we start building new flavor profiles from fractions, exponents, graphs and equations….that’s when we are doing math.

And now, off for some jogging to burn off the cake….

 

 

 

 

 

Getting close to the end of the semester here, with just a few days remaining before final exams. In Algebra 2, one of the last units of the course is on parametric equations. With changes to our HS courses, and having not taught Algebra 2 for many years, I have never taught parametrics before. But a new unit to me became fun and hopefully relevant to my students through technology and some great class discussions.

HOW JERRY JONES’ OBNOXIOUS FOOTBALL STADIUM PROVIDED A FUN ACTIVITY….FROM AN EAGLES’ FAN

In this short unit, my ultimate goal was to expose students to projectile motion expressed parametrically.  This was a contrast to material learned earlier in the semester, where quadratics are often used to model motion and explore max/min of projectiles.  Now we had the ability to think about how the horizontal and vertical position of an object were both dependent on the time.  After a quick trig review, we used these equations for our projectile problems.

I wanted a culmuniating assignment for this unit which would allow students to work collaboratively, and demonstrate their comfort with utilizing the parametric equations.  To start the day, teams had netbook computers to use, and I showed the following video to start discussion:

After a brief discussion about football and the uniqueness of the Cowboys Stadium scoreboard, I told they class they had two questions to answer:

• Is the Cowboys Stadium scoreboard too low?
• How far could a punter kick a football?

Groups were allowed to use the internet to research any information they needed, as well as the Desmos calculator to help model their ideas.  Most teams found information they needed:

• What angles are ideal for punting a football?
• What are plausible initial velocities?

I did have to stop class at one point to explain that a good angle for hitting the scoreboard may NOT be ideal for hitting a punt as long as possible; the two questions I am asking are separate from each other.  I was happy with my students’ arguments and hope to do this activity again.  Below, one group turned in most of their work in a Desmos file, which shows an “average” punt, a punt which would hit the scoreboard, and an “ideal” punt for distance.  Click below to open the graph.

HOW THE UNIT STARTED

Desmos was also used to start the unit, and allow students to see how parametrics behave.  With netbooks, partners opened the Desmos file below, and were given the instructions:

Thanks to the Desmos folks for adding the path trails to the file.  Cool addition.  Many students were able to desribe the paths they observed, often writing a rectangular equation to describe, but could not quite explain the motion.  Some student paraphrasing:

I think it is an equation, and the slider somehow controls the domain.

Note, this falls apart some when we get to example 4, the circle.

I think the slider controls the value of x, and this controls y.

Getting closer

I think the slider somehow controls x and y.

Now we are there.  A parameter, here “a”, controls the values of both x and y.  After discussion, students opened the folders and observed the structure and behavior of the equations.  This opener made the “buy-in” for the need for parametrics so much stronger, and I hope to use this again!

You can find more Desmos class files in my Desmos Virtual Filing Cabinet!